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Triples, Fluxes, and Strings



Revised April 13, 2001

hep-th/0103170 ITFA-2001-11, HUTP-01/A009, EFI-01-06, DUKE-CGTP-01-05 PAR-LPTHE-01-10, LPTENS 01/07, NSF-ITP-01-12

Triples, Fluxes, and Strings

arXiv:hep-th/0103170v3 20 Jul 2003

J. de Boer1 , R. Dijkgraaf1,2, K. Hori3 , A. Keurentjes4 , J. Morgan5 , D. R. Morrison6 , and S. Sethi7
1

Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

2

Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands Department of Physics, Harvard University, Cambridge, MA 02138, USA LPTHE, Universit? e Pierre et Marie Curie, Paris VI, Tour 16, 4 place Jussieu, F-75252 Paris Cedex 05, France Laboratoire de Physique Th? eorique de l’Ecole Normale Sup? erieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France
4

3

5

Department of Mathematics, Columbia University, New York, NY 10027, USA
6 7

Department of Mathematics, Duke University, Durham, NC 27708, USA Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA

Abstract We study string compacti?cations with sixteen supersymmetries. The moduli space for these compacti?cations becomes quite intricate in lower dimensions, partly because there are many di?erent irreducible components. We focus primarily, but not exclusively, on compacti?cations to seven or more dimensions. These vacua can be realized in a number ways: the perturbative constructions we study include toroidal compacti?cations of the heterotic/type I strings, asymmetric orbifolds, and orientifolds. In addition, we describe less conventional M and F theory compacti?cations on smooth spaces. The last class of vacua considered are compacti?cations on singular spaces with non-trivial discrete ?uxes. We ?nd a number of new components in the string moduli space. Contained in some of these components are M theory compacti?cations with novel kinds of “frozen” singularities. We are naturally led to conjecture the existence of new dualities relating spaces with di?erent singular geometries and ?uxes. As our study of these vacua unfolds, we also learn about additional topics including: F theory on spaces without section, automorphisms of del Pezzo surfaces, and novel physics (and puzzles) from equivariant K-theory. Lastly, we comment on how the data we gain about the M theory three-form might be interpreted.

Contents
1 Introduction and Summary 2 The Heterotic/Type I String on a Torus 2.1 Gauge bundles on a torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 3 9 9

Bundles on S 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Bundles on T 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Bundles on T 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Bundles on T 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Bundles on T 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Anomaly cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 A perturbative argument . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 An M theory argument . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Gauge bundles in string theory . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 Holonomy in string theory I: Narain compacti?cation . . . . . . . . . 19 The topology of subgroups in string theory . . . . . . . . . . . . . . . 21 Holonomy in string theory II: asymmetric orbifolds . . . . . . . . . . 24 Triples in string theory . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Anomaly cancellation and winding states . . . . . . . . . . . . . . . . 29

2.4 Moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.1 Lattices for the orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.2 2.4.3 2.4.4 Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Degeneration limits: connections to other models . . . . . . . . . . . 43 A strong coupling description of the Z2 triple . . . . . . . . . . . . . 45

3 Orientifolds 46 3.1 Background and de?nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.1 The closed string perturbation expansion . . . . . . . . . . . . . . . . 47 3.1.2 Some pathologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 The classi?cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 D = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 D = 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 D = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 D = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 D = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1

3.2.6

D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Compacti?cations of M and F Theory 55 4.1 Some preliminary comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Six-dimensional M theory compacti?cations without ?uxes . . . . . . . . . . 58 4.3 F theory compacti?cations without ?ux . . . . . . . . . . . . . . . . . . . . . 61 4.3.1 From F theory to type I′ . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3.2 Automorphisms of del Pezzo surfaces . . . . . . . . . . . . . . . . . . 69 4.4 Type IIA compacti?cations with RR one-form ?ux . . . . . . . . . . . . . . 75 4.4.1 4.4.2 Equivariant ?at line bundles on T 4 . . . . . . . . . . . . . . . . . . . 75 Local holonomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.3 Singular K 3 manifolds with one-form ?ux . . . . . . . . . . . . . . . 84 4.5 Fluxes and K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 K-theory description of RR ?uxes . . . . . . . . . . . . . . . . . . . . 86 K-theory on orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 One-form ?uxes in K-theory . . . . . . . . . . . . . . . . . . . . . . . 89 Three-form ?uxes in K-theory . . . . . . . . . . . . . . . . . . . . . . 90 An alternate method of computation . . . . . . . . . . . . . . . . . . 95

4.6 M theory compacti?cations with three-form ?ux . . . . . . . . . . . . . . . . 96 4.6.1 Frozen singularities and new dualities . . . . . . . . . . . . . . . . . . 97 4.6.2 4.6.3 4.6.4 4.6.5 4.6.6 4.6.7 Three-form ?ux as equivariant cohomology . . . . . . . . . . . . . . . 103 Three-form holonomies . . . . . . . . . . . . . . . . . . . . . . . . . . 104 An M theory dual of the CHL string? . . . . . . . . . . . . . . . . . . 108 Some comments on type IIA versus M theory . . . . . . . . . . . . . 110 The geometry of the three-form . . . . . . . . . . . . . . . . . . . . . 114 F theory compacti?cations with ?ux . . . . . . . . . . . . . . . . . . 116 117 117

5 Acknowledgements A Lattice Conventions and Some Useful De?nitions

B Heterotic-Heterotic Duality 118 1 B.1 Duality on S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 B.2 Duality on T n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 C Classifying Orientifold Con?gurations 122

2

C.1 Con?gurations on T 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 C.2 Con?gurations on T 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 C.3 Con?gurations on T 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 C.4 Con?gurations on T 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 D Equivariant Bundles and Equivariant Cohomology 126 D.1 Equivariant cohomology via spectral sequences . . . . . . . . . . . . . . . . . 126 D.2 The case of T 4 /G for more general G . . . . . . . . . . . . . . . . . . . . . . 129 2 D.3 Explicit generators for HZ (T n , Z2 ) . . . . . . . . . . . . . . . . . . . . . . . 136 2 —————————————————————————–

1

Introduction and Summary

The moduli space of supersymmetric string compacti?cations is an immensely complicated object. One of the aspects that we might hope to understand are the discrete choices that characterize disconnected components of the moduli space. We shall focus on string compacti?cations with sixteen supersymmetries. Familiar examples of such compacti?cations are the heterotic string on a torus and M theory on a K 3 surface. With this much supersymmetry, the moduli space cannot be lifted by space-time superpotentials. The number of distinct components in the string moduli space can, however, change as we compactify to lower dimensions. For example, when compacti?ed on a circle, there is a new component in the moduli space of the heterotic string which contains the CHL string [1, 2]. We shall primarily, but not exclusively, focus on compacti?cations to 7 or more dimensions. Our goal is to describe the di?erent components of the string moduli space in each dimension. We also describe the various dual ways in which a given space-time theory can be realized in string theory or in M theory, its non-perturbative, mysterious completion. Down to 7 dimensions, it seems quite likely that our study of the string moduli space captures all the distinct components. However, a proof must await a deeper understanding of M theory. As we analyze the various components of the string moduli space, we will learn about new phenomena in string theory and some interesting mathematical relations. Many of the results we describe call for a deeper analysis, or suggest natural paths for further study. There are also tantalizing hints of how we might correctly treat the M theory 3-form. Some of these hints suggest relations between the 3-form and E8 gauge bundles which are somewhat reminiscent of [3, 4]. In the remainder of the introduction, we shall outline our results. 3

In the following section, we begin by describing the classi?cation of ?at bundles on a torus. While this might seem trivial at ?rst sight, there are actually interesting new components in the moduli space of ?at connections on T 3 , and on higher-dimensional tori. On T 3 , these new components correspond to “triples” of commuting ?at connections which are not connected to the trivial connection—the case with no Wilson line. If we pick a connection in a given component of the moduli space, we can compute its Chern–Simons invariant. This is constant over a given component of the moduli space and actually uniquely characterizes each component of the moduli space. These new components in the gauge theory moduli space are the basis for a new set of components in the moduli space of the heterotic string on T 3 . Our discussion of these heterotic/type I toroidal compacti?cations begins with a discussion of anomaly cancellation conditions, and continues with a study of asymmetric orbifold realizations together with the structure of the moduli space for these new components. Let us summarize our ?ndings: in 9 dimensions, we ?nd only the 2 known components in the heterotic/type I string moduli space. The “standard” component uni?es the conventional E8 × E8 string together with the Spin(32)/Z2 heterotic/type I string. It is important

for us to note that the gauge group for the E8 × E8 string is actually (E8 × E8 ) ? Z2 , as explained in section 2.1.1.1 The other component contains the CHL string. In 8 dimensions, we still have the standard component. There is still only one other component which now contains both the CHL string and the compacti?cation of the type I string with no vector structure. The interesting new physics appears in 7 dimensions where we ?nd 6 compo-

nents in the moduli space. These components can be labeled by a cyclic group Zm for m = 1, 2, 3, 4, 5, 6. This cyclic group appears naturally in the construction of the Zm -triple for an E8 gauge bundle. Like the CHL string, these new components have reduced rank and interesting space-time gauge groups like F4 and G2 . The m = 1 case is just the standard compacti?cation of the heterotic string, while the Z2 -triple contains the CHL string in its moduli space. We then proceed in section 2.4.2 to describe duality chains relating heterotic compacti?cations with di?erent gauge bundles. These chains generalize the usual T-duality relating the Spin(32)/Z2 and E8 × E8 string on S 1 . In some cases, we follow these chains all the way down to 5 dimensions ?nding new relations as we descend. The relations involve “quadruples” and “quintuples” which are analogues of triples for T 4 and T 5 . Unfortunately, the
1

classi?cation of gauge bundles on tori of dimension greater than 3 is unknown. This is an
Nevertheless, throughout this paper we use the common nomenclature, “E8 × E8 string.”

4

outstanding open question. In section 2.4.4, we conclude our discussion of the heterotic string by describing an intriguing connection between the Z2 -triple and a Hoˇ rava–Witten style construction of the E8 × E8 string with background 3-form ?ux. In section 3, we turn to orientifold string vacua. Proceeding again dimension by dimension, we ?nd only the two previously known components in 9 dimensions: the ?rst is the standard component containing the type I string. The second is the (+, ?) orientifold which contains no D-branes and has no enhanced gauge symmetry. We use + to refer to an O + plane, ? to refer to an O ? plane, and ?′ to refer to an O ? plane with a single stuck D-brane. This notation and our conventions are explained more fully in section 3. This is a new component in the string moduli space beyond those with a dual heterotic description. In 8 dimensions, we ?nd three components: the standard one, the orientifold realization of type I with no vector structure and the compacti?cation of the (+, ?) orientifold which is the (+, +, ?, ?) orientifold. Again, there is only one new component beyond those already described. In 7 dimensions, we again ?nd new physics. The compacti?cations of the 8-dimensional constructions give three components. However, there is now an interesting subtlety with the case of (+4 , ?4 ). We can imagine arranging the 8 orientifold planes on the vertices of a cube. However, there are 2 distinct ways of arranging the orientifold planes which are not di?eomorphic. The ?rst arrangement is the one obtained by compactifying (+, ?) on T 2 . The four + planes lie on a single face of the cube. The ? planes lie on the four vertices of the opposite face. If we exchange one adjacent pair of + and ? planes,

we ?nd an inequivalent con?guration. As perturbative string compacti?cations, we show that these two con?gurations are inequivalent.2 Whether these orientifolds are distinct

non-perturbatively is more subtle to determine, and we comment on this in section 4.6.1. This question of how we order the orientifold planes continues to be important in lowerdimensional compacti?cations. Therefore, there are two new components in the moduli space of perturbative string compacti?cations to 7 dimensions. We also give evidence against the existence of an O 6? plane—a conclusion arrived at independently using di?erent arguments in [6]. In dimensions below 7, our classi?cation of orientifold con?gurations is no longer complete. However, we ?nd evidence for a number of interesting relations including a 6dimensional duality between (+′ 4 , ?′ 12 ) and a quadruple compacti?cation of type I with
2


no vector structure. We also ?nd evidence for a 5-dimensional equivalence between (?′ 32 )
An interesting paper with a similar conclusion appeared shortly after our paper [5].

5

and (+16 , ?16 ). There are a host of open questions concerning the complete classi?cation of orientifold con?gurations below 7 dimensions, the action of S-duality etc. In section 4, we turn to M and F theory compacti?cations. Our starting point is 6dimensional M theory compacti?cations without ?ux. The compacti?cations we study are on spaces of the form (Z × S 1 )/G where Z = K 3 or T 4 and G is a discrete group acting

freely. For Z = K 3, the choice of groups G has been classi?ed by Nikulin. In our M theory context, the possible choices are G = Zm with m = 1, . . . , 8, while for Z = T 4 , G = Zn with

n = 2, 3, 4, 6. We describe both the lattices for these compacti?cations and the singularities of Z/G. Only some of these M theory compacti?cations can be lifted to 7-dimensional F theory compacti?cations. For Z = K 3, the cases m = 1, . . . , 6 lift to new 7-dimensional theories which are dual descriptions of the heterotic triples constructed in section 2. It seems worth mentioning that studying D3-brane probes on these backgrounds, along the lines of [7–12], should be interesting. All of the Z = T 4 theories lift to 7 dimensions. The case G = Z2 is another description of the compacti?ed (+, ?) orientifold while the 3 remaining cases are new components in the string theory moduli space. We also point out the existence of a new F theory vacuum in 6 dimensions associated with G = Z2 × Z2 . In studying these vacua and their dual realizations, we arrive at a natural interpretation of F theory compacti?cations without section [13]: the type IIB circle which should decompactify under M theory/F theory duality as the volume of the elliptic ?ber on the M-theory side goes to zero has a non-trivial twist. On decompactifying the circle, the twist becomes irrelevant and we gain additional degrees of freedom beyond those that we might have expected. The F theory compacti?cation then “attaches” to a larger moduli space. We proceed in section 4.3.2 to study del Pezzo surfaces with automorphisms. We show that the list of possible automorphisms of del Pezzo surfaces is classi?ed by exactly the same data that classi?es commuting triples of E8 . This is naturally suggested by the existence of F theory duals for the heterotic triples, and con?rmed by direct analysis. This also suggests a possible way of classifying E8 bundles on higher-dimensional tori using a purely geometric analysis. We also recover our heterotic anomaly matching condition directly from the geometric analysis. Compacti?cations with ?ux are the next topic of discussion. In section 4.4, we start by describing type IIA compacti?cations on quotient spaces Z/G with RR 1-form ?ux. These arise by reducing M theory on (Z × S 1 )/G to type IIA on the S 1 ?ber. These models generalize the 6-dimensional Schwarz–Sen model which is dual to the 6-dimensional CHL string [14]. We begin by describing equivariant line bundles on T 4 and the computation of 6

the relevant equivariant cohomology groups. This approach is naturally suggested from our geometric M theory starting point. We then proceed to explain in what sense the 1-form ?ux is actually localized at the singularities of T 4 /G by studying the local holonomies for these bundles. We then generalize our discussion to the case of singular K 3 surfaces. This gives us a technique for ?nding the group of 1-form ?uxes given the sublattice of vanishing cycles of the singular K 3 surface. The description of RR charges and ?elds in type II string theory seems to involve Ktheory rather than cohomology, at least at zero string coupling. In section 4.5, we study torsion RR 1-form and 3-form ?uxes on orbifolds from the perspective of equivariant Ktheory. Our analysis is for local singularities of the form C2 /G. As usual, to preserve supersymmetry, G should lead to singularities of ADE -type. A torsion 1-form RR ?ux can be measured by a D0-brane, while a 3-form RR ?ux can be measured by a D2-brane. In both cases, the D-brane acquires an additional phase factor in the string theory pathintegral. We describe how this phase can be computed for a given ?ux in terms of a reduced eta-invariant for the virtual bundle representing the ?ux. The group of RR 1-form ?uxes (modulo higher ?uxes in a sense explained in section 4.5) is given by H 1 (G, U (1)) which agrees with the result from equivariant cohomology. This is reassuring since we expect to be able to trust a straightforward analysis of ?uxes for type II backgrounds that descend from purely geometric M theory compacti?cations. The case of RR 3-form ?ux is more interesting: with vanishing 1-form ?ux, we ?nd that the group of 3-form ?uxes is given by H 3(G, U (1)). However, the full group of 1-form and 3-form ?uxes exhibits an unusual additive structure. The physical interpretation of this e?ect is that 3-brane ?ux can be induced by the presence of 1-brane ?ux: the 3-brane ?ux has a shifted quantization law. It might be possible to verify this from a dual description, perhaps one involving branes along the lines of [15]. This is quite critical because our later results suggest that it is far from clear that equivariant K-theory is the right framework even in string theory. For example, from equivariant K-theory, we ?nd Z120 as the group of RR 3-form ?uxes supported by an E8 singularity. Are all of these ?uxes actually possible, or are some choices inconsistent? In section 4.5.5, we present an alternate algebraic method for computing the desired K-theory quotients. The groups arrived at via this method con?rm the results obtained from the reduced eta-invariant approach. In section 4.6, we turn to the issue of M theory compacti?cations with ?ux. We are immediately met by the challenge of not knowing the correct framework in which to study the M theory 3-form. This is a basic problem for smooth compacti?cations. In our case, 7

the problem is only compounded by the fact that our compacti?cations involve singular geometries. The only previously known case is that of a D4+n singularity which comes in two ?avors: a conventional resolvable singularity with space-time gauge group SO (8 + 2n), and a partially frozen variety with gauge group Sp(n) [16, 17]. The D4 frozen singularity appears in the M theory description of O 6+ planes. In section 4.6.1, we argue that our new 7-dimensional components in the string moduli space imply the existence of frozen variants of E6 , E7 and E8 singularities. Each of these singularities can support a variety of ?uxes with di?erent associated space-time gauge groups. For example, E8 comes in 5 frozen, or partially frozen, variants. This result is starkly di?erent from what we might expect, for example, from equivariant K-theory. We propose M theory duals for our new 7-dimensional heterotic models, and for our new 7-dimensional F theory models. The M theory duals are on singular K 3 surfaces with various combinations of frozen D and E singularities. We then proceed to argue for the existence of dualities that map type IIA compacti?cations on singular spaces with RR 1-form ?ux to type IIA compacti?cations on spaces with completely di?erent sets of singularities and RR 3-form ?ux. In section 4.6.2, we turn to the possibility that 3-form ?ux could be described by equivariant cohomology—perhaps with additional consistency conditions from equations of motion, or anomalies. We describe the computation of the relevant equivariant cohomology group using T 4 /Z2 as an example. Section 4.6.3 extends our discussion of 1-form holonomies to torsion 3-form ?uxes. Working under the premise that the physical choices for 3-form ?ux form a subset of choices predicted by equivariant cohomology, we study the global orbifold T 4 /D4 in section 4.6.4. This orbifold has 2 D4 singularities, and we show that there is a choice of ?ux with holonomies localized at those singularities. This is a natural concrete proposal for the M theory dual of the 7-dimensional CHL string.

We turn to some puzzles in matching M theory with type IIA in section 4.6.5. These puzzles involve the spectrum of 2-branes computed both in M theory and type IIA. A generalization of the Freed–Witten anomaly [18] for D2-branes resolves the puzzle and leads us to speculate about a generalization of the anomaly in the context of K-theory. In section 4.6.6, we present some comments on the framework in which the M theory 3-form should be studied. Using a line of reasoning suggested by anomalies in wrapped branes, we are actually able to reproduce our list of frozen singularities. This is quite exciting, although the arguments are preliminary, and leave many (interesting) unresolved questions. The ?nal section concludes with a brief summary of known F theory compacti?cations with ?ux. We ?nd no new models beyond those previously studied. 8

Heterotic description “standard component” Z2 triple CHL string no vector structure Z3 triple Z4 triple Z5 triple Z6 triple

Orientifold description (?8 ) (?6 , +2 )

M theory on K3 with frozen singularities of type smooth K 3 D4 ⊕ D4

F theory compacti?ed on K3 × S1

(K 3 × S 1 )/Z2

E6 ⊕ E6 E7 ⊕ E7 E8 ⊕ E8 E8 ⊕ E8 (?4 , +4 )1 (?4 , +4 )2 (E6 )3 D4 ⊕ E7 ⊕ E7 D4 ⊕ E6 ⊕ E8 ( D4 ) 4

(K 3 × S 1 )/Z3 (K 3 × S 1 )/Z4 (K 3 × S 1 )/Z5 (K 3 × S 1 )/Z6 (T 4 × S 1 )/Z2 (T 4 × S 1 )/Z3 (T 4 × S 1 )/Z4 (T 4 × S 1 )/Z6

Table 1: A summary of 7-dimensional string theories with 16 supercharges.

As a guide for the reader, we summarize our results on the moduli space of 7-dimensional string compacti?cations in table 1. This includes a listing of all (known) dual ways of realizing a given component of the moduli space.

2
2.1

The Heterotic/Type I String on a Torus
Gauge bundles on a torus

Let us begin by reviewing the choice of gauge bundles on tori. While we need speci?c results only for the case of an E8 or Spin(32)/Z2 bundle, we shall include some general comments independent of the choice gauge group. For a more detailed review of this topic as well as further references, see [19]. We want our gauge ?elds to have zero curvature. This ensures that when we turn to string theory, they contribute nothing to the energy. A ?at connection of Yang–Mills theory with gauge group G on T n is speci?ed by a set of n 9

commuting elements of G, denoted ?i . These Wilson lines, which specify the holonomies around the n non-trivial cycles of the torus, are not unique. The same classical vacuum is also described by any other choice ?′i obtained by a global gauge transformation, ?′i = g ?i g ?1. Classifying all ?at connections on T n with gauge group G therefore amounts to classifying all sets of commuting elements in G up to simultaneous conjugation in G. The simplest way to construct a set of commuting elements is as follows: exponentiating the Cartan subalgebra of G gives a maximal torus TG , which is an abelian subgroup of G. By choosing our ?i ∈ TG , we obtain a ?at connection on T n . For particular groups like G = SU (N ) or G = Sp(N ), all ?at connections are gauge equivalent to a ?at connection with holonomies on a maximal torus, for any n. However, in general the moduli space of ?at connections contains additional components beyond the one containing the trivial connection. This insight was crucial in resolving some puzzles about counting vacua in four-dimensional gauge theory [17, 20–25]. How do we describe the component of the moduli space containing the trivial connection? With all n holonomies on a maximal torus, we can use a gauge transformation to set the corresponding gauge potentials Ai to constant elements of the Cartan subalgebra. The centraliser of this connection—the subgroup of G commuting with each ?i —clearly contains the maximal torus as a subgroup. Therefore the rank of the centraliser of this ?at connection equals the rank of G. We can characterize elements of the Cartan subalgebra by vectors on the space Rr with r the rank of G. To represent our n holonomies, we can therefore choose n vectors ai where we identify vectors that di?er by elements of the coroot lattice. This identi?cation simply corresponds to quotienting out periodic gauge transformations. The resulting moduli space is then compact. Lastly, we can conjugate each ?i simultaneously by elements of the normalizer of TG . This corresponds to further quotienting our moduli space by the action of the Weyl group W on each vector ai , simultaneously.

In later applications to string theory, we shall deal exclusively with simply-laced groups √ where we can normalize the roots to have length 2. The roots and coroots can then be identi?ed. As an example, let us take the familiar case of SU (N ) for which the moduli space is (T N ?1 )n /W .

For other components of the moduli space, we typically have a reduction of the rank of the centraliser of a ?at connection. It is clear in this case that we cannot simultaneously conjugate all holonomies into a maximal torus. However, it is possible to gauge transform to a set where each holonomy ?i can be written as the product of two commuting elements. 10

One element is on a maximal torus while the second element implements a discrete transformation: either an outer automorphism, or a Weyl re?ection. Let us now consider the possibilities for various choices of n. 2.1.1 Bundles on S 1

A ?at connection on a circle is speci?ed by a single holonomy ?. The topological types of bundles over S 1 are in natural one-to-one correspondence with π0 (G). If ? is in a component Gc of G connected to the identity, then we can always choose a maximal torus TG containing ?. The rank of the centraliser of ? then equals the rank of G. To ?nd something new, we require a component of G not connected to the identity. It is clear that conjugation with ? maps Gc to itself. Therefore ? represents an automorphism of Gc and because ? ∈ / Gc , it is an outer automorphism. In order to realize a holonomy which acts as an outer automorphism of Gc , the gauge group G must be disconnected. The gauge group G typically takes the form G = Gc ? Γ where Γ is a ?nite group (acting by outer automorphisms) and ? denotes semi-direct product. The outer automorphisms of a compact, simple, connected, and simply-connected Lie group are in correspondence with the symmetries of its Dynkin diagram. The only compact, connected, and simply-connected simple Lie groups with outer automorphisms are SU (N ), Spin(2N ) and E6 . These outer automorphisms permute the nodes of the Dynkin diagram. Thus, gauge theories with gauge groups SU (N ) ?Z2 for N > 2, Spin(8)?S3 , Spin(2N ) ?Z2 for N > 4, and E6 ? Z2 all admit non-trivial bundles over S 1 . The abelian group U (1) = SO (2) admits an outer automorphism. The group manifold U (1) is a circle and the outer automorphism acts by re?ection on the circle. It can be represented as complex conjugation on U (1), or as an element of O (2) with det = ?1 when Gc = SO (2). For the gauge group we take G = U (1) ? Z2 = O (2). A group G with a subgroup containing multiple isomorphic factors gives another example. There are outer automorphisms which permute the isomorphic factors.

Turning to the cases of interest to us, we note that Spin(32)/Z2 does not have an outer automorphism, although Spin(32) does. The group Spin(32) has two isomorphic spin representations that are interchanged by its outer automorphism. Only one of these spin representations is present in Spin(32)/Z2 , and therefore the outer automorphism of Spin(32) does not descend to a symmetry of Spin(32)/Z2 . Although E8 itself does not admit any outer automorphisms, the product E8 × E8 has two isomorphic factors and therefore has an outer automorphism exchanging the two E8

11

factors. Compactifying (E8 × E8 ) ?Z2 gauge theory on a circle with holonomy interchanging the two E8 factors leads to a theory that has rank reduced by 8 because the group elements invariant under the holonomy must be symmetric in the two E8 factors. This construction is the one employed in [2] in their realisation of the 9-dimensional CHL theory [1]. 2.1.2 Bundles on T 2

A ?at connection on T 2 is speci?ed by two commuting holonomies. Let us ?rst dispense with some simple extensions of our prior discussion. We can always pick two holonomies, ?i , on a maximal torus of G. A second possibility is to have an outer automorphism, as in our S 1 discussion, as one holonomy and a group element left invariant by this automorphism as a second holonomy. The next question we should ask is whether gauge bundles can have any non-trivial topological types on T 2 . The ?rst obstruction is measured by an element of H 1 (T 2 ; π0 (G)) as we discussed in section 2.1.1. Let us assume that G is connected so this obstruction is trivial. A non-trivial topological type then requires a non-simply-connected group. For E8 , there are therefore no non-trivial choices. However, for Spin(32)/Z2 there is a non-trivial choice. We begin with a general discussion about how this topological choice comes about. Take G to be connected but pick a holonomy ?1 for one cycle that has a disconnected centraliser Z (?1 ). Elements of the disconnected part of Z (?1 ) map the connected part to itself, and are allowed choices for the second holonomy ?2 . For simplicity, take T 2 = S 1 ×S 1 . By dimensional reduction on the ?rst circle with holonomy ?1 , we may regard this as a theory with gauge group Z (?1 ) on the remaining S 1 . Therefore this is to some extent the same as our previous example. This is the situation that occurs for ’t Hooft’s twisted boundary conditions [26]. The group G should be non-abelian since we require a disconnected centraliser for ?1 ∈ G. Let us assume that G is simple. A theorem by Bott, as quoted in [22, 24], states that the centraliser of any element from a simple and simply-connected group is connected. Therefore G should be a non-simply-connected group. Examples of non-simply-connected groups include SU (n), Sp(n), Spin(n), E6 and E7 quotiented by a non-trivial subgroup ? . The allowed Z of their centers. Let us denote the simply-connected cover of G by G representations of the gauge group are then restricted to those which represent Z by the identity element. We can now choose holonomies which commute in G but commute to a ? → G [27]. The obstruction for lifting G bundles to G ? non-trivial element in the kernel of G bundles is measured by a characteristic class w2 ∈ H 2 (T 2 , ZG ? /ZG ), where ZG ? and ZG are 12

? and G, respectively. the centres of G More explicitly for the case of Spin(32)/Z2 , there is one choice, measured by a generalized second Stiefel–Whitney class, which determines whether the compacti?cation does or does not have “vector structure” [17, 28]. The case of no vector structure corresponds to taking Wilson lines, (?1 , ?2 ), which commute to the non-trivial element in the kernel of the map Spin(32) → Spin(32)/Z2 . In the component without vector structure, the rank is reduced by 8. This is our only discrete choice on T 2 . 2.1.3 Bundles on T 3

The simplest way to construct commuting triples is to pick an element of the maximal torus that commutes with two holonomies constructed with the methods that we just described. Again, this is essentially a dimensional reduction of our prior discussion. However, we shall meet new possibilities on T 3 . For the groups E8 and Spin(32)/Z2 , compacti?cation on T 3 introduces no additional topological choice beyond the choice of the generalized Stiefel–Whitney class in H 2 (T 3 , Z2 ). Up to automorphisms of T 3 , there are two topological types for the case of Spin(32)/Z2 : Bundles of trivial class which are liftable to Spin(32), and non-liftable bundles. If we choose coordinates (x1 , x2 , x3 ) for T 3 , we can always choose these non-trivial bundles to be unliftable on the T 2 parametrized by (x1 , x2 ) and liftable on all other two-tori. For E8 bundles, there are no non-trivial topological choices. Even after ?xing this topological choice, there is the possibility of additional components in the moduli space of ?at connections. For example, in the case with trivial generalized Stiefel–Whitney class, these additional components consist of connections with three holonomies, ?i , which commute but which are not connected by a path of ?at connections to the trivial connection. Let us again begin by framing our discussion in more general terms, before turning to the special groups of interest to us. Let G be simply connected. Pick an element ?1 with centraliser Z (?1 ) so that Z (?1 ) contains a semisimple part Zss (?1 ) that is not simplyconnected. We can then choose holonomies ?2 and ?3 from Zss (?1 ) that obey twisted ? 2 and ? ? 3 to the simplyboundary conditions: they commute in Zss (?1 ) but their lifts ? ?ss (?1 ) do not commute. In this way, we achieve rank reduction even in a connected cover Z connected and simply-connected group. The groups Spin(n > 7) and all exceptional groups have non-trivial triples of this kind [17, 20–24]. If G is not simply-connected (but still semisimple), it is also possible in speci?c cases

13

to choose an element ?1 with centraliser Z (?1 ) such that the semisimple part Zss (?1 ) has a fundamental group strictly larger than the fundamental group of G. Then we can pick ? 2 and ? ? 3 to the simply-connected elements ?2 and ?3 from Zss (?1 ) so that their lifts ? ?ss (?1 ) commute up to an element that is not contained in the fundamental group cover Z of G. Compactifying with ?1 , ?2 and ?3 as holonomies leads to rank reduced theories, but the rank reduction can be larger than would be the case for a compacti?cation with twisted boundary conditions on a two-torus and a third holonomy from the maximal torus. The various possible ?at connections are characterised by two sets of data: ?rst, the topological type of the bundle measured by the generalized Stiefel–Whitney class. Second, for a ?xed topological choice, there can be di?erent components of the moduli space. An important characteristic of a connection A in some component of the moduli space is its Chern–Simons invariant which is de?ned by, CS (A) =
T3

1 16π 2 h

2 tr AdA + A3 , 3 T3

(1)

where h is the dual Coxeter number. The Chern–Simons (CS ) invariant is well-de?ned in R/Z and is constant over a connected component of the moduli space. These invariants, which have been computed for all simple groups in [24], are typically rational for nontrivial components3 . A key result is that these components can be distinguished by their CS invariant [24]. In fact, ?xing the generalized Stiefel–Whitney class, CS embeds the set of components of the moduli space of a ?xed type into Q/Z. By the order of a component k , we mean the order of its CS invariant in Q/Z. Tables 2 and 3 summarize the structure of the moduli space for E8 and Spin(32)/Z2 . In the latter case, we include both bundles with and without vector structure. Note that there are 12 distinct components for E8 and 6 for Spin(32)/Z2 . The Chern–Simons invariants of a component of order k are of the form n/k with 1 ≤ n ≤ k and n relatively prime to k . There is exactly one component of order k for each such n. For example in the E8 case, there are 2 components with k = 6. We can distinguish these two components by their CS invariants which are 1/6 and 5/6 mod Z, respectively. Denoting the moduli space of ?at connections on T 3 of a given topological type by M and letting X be a component of M, we note that [24] 1 ( dim(X ) + 1) = h. 3 X ∈M Since there are no topological choices for E8 × E8 , all solutions are characterized by the CS invariant for each E8 factor. The case of integer CS invariant corresponds to no rank
3

Those that do not contain the trivial connection.

14

Order of the Component 1 2 3 4 5 6

Maximal Unbroken Gauge Groups Degeneracy E8 1 F4 , C4 1 G2 2 A1 2 {e} 4 {e} 2

Dimension 24 12 6 3 0 0

Table 2: The structure of the moduli space for E8 . Order of Component 1 2 2 4 Maximal Unbroken Gauge Groups Degeneracy Dimension D16 1 48 B12 1 36 Dn × C m , n + m = 8 2 24 Bn × Cm , n + m = 5 2 15 (No) Vector Structure VS VS NVS NVS

Table 3: The structure of the moduli space for Spin(32)/Z2 .

reduction. For CS = 1 , the rank is reduced by 4, CS = 2
1 3 , 4 4

give a rank reduction of 7, while CS = of CS = 8. For E8 × E8 we therefore have 144 di?erent combinations with possible rank reductions

1 2 , give a rank reduction of 3 3 1 2 3 4 1 5 , , , , , give a rank reduction 5 5 5 5 6 6

6,

of 0, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15 and 16. Note that for an E8 × E8 bundle where both factors have identical triples, we can further impose the CHL outer automorphism on one of the holonomies leading to rank reductions of 12, 14, 15 and 16. The moduli space of Spin(32)/Z2 is slightly more intricate, but as we shall see in section 2.2, we do not require a detailed description of the non-trivial components in the moduli space beyond the usual no vector structure compacti?cation with CS = 0. 2.1.4 Bundles on T 4

As usual, we can extend our prior discussion to the case of T 4 in a simple way. We add a circle to T 3 and choose the holonomy around the circle to lie in the maximal torus of G, and commute with the other holonomies. However, there are again new possibilities that cannot be obtained this way. Beyond T 3 , there is no complete analysis for the general case so we shall restrict ourselves to examples which naturally arise in string theory. Let us begin our discussion with Spin(32). We note that the group Spin(32) admits

15

holonomies that break the group to a subgroup of the form Spin(2N ) × Spin(2N ′ ) × G, where N, N ′ ≥ 4 and G is some product of U (n)-factors, possibly not semisimple, of rank

16 ? N ? N ′ . In fact the semisimple part of this subgroup is two-fold connected so we should include a quotient by some Z2 ; however, we will ignore this subtlety since it plays no role in our current considerations. The point is that both Spin(2N ) and Spin(2N ′ ) have non-

trivial triples. We may therefore construct Spin(32) holonomies that implement a triple in each subgroup. This leads to a non-trivial quadruple. It results in a rank reduction of 8 which is twice the reduction of a triple. We also note that the CS invariants de?ned for any sub-three-torus of T 4 are always integer. A similar construction can be applied to the case of Spin(32)/Z2 without vector structure. Pick a holonomy that breaks the group to a subgroup Spin(2N ) × Spin(2N ′ ) × G,

beyond the reduction that follows from no vector structure [24, 29]. Constructing such a triple in each of the factors of Spin(2N ) × Spin(2N ′ ) leads to two possibilities with rank

where N, N ′ ≥ 6 and G is some (possibly not semisimple) group of rank 16 ? N ? N ′ . We have again ignored the topology of the subgroup. We imposed the restriction N, N ′ ≥ 6 because Spin(2N ≥ 12) admits triples without vector structure that lead to rank reduction

reduction 14. One of these has a half-integer CS invariant on a three-torus, but the other has integer CS on all sub-three-tori. We shall meet these gauge bundles when we discuss orientifold constructions in section 3. A new topological possibility that occurs when compactifying Spin(32)/Z2 without vector structure on T 4 was brie?y described in [17]. Recall that the class w2 is an element of H 2 (T 4 , Z2 ) = Z6 2 . The new possibility appears when w2 , viewed as an antisymmetric 2 4 × 4 matrix, has maximal rank. This happens precisely when w ?2 = 0. Such bundles have

no vector structure over two complementary two-tori. The orientifold realization of this type of bundle was discussed in [17]. Thus for Spin(32)/Z2 bundles on T 4 , we encounter altogether three new types of bundles. On the other hand, for E8 or E8 × E8 no new bundles appear, since neither E8 nor Bundles on T 5

E8 × E8 admit non-trivial quadruples. 2.1.5

A T 5 compacti?cation can be achieved in the usual trivial way: add a circle to T 4 with a holonomy from the maximal torus chosen to commute with the other holonomies. For the groups of interest to us, there are some interesting new possibilities to which we now turn. The group Spin(32) has an element with centraliser (Spin(16) × Spin(16))/Z2 . The 16

group Spin(16) admits a non-trivial quadruple [22]. The argument proceeds along the lines sketched for Spin(32): the group Spin(16) has elements that have as centraliser (Spin(8) ×

leads in turn to a non-trivial quintuple for Spin(32). For this case, the rank reduction is complete and equals 16.

Spin(8))/Z2 . Since Spin(8) is among the groups that have a non-trivial triple, we construct one in each factor of Spin(8) × Spin(8). This results in a non-trivial quadruple of Spin(16). Constructing a non-trivial quadruple in each factor of the Spin(16) × Spin(16) ? Spin(32)

On the other hand, the group E8 has an element that has as its centraliser Spin(16)/Z2 . Constructing a non-trivial quadruple in Spin(16) leads to a non-trivial quintuple in E8 , with a complete rank reduction of 8. For an E8 × E8 bundle on T 5 , we can embed a quintuple in one or both E8 -factors. This leads to rank reductions of 8 and 16, respectively. 2.2 2.2.1 Anomaly cancellation A perturbative argument

Our primary interest is in constructing consistent string compacti?cations. We need to know which of the many possible gauge bundle con?gurations actually give anomaly-free compacti?cations. The issue can be addressed from multiple perspectives. Let us begin with the familiar heterotic/type I anomaly cancellation conditions. Let us phrase our discussion in the language of the heterotic string. Up to irrelevant coe?cients, the NS-NS H -?eld of the heterotic string satis?es, H = dB + CS (ω ) ? CS (A), (2)

where ω is the spin connection, and A is the connection for either an E8 × E8 or Spin(32)/Z2 bundle. For a toroidal compacti?cation, CS (ω ) vanishes. For a ?at geometry to remain a solution of string theory, the H -?eld cannot have energy which would warp the background geometry. This requirement can only be satis?ed if the H -?eld is torsion or trivial in cohomology. On T 3 , there is no possibility for torsion so the H -?eld must be trivial. Integrating equation (2) over T 3 leads to the requirement that the total CS invariant for A vanish. Anomaly cancellation therefore rules out all the components in table 3 with nonvanishing CS invariant. We are left with two Spin(32)/Z2 compacti?cations: the component of order one with vector structure, and the component of order two without vector structure but with vanishing CS invariant. Of the 144 possible choices of E8 × E8 gauge bundle, only 12 survive. These choices correspond to taking two E8 gauge bundles with opposite CS invariants. 17

2.2.2

An M theory argument

For the case of the E8 ×E8 string, we can revisit anomaly cancellation from the perspective of the strong coupling Hoˇ rava–Witten description of M theory on S 1 /Z2 [30]. Let us sketch the argument without worrying about overall constants that are not needed for this argument.

In the presence of boundaries at x11 = 0 and at x11 = π , the de?nition of the M theory four-form G is modi?ed. The component of G with legs on the torus and a leg on x11 satis?es [30], G11xyz ? δ (x11 )CS (A1 ) + δ (x11 ? π )CS (A2 ) + . . . , (3)

where A1 and A2 are connections for the E8 × E8 bundle. The terms omitted involve the C -?eld which is constant on T 3 (see section 2.4.4) . For a ?at geometry like T 3 , we require that G/2π be an integral cohomology class [31]. Integrating eq. (3) over T 3 × S 1 /Z2 then implies that the total CS invariant must cancel between the two E8 bundles. This is just the strong coupling version of perturbative anomaly cancellation. 2.3 Gauge bundles in string theory

We now turn to a detailed discussion of toroidal compacti?cations of the heterotic string. While our discussion is in the context of the weakly coupled string, supersymmetry should guarantee that results about moduli spaces remain uncorrected at strong coupling. Since the data specifying perturbative type I compacti?cations is identical to the data specifying Spin(32)/Z2 heterotic compacti?cations, our results also apply to the type I string. From our prior discussion, we saw that all gauge bundles on a torus can be characterised by commuting holonomies ?i which we can write in the form, ?i = exp (2πiai ) Θi , (4)

where ai an element of the Cartan subalgebra. The second factor Θi implements a discrete transformation (but may be set to the identity on the group). For the decomposition of the holonomy given in equation (4) to be unambiguous, we demand that the two factors commute with each other. The Θi implement automorphisms of the group lattice. These can be either inner automorphisms which constitute the Weyl group, or outer automorphisms. Let us denote the automorphism implemented by Θi by θi , and note that our requirement of commutativity allows us to choose,
1 exp (2πiai ) Θi = exp (2πiθi (ai )) ? ai = θi (ai ). Θ? i

(5)

18

In string theory, this decomposition of the holonomy into a component on the maximal torus and a discrete part is convenient since the two factors are treated di?erently in the world-sheet conformal ?eld theory. The factor representing the maximal torus contribution can studied within the usual framework of Narain compacti?cation [32,33]. The discrete factor Θi can be implemented by the asymmetric orbifold construction [34,35]. Let us ?rst turn to Narain compacti?cations, postponing the asymmetric orbifold discussion until later in this section. 2.3.1 Holonomy in string theory I: Narain compacti?cation

For simplicity, we will consider the heterotic string theory on a rectangular torus. We therefore set the metric on the torus gij = δij , and display the radii Ri explicitly. The heterotic NS-NS two-form ?eld B will not enter our considerations so we will set it to zero. The Regge slope α′ will eventually enter our discussion so we shall keep it explicit. With these conventions, the momenta of the heterotic string are denoted by: k = (q +
i

wiai )

2 , α′
wj j 2 ai

(6) · aj ± wi Ri . α′ (7)

kiL,R

ni ? q · ai ? = Ri

In these formulae, no summation is implied unless explicitly stated. The vector q takes values on the lattice Γ8 ⊕ Γ8 or Γ16 for the E8 × E8 or the Spin(32)/Z2 heterotic string, respectively. The ni and wi are integers corresponding to the momentum and winding numbers, respectively. When rescaled by α′ /2, these momenta are dimensionless and take values on an even self-dual lattice Γ16+d,d . Restricting to states with wi equal to zero, the spectrum exhibited in eqs (6) and (7) is that of compacti?ed gauge theory with holonomies parametrised by ai . We may take the ai used for a speci?c compacti?cation of Yang–Mills theory as a starting point for discussing a similar compacti?cation of heterotic string theory. However, the string theory spectrum is richer because states with non-zero winding have to be added to the spectrum. In the decomposition of eqs (6) and (7), the vectors k correspond to quantum numbers re?ecting the group lattice. In gauge and string theory, the group is broken to subgroups by expectation values for the Wilson lines ai . In gauge theory, the weight lattices for these subgroups correspond to sublattices of the group lattice—an observation which is crucial to the analysis of [22, 24]. This essential point however is not true for string theory! Winding heterotic strings have bosons on their world-sheet that are charged with respect to the 19

holonomies. Therefore the momenta of these bosons receive corrections re?ected in (6). This implies the existence of representations of the gauge group that would not be present in pure gauge theory, where winding modes are absent. In particular, the topology of the unbroken subgroup will be di?erent from the topology that would be deduced from an analysis of the low energy gauge theory. With respect to the use of the phrase “low energy theory,” note that the extra representations can be arbitrarily heavy because they arise from strings that may wrap arbitrarily large circles. Which compacti?cations feel this di?erence between gauge and string theory? Those compacti?cations with holonomies implementing outer automorphisms are not a?ected. By construction all ai should commute with the outer automorphism. This implies that although there will be extra representations, these representations are invariant under the outer automorphism and therefore cannot obstruct the compacti?cation. In particular, the construction of the CHL string in [2] completely parallels the construction in gauge theory described earlier. For compacti?cations with twisted boundary conditions, at least one of the ai is not invariant under the action implemented by one of the Θj [27] (with i = j ). Instead, we have θj (ai ) = ai + zij ?
1 Θ? j exp (2πiai ) Θj exp (?2πiai ) = exp (2πizij ) .

(8)

This equation implies that zij is a vector on the (co)weight4 lattice of the original group, since the commutator on the left hand side should be equal to the identity. The group element Θj implements a length preserving (orthogonal) automorphism, and hence ai has the same length as θj (ai ). Putting these facts together, we see that the insertion of winding states not only leads to the introduction of states with group quantum numbers ai , but also automatically generates images for these states. Therefore Θj is also a symmetry of the string theory. Although the explicit representation content of string and gauge theory are distinct, the constructions are completely parallel so the arguments used in [17] and [36] are not a?ected. The construction of triples is a?ected by the extra winding states. Recall that to construct a non-trivial triple, we have to turn on holonomies that leave a non-simply-connected subgroup unbroken, or a subgroup with fundamental group larger than the fundamental group of the original group. The topology of the unbroken subgroup can be deduced from the representation content of the theory, and this is modi?ed by the presence of winding
Actually it is the coweight lattice [27], but since we will discuss string theory with simply-laced groups only, we can identify the coweight lattice with the weight lattice.
4

20

states. We will return to this point momentarily. For constructions of quadruples and quintuples, it is hard to make a general statement. The construction of quadruples and quintuples may be viewed as an inductive process, where triples are constructed in an intermediate step. These triple constructions may be restricted by winding states, but one has to check the topology of the gauge (sub)group case by case. 2.3.2 The topology of subgroups in string theory

We have seen that some gauge theory vacuum con?gurations cannot be reproduced in string theory. We will now investigate which con?gurations remain in E8 × E8 and Spin(32)/Z2 string theory for the speci?c case of triples. This will provide an alternate derivation of the constraints obtained in section 2.2. Recall that we should look for group elements, to be used as holonomies, with a non-simply-connected centraliser. Our analysis is based on theorem 1 of [22]. Equivalent results can be found in [24]. This theorem states that any element of a simple group can be conjugated into the form,
r

exp 2πi
j =1

sj ω j

,

(9)

where r is the rank of the group, and the ωj are the fundamental coweights of the group. The sj are a set of non-negative numbers satisfying sj g j = 1
j =0

with gj the root integers. This last relation determines the number s0 . The theorem further states that the centraliser of this element is obtained by erasing all nodes i for which si = 0 from the extended Dynkin diagram, and adding U (1) factors to complete the rank of the group. The fundamental group π1 of the centraliser contains Z factors for the added U (1) factors. In addition, there is a Zn where n is the greatest common divisor (gcd) of the coroot integers of the erased roots. For the non-simple group E8 × E8 any element can be conjugated to an element that

is a product of two elements of the form (9). Since the group is simply-laced, we will drop the distinction between root and coroot, weight and coweight. Let us ?rst consider one of the E8 -factors, setting the group element for the other factor to the identity. If we desire that the centraliser of an element contains an m-fold connected factor with m = 1, we cannot erase the extended root α0 which has root integer 1. Therefore, we have 21

α α1 2 α 4 3
2

8 3

α 6

α 5

4

α 4

5

α 3

6

α 2

7

α 1

0

Figure 1: The extended Dynkin diagram of E8 . The integers are the (co)root integers associated to the respective nodes.

to set s0 = 0. The extended root α0 will now survive as a root of the subgroup. The simple factor of which α0 is a root is either SU (n) with 2 ≤ n ≤ 9 or Spin(16). In heterotic string theory, there are winding states on the weight lattice,
8

sj ωj + roots,
j =1

of the unbroken gauge group. We easily ?nd that,
8

<
j =1

sj ωj , α0 >= s0 ? 1 = ?1,

(10)

and therefore

8 j =1

sj ωj projected onto the subgroup containing α0 is minus the weight

corresponding to the simple root α0 in the unbroken gauge group. Therefore, there is at least one state, with winding number 1, which transforms in the n ? irreducible representation (the anti-fundamental representation) of an SU (n) factor, or the 16 (the vector representation) of Spin(16). This state is a singlet with respect to other simple factors in the centraliser because < 8 j =1 sj ωj , αi >= si = 0 when αi is a root of a surviving subgroup. Note that this then implies that the relevant state transforms in a simply-connected representation, unless it transforms in the 16 of Spin(16). In this single exceptional case, the vector lattice of Spin(16) has to be added to a group lattice that already contains a spin weight lattice and the root lattice. Again, the result is the lattice of a simply-connected group. The conclusion then is that in E8 × E8 string theory the holonomies that are trivial in one E8 factor do not give non-simply-connected subgroups at all. Closer inspection shows that the only way to get non-simply-connected subgroups in string theory, is to use holonomies that in gauge theory would break each E8 factor to a group containing a semisimple, non-simply-connected factor. In gauge theory, this would result in a semisimple part with fundamental group Zn1 × Zn2 . An analysis of the group elements that give rise to such a centraliser shows that in string theory, the semisimple part has fundamental group Zn with n = gcd(n1 , n2 ). 22

in string theory. This is in complete accord with our earlier anomaly cancellation results. A similar analysis can be performed for the Spin(32)/Z2 string. As the techniques involved are the same as for the E8 × E8 heterotic string, we will give fewer details.
α1 1 α 1 α
0 2

It is therefore possible to construct triples in the E8 × E8 theory, but of the 144 components of the E8 × E8 gauge theory only a set of 12 “diagonal” constructions can be realized

α 2

3

α 2

4

α 2

5

α 2

6

α 2

7

α8 α9 α10 α11 α12 α13 2 2 2 2 2 2

α14 2

α15 1 α16 1

2

Figure 2: The extended Dynkin diagram of D16 (Spin(32)). The integers are the (co)root integers associated to the respective nodes.

As remarked before, the standard compacti?cations with and without vector structure are not obstructed in any way. Let us ?rst consider the gauge theory triple in Spin(32) with vector structure. It is not hard to show that it is impossible to construct this triple in string theory because there is no element that gives a centraliser with non-simply-connected semisimple part in Spin(32). There are elements that have a centraliser with non-simplyconnected semisimple part in Spin(32)/Z2 , but this is because the group itself is not simplyconnected, and therefore results in compacti?cations without vector structure rather than a triple with vector structure. Moving to compacti?cations without vector structure, let us ?rst note that Spin(32)/Z2 has a non-trivial center which is isomorphic to Z2 . We need two holonomies to encode the absence of vector structure. The third holonomy has to commute with the other two, but is otherwise unrestricted. In particular, if ? is an allowed choice, then so is z ?, z being the non-trivial centre element in the centre of Spin(32)/Z2 . These two choices are represented as points in two disconnected components which have an identical structure. The element z is represented by the identity in the vector representation of Spin(32). In particular, the two components mentioned above cannot be distinguished by their holonomies in SO (32), and have an identical orientifold description [29]. This degeneracy should not persist in a consistent string theory, and indeed it does not. As an example, set ? to the identity. Then ? has Spin(32)/Z2 as its centraliser. On the other hand, for ? = z = exp (2πiai ), with ai on the vector weight lattice, the centraliser in gauge theory would also be Spin(32)/Z2 ; but in string theory it turns out to be5 Spin(32)—a simply
5

We are ignoring the Kaluza–Klein gauge bosons here.

23

connected group! It therefore destroys the possibility for absence of vector structure. By continuation, we see that the degeneracy is completely lifted, and only one of the degenerate components survives. For gauge theory on T 3 with Spin(32)/Z2 gauge group, and absence of vector structure, two more components exist. One may construct holonomies parametrising these components by the methods of [24] as mentioned earlier, or with an alternative approach [29]. Attempting to construct these holonomies in heterotic string theory does not lead to ?at bundles. These choices therefore are not realized in toroidally compacti?ed string theory. 2.3.3 Holonomy in string theory II: asymmetric orbifolds

Having dealt with the intricacies of Narain compacti?cation, we should now implement the discrete transformations Θj appearing in equation (4). This can be done by means of the asymmetric orbifold construction [34, 35]. We will brie?y review the general formalism and then apply it to our problem. Essentially by de?nition, an asymmetric orbifold uses the fact that the left and rightmoving degrees of freedom on the world-sheet are largely independent. The orbifold group can therefore have a di?erent action on the left and right-movers. For heterotic strings, where left and right-movers live on di?erent spaces, this possibility is quite natural. Let us use PL (PR ) to denote the left-moving (right-moving) momenta of the heterotic string. The group elements g of the orbifold group act separately on the left and right-moving momenta since mixing left and right-movers typically leads to inconsistencies. The action of g consists of a combination of a rotation θL and a translation aL acting on the left-moving sector. Similarly, a rotation θR and translation aR acting on the right-movers. The action of g on states in the Hilbert space takes the form, g |PL, PR = e2πi(PL ·aL ?PR ·aR ) |θL PL , θR PR . (11)

The orbifold construction leads to untwisted and twisted sectors in the theory. In the untwisted sector, describing states invariant under the orbifold action, we encounter (in the partition function) a sum over a lattice I which is the sublattice of Γ16+d,d invariant under rotations by θ. In the twisted sector, we ?nd a lattice I ? + a? , where I ? is the lattice dual to I and a? is the orthogonal projection of a = (aL , aR ) onto I [34]. The left- and right-moving sectors of the closed string are not completely independent. The constraint of level matching connects both sectors. This constraint leads to consistency conditions on the asymmetric orbifold. Let the group element g have ?nite order n. The eigenvalues of θL are then of the form exp (2πiri /n), i = 1, . . . , 19, while exp (2πisi /n), 24

i = 1, 2, 3 are the eigenvalues of θR . The consistency conditions for n odd are:
2 ri = (na? )2 i

mod n.

(12)

For even n, this condition is replaced by a more stringent one. There are supplementary conditions,
2 ri = (na? )2 i

mod 2n, mod 2.

(13) (14)

si = 0 mod 2,
i

v θB v = 0

n/2

with θL and θR on the diagonal. In our applications, the asymmetric orbifold construction will be used to implement

The last condition should hold for any v ∈ Γ16+d,d , where θB is a block diagonal matrix

outer automorphisms, or Weyl re?ections on the gauge group. Since the gauge group comes from left-moving excitations, we set θR = 1 and drop the subscript on θL = θ. This conforms with notation used in previous sections. Notice that the ?rst condition in (14) is trivialized. The shifts aL,R will be interpreted as physical translations. We therefore take a minimal lightlike vector in a Γ1,1 sublattice, divide it by n, and identify the shifts (aL , aR ) with the components of that resulting vector. There is only one ambiguity in this prescription: there is the possibility of a relative sign between the components aL and aR (there is another overall sign corresponding to a parity transformation). The di?erence in sign comes from the choice of fractionalizing either the winding numbers, or fractionalizing the momenta. Both choices are related by a T-duality. We shall return to this point later. In the following discussion, we fractionalise the winding numbers because this has an obvious space-time interpretation. After quotienting string theory by g , we obtain a theory with a clear geometric interpretation. Traversing the cycle on the spatial torus in the direction of (aL , aR ) gives a holonomy implementing a Weyl re?ection or an outer automorphism (see also [2, 36]). We have now gathered all the elements needed to construct the orbifolds. 2.3.4 Triples in string theory

In this section, we present an analysis of some non-trivial heterotic compacti?cations on a 3-torus. For non-trivial compacti?cations on lower-dimensional tori, we refer the reader to [2, 36]. 25

Since we earlier ruled out the existence of triples in Spin(32)/Z2 string theory, we deal exclusively with the E8 × E8 theory in this section. Our previous analysis combined with

results in E8 gauge theory [19, 21–24] lead us to expect non-trivial triples for which one holonomy implements a Weyl re?ection generating a cyclic group Zm with m = 2, 3, 4, 5 or 6. In this section, we will construct asymmetric orbifolds for special choices of holonomy. The extension to the general case will be delayed until section 2.4. Our choice will be to embed the holonomies for the Zm -triples in “minimal” subgroups of E8 . These are the smallest simply-laced subgroups that contain a Zm -triple. This choice can be motivated along the lines sketched in [22]. The maximal torus of the group has a subtorus T that commutes with the triple. The centraliser of T is a product of T with a semisimple group C . This semisimple group C is the “minimal” subgroup which we require. For E8 gauge theory, C = D4 , E6 , E7 , E8 or E8 for the Zm -triple with m = 2, 3, 4, 5 or 6, respectively. For E8 × E8 string theory, we ?nd that C = D4 × D4 , etc. The subtorus T corresponds to surviving moduli, so we can interpret the group C as representing eliminated moduli. Our choice is dictated by the desire to make C manifest in the construction. The entries from

the list of possible C ’s will reappear later in our paper. The holonomies are essentially ?xed by the decision to embed them in a “minimal” subgroup. The only remaining freedom corresponds to global gauge transformations, or equivalently, lattice symmetries of the heterotic string. Triples embedded in these subgroups have a number of convenient properties. One is that all three holonomies are conjugate to each other [24], which implies that they have the same set of eigenvalues in every representation of the gauge group. Further, from the minimality property, it follows that these eigenvalues are of the form exp (2πin/m) where n ∈ Z [21, 23]. This is convenient for checking the asymmetric orbifold consistency conditions (12), (13) and (14). Additional properties will emerge in the construction. Let us now specify values for some of the quantities appearing in the formulae for the momenta, (6) and (7). We work on a 3-torus so i = 1, 2, 3. We will turn on holonomies in the 1 and 2-directions, and use direction 3 for the shift accompanying the orbifold projection. The holonomy in this direction, a3 , will be set to zero for the moment. In equation (7), the inner products ai · aj appear. Since the holonomies at the relevant point in moduli space are conjugate to each other, we see that
2 a2 1 = a2 .

Let us introduce the notation ai = (? aI ?II i,a i ) to display the Wilson lines in the ‘?rst’ (I), 26

and ‘second’ (II) E8 factor. It is convenient to set a1 = (? a1 , a ?1 ) and a2 = (? a2 , ?a ?2 ). This eliminates the inner product a1 · a2 from our formulae, leaving only diagonal terms in the spatial momenta. We will use an orbifold projection that is symmetric in both E8 factors. Notice that in this way, we implement the prescription that the contributions of each E8 factor to the Chern–Simons invariant cancel each other. There are other ways

of implementing this constraint; for example, by choosing a1 and a2 symmetric in both factors, and choosing opposite orbifold projections in the two E8 ’s. This would leave us with o?-diagonal terms in the momenta, and we consider this less convenient. Nevertheless, it should provide equivalent results.
2 The value of a2 1 = a2 can be found in various ways. In the setup we have chosen, the holonomy parametrized by a1 eliminates only one node from each E8 extended Dynkin dia1 ?1 gram, and from the discussion around eq. (9), it follows that a1 is of the form (ωj h? j , ωj hj ). Here ωj is the (co)weight and hj the (co)root integer associated to the node.6 We now easily ?nd a2 i by noting that the weight can be expanded in the simple roots,

ωi =
k

pk i αk .

i 2 i It is then trivial to show that a2 i = 2pi /(hi ) (no summation implied), where pi is a diagonal

element of the inverse Cartan matrix. For the cases under consideration, we ?nd that 7 a2 i = 2(m ? 1)/m. Combining these conventions and results, we ?nd the momenta for the compacti?ed heterotic string before the orbifold projection : k = (q +
i=1,2

wi ai )

2 , α′ i = 1, 2

(15) (16) (17)

kiL,R = k3L,R

mni ? mq · ai ? wi (m ? 1) wi Ri ± , mRi α′ n3 w3 R3 = ± . R3 α′

Because of the eigenvalues of the holonomies, q · ai is always a multiple of 1/m. Therefore,

the combination mni ? mq · ai ? wi (m ? 1) is always an integer, and actually can take any integer value. We have now arrived at the point where we want to perform the orbifold construction. From the gauge theory interpretation of the theory, we should be con?dent that orbifolding

For m < 5, there seem to be more options but only one corresponds to a minimal triple. It can be proven that this is the minimal value that a2 i can have for an m-triple. This provides another invariant way of characterising these holonomies.
7

6

27

will lead to a consistent theory. Nevertheless, we shall verify that the theory given by equations (15), (16) and (17) has the right symmetries, and that the orbifold operation obeys the consistency conditions (12), (13) and (14). The orbifolding operation consists of a shift a, and a transformation θ acting on the gauge part of the lattice. The transformation θ is an element of the Weyl group of E8 × E8 .

Since the rank of E8 × E8 is 16, the Weyl group is a discrete subgroup of the orthogonal group O (16). As discussed before, the requirement of commuting holonomies does not

necessarily mean that θ(ai ) is equal to ai , but rather implies the weaker condition (8) where zij is some lattice vector. There is some ambiguity in the choice of zij , but in the cases of non-trivial commuting triples, the lattice vector cannot be set equal zero. The choice zij = 0 corresponds to a trivial triple, which should be equivalent to a conventional Narain compacti?ed theory. To see that the lattice has the right symmetry, we construct the image of a vector ′ ′ with labels (q, ni , wi , n3 , w3 ). There should exist a vector labelled by (q′ , n′i , wi , n′3 , w3 ) with q′ = θ(q + w1 a1 + w2 a2 ). We expect existence for generic radii of the spacial torus which ′ ′ implies wi = wi , w3 = w3 and n3 = n′3 . We are therefore led to the equations, q′ = θ(q) + w1 (θ(a1 ) ? a1 ) + w2 (θ(a2 ) ? a2 ), i = 1, 2 . (18) (19)

n′i ? q′ · ai = ni ? q · ai

Equation (18) is consistent by construction since both the left and right hand side contain lattice vectors only. We still have to verify that (q′ ? q) · ai is an integer for i = 1, 2. We will show that both (θ(q) ? q) · ai and (θ(ai ) ? ai ) · aj are integers, and hence (19) always

that ai · zi is an integer. Finally, rewriting (θ(q) ? q) · ai as (θ?1 (ai ) ? ai ) · q, we notice that this is an inner product between two lattice vectors, and hence also integer. Therefore an image point always exists. For the orbifold consistency conditions (12) and (13), we need the eigenvalues ri of θ. These can be obtained from group theory [21–24] and are listed in table 4. We set the shift a to a light-like vector so (a? )2 is always zero. In all cases, the orbifold consistency conditions are satis?ed. For future use, we remark that the eigenvalues and multiplicities appearing in table 4 are identical to the eigenvalues and multiplicities of automorphisms 28

of a1 , a2 , and because θ is symmetric in both E8 factors. We remarked previously that 2 2 θ(ai ) ? ai = zi for some lattice vector zi . Then (θ(ai ))2 = a2 i = zi + 2ai · zi + ai , where use was made of the fact that θ ∈ O (16). Since zi is on an even lattice, it immediately follows

has a solution. The quantity (θ(ai ) ? ai ) · aj is actually zero for i = j because of the speci?c choice

m ri multiplicity

2 1 8

3 1 2 1 6 6 4

4 2 3 6 4

5 1 2 3 4 4 4 4 1 4 2

6 2 3 4 4 4 4 5 2

Table 4: Eigenvalues ri = 0 for the Zm orbifolds

K3 [37]. The last condition that we need to check is the second condition of eq. (14). It can be shown that this leads to the same condition for n = 2, 4, 6 (compare with table 4). In all cases θB is a matrix that re?ects 8 orthogonal roots. It is then easily checked that (14) is satis?ed. With the table of eigenvalues, it is also easy to calculate the zero-point energies for the twisted sector(s). An eigenvalue ri contributes, 1 ri 1 2 ?1 ? 48 16 m
2 n/2

(20)

the twisted sector(s) of the Zm -orbifold. With all the requirements checked, we found—as expected—that the asymmetric orbifolds are consistent. 2.3.5 Anomaly cancellation and winding states

1 . Summing to the zero-point energy. A periodic boson has ri = 0, and hence contributes ? 24 all contributions leads to a remarkably simple result: the zero point energies are ?1/m for

In the previous section we barely mentioned the Chern–Simons invariant, which provides another way to decide which orbifolds are consistent. Nevertheless, both orbifold analysis and the Chern–Simons analysis lead to identical results. Let us examine the relation between the approaches in more detail. According to our analysis of the topology of subgroups in string theory, it is the presence of winding states that rules out particular gauge theory compacti?cations in string theory. Let us consider such a state with w1 = 1 and w2 = w3 = 0, q = 0. We have seen that such a state carries a gauge group representation vector equal to a1 . We denote this state by |a1 . Consider parallel transport of this state along the following path: we start by going

around a closed cycle in the 2 direction then a closed cycle in the 3 direction, around the 2 direction with the opposite orientation, then around the 3 direction with the opposite

orientation. Because of the background gauge ?elds, the state transforms in the following

29

way, |a1 → e2πi(a1 ·a2 ) |a1 → → e2πi(a1 ·a2 ) |θ(a1 )

→ e2πi((a1 ?θ(a1 ))·a2 ) |θ(a1 )

e2πi((a1 ?θ(a1 ))·a2 ) |a1 .

(21)

With the results from [24], the Chern–Simons invariant can be expressed in terms of the gauge ?elds as (θ(a1 ) ? a1 ) · a2 . On the other hand, we transported a state around a contractible curve in a ?at background, and consistency requires that the ?nal phase factor appearing in (21) equal unity. The conclusion is that the Chern–Simons invariant must be integer, precisely as was argued in section 2.2. Analogous arguments apply to other winding states. To complete the connection, we remark that (21) only expresses the change in phase caused by the gauge ?elds. The state |a1 is due to a winding string, and in the transport process sketched above, it sweeps out a two-dimensional world sheet. It therefore also picks up another contribution exp(2πi B ) to the phase, where the integral is over the world sheet area sweeped out. The surface integral can be converted to a volume integral giving the total phase change: exp 2πi {dB ? CS (A)} = exp 2πi H .

The right hand side states that the total phase change should be attributed to the gauge ?eld strength to which the string couples. This equation is just a global version of anomaly cancellation (2). 2.4 Moduli spaces

Our previous discussion focused on orbifold descriptions appearing at speci?c points in the orbifold moduli space. In this section, we extend the discussion to cover the whole moduli space of asymmetric orbifold theories. 2.4.1 Lattices for the orbifolds

In the standard toroidal or Narain compacti?cation of the heterotic string, a central role is played by the even self-dual lattice Γd+16,d [32, 33]. The momenta lie on this lattice. In the construction of the moduli space for a Narain compacti?cation, we further divide out by a discrete subgroup corresponding to the symmetries of Γd+16,d . In an attempt to set up a similar structure for the CHL string and its compacti?cations, Mikhailov introduced 30

lattices for these theories [38]. In a somewhat more laborious construction, the same can be done for the asymmetric orbifolds of the previous section. Recall that for all orbifolds corresponding to triples, we had a transformation θ which has order m. For each θ, we can de?ne a projection Pθ acting on R3,19 by, Pθ = 1 m
m?1

θn .
n=0

(22)

From θPθ = Pθ θ = Pθ , we see that Pθ projects all lattice vectors in Γ3,19 onto the space invariant under θ. In particular, for the holonomies introduced in the previous section we have Pθ (ai ) = 0. As our starting point, we return to the momenta (15), (16) and (17) of the heterotic string prior to orbifolding. In the orbifolded theory, the untwisted sector consists of those states that are left invariant under the projection. These are of the form
m?1

N
n=0

exp (2πina · p) |θn (k), p .

(23)

Here, we symbolically denote the spatial momenta by p, the group quantum numbers by k, while a and θ are the shift and rotation of the orbifold symmetry. There is also a normalization constant N . To these states, we associate a lattice in the following way: ?rst we de?ne √ qinv = mPθ (q), where the reason for the factor of √ m will soon become clear. We also set

n ? i = mni ? mq · ai ? wi (m ? 1),
′ for i = 1, 2. We will rescale the radii for the 1 and 2 directions by de?ning Ri = mRi . We √ ′′ ′ also de?ne α = mα . Note that the invariant radii Ri / α′ are only rescaled by a factor √ m. Now de?ne a lattice by projecting the Narain lattice (15), (16) and (17) onto the invari?. With the reparametrisations introduced above, ant subspace of θ. We call this lattice I

its vectors are given by: v = (qinv ) 2 , α′′ viL,R =
′ wi Ri n ?i ± , ′ Ri α′′

i = 1, 2

v3L,R =

mw3 R3 n3 ± . R3 α′′

(24)

The vectors viL,R and v3L,R form a lattice, which when rescaled by

α′′ /2 may be called

Γ2,2 ⊕ Γ1,1 (m). Here we follow the notation of Mikhailov [38], de?ning the lattice Γ1,1 (m) 31

to be a lattice of signature (1, 1) generated by 2 vectors e and f with scalar product (e · f ) = m. For a summary of our lattice conventions, see Appendix A. This lattice arises as an intermediate step because we have not yet included the twisted sectors. The vectors v are the vectors of Γ8 ⊕ Γ8 projected onto the subspace invariant under θ and suitably rescaled. This de?nes a di?erent lattice for every m, which can be deduced

from group theory. The lattices are D4 ⊕ D4 , A2 ⊕ A2 , A1 ⊕ A1 for m = 2, 3, 4, respectively. The lattice for the cases m = 5 and m = 6 is the empty lattice. These lattices are usually √ de?ned so their roots are normalized with length 2. In the symmetry groups which arise in gauge theory, these form the short roots of non-simply-laced algebras at level 1. For example, at the point in moduli space constructed here the gauge group is F4 × F4 for the √ m = 2 case, and G2 × G2 for the m = 3 case, with long roots which have length 2 and 6, √ respectively. The gauge group SU (2) in the m = 4 case has roots of length 8 (it is at level √ √ 4). The vectors with length 2 = 8/2 are on the weight lattice of SU (2). Although there is no simple 4-laced algebra, there is a 4-laced a?ne Dynkin diagram that plays a role in the description of the group theory [24]. Interestingly, the lattices of F4 , G2 and A1 at level 4 all satisfy a ‘generalised self-duality’ in the sense that their weight lattice is identical to the original lattice. For the simply-laced E8 lattice, this notion of ‘generalised self-duality’ coincides with self-duality. There is an interesting connection between this observation and S-duality of four-dimensional theories, which we shall discuss later. In the twisted sectors, the momenta lie on the lattice I ? , which is dual to the lattice I of invariant vectors. As in the case of the untwisted sector, we will treat the parts of the lattices that represent the group quantum numbers, and the part that represents the space quantum numbers separately. ?. It can be Obviously the lattice I of invariant vectors is a sublattice of the lattice I veri?ed that the group part of the lattice I of invariant vectors is the lattice which we will √ √ ? ? ? ? ? ⊕ D4 ), 3(A? denote 2(D4 2 ⊕ A2 ), 2(A1 ⊕ A1 ) for m = 2, 3, 4, respectively. As usual, the star denote the dual lattice which is, of course, the (co)weight lattice. The stars arise ?, which forces us to keep track of because we de?ne the lattices I relative to the lattices I relative orientations. It is now trivial to construct the group part of the lattices I ? : these √ √ are (D4 ⊕ D4 )/ 2, (A2 ⊕ A2 )/ 3, (A1 ⊕ A1 )/2, for m = 2, 3, 4. We now construct the spatial part of the invariant lattice I . First note that for invariant vectors, P θ (q +
i

wi ai ) = Pθ (q) = q +
i

wiai .

Since Pθ (q) is a sum of elements of the root lattice, it again lies on the root lattice. It then 32

follows that wi ai is on the root lattice. Because ai is on the weight lattice, we deduce that for invariant vectors, wi has to be a multiple of m, say li m. Another way to see this is from the value of a2 i = 2(m ? 1)/m. Also, if q + product with either aj has to vanish: (q +
i

wi ai is on the invariant lattice then its dot

wi ai ) · aj = Pθ (q +

i

wi ai ) · aj = (q +

i

wi ai ) · Pθ (aj ) = 0.

This leads immediately to q · ai = ?2li (m ? 1). The spatial momenta on the invariant lattice are thus given by, ni + li (m ? 1) mli Ri ± , Ri α′ i = 1, 2 n3 w3 R3 ± . R3 α′ (25)

Note that ni + li (m ? 1) can take any integer value, while li m is always a multiple of m. The momenta on the dual to the spatial part of the invariant lattice are then given by the vectors, w ′ Ri n′3 w ′ R3 n′i ± i ′ , i = 1, 2 ± 3′ . (26) mRi α R3 α We complete the construction of I ? with the same reparametrisations as in the untwisted √ ′ sector: multiply the group parts of the lattices by m, de?ne Ri = mRi for i = 1, 2 and ′′ ′ ? = I ? , con?rming a result set α = mα . Note that in all cases, we have the simple result I from Appendix A of [34] for our speci?c case. To construct the twisted sectors, we still need the shift a? . It is given by multiples of ?v3L,R = ± R3 R3 = ± ′′ . ′ mα α (27)

We remind the reader that the sign choice between the left and right-moving parts of the shift re?ects our choice of fractionalizing the winding numbers. Because the lattices I ? are ?, and because the momenta in the nth twisted sector are given by I ? + n?v3L,R identical to I with n = 1, . . . , (m ? 1), we can assemble the lattices into a single lattice Λ. In the process

of assembly, the spatial part of the lattice is completed to Γ3,3 . We only computed the lattices for very speci?c orbifolds with special values of the

holonomies and other background ?elds. To extend to the general case, ?rst note that the metric and antisymmetric tensor ?eld did not play any role so far, and the moduli corresponding to these ?elds survive the orbifold projection. For the holonomies, we took special values that had Pθ⊥ (ai ) = (1 ? Pθ )(ai ) = ai . 33

For the general case, we take holonomies parametrized by a′i , subject to Pθ⊥ (a′i ) = ai . (28)

The possible moduli for varying the holonomies are then given by Pθ (a′i ). We may use general formulae from [32, 33, 39] to show that this results in a moduli space that locally has the form, O (19 ? ?r, 3)/ (O (19 ? ?r ) × O (3)) , where ?r is the rank reduction for the Zm orbifold. As usual, we should also divide on the left by a discrete group of lattice symmetries. Following Mikhailov [38], we propose that this discrete group is formed by the symmetries of the lattice Λ constructed above. This is a non-trivial statement with regard to those symmetries in Λ that connect di?erent twisted and untwisted sectors. Mikhailov demonstrates this explicitly for his lattice. For our cases, we will not attempt to prove this. However, we note that from a gauge theory point of view, all holonomies are on equal footing (and in special situations, even conjugate to each other). We stress that the asymmetry in our treatment of the various holonomies is purely technical, caused by the fact that we are working at the level of the algebra rather than the group. There is therefore every reason to expect symmetry transformations that connect the various sectors, and lift the apparent asymmetry between the holonomies. We therefore propose that the moduli space of these asymmetric orbifolds is given by, O (Λ) \ O (19 ? ?r, 3) / (O (19 ? ?r ) × O (3)) . (29)

The lattices Λ and the rank reduction ?r are collected in table 5, where other relevant results are summarized. We have included the standard Narain compacti?cation, denoted by Z1 , which ?ts perfectly in the picture when we take trivial holonomies and m = 1. In a separate column, we list the lattices Λ⊥ which are the lattices of vectors orthogonal to the Λ sublattice of Γ19,3 . Not surprisingly, the lattices are those of the ‘minimal subgroups’ C for m-triples. We note that the entries in the Z2 row are identical to those for the CHL string: the same Mikhailov lattice, the same rank reduction and the same zero-point energy. It can indeed be proven that the Z2 -triple and the CHL string are equivalent. We will encounter these and many other dualities in the next section. To conclude this discussion of the moduli spaces for these compacti?cations, let us make a preliminary count of the number of distinct 7-dimensional heterotic compacti?cations that we have constructed. We begin 34

Λ Z1 Z2 Z3 Z4 Z5 Z6 Γ3,3 ⊕ E8 ⊕ E8 Γ3,3 ⊕ D4 ⊕ D4 Γ3,3 ⊕ A2 ⊕ A2 Γ3,3 ⊕ A1 ⊕ A1 Γ3,3 Γ3,3

Λ⊥ ? D4 ⊕ D4 E6 ⊕ E6 E7 ⊕ E7 E8 ⊕ E8 E8 ⊕ E8

?r 0 8 12 14 16 16

Et ?1 ?1/2 ?1/3 ?1/4 ?1/5 ?1/6

Table 5: Lattices Λ, complements Λ⊥ , rank reduction ?r and zero-point energies in the twisted sector Et for the Zm asymmetric orbifolds corresponding to triples.

by noting an obvious discrete symmetry: for example in the Z3 case, we can embed a bundle with CS = 1/3 in one E8 factor and a bundle with CS = 2/3 in the other E8 . Flipping the choice of embedding does not generate a new theory. There is a single theory associated to the Z3 orbifold. This counting gives us a single component for Z1 , Z2 , Z3 , Z4 and Z6 . At 1 4 ?rst sight, the case of Z5 seems to give two distinct embeddings ( 5 , 5 ) and ( 2 , 3 ). However, 5 5 the two theories are actually equivalent. This can be argued directly using discrete gauge symmetries, but is more easily seen in the dual description which we shall meet in section 4. Therefore, we ?nd 6 distinct components in the moduli space. 2.4.2 Dualities

heterotic Spin(32)/Z2 string compacti?ed on a circle. This may be deduced from the form of their moduli spaces [32], and can be made explicit by constructing a map between the two theories at a particular point in the moduli space. The rest of the moduli space is covered by continuation [39]. This duality can be shown to imply a duality between the CHL string, and a compacti?cation of the Spin(32)/Z2 string without vector structure [17,36,40], which are Z2 -asymmetric orbifolds of the E8 × E8 string and the Spin(32)/Z2 string, respectively. We now present a list of dualities between heterotic theories with various bundles. The previously mentioned duality between the CHL string and the Spin(32)/Z2 compacti?cation without vector structure is included as part of a much larger chain. We also ?nd new dualities for theories with rank reduction bigger than 8. These dualities should be expected on general grounds, such as the structure of the orbifold groups and the moduli spaces of 35

It is well known that the heterotic E8 × E8 string compacti?ed on a circle is equivalent to the

various heterotic asymmetric orbifolds. A more detailed study of T-duality for the heterotic string, performed in appendix B, gives us tools that allow us to make these statements more precise. The Z2 -chain Our ?rst example starts with the CHL string and will feature toroidal compacti?cations down to 5 dimensions. We take the heterotic E8 × E8 string compacti?ed on a circle of radius R1 . We will use standard coordinates for the E8 × E8 lattice, giving 16 numbers ui i = 1, . . . , 16 of which the ?rst 8 denote the ?rst E8 factor, and the second group of 8 denotes the other E8 factor. The construction from [2] involves modding out this theory by a shift over πR1 combined with the following transformation on the group lattice: θ(u1 , . . . , u16 ) = ?(u16 , . . . , u1). (30)

The transformation θ interchanges the two E8 factors so we end up with a theory with gauge group (E8 )2 on a circle with radius R1 /2. The extra subscript denotes that the gauge group is at level 2. We compactify this theory on a second circle with radius R2 . We may turn on a holonomy provided it is invariant under θ. Therefore, we can smoothly deform the theory and introduce a holonomy parametrised by, a2 = (1, 014 , ?1). (31)

The notation 014 denotes 14 subsequent entries of zero. Introducing this holonomy breaks the gauge group to Spin(16)2 . This theory is interpreted as the CHL string on a torus with radii (R1 /2, R2 ). We can now ?nd an element of the T-duality group that inverts R2 (for details, see appendix B)
′ to obtain a compacti?cation of the Spin(32)/Z2 theory on a torus with radii (R1 /2, R2 = ′ α /2R2 ) with holonomy given by,

a′2

=

1 ? 2

8

, 08 .

(32)

Of course, the gauge group is still Spin(16)2 . The vector a′2 is no longer invariant under θ. However, we note that θ(a′2 ) ? a′2 = 1 16 , 2

36

is a lattice vector of the Spin(32)/Z2 lattice. That it lies on which is allowed since 1 2 the spin-weight lattice indicates that we are dealing here with a compacti?cation without vector structure. We have rederived the result of [36], which was also discussed in [17, 40]. We compactify this theory on a third circle with radius R3 , and turn on a holonomy parametrised by 2 2 2 12 1 1 8 1 a3 = , ? . (33) ,0 , , ? 2 2 2 2 This breaks the group to (Spin(8)2 )2 . We have chosen a3 to be invariant under θ, and a′2 · a3 = 0. We can dualize again in the 3 direction to obtain an E8 × E8 theory on a 3-torus ′ ′ ′ with radii (R1 /2, R2 , R3 ) (with Ri = α′ /2Ri ), and holonomies given by a′2 , (32), and a′3 = ?1, 03 , 1, 011 . (34)

16

Along the ?rst circle, there is still the action of θ given by (30). However, in the present E8 × E8 theory, the root lattice is organized di?erently. Here one of the E8 root lattices is denoted by the 8 coordinates ui with i = 5, . . . , 12, while the second E8 resides in the remaining 8 positions. The transformation θ therefore acts within each E8 factor and on both factors simultaneously. This is a particular instance of the Z2 -triple construction described previously. Note that θ(a′i ) ? a′i = 0 for either i = 2 or i = 3, but that in both

cases the di?erence is a root vector of E8 × E8 . We see that the CHL string, Spin(32)/Z2 compacti?cation without vector structure and

the Z2 -triple are indeed equivalent, as claimed. Our duality chain does not end here, but by proceeding straightforwardly, we would end up with non-standard coordinates on grouplattices for the theories that we encounter. To avoid possible confusion, let us perform a coordinate transformation on the group lattice of E8 × E8 so that the ?rst E8 ends up in the ?rst 8 positions again, and the second in the second 8. Furthermore, the coordinates are chosen such that, θ(ui ) = ?ui , i = 5, . . . , 12, 12 4 14 4 12 ,0 , ,0 , , a′2 = 2 2 2 a′3 = 0, 1 2
4

θ (u i ) = u i ,

i = 5, . . . , 12

(35) (36)

1 , ? ,0 2

4

.

(37)

We stress that we have not changed anything in the theory. It is easy to check that the coordinate transformation can be chosen so that it corresponds to a lattice symmetry, or 37

equivalently, a gauge transformation. To emphasize this, we will continue using the symbols θ and a′i since we are working with the same theory as before.
′ ′ Let us continue our study of the Z2 -triple in E8 ×E8 on a 3-torus with radii (R1 /2, R2 , R3 ) and gauge group (Spin(8)2 )2 . We compactify the theory on a fourth circle with radius R4 and turn on a holonomy parametrised by,

a4 = (1, 014 , ?1).

(38)

This breaks the gauge group to (Spin(4)2 )4 = (SU (2)2 )8 . We dualise along the 4-direction ′ to a Spin(32)/Z2 theory on a fourth circle with radius R4 = α′ /2R4 and holonomy parametrised by, 2 2 2 2 1 1 1 1 . (39) , 0, , 0, ? , 0, a′4 = 0, ? 2 2 2 2 We have discussed the physical interpretation of this theory brie?y in the section on gauge bundles over the 4-torus. The transformation θ does not leave any of the a′i invariant, but now θ(a′i ) ? a′i are roots for all i. This theory is therefore not a compacti?cation without

vector structure. Instead, we should interpret this theory as a particular case of a quadruple of Spin(32)/Z2 on a 4-torus.

To arrive at the ?nal theory on this chain, we compactify on a ?fth circle with radius R5 , and turn on a holonomy parametrised by, a5 = 1 12 , ? 2 2
2

1 12 ,0 , , ? 2 2
8

2

.

(40)

The resulting gauge group is U (1)8 . We dualize along the 5 direction to obtain an E8 × E8
′ theory where the ?fth circle has radius R5 = α′ /2R5 and holonomy,

a′5 = (02 , ?1, 0, 1, 011).

(41)

The ?rst E8 factor is in the positions i = 5, . . . , 12, while the second occupies the positions i = 1, . . . , 4, 13, . . . , 16. None of the ai are invariant. The transformation θ inverts all coordinates of one of the E8 ’s; this compacti?cation should therefore be interpreted as a quintuple embedded in one of the E8 gauge factors. This chain of 5 theories shows again that the list of toroidal compacti?cations in string theory is much shorter than the list of gauge theory compacti?cations. Among the gauge theory compacti?cations that can be implemented consistently in string theory, there are many equivalences. Various choices of bundles just correspond to di?erent limits in the 38

string moduli space. It also shows that the topology of the gauge bundle, although entering crucially in our analysis, is a not a sharp notion to a string. For example, note that according to our preceeding analysis, certain compacti?cations of Spin(32)/Z2 without vector structure are connected to compacti?cations with vector structure and another mechanism of rank reduction. As we will see later, this provides connections between seemingly di?erent dual theories. In particular, between choices of bundles in con?gurations of D-branes on orientifolds. √ In the 5 dual descriptions obtained above, ai and a′i are always of length 2. This is not the minimum value. As we remarked in the previous section, the minimal value for the Z2 -theory is 1. To reach this minimal value, we can deform the a′i by adding vectors in R16 which are left invariant by θ. By doing so, we can ?ow to a ‘canonical’ point in the moduli space. At such a point, there is non-abelian gauge symmetry. We have listed the gauge symmetries appearing at these ‘canonical points’ in table 6, where we have also included the Kaluza–Klein gauge bosons. We have also listed the Mikhailov lattices Γ(10?n) derived in [38]. The expression for Γ(5) is not found in [38], but is easily derived with our methods. Theory E8 × E8 E8 × E8 E8 × E8 bundle CHL NVS triple quadruple quintuple n 1 2 3 4 5 Symmetry group (E8 )2 × U (1) × U (1) Mikhailov lattice E8 ⊕ Γ1,1 D8 ⊕ Γ2,2 D4 ⊕ D4 ⊕ Γ3,3
? D8 (2) ⊕ Γ4,4

Spin(32)/Z2 Spin(32)/Z2

(Sp(8)/Z2 ) × U (1)2 × U (1)2 Spin(17) × U (1)4 × U (1)4 E8 × U (1)5 × U (1)5 F4 × F4 × U (1)3 × U (1)3

E8 (2) ⊕ Γ5,5

Table 6: Asymmetric Z2 -orbifolds of heterotic theories on an n-torus.

On the canonical points in the moduli space, the connection between the groups and in particular their lattices, and the lattices of Mikhailov is clear: The lattices are related to the short roots of the symmetry groups. This provides a physical link to aspects of Mikhailov’s mathematical constructions. Mikhailov argues that his lattices are also useful for understanding aspects of dual theories. This point will be revisited later in our paper. More quintuples Let us now set aside compacti?cations of the CHL string, and turn to other theories that admit dual realizations. The endpoint of our Z2 -chain was given by the E8 × E8 theory with 39

a quintuple in one E8 . Actually, with an E8 theory on a 5-torus and holonomies a′2 , a′3 , a′4 and a′5 given by eqs. (36), (37), (39) and (41), it is possible to have quintuples in both E8 factors. For this purpose, we construct the asymmetric orbifold obtained by shifting over πR1 combined with the complete re?ection on the gauge group lattice: θ(ui ) = ?ui . (42)

Only discrete gauge symmetries survive in this construction. All continuous gauge symmetry is broken. This E8 × E8 theory on a 5-torus with radii (R1 /2, R2 , R3 , R4 , R5 ) with quintuples in each

E8 may be easily dualized in either the 2, 3, 4 or 5 directions, since these are all equivalent. For ease of notation, we will dualize in the 5 direction with, R5 → α′ 2R5 a′5 → a5 ,

(43)

and where a5 is given in eq. (40). We have obtained a Spin(32)/Z2 theory on the 5-torus with completely broken gauge group. Checking the action of θ on the a′i and a5 reveals that this is a case of a quintuple in Spin(32)/Z2 . Notice that this duality connects two intrinsically ?ve dimensional theories: decompacti?cation of dimensions on either side of the duality would restore some gauge symmetry. The Z4 chain Take the heterotic E8 ×E8 theory with a Z4 -triple on a 3-torus. We use standard coordinates on E8 × E8 so that the ?rst E8 is in the ?rst 8 positions, and the second E8 in the remaining

8. We turn on holonomies a2 and a3 . To obtain the Z4 -triple, we divide by a shift over πR1 /2 combined with a θ of order 4. In a special case, the expressions for θ and ai can be

taken to be, θ(ui , ui+1, ui+2 ) = (?ui+2 , ?ui+1 , ui ), a2 = a3 = 1 13 5 ,0 , ? 2 2
3

i = 1, 9 i = 4, 13

(44) (45) (46) 1 4
3

θ(ui , ui+1, ui+2 , ui+3, ui+4 ) = (?ui+4 , ?ui+3 , ?ui+2 , ?ui+1 , ui ), , 05 1 4
3

, ,0 .

1 1 1 , , 0, , 2 4 2

1 1 1 , 0, , , 0, , 2 4 2

(47)

To clarify these formulae a little, let us elaborate somewhat on the steps in the triple construction. Starting with a torus without holonomies, turning on a2 will break E8 × E8 to 40

(Spin(6)2 × Spin(10)2 )/Z4 . The Z4 is generated by the products of the generators of the Z4 centers of Spin(6) = SU (4) and Spin(10). Therefore, it acts diagonally on all factors. The reader may anticipate that (Spin(6)2 × Spin(10)2 )/Z4 is a group that can also be obtained by compactifying Spin(32)/Z2 with two holonomies. The remaining holonomies embed twisted boundary conditions: a3 breaks Spin(6) to U (1)3 and Spin(10) to SU (3) × U (1)3 .

Finally, the θ quotients eliminate all U (1)’s and mod SU (3) by its outer automorphism to SU (2) which is the surviving gauge group. The holonomies de?ned above are equivalent to the ones described abstractly in the previous section. The theory is at what we have called a canonical point in the moduli space. We will not dualize this theory directly, but instead make a slight detour to stress some subtle points. We start by compactifying on a fourth circle, with radius R4 , without turning on a holonomy on this circle. Instead we will perform an SL(4, Z) transformation on the 4-torus: x4 → x4 ? 2x2 xi → xi i = 4. (48) In this way, we obtain a theory with identical spectrum but with holonomies parametrised by θ, a2 , a3 and a holonomy around the fourth circle given by a4 = 2a2 . Although none of the ai is invariant under θ, this theory on the 4-torus can be decompacti?ed to one on the 3-torus by decompactifying in the x4 ? 2x2 direction.

We now turn on a B -?eld, which has as its only non-zero components B24 = ?B42 = 3/2. The purpose of this, as may be veri?ed with the aid of the formulae given in appendix B, is so we can use lattice symmetries to re-express this as an equivalent theory with Bij = 0

and holonomies parametrised by θ, a2 , a3 . The holonomy around the fourth circle is now given by, ?4 = 07 , 1, 07 , ?1 . a (49) We have arrived at the theory we wish to dualize. Doing so gives a theory where the
′ fourth radius is R4 = α′ /2R4 . There is a holonomy parametrised by a′4 around the 4 direction, and there is a non-zero B -?eld. The non-zero components are given by,

a′4 =

1 12 1 14 08 , ? , , ? , 2 2 2 2

1 B24 = ?B42 = . 4

(50)

The remaining moduli are given by a2 and a3 . We may ignore the B -?eld, which does play a role in the dualities, but not in the gauge ?eld interpretation.8 We have found a Spin(32)/Z2 theory. It is not too hard to verify
8

Besides, we can always deform the B -?eld away.

41

that θ(a3 ) ? a3 is on the spin lattice of Spin(32)/Z2 , indicating that we are dealing with a compacti?cation without vector structure. The rank reduction is, however, not equal to 8 but to 14. This is a realization of a quadruple without vector structure. All the Chern– Simons invariants that can be de?ned over sub-three-tori of the four-torus are integer. The existence of a Spin(32)/Z2 description of this theory implies the existence of a type I theory on the 4-torus with the same bundle. By T-dualities, this translates into orientifolds of type II theories to be described in section 3. Z2 × Z2 asymmetric orbifolds Consider an E8 × E8 theory on a 4-torus with holonomies parametrised by, a3 = a4 = 1, 014 , ?1 , 04 , (51)
4

1 14 , ? 2 2

, 04 .

(52)

We construct a Z2 -triple in this theory by dividing this theory by a πR1 shift over the ?rst circle combined with θ1 : (u1 , . . . , u8 , u9, . . . u16 ) → ?(u8 , . . . , u1 , u16 , . . . u9 ).

We have called this Weyl re?ection θ1 , because there is a second Z2 -symmetry that we wish to use in an orbifold construction. Consider θ2 : (u1 , . . . , u16 ) → ?(u16 , . . . , u1). (53)

Note that a3 and a4 are invariant under θ2 , which is an outer automorphism. Hence it is consistent to divide this theory by a Z2 -shift over πR2 combined with θ2 . The resulting theory combines the CHL construction with the Z2 -triple. The rank of the gauge group is reduced by 12. We may dualize, for example, in the 3 direction to obtain a Spin(32)/Z2 theory with a3 replaced by: a′3 = 1 ? 2
2

1 12 , , ? 2 2

2

12 8 , ,0 . 2

(54)

Note that a′3 is invariant under θ1 , while θ2 (a′3 ) ? a′3 lies on the spin weight lattice. On the other hand a4 is invariant under θ2 , while θ1 (a4 ) ? a4 lies on the spin weight lattice. 2 The resulting theory is interpreted as a theory without vector structure, which has w ?2 = 0. Notice that we have now encountered all three types of bundles for Spin(32)/Z2 on T 4 that we met in section 2.1.4. They appear for respectively for G = Z2 , G = Z4 , and G = Z2 × Z2 . 42

2.4.3

Degeneration limits: connections to other models

We have constructed moduli spaces for a number of asymmetric orbifold theories. These moduli space are non-compact and the in?nities correspond to decompacti?cation limits. The moduli space of the D = 9 CHL string is [38], O (Γ1,1 ⊕ E8 ) \ O (9, 1) / (O (9) × O (1)) . (55)

We have omitted a factor R+ corresponding to the expectation value of the dilaton φ. Moving to the end points of R+ corresponds to the weak and strong-coupling limits of the theory. Our preceeding discussion has been limited to weak coupling, where eφ → 0.

descriptions of the strongly-coupled heterotic string. We shall discuss some of these duals in section four.

There are various strong coupling limits which depend on how the string scale is treated as eφ → ∞. At least in certain regions of the moduli space, there are M and F theory dual

The remaining in?nities correspond to decompacti?cation limits. In [38], it is shown that there exists only one light-like vector in Γ1,1 ⊕ E8 , modulo the symmetry group. One of the elements of the symmetry groups inverts this vector. Therefore, there are two directions in which one can decompactify—roughly by taking R → ∞ or R → 0. These two limits appear to correspond to physically distinct theories, because the orbifold projection involves a shift over the compacti?cation circle, and therefore explicitly involves R. In the standard picture [2] that we have also used so far, the winding numbers are fractionalized. The masses of excitations from the twisted sector are then o?set by a contribution that is linear in R for large R. As R → ∞, the states in the twisted sector

This also exchanges momenta and winding numbers so that now we have a theory with fractionalized momenta and integer winding numbers. Fractionalized momenta are common in theories with background gauge ?elds. This may seem unusual for the CHL theory, but is an appropriate way to think about compacti?cations without vector structure, triples, quadruples and quintuples. Indeed, since the CHL outer automorphism is an orbifold action, 43

twisted sector and one from the untwisted sector. This limit is therefore also smooth. A perhaps more apropriate way to describe this is by T-dualizing, taking R → α′ /R.

massless. These massless states from the twisted sector transform in the adjoint of E8 , and as R goes to zero, these enhance the gauge group to E8 × E8 with one E8 coming from the

become in?nitely heavy, and therefore decouple. In this limit, the shift becomes irrelvant and so we expect to recover the conventional ten-dimensional E8 × E8 heterotic string. On the other hand, in the limit R → 0, there are states in the twisted sector that become

it is a discrete gauge symmetry. Associated to this discrete gauge symmetry is a discrete gauge-?eld modulus which is fractionalizing the momenta. At least in one limit, the asymmetric orbifold, which is often deemed non-geometric, has a natural interpretation in terms of a gauge bundle, which is a geometric concept. It is an interesting observation that asymmetric orbifolds can be interpreted in terms of bundles. Combining such bundles with symmetric orbifolds may o?er geometric interpretations for at least a particular class of asymmetric orbifolds. Having settled this subtlety, the rest of our discussion is parallel to the last section of [38]. Large volume limits can be identi?ed from our discussion of dualities. The 8-dimensional CHL string has various decompacti?cation limits in which a two-torus becomes large. One of these corresponds to the Spin(32)/Z2 compacti?cation without vector structure, and one to the CHL string. The 7-dimensional CHL string has various limits which we have identi?ed as the CHL string, Spin(32)/Z2 without vector structure and the E8 × E8 Z2 triple. This is in disagreement with [38] where two Spin(32) and one E8 × E8 degenerations

were found. That analysis, however, is far less concerned with identifying the theories that appear in these limits. The 6-dimensional CHL string brings us one new limit where a 4-torus becomes large, which is identi?ed as a quadruple, which has vector structure. Another new limit is found in 5-dimensions in terms of a quintuple in one E8 . Also, in the

cases of Z4 and Z2 × Z2 , we have identi?ed the various limits. The duality chains of this section connect a diverse set of theories encountered later in this paper. Note in particular the Spin(32)/Z2 theories in various chains. By S and T-dualities these will translate into type I theories and type II orientifolds, which make their appearance in section 3. We will make a brief excursion to four dimensions where it is believed that the heterotic string theory is S-dual. In [38], this translates into a property of the Mikhailov lattice: √ namely, that it is isomorphic to its dual lattice up to a rescaling by a factor of 2. One way √ to heuristically understand the factor of 2 is that it is related to the possible appearance of non-simply-laced groups in CHL theories. For non-simply-laced groups G, the gauge group in the S-dual theory is given by the dual group GD . The roots of GD are the coroots of G up to a suitable rescaling [41], while the weight lattice of GD is the coweight lattice of G, √ also rescaled. The rescaling is precisely the factor of 2 for the 2-laced theories appearing in CHL compacti?cations. By analogy the asymmetric Zm -orbifolds with m = 3, 4, 5, 6 should have lattices for their 4-dimensional theories that are isomorphic to their dual lattices up to rescaling by a √ factor of m. The reader may verify that using the lattices Λ from table 5, the lattices 44

Γ3,3 (m) ⊕ Λ indeed have this property. This is again closely related to the fact that the F4 , G2 and A1 at level 4 are their own dual groups up to rotations [41]. It is amusing to see structure in a theory compacti?ed on a 6-torus re?ected in a similar theory compacti?ed on the 3-torus. Also the table 6 suggests that the 7-dimensional theory has special status because the groups and lattices listed there for (7 + d) are the duals of those in (7 ? d). This is somehow related to the Z2 nature of the duality operation, and to the fact that 3 = 6/2, but a more precise understanding of this relation is in order. We also wish to brie?y comment on other heterotic theories appearing in the literature. In particular we note that Z2 , Z3 , Z4 Z5 , Z6 and Z2 × Z2 asymmetric orbifolds in dimensions less than 7 do appear in [42,43]. The models in these papers are similar to the construction of [14], and originate in exploiting symmetries of K3. That heterotic duals should exist is obvious from the constructions, but an explanation for their existence has been lacking. The gauge theory based analysis presented here ?lls that gap. It also makes clear that these models can be traced back to constructions in higher dimensions. This again presents new challenges for ?nding dual descriptions which we take up in the following sections. 2.4.4 A strong coupling description of the Z2 triple

Let us conclude our discussion of heterotic string theory by pointing out an intriguing relation between the Z2 triple and a discrete 3-form ?ux appearing in its strong coupling description. The type I string on T 2 without vector structure can be viewed as an orientifold of type IIB with a half-integral NS-NS B -?eld ?ux through the T 2 [44,45]. It is natural to ask whether the strong coupling Hoˇ rava–Witten description of the E8 ×E8 heterotic string might

permit a similar discrete ?ux. Consider M theory on S 1 /Z2 × T 3 . The Z2 action projects out the M theory 3-form C -?eld. After all, there are no membranes in heterotic string theory. The component of the C -?eld with a leg on S 1 /Z2 , however, survives projection and couples to the perturbative heterotic string. It is natural to ask whether we can turn on a half-integral C -?eld on T 3 and then

quotient by Z2 . It is not clear that such a compacti?cation is consistent but the following chain of dualities suggests that it exists and is an alternate description of the Z2 -triple. Let us perform a 9 ? 11 ?ip and reduce from M theory to string theory on a circle of the T 3 . This gives an orientifold of IIA on S 1 /Z2 × T 2 with a half-integral B -?eld through T 2 . A further T-duality on S 1 /Z2 turns this compacti?cation into type I on S 1 × T 2 with 45

a half-integral B -?eld. As we recalled above, this is just type I with no vector structure which is in the same moduli space as the Z2 -triple. This suggests an intimate connection between background 3-form ?uxes and non-trivial Chern–Simons invariants for the E8 × E8 gauge bundle.

3
3.1

Orientifolds
Background and de?nitions

We begin our discussion of orientifolds with some background and some words on our notation. We use I9?p to denote the sign ?ip of the last 9 ? p spatial coordinates of R10 , (x0 , . . . , xp , xp+1 , . . . , x9 ) → (x0 , . . . , xp , ?xp+1 , . . . , ?x9 ). (56)

Type II string theory on R10 is invariant under the action of I9?p when combined with world-sheet orientation reversal ?, where p is even for type IIA and odd for type IIB. As is standard, we use (?1)FL to denote the symmetry that ?ips the sign of all R-NS and RR states. It is not hard to check that, I9?p ?, is an involution, i.e., squares to the identity, for p = 0, 1, 4, 5, 8, 9 while, I9?p (?1)FL ?, (58) (57)

is an involution for p = 7, 6, 3, 2. We can then consider an orbifold of Type II string theory by the Z2 symmetry group generated by this involution. This is called a Type II orientifold on R10 /I9?p , or just R10 /Z2 when there is no room for confusion. The Z2 ?xed plane xp+1 = · · · = x9 = 0 is called an orientifold p-plane, or Op plane for short. We can also extend this construction to the case where R10 is replaced by a non-trivial ten-manifold M 10 with an involution I , as we will do in the case of T 9?p × Rp+1 where I acts by inversion on the (9 ? p) coordinates of T 9?p . There are two kinds of orientifold planes, Op? and Op+ , which are distinguished by the sign of the closed string RP2 diagram surrounding the plane—Op+ has an extra (?1) factor

when compared to Op? . When N Dp-branes are placed on top of Op? (resp. Op+ ), they support an SO (N ) (resp. USp(N ) = Sp(N/2)) gauge group, where we count the number of D-branes on the double cover. An Op plane carries Dp-brane charge, and the two kinds of Op planes are also distinguished by the sign of this charge—Op± carries Dp-brane charge 46

±2p?5 when counted on the double cover. The superscript has its origin here.

+

or

?

in the name of the plane

An Op? plane together with an even number of Dp-branes is quite di?erent in character from an Op? plane together with an odd number of Dp-branes. We distinguish these two ′ cases by using the notation Op? for an Op? plane with a single Dp-brane stuck to it. It has been shown that Op? has a non-trivial Z2 ?ux associated with the RR (6 ? p)-form G6?p , while the ?ux is trivial for Op? . It has also been shown that there are two kinds


of Op+ planes distinguished by the same ?ux; we denote the trivial one by Op+ and the ′ non-trivial one by Op+ . In total, there are four kinds of orientifold planes: Op? , Op? , Op+ , Op+ .
′ ′ ′ ′

(59)

These four kinds of orientifold planes—especially the planes Op? and Op+ with non-trivial Z2 ?ux—have been identi?ed for the cases p = 5, 4, 3, 2, 1, 0 [15, 46–51]. We refer to these references for a fuller discussion of the Z2 ?ux associated with G6?p . 3.1.1 The closed string perturbation expansion


As mentioned above, Op+ and Op+ have an extra (?1) sign when compared to Op? and Op? for the fundamental string RP2 diagram. The meaning of an ‘RP2 surrounding the plane’ is clear only if the transverse dimension (9?p) is greater or equal to 3. We can actually


make this statement more precise so that it also applies to the cases (9 ? p) = 1, 2. We note that the phase of the closed string diagram is given by the B -?eld, and therefore it is natural to use equivariant cohomology to determine the allowed B -?eld con?gurations. Let us consider the closed string perturbation expansion for a type II orientifold on M/I following [17]. The world-sheet is an orientable Riemann surface Σ with orientation reversing freelyacting involution IΣ . Note that the quotient Σ/IΣ is a smooth unorientable surface if Σ is connected. However, it is smooth and orientable but not oriented when Σ consists of two identical components exchanged by IΣ . The world-sheet path-integral is over maps φ : Σ → M which commute with the involution, I ? φ = φ ? IΣ . (60)

We would like to assign a phase factor in a way that is consistent with all physical consistency conditions, such as factorization etc. This can be accomplished in the following way. Let us ?x an element of the equivariant cohomology group,
2 y ∈ HZ (M, Z2 ). 2

(61)

47

This means choosing an element of H 2 (MZ2 , Z2 ) = H 2 (S N ×Z2 M, Z2 ) for su?ciently large N , where Z2 acts on S N via the antipodal map and on M by I .9 The map φ, obeying the ? : ΣZ → MZ . Therefore the element y ∈ H 2 (MZ , Z2 ) can condition (60), induces a map φ
2 2 2

?? y ∈ H 2 (ΣZ , Z2 ). We denote this element simply by be pulled back to an element φ 2
2 φ? y ∈ HZ (Σ, Z2 ). 2

Since IΣ acts on Σ freely, φ? y can be viewed as an element of H 2 (Σ/IΣ , Z2 ) which can be integrated over Σ/IΣ : Σ/IΣ , φ? y :=
Σ/IΣ

φ? y ∈ Z2 .

(62)

We can then assign the following sign factor to the world-sheet path-integral, (?1) Σ/IΣ ,φ
?y

.

(63)

Now, we can make precise the meaning of the sign of the RP2 diagram. Suppose I acting on M has a ?xed point P . The constant map φ0 from S 2 → P trivially obeys condition (60). ?0 : S 2 → MZ then collapses to the identity map RPN → RPN and the integral The map φ 2 Z2 2 ? RP , φ y is simply evaluating y on the 2-cycle S 2 ×Z {P } ? = RP2 of MZ = S N ×Z M .
0
2 2 2

This de?nition agrees with the standard one when I acts around P by the action I9?p with

Z2 acts on M ? freely, y ? can be viewed as an element of H 2 (M ? /I , Z2 ). We now note that ?0 : S 2 → MZ can be continuously deformed to a map φ ?? where φ? is a map of the map φ 2 Z2

(9 ? p) ≥ 3. To see this, ?rst consider the space M ? obtained from M by deleting all the 2 2 Z2 ?xed points. By restriction, y ∈ HZ (M, Z2 ) yields an element y ? ∈ HZ (M ? , Z2 ). Since 2 2

S 2 to a small Z2 invariant 2-sphere surrounding P and lying in M ? . Or equivalently, a map from RP2 to a small RP2 surrounding P and lying in M ? /I . It is then clear that,
2 ? RP2 , φ? 0 y = RP , φ? y = RP2
9

? φ? ?y ,

(64)

For a discussion of equivariant cohomology, see appendix D, and section 4.4. Here, we provide a brief ? summary of some basic de?nitions. The equivariant cohomology HG (X, R), with G a group acting on a space X and R a coe?cient ring, is de?ned to be the ordinary cohomology H ? (MG , R) where MG is the ?bre product MG = EG ×G M := (EG × M )/G. Here EG is the universal G-bundle over the classifying space BG. In the context of our orientifold discussion, G = Z2 = {1, I} for X = M and R = Z2 . For G = Z2 , we can take EG = S N so that BG = RPN with N → ∞ strictly speaking. However, it is enough to take large but ?nite N for most of the practical purposes. If G acts on X freely (as is the case ? for X = Σ and G = {1, IΣ }), then HG (X, R) = H ? (X/G, R) since EG is contractible.

48

2 orientifolds can be classi?ed from the closed string perspective by the group HZ (M, Z2 ). As 2 we shall see below, there can be further constraints.

which is the standard meaning of the sign of the RP2 diagram surrounding P . To summarize, the possible con?gurations of plus (+ or +′ ) versus minus (? or ?′ )

3.1.2

Some pathologies

We note that both ?′ and +′ orientifolds are inconsistent for p = 9, 8, 7, 6 if we insist, as we shall, on backgrounds preserving sixteen supersymmetries. There are a number of ′ ways to see this. For example, for p = 6 we can probe the O 6? with a D2-brane. The theory on the probe is an N = 4 Sp(1) gauge theory with a half-hypermultiplet in the fundamental representation. This three-dimensional gauge theory has a Z2 anomaly, and is therefore inconsistent. The only way to cancel such an anomaly is by including a Chern– Simons term which makes the gauge-?eld massive. However, Chern–Simons terms can only be written down for N ≤ 3 supersymmetry in d = 3 [52]; N = 4 supersymmetry does not allow a short massive vector multiplet. Rather, there are arguments given in [6] which suggest that O 6? planes require a non-zero cosmological constant, and so can only be found ′ in massive type IIA supergravity. Similarly, if O 6+ is distinguished from O 6+ by a ?ux associated with G0 , it is expected to be in a theory with a cosmological constant. In our subsequent discussion, we shall classify ? and ?′ con?gurations by using the fact that we can T-dualize these con?gurations to type I. The type I requirement that the O (32) gauge bundle lift to a Spin(32)/Z2 bundle with vanishing Chern–Simons invariant constrains the possible ? and ?′ con?gurations. This actually gives an alternative and stronger argument for excluding O 6? in a T 3 /Z2 orientifold. The details of the derivation are found in appendix C. This anomaly cancellation argument and the D2-brane probe
′ ′

brane argument are not completely independent. Suppose that we na¨ ?vely construct the 3 ?′ T /Z2 orientifold with 8 O 6 planes. Somewhere, we should encounter an inconsistency. From the chain of dualities connecting this type IIA orientifold to the heterotic string on T 3 , we ?nd a hint of where to look. In the heterotic theory, we argued that obstructions to certain compacti?cations arise from states associated to winding strings. After S-duality, these correspond to winding D-strings in a type I compacti?cation. The states on these winding strings transform non-trivially under a Z2 that should be divided out in the construction of the bundle. As a consequence, the phases of these states have a Z2 ambiguity, and the theory is not well-de?ned. Nevertheless, proceeding with this inconsistent theory ′ and T-dualizing 3 directions leads to the T 3 /Z2 orientifold with 8 O 6? planes. In the

49

process, the D-string becomes a D2-brane, and the Z2 ambiguity in the phases of winding states becomes the Z2 anomaly in the gauge theory on the probe brane. The anomaly therefore has multiple dual realizations. 3.2 The classi?cation

Our discussion below focuses on type II orientifolds of T 9?p × RD=p+1. We note that there are 29?p Z2 ?xed points, and the type of orientifold plane has to be speci?ed at each ?xed point. We use the notation {?, ?′ , +, +′ } for Op? , Op? , Op+ and Op+ , respectively. For example, the notation (?, +, +, ?′ ) speci?es a (possible) con?guration in eight dimensions, or equivalently, a particular orientifold of T 2 . For low enough dimensions, ordering is also required to completely specify the con?guration. In what follows we provide the complete list of possible orientifolds in dimensions D := p + 1 = 10, 9, 8, 7. We also give the complete list of orientifolds involving only ? and ?′ in dimensions D = 6, 5. In D = 6, 5 we do not attempt to completely classify con?gurations involving + and +′ but we will comment on some of the new issues that arise. To see what kind of {+, +′ } con?gurations are possible from the closed string point of view, we have 2 included an explicit computation of the equivariant cohomology HZ (T n , Z2 ) for n = 1, 2, 3 2 in appendix D. The results are stated in the main text below. 3.2.1 D = 10
′ ′

2 The equivariant cohomology HZ (R10 , Z2 ) is the cohomology of the classifying space B Z2 = 2

this is the standard type I string on R10 which indeed preserves 16 supersymmetries. However, O 9+ has positive D9-brane charge which is cancelled by introducing anti-D9-branes. Since O 9+ and the anti-D9-branes preserve di?erent supersymmetries, the system is not supersymmetric. 3.2.2 D=9

RP∞ which is H 2 (RP∞ , Z2 ) = Z2 . The two choices correspond to O 9? and O 9+ . O 9? has D9-brane charge ?32 and can be cancelled by introducing 32 D9-branes. Of course,

2 The equivariant cohomology has rank 2, HZ (T 1 , Z2 ) = Z2 ⊕ Z2 . The two generators are 2 non-zero on RP2 at the two ?xed points of S 1 in the sense described above. The four

elements correspond to (?, ?), (+, ?), (?, +), (+, +). 50

The con?guration (+, +) is not supersymmetric for the same reason that O 9+ is not supersymmetric. The two con?gurations (+, ?) and (?, +) are related by a di?eomorphism.

Therefore in 9 dimensions, there are essentially 2 possible orientifold con?gurations. The ?rst, (?, ?), has 32 D8-branes and is simply T-dual to type I compacti?ed on a circle. This component also contains the E8 × E8 string [39].

The second component, (?, +), has no net D-brane charge. Therefore no branes can be added while preserving supersymmetry so there is no enhanced gauge symmetry. This

compacti?cation is T-dual to type IIB on S 1 /δ ? where δ is a half-shift along the circle and ? is world-sheet parity. For a more detailed discussion, see [17,53]. Together with the CHL string, this gives a total of 3 distinct components in the moduli space of perturbative string compacti?cations. 3.2.3 D=8

2 The equivariant cohomology now has rank 4, HZ (T 2 , Z2 ) = (Z2 )4 . The four generators are 2 in one-to-one correspondence with the four Z2 ?xed points—each has a non-trivial value on

RP2 at the corrsponding ?xed point and is trivial at the other three points. The possible orientifolds up to di?eomorphisms are (?, ?, ?, ?), (+, ?, ?, ?), (+, +, ?, ?).

The last is obtained from the sum of two generators. In addition, there are non-supersymmetric con?gurations. The ?rst component (?, ?, ?, ?) has 32 D7-branes and is T-dual to the usual type I string on T 2 . The second component (?, ?, ?, +) is more interesting. The orientifold

planes have total of ?16 units of D7-brane charge and therefore require 16 D7-branes. This orientifold is T-dual to type IIB on T 2 modded out by the world-sheet parity, ?, in the presence of a half-integral background NS-NS B -?eld [44, 45]. Although world-sheet parity sends B → ?B , a half-integral value of B is permitted since B takes values in a

torus. Orientifolds with quantized B -?eld backgrounds have been explored, for example, in [54,55]. This orientifold is equivalent to a compacti?cation without vector structure [45]. By the T-duality argument of section 2.4.2, we know that this component containing type I without vector structure also contains the 8-dimensional CHL string. The third and ?nal component (?, ?, +, +) is T-dual to the compacti?cation of the 9-dimensional (?, +) orientifold to eight dimensions. This has no D-branes and therefore no gauge symmetry. 51

3.2.4

D=7 The seven generators

2 The equivariant cohomology has rank 7, HZ (T 3 , Z2 ) = (Z2 )7 . 2

y1 , . . . , y7 having the following property: let us use P1 , . . . , P8 to denote the eight Z2 ?xed points of T 3 . Then yi has a non-trivial value on both the RP2 surrounding Pi and the one surrounding P8 , while it is trivial at the other six points. This shows that the allowed orientifolds are (?8 ), (?6 , +2 ) and its permutations, and (?4 , +4 ) and its permutations. We

exclude those cases with more + than ? planes since they require anti-D6-branes. That an odd number of O 6+ planes is not allowed can also be understood by an elementary argument. On T 3 /Z2 , we can enclose each O 6 plane with an RP2 . The eight of them together correspond to a trivial cycle, and therefore the product of the eight RP2 diagrams must have sign +1. Therefore, the number of O 6+ planes must be even. The ?rst case, (?8 ), has 32 D6-branes. It is T-dual to the standard compacti?cation of type I on T 3 . The second case, (?6 , +2 ), requires 16 D6-branes. All permutations are

tifold planes on a single face of the cube. The remaining four + planes are found at the remaining vertices. This is the con?guration that follows from a dimensional reduction

there is an interesting subtlety. The various permutations are not necessarily di?eomorphic to each other. There are essentially 2 distinct ways to place the orientifold planes at the vertices of the cube. The ?rst can be characterized by placing the four ? orien-

circle and T-duality on that circle. The third case (?4 , +4 ) has no D6-branes and therefore no open strings. In this case,

di?eomorphic to each other so there is essentially one con?guration of this type. This is obtained from the (?, ?, ?, +) orientifold on T 3 /Z2 by compacti?cation along another

of the (?, ?, +, +) case. It is therefore automatically consistent. However, we could also consider the case where a single adjacent pair of + and ? planes are interchanged. This

gives a distinct con?guration. For a ?xed ordering of the Z2 ?xed plane, let us denote these 2 cases by (?, ?, ?, ?, +, +, +, +) and by (?, ?, ?, +, +, +, +, ?). Both possibilities are 2 allowed from the closed string point of view. Both are realized as elements of HZ (T 3 , Z2 ). 2 This is easy to see because the equivariant cohomology is a group. Noting that all permutations of (?, ?, ?, ?, ?, ?, +, +) are realized as the elements of the group, we see that

the ?rst case is simply the sum of (?, ?, ?, ?, +, +, ?, ?) and (?, ?, ?, ?, ?, ?, +, +). The second possibility is the sum of (?, ?, ?, +, +, ?, ?, ?) and (?, ?, ?, ?, ?, +, +, ?).

Thus, both are consistent con?gurations in perturbative closed string theory and they are distinct. Although the two are distinct con?gurations perturbatively, it is possible that they are equivalent non-perturbatively (if they are both consistent). We shall mention how

52

this possibility can be checked in section 4.6.1 on M theory compacti?cations with ?ux. This subtlety involving the ordering of orientifold planes also appears for lower-dimensional orientifold con?gurations. Therefore, we have a total of 4 distinct orientifold compacti?cations, and an additional 4 components in the E8 × E8 string moduli space. The standard E8 × E8 compacti?cation D=6

together with the CHL string/Z2 -triple are contained in the orientifold moduli spaces. 3.2.5

In this dimension, all four kinds of orientifold planes are possible, and indeed each is realized in a particular T 4 /Z2 orientifold. Let us ?rst restrict our attention to O 5? and O 5? planes only. With this restriction, there are only two possibilities: one is (?16 ) and the other is


(?′ 16 ). The latter case is quite interesting. If we coalesce the 8 D-brane pairs on one of the ′ O ? planes, we get an SO (17) maximal gauge group. The rank reduction is therefore 8. As discussed in appendix C, the Chern–Simons invariant for any sub-three-torus is integer as required by anomaly cancellation. However, this compacti?cation clearly has vector structure. The type I dual of this orientifold compacti?cation therefore has a gauge bundle with vector structure which is not connected to the trivial bundle. We met this bundle in section 2.1.4, it is a non-trivial quadruple. Here we have found its orientifold realization. In addition to these two cases, we have the dimensional reductions of the higherdimensional cases which are (?12 , +4 ) and (?8 , +8 ). Recall that the ?rst case includes

the CHL string, the compacti?cation with no vector structure and the Z2 triple in its moduli space. However, the duality chain of section 2.4.2 showed that the quadruple found above is in the same moduli space as the CHL string on T 4 . Somewhat surprisingly, this implies that the (?12 , +4 ) and (?′ 16 ) orientifolds are in the same moduli space. We can give further motivation for this inference in a simple way. Compactify both con?gurations ′ to four dimensions on T 2 . This gives (?48 , +16 ) and (?′ 16 , ?48 ). S-duality maps O ? ? O + and O ? to itself. These con?gurations are therefore S-dual and their moduli spaces must necessarily agree. Unlike the prior cases, we will not attempt a complete analysis of orientifold con?gurations in D = 6. There are two other cases worth mentioning, however. The ?rst is the case of (?10 , +6 ) which made an appearance in [17]. This orientifold corresponds to a gauge
2 bundle with w2 non-zero. The last case is (+′ 4 , ?′ 12 ). In this case, we need an additional 2 pairs of D-branes. The maximal gauge group is then Sp(2) and the rank reduction is 14. Based on the structure of its moduli space, it is natural to conjecture that this orientifold is

53

dual to the quadruple with no vector structure heterotic/type I compacti?cation described in section 2.1.4. It would certainly be interesting to analyze this case further. 3.2.6 D=5

In this dimension, all four ?avors of orientifold plane can again be realized by T 5 /Z2 orien′ tifolds. If we restrict our attention to O 4? and O 4? , there are only three possibilities up to di?eomorphisms: (?32 ), (?′ 32 ) and (?16 , ?′ 16 ). We need a few additional words to actually describe these orientifolds. On a torus of su?ciently high dimension, the number of ? and ?′ planes does not in general completely specify the con?guration up to di?eomorphisms.

in natural coordinates) of the S 1 . All the O 4? planes reside at the 16 ?xed points in the “middle” of the S 1 (θ = π ).

case, the only allowed con?guration corresponds to the one where the T 5 can be factorized into S 1 × T 4 so that all the O 4? planes sit at the 16 ?xed points at the “origin” (θ = 0


The actual pattern of the distribution must be speci?ed. There is no room for such an ambiguity for (?32 ) and (?′ 32 ) but there are several possibilities for (?16 , ?′ 16 ). In this

We can ?nd M theory duals for each of these cases. Recall that a single O 4? plane at the origin of R5 /Z2 is dual to M theory on R5 /Z2 × S 1 . Note that the Z2 action acts on ′ the 3-form C of M theory by inversion. On the other hand, O 4? is dual to M theory on (R5 × S 1 )/Z2 where the Z2 acts on the last circle by a shift of a half period [47]. The dual of (?32 ) is then M theory on T 5 /Z2 × S 1 as we would naturally expect. The dual of (?′ 32 )

is M theory on (T 5 × S 1 )/Z2 . The dual of (?16 , ?′ 16 ) is given by M theory on (T 5 × S 1 )/Z2 . The Z2 now acts on the coordinates x1 , x2 , x3 , x4 , x5 , x11 of T 5 × S 1 , where xi ≡ xi + 2π , by (x1 , x2 , x3 , x4 , x5 , x11 ) ?→ (?x1 , ?x2 , ?x3 , ?x4 , ?x5 , x11 + x1 ). (65)

in D = 6, these should be part of the same moduli space. In addition, we have (+16 , ?16 ) and (?20 , +12 ). Note that (?′ 32 ) has no enhanced gauge symmetry so the rank reduction is 16. Further, it does admit vector particles. This suggests that it is dual to type I with a non-trivial quintuple. This particular bundle made an appearance in section 2.1.5. By 54

As in the D = 6 case, we shall only discuss select additional examples. From dimensional reduction, we obtain (?24 , +8 ) and (?16 , ?′ 16 ). However, because of the duality explained

that we would not be able to construct an M theory dual if other distributions of 16 ? and 16 ?′ planes were permitted.

In the neighborhood of the “origin” x1 = 0 of the ?rst circle, the Z2 action is of the type R5 /Z2 × S 1 and the O4-planes are all ?. In the neighborhood of the “midpoint,” x1 = π , the Z2 action is of the type (R5 × S 1 )/Z2 and indeed the O4-planes are all ?′ . We note

the chain of dualities in section 2.4.2, we see that this orientifold is further equivalent to a compacti?cation of the E8 string with a quintuple in both E8 factors. At ?rst sight, it also seems plausible that (?′ 32 ) could be identi?ed with (+16 , ?16 ). By (+16 , ?16 ), we mean the con?guration obtained by toroidally compactifying (+, ?) in nine dimensions. As support for this conjecture, note that on compacti?cation to four dimensions, we ?nd two con?gurations (?32 , ?′ 32 ) and (?32 , +32 ) which are S-dual. Of course, this alone does not demonstrate the equivalence. We can also study the M theory description of (+16 , ?16 ). Recall that the D = 9 (+, ?) orientifold is described, after Tduality, by IIB on S 1 /δ ? where δ is a half-shift along the circle [17, 53]. Let us compactify this con?guration on a further T 4 . We want to determine the corresponding M theory description. It is convenient to ?rst compactify on one additional circle S 1 . We can then T-dualize ?ve times on the T 5 which leaves us in type IIA sending δ? → δ ?Z2 ., (66)

where the Z2 acts by inversion on the T 5 . The operation ?Z2 lifts in M theory to inversion of the T 5 and the 3-form C [53]. This leaves us with M theory on (T 5 × S 1 )/Z2 × S 1 where the Z2 acts as δ on the S 1 factor and by inversion on T 5 . It also inverts the 3-form C . This suggests that the M theory description of (+16 , ?16 ) is the same as the description of (?′ 32 )

which is further evidence in favor of their equivalence.

4
4.1

Compacti?cations of M and F Theory
Some preliminary comments

In prior sections, we have discussed aspects of perturbative string compacti?cations: either heterotic/type I or type II orientifolds. These descriptions are valid when the string coupling constant is small, regardless of the size of the compacti?cation space. It is natural to ask what kind of description is valid when the string coupling constant is large. The answer to this question depends on how we treat the string scale α′ and the volume of the compacti?cation space as gs → ∞. As an example, let us take the CHL string in 9
2/3

dimensions. If we wish to hold the 11-dimensional Planck scale ?xed then α′ gs must be held constant. In this limit, the strong coupling description will involve M theory

compacti?ed on a space which has been argued to be the M¨ obius strip [56]. This is analogous to the Hoˇ rava–Witten description of the strongly-coupled E8 × E8 heterotic string. On the other hand, another strong coupling description of the E8 × E8 string on T 2 is given by F theory on K 3. When is this a valid description? Like M theory, F theory 55

generically has no perturbative expansion and the condition for validity is that the base B of the elliptic ?bration (in this case K 3) be large in string units. It is convenient to analyze this relation at the point in the moduli space where the gauge group is broken to (Spin(8)4 × U (1)4 )/Z3 2 . Let the torus be square with volume V , and let gH denote the heterotic string coupling. T-duality along one cycle of the torus takes us to the Spin(32)/Z2
′ 2 2 ′ heterotic string with ten-dimensional coupling (gH ) = gH α /V on a torus of volume α′ . S-duality then takes us to the type I string with, 2 gI =

V
2 ′ gH α

,

VI =

V . 2 gH

(67)

Two further T-dualities on the resulting torus take us to the type IIB on T 2 /?(?1)FL Z2 , with couplings:
2 gB 2 α ′ gH = , V

VB =

2 (α ′ )2 g H . V

(68)

This is an orientifold limit of F theory on K 3 [57]. We see that F theory is a good description when gH becomes large with the volume V ?xed in string units. In this regime of the moduli space, we can use F theory to describe the physics. A more general statement goes as follows. With sixteen supersymmetries, each component of the moduli space is highly constrained; for example, the moduli space metric does not receive quantum corrections. If the e?ective theory is formulated in d space-time dimensions, then a given component of the moduli space can be described in terms of an even lattice L of signature (s + 10 ? d, 10 ? d). (We use the convention in which there are more positive than negative eigenvalues in the lattice.) When d > 4, this space takes the form ML := O (L)\DL × R+ (69)

where DL = O (s + 10 ? d, 10 ? d)/(O (s + 10 ? d) × O (10 ? d)) is the symmetric space associated to the lattice, and O (L) is the orthogonal group. This must be modi?ed somewhat in low dimension: when d = 4, the universal cover of the given component of the moduli space is DL × h where h is the upper half plane, and

when d = 3, the universal cover of the given component is DL , where L is the direct sum of L and a lattice of signature (1, 1). A given moduli space has boundaries that correspond to the various ways in which a theory can degenerate. A di?erent physical description is typically valid as we approach 56

a boundary of the moduli space. Our goal in this section is to study a class of M and F theory compacti?cations which naturally include dual descriptions for the perturbative compacti?cations described in earlier sections. These include purely geometric models and also models with background ?uxes, as in the case studied by Schwarz and Sen [14]. We now require a more general discussion of the boundaries of moduli spaces than appeared in our initial discussion of section 2.4.3. The possible boundary components of ML are determined as follows (setting d ≥ 5 for simplicity). One type of boundary is

given by approaching one end or the other of the R+ factor. These include non-stringy limits: for example, in a conventional heterotic or type II compacti?cation one of these limits is the zero-coupling limit which yields a conformal ?eld theory rather than a string theory. This is the kind of description that we studied in prior sections. As discussed

that is, every x ∈ M satis?es q (x) = 0. The lattice associated to the boundary component is then given by LM := M ⊥ /M , and the boundary component takes the form O (LM )\DLM , where we suppress the R+ factor. To determine all boundary components, all isotropic sublattices M must be found, modulo the action of the orthogonal group O (L). If the sublattice M has rank m, the limiting theory will have e?ective dimension d + m. In general, LM only determines part of a component of the moduli space of the limiting theory in e?ective dimension d + m. That is (suppressing the R+ factors), the space O (L)\DL is glued to a space O (L′ )\DL′ along the boundary component O (LM )\DLM , where L′ is a lattice of signature (s′ + 10 ? (d + m), 10 ? (d + m)) which represents the limiting

theory whose e?ective dimension is greater than d. A boundary component of O (L)\DL is determined by an isotropic sublattice M ? L,

above, the strong coupling limit will typically have an M or F theory description. Let us here instead focus on the other class of limits, given by boundary components of the O (L)\DL factor. These boundary components typically correspond to a limiting stringy

e?ective theory. The gluing is speci?ed by an inclusion LM ? L′ with L′ /LM a positive de?nite lattice of rank s′ ? s ≥ 0. The lattice L′ must be determined by analyzing the

physics of the limiting process; it agrees with LM for some boundary components but is larger than LM for others. For example, the CHL string in nine dimensions has lattice [38] L ? = Γ1,1 ⊕ E8 . There ? is a unique boundary component, corresponding to Lx = E8 . However, as we discussed in section 2.4.3, in the decompacti?cation limit we actually obtain the heterotic string in ten dimensions with lattice L′ ? = E8 ⊕ E8 . 57

Decompacti?cation limits of components corresponding to non-trivial discrete choices of Wilson lines exhibit a similar phenomenon: viewed from the perspective of the higherdimensional theory, the non-trivial types of Wilson lines can only be turned on for special values of the moduli. Thus, it is along a subspace O (LM )\DLM of O (L′ )\DL′ that the moduli space of the lower dimensional theory “attaches,” and one recovers additional degrees of freedom in the decompacti?cation limit. A similar phenomenon occurs in F theory [13], where compacti?cation along an additional circle gives theories which are dual to M theory on elliptically ?bered manifolds. The discrete Wilson line degrees of freedom correspond to the possibility of compactifying M theory on an elliptically ?bered manifold without a section, which typically is only possible for special values of the moduli. 4.2 Six-dimensional M theory compacti?cations without ?uxes

Our starting point is M theory compacti?ed to 6 dimensions, but we will also include remarks about lower-dimensional compacti?cations.10 Let us begin by excluding any background ?ux so that this is a purely geometric compacti?cation. We shall also restrict to supersymmetric compacti?cations which take the form of a compact Riemannian manifold times Minkowski space. The metric on the compact part must then admit a covariantly constant spinor, which leads to restrictions on the holonomy. In fact, the list of possibilities can be determined by examining the holonomy classi?cation of Riemannian metrics, which (when formulated carefully [58]11 ) implies that every compact Riemannian manifold Y admitting a covariantly constant spinor takes the form Y = (T k × (X1 × · · · × Xm )/Γ)/G, (70)

manifold whose holonomy is either SU (ni ), Sp(ni ), G2 , or Spin(7), and Γ and G are ?nite groups which act without ?xed points. The e?ective dimension of the physical theory is d = 11 ? dim Y . In order to guarantee at least 16 supercharges in the e?ective theory, there must be at least half as many holonomy-invariant spinors on this manifold as there are on ?at space. Because each Xi in eq. (70) reduces the set of holonomy-invariant spinors by at least a factor of two, and the factor is greater than two except in the case of holonomy SU (2) = Sp(1),
10 11

where T k is a torus of dimension k ≥ 0, each Xi is a compact simply-connected Riemannian

The corresponding type IIA string compacti?cations were studied in detail in [42, 43]. We thank B. McInnes for helpful correspondence on this point.

58

there are two cases: either (1) there is a single Xi with holonomy SU (2) = Sp(1) (i.e., a K3 surface), the group Γ is trivial, and the group G preserves all of the spinors on T k × X , or (2) there is no Xi at all (and hence no Γ) and the group G preserves one-half of the spinors on T k . In the second case, possibly after replacing the torus by a ?nite cover or a ?nite quotient, we can assume that [59] T k = T 4 × T k?4 with the group action preserving the

1-forms on T k?4 and the holomorphic 2-form on T 4 , but leaving no invariant 1-forms on T 4 . To get the correct holonomy, the image of G in SO (k ) must lie in an SU (2) subgroup corresponding to a complex structure on the T 4 factor. In both cases, then, we can write Y = (T ? × Z )/G where Z is either a K3 surface or a complex 2-torus (that is, a real 4-torus on which a complex structure has been speci?ed). The lattice L can be directly determined from the cohomology of Y . When d > 4, the

possible gauge charges for the theory are described by H 1 (Y, Z) ⊕ H 2 (Y, Z) ⊕ H 5 (Y, Z), (71)

(with the ?rst factor coming from Kaluza–Klein modes, and the latter two coming from the M theory three-form and its dual six-form). This cohomology group comes equipped with a natural quadratic form, to be described below. Bearing in mind the sign conventions, we can identify the free part of eq. (71) with the lattice L(?1) if d > 5. (There is also the

possibility of torsion in eq. (71), which we will not explore in any detail.) When d = 5, the free part of the gauge lattice in eq. (71) takes the form L(?1) ⊕ x with q (x) = 0; the

element x is unique up to ±1, so L(?1) can be recovered by modding out the free part of the gauge lattice by the span of x. When d = 4, the free part of the gauge lattice is simply H 1 (Y, Z) ⊕ H 2 (Y, Z) (72)

due to the self-duality of gauge ?elds in this dimension, and this coincides with L(?1). The description of the quadratic form on L depends on the dimension d of the e?ective theory. If d = 6, both H 1 (Y ) and H 5 (Y ) are 1-dimensional and this part of the lattice is isomorphic to Γ1,1 . The quadratic form on H 2 (Y, Z)/torsion is inherited from the intersection form on the resolution Z/G of Z/G via the isomorphism H 2 (Y, Z)/torsion ? = H 2 (Z/G, Z)/torsion ? = the orthogonal complement of the exceptional divisors in H 2 (Z/G, Z). Thus, although the action of G on S 1 × Z has no ?xed points and a smooth quotient, keeping 59

(73)

track of the singular points on Z/G provides a convenient bookkeeping device for analyzing

k 0 1 2

G {e}

Maximum dimension of e?ective theory 7 6

Zm , m = 2, 3, 4, 5, 6, 7, 8 Z2 × Zm , m = 2, 4, 6 or Zm × Zm , m = 3, 4

5 4 3

3 4

(Z2 )3 (Z2 )4

Table 7: Automorphisms of K3 surfaces and the resulting M theory vacua.

the lattice associated to (S 1 × Z )/G. There is one subtlety associated to this, however. If E ? H 2 (Z/G, Z) denotes the lattice spanned by the exceptional divisors, then (E ⊥ )⊥ will

be larger than E : there are Q-linear combinations of exceptional divisors which belong to H 2 (Z/G, Z). In fact, the ?nite group (E ⊥ )⊥ /E provides another important invariant in this situation. Consider ?rst the case in which Y = (T ? × Z )/G where Z is a K3 surface. The action of G preserves the two factors T ? and Z . In order to preserve the invariant spinors on T ? it must act by translations on that factor, and in order to preserve the invariant spinors on Z it must preserve the holomorphic 2-form on Z . Abelian group actions which preserve the holomorphic 2-form on Z were classi?ed by Nikulin [37]; there are 15 cases, including the trivial group. The resulting vacua are displayed in table 7, where k denotes the number of generators in the group, and hence the minimal dimension of a torus factor in Y . The lattices H 2 (Z/G) in these cases are also known [37]. In table 8 we exhibit the lattices

L associated to six-dimensional e?ective theories built from M theory on Y = (S 1 × K 3)/G when G is trivial or cyclic;12 we also describe the singularities which are found on Z/G itself, using the ADE notation for rational double points, and the ?nite group (E ⊥ )⊥ /E where E is the sublattice of H 2 (Z/G, Z) spanned by the exceptional divisors. The corresponding facts about non-cyclic groups G (where the e?ective theory has lower dimension) are given in table 9. (The “discriminant group” which appears in that table is discussed in appendix A.) Turning to the case where Y = (T ? × Z )/G with Z a complex 2-torus, we need an
The descriptions we give of the lattices can be inferred from the descriptions in [37] using techniques from [60].
12

60

G Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8

Lattice L Γ4,4 ⊕ E8 ⊕ E8 Γ1,1 (2) ⊕ Γ3,3 ⊕ D4 ⊕ D4 Γ1,1 (3) ⊕ Γ3,3 ⊕ A2 ⊕ A2 Γ1,1 (4) ⊕ Γ3,3 ⊕ A1 ⊕ A1 Γ1,1 (5) ⊕ Γ3,3 Γ1,1 (6) ⊕ Γ3,3 ? ? 28 7 ? Γ2,2 ⊕ ? 7 2 ? ? 4 0 ? Γ2,2 ⊕ ? 0 2

Singularities on K3/G (E ⊥ )⊥ /E none 8A1 6A2 4A3 + 2A1 4A4 2A5 + 2A2 + 2A1 3A6 {e} Z2 Z3 Z4 Z5 Z6 Z7 Z8

2A7 + A3 + A1

Table 8: Choices for G together with their associated lattices for Y = (S 1 × K 3)/G. abelian group G acting on Z in such a way that the holomorphic 2-form is preserved by the G-action. Moreover, the group action must not leave any holomorphic 1-form invariant, so G cannot act entirely by translations. Such group actions were classi?ed in modern language by Fujiki [59] (although the classi?cation was essentially done more than ninety years ago by Enriques and Severi [61]). The computations of the corresponding lattices in the case of abelian group actions were made in [62–64] and were applied in the physics literature in [65]. The only possibilities are cyclic groups G = Zm with m = 2, 3, 4, or 6, and all of these lead to theories of e?ective dimension six. The associated lattices are described in table 10,13 where we also give the singularities on Z/G and the ?nite group (E ⊥ )⊥ /E . 4.3 F theory compacti?cations without ?ux

It is common to describe an F theory vacuum in terms of a Ricci-?at manifold Y together with an elliptic ?bration π : Y → B . However, to specify an F theory vacuum, we actually only need 1. the manifold B with a subset ? of real codimension 2 (where ? speci?es the location
13

Again, matching the descriptions in table 10 with those in [62–64] requires techniques from [60].

61

Compacti?cation dimension 5 5 5 5 5 4 3

G Z2 × Z2 Z2 × Z4 Z2 × Z6 Z3 × Z3 Z2 × Z2 × Z2 Z4 × Z4

Rank of H 2 (K 3/G) 10 6 4 6 4 8 7

Discriminant group (Z2 )8 (Z2 )2 × (Z4 )2 (Z2 ) × (Z6 ) (Z3 )4 (Z4 )2 (Z2 )8 (Z2 )7

Z2 × Z2 × Z2 × Z2

Table 9: Choices for G for lower-dimensional compacti?cations.

of the singular ?bers in the ?bration π ), 2. a monodromy representation π1 (B ??, p) → SL(2, Z) and a “j -function” j : B → CP1 compatible with the monodromy (which are speci?ed by the complex structure on the ?bers of π ), and 3. a metric on B ? ? whose asymptotics near ? are described by the Greene–Shapere–

Vafa–Yau ansatz [66] (which can be seen as a limit of metrics on Y as the area of the elliptic ?ber approaches zero [67]).

The F theory vacuum is then described as type IIB string theory compacti?ed on B with the given metric and with branes along ?, using the S-duality of type IIB theory to compensate for the SL(2, Z) monodromy. If we begin from M theory compacti?ed on Y , and take the limit as the area of the ?bers of π approaches zero, then one dimension of the e?ective theory decompacti?es [43,68] and we obtain the F theory vacuum in the limit. This is sometimes referred to as “F theory compacti?ed on Y ” although the full data of Y is not needed. Conversely, if the elliptic ?bration on Y has a section, then the standard M theory/F theory duality [69] asserts that F theory on Y × S 1 (with a trivial Wilson line) is dual to M theory on Y . When the elliptic ?bration on Y does not have a section, there is always an associated manifold J (Y ), the Jacobian of the ?bration, which has an elliptic ?bration with a section that gives rise to the same monodromy and j -function data as the elliptic ?bration on the

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G Z2 Z3 Z4 Z6

Lattice L Γ3,3 (2) ⊕ Γ1,1 ? 2 Γ1,1 (3) ⊕ Γ1,1 ⊕ ? 1 ? 2 Γ1,1 (4) ⊕ Γ1,1 ⊕ ? 0 ? 6 Γ1,1 (6) ⊕ Γ1,1 ⊕ ? 0 ? ? ? ? ? ?

Singularities on T 4 /G 16A1 9A2

(E ⊥ )⊥ /E (Z2 )5 (Z3 )3 Z4 × (Z2 )2 Z6

1 2 0 2 0 2

4A3 + 6A1

A5 + 4A2 + 5A1

Table 10: Choices for G together with their associated lattices for Y = (S 1 × T 4 )/G. original manifold. Thus, F theory cannot distinguish between the compacti?cation on Y and the compacti?cation on J (Y ). The M theory/F theory duality can be extended to cover this case [13], where it becomes the assertion that when Wilson line data is included, F theory on J (Y ) × S 1 is dual to the union of the M theory moduli spaces on Yk for all manifolds Yk with the same Jacobian ?bration J (Yk ) = J (Y ).14 Thus, discrete choices for Wilson lines in F theory correspond in M theory to di?erent elliptic ?brations with the same Jacobian ?bration. Typically, such discrete choices are only present for special values of moduli. An elliptic ?bration on a Ricci-?at manifold Y always determines a class x ∈ H 2 (Y, Z) with q (x) = 0. Thus, in the case of 16 supercharges, the boundary lattice associated to taking the F theory limit is Lx = x⊥ / x . If the elliptic ?bration admits a section, then this lattice can be used to describe the entire component of the F theory moduli space. On the other hand, if the elliptic ?bration does not admit a section, then the lattice L′ for the component of F theory moduli space associated to J (Y ) is typically larger than Lx .

To ?nd these components in detail, we need to examine possible elliptic ?brations on the Ricci-?at manifolds Y = (T ? × Z )/G (with Z either a K3 surface or a T 4 ). An elliptic ?bration π : (T ? × Z )/G → B will lift to an elliptic ?bration π : T ? × Z → B which is G-invariant. If π has a section then its inverse image will be a G-invariant section of π . The group G acts on the base B and B = B/G. We let G0 be the subgroup of G which acts

trivially on the base B . It is easy to see that if G0 is non-trivial, there cannot be a section for the ?bration π : the group G0 acts by translations on the ?bers, so cannot preserve a
14

For other comments on F theory compacti?cations without section, see [70].

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section of π , but π must have a G-invariant section when π has a section. Thus, when π has a section the action of G on B must be faithful. Let us ?rst analyze the cases in which π does not have a section. As indicated above, the lack of a section can be attributed to a non-trivial group G0 which acts trivially on the base B . Let us begin with the case Z = K 3 and return to the case Z = T 4 in a little while. In fact, it is easy to see that the Jacobian of the elliptic ?bration on (T ? × Z )/G0 is just T ? × (Z/G0 ). The manifold Z/G0 is a singular K3 surface, and this is in fact part of a larger family of manifolds of the form T ? × K 3. The possible G0 ’s which can occur here can be classi?ed: we need to know which ?nite abelian groups G0 could act as a group of

translations on the ?bers of an elliptic ?bration on a K3 surface. This classi?cation was carried out by Cox [71], who found almost exactly the same list as Nikulin’s classi?cation of abelian automorphism groups [37], except that (Z/2Z)3 and (Z/2Z)4 cannot occur as translations on the ?bers. Thus, in all of these cases, there is a limit of the M theory moduli space in which the limiting F theory vacua gain additional degrees of freedom which allow them to be part of the “standard component” of the F theory moduli space on S 1 × K 3. Since the lattice of the component of M theory moduli space takes the form L(?1) ? = Γ1,1 ⊕ H 2 (Z/G0 , Z), we must have Lx (?1) = H 2 (Z/G0, Z) in order to allow the lattice Lx to be embedded into the “standard lattice” Γ3,3 ⊕ E8 ⊕ E8 . This is exactly the type of lattice limit which is found for the standard component. There can be “mixed” cases as well, in which both G0 and G1 := G/G0 are non-trivial. In such a case, the Jacobian of the elliptic ?bration on (T ? × Z )/G will be an elliptic ?bration on (T ? × (Z/G0))/G1 . The limiting theory “attaches” to the moduli space (T ? × Z ′ )/G1 , and the lattice decomposition L ? = M1 ⊕ Lx should use the rank two lattice M1 associated with G1 . (We will determine those lattices below.) Turning now to the cases in which π has a section, there will be a component of the F

theory moduli space for each case in which π has a section. When Z is a K3 surface, both B and B are isomorphic to CP1 so G must have a faithful action on CP1 , that is, there must be an injective homomorphism G → SO (3). Since G is abelian, the only possibilities are that G is trivial, or G is cyclic, or G ? = (Z2 )2 . When G is trivial, we can take ? = 0 and we get the “standard” F theory component in 8 dimensions. This leads to standard components in lower dimensions as well, which can be treated as F theory on T ? × K 3. When G ? = Zm is cyclic, there will be two ?xed points for the action of G on B = CP1 . All of the ?xed points for the action of G on Z must lie in one of the two elliptic curves ?xed by the G-action, and the quotient Z/G will have an elliptic ?bration which degenerates to 64

2

3

4

5

6

4A1

3A2

2A3 + A1

2A4

A5 + A2 + A1

Figure 3: The singular ?bers on Z/G.

have a irreducible ?ber of multiplicty m at each of the two ?xed points. Note that Z/G has singularities along these two ?bers as well, and that the resolved surface Z/G will have a more conventional elliptic ?bration, with section (since there is a section up on Z by assumption). Since the only multiplicities which can occur within ?bers in such a ?bration are m ≤ 6, we learn that the only possible cyclic group actions in this case are Zm with 2 ≤ m ≤ 6.

In fact, it is possible to see the geometry of these group actions quite explicitly. We have already enumerated the possible singular points on Z/G in table 8. These singular points are grouped together into elliptic ?bers as indicated in ?gure 3. In each case, the elliptic curves on Z/G degenerate to an irreducible curve passing through 2, 3, or 4 singular points, and the irreducible curve has multiplicity m in the elliptic ?bration on Z/G. All of the values 2 ≤ m ≤ 6 do occur, as shown in ?gure 3.15 On each ?ber in the ?gure, the thick (blue) line represents the irreducible curve (which is labeled by its multiplicity m in the ?ber), and the thin (red) lines represent the curves in the resolutions of the various singularities. We have labeled each ?ber with the types of singular points that occur on it. We remind the reader that the singularities of Z/G do not actually occur in our M theory and F theory vacua, which are compacti?ed on (S 1 × Z )/G; the surface Z/G is just

a convenient device for determining the lattice L of the M theory compacti?cation. To determine the lattice of the F theory compacti?cation, we should use a di?erent birational model of Z/G: either the nonsingular surface Z/G, or the Weierstrass model obtained from
A related geometric structure appears in [72]. It would be interesting to understand how this is related to the frozen singularities that we will later discuss.
15

65

? I0

IV ?

III ?

II ?

II ?

Figure 4: The singular ?bers for F theory on Z/G. G Z1 Z2 Z3 Z4 Z5 Z6 Lattice Lx Γ3,3 ⊕ E8 ⊕ E8 Γ3,3 ⊕ D4 ⊕ D4 Γ3,3 ⊕ A2 ⊕ A2 Γ3,3 ⊕ A1 ⊕ A1 Γ3,3 Γ3,3 Singular ?bers on K 3/G none
? ? I0 + I0

IV ? + IV ? III ? + III ? II ? + II ? II ? + II ?

Table 11: F theory lattices and singular ?bers for Y = (S 1 × K 3)/G. Z/G by blowing down all components of ?bers other than the ones meeting a section.16 Each singular ?ber can then be labeled by its Kodaira type; the labels for the ?bers for di?erent values of m are shown in ?gure 4. The ?bers in that ?gure are in one-to-one correspondence with the ?bers in ?gure 3, and for each ?ber in ?gure 4, the thick (blue) line represents the ?ber component which meets the section, and the thin (red) lines represent the components which are blown down to give the Weierstrass model. The con?gurations of singular points on Z/G from table 8 are thus collected into Kodaira ?bers, giving the results in the right hand column of table 11.17 From the Kodaira ?bers,
Again, the singularities on the Weierstrass model do not directly show up in our F theory vacuum, but they will reappear as “frozen singularities” in an M theory limit with 3-form ?ux described in section 4.6.1. 17 Note that the free part of the lattices for the Z5 and Z6 cases appearing table 11 are identical. It would be interesting to check whether the full cohomology lattices di?er by torsion classes.
16

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G (Z2 )2

Lattice for K 3/G Γ2,2 ⊕ Γ1,1 (2) ⊕ D4

Singular ?bers on K 3/G
? ? ? I0 + I0 + I0

Table 12: The additional F theory vacuum in dimension 6. G Z2 Z3 Z4 Z6 Lattice Lx Γ2,2 (2) ⊕ Γ1,1 2 1 2 0 6 0 1 ⊕ Γ1,1 2 0 ⊕ Γ1,1 2 0 ⊕ Γ1,1 2 Singular ?bers on T 4 /G
? ? ? ? I0 + I0 + I0 + I0

IV ? + IV ? + IV ?
? III ? + III ? + I0 ? II ? + IV ? + I0

Table 13: F theory lattices and singular ?bers for Y = (S 1 × T 4 )/G. a lattice can easily be computed, as shown in the middle column of table 11, and we claim that this is the lattice which describes the moduli space for the corresponding F theory component. In fact, this same lattice can be arrived at in two ways: either directly in terms of the Kodaira ?bers for the elliptic ?bration on Z/G, or by using the x⊥ / x construction, using the fact that L = Γ1,1 (m) ⊕ Lx in the case that Y = (S 1 × Z )/G. (This latter result becomes obvious when comparing tables 8 and 11.) This gives six di?erent F theory vacua. These vacua are dual to the six heterotic asymmetric orbifolds in 7 dimensions that we constructed in section 2. Note that in the two remaining cyclic cases for M theory vacua, namely Z7 and Z8 , it is not possible to split o? a factor of Γ1,1 (m) from the lattice L as given in table 8. This gives further con?rmation that there are no F theory vacua associated with these cases. The one remaining F theory vacuum with Z = K 3 is associated to the group G = Z2 × Z2 ; see table 12. The lattice we list is for K 3/G. To this lattice, we must add a lattice

of signature (2, 2) for the 5-dimensional M theory compacti?cation and a lattice of signature (1, 1) for the F theory compacti?cation. The ?xed points for the action of G on B = CP1 have stabilizer Z2 in each case, so on the quotient Z/G we will ?nd ?bers with multiplicity

? 2 which become I0 Kodaira ?bers on Z/G. Once again the lattices satisfy L = Γ1,1 (2) ⊕ Lx . Turning to the case in which Z = T 4 , we ?nd a similar story. The base B of the elliptic

?bration on Z/G is still CP1 , but the base B of the elliptic ?bration on Z is now an elliptic 67

curve. The action of G on B must be faithful, and in fact it su?ces to consider the case in which G does not contain any translations (else we could mod out by the translations ?rst). As an action on an elliptic curve with a ?xed point, the only possibilities are G = Zm with m = 2, 3, 4, or 6. In fact, these are exactly the group actions that we have (in table 10)! As in the previous case, the singular points on Z/G can be collected into Kodaira ?bers for Z/G; the results of this are displayed in table 13. The lattices once again satisfy L? = Γ1,1 (m) ⊕ Lx .

The entries in tables 11, 12, and 13 thus describe the possible F theory vacua with 16 supercharges. The ?rst entry in table 11 gives the “standard” F theory vacuum in eight dimensions, the remaining ?ve entries in table 11 together with the four entries in table 13 give the new F theory vacua in seven dimensions, and table 12 shows the one new F theory vacuum in six dimensions. (Of course the higher dimensional theories can be reduced to lower dimensional ones by compacti?cation on additional circles.) 4.3.1 From F theory to type I′

Decompacti?cation limits of the components of the F theory moduli space which have D dimensional e?ective theories should lead to (D +1)-dimensional e?ective theories (using an isotropic sublattice of rank 1). It would be desirable to give a description of these directly in terms of type I′ theory. However, despite the interesting pictures presented in [73, 74], at present we do not have enough control over type I′ vacua to be able to do this. So our analysis will be somewhat indirect. In this section, we describe the overall picture leaving a detailed discussion for the following section. The most common decompacti?cation limit from these components involves a decomposition of the lattice in the form L? = Γ1,1 ⊕ Lx . (74)

We saw in the previous section that when going from six dimensions up to seven dimensions, a lattice decomposition of this form corresponded to a boundary component which gained additional degrees of freedom in the limit, and which “attached” to the standard boundary component. To see this, we will give a geometric construction of these components and their decompacti?cation limits, and note that for the vast majority of components in dimension seven, there is only one decompacti?cation limit, which must therefore be of this type. We thus argue by analogy that all limits of this type must have the lattice decomposition given in 68

eq. (74); this leaves only a few additional components in dimension eight for which we must account. Our geometric construction—to be described in section 4.3.2 for Z = K 3—is designed for easy comparison with the heterotic duals of these vacua, where this phenomenon is known: the seven-dimensional components we have found are dual to non-trivial triples of Wilson lines on the heterotic side, and their eight-dimensional limits all attach to the standard eight-dimensional component. The exceptional case is G = Z2 which contains the CHL string. This case does have a non-trivial 8 and 9-dimensional limit. A similar discussion applies to the 8-dimensional limits of the F theory moduli spaces coming from Y = (S 1 ×T 4 )/G. Since the lattices for these models also have a Γ1,1 summand, 8-dimensional limits exist. However, in each case, again with the exception of G = Z2 , these boundaries attach to a conventional toroidal type IIB compacti?cation. The case of G = Z2 has non-trivial 8 and 9-dimensional limits as re?ected in its lattice given in table 11. This is natural since this model has a dual description as the compacti?cation of the (+, ?) orientifold to 7 dimensions. This duality can be seen immediately by studying the geometric description of the compacti?ed (+, ?) orientifold obtained in section 3.2.6.

The last case we need to discuss is the 6-dimensional F theory vacuum associated to G = Z2 × Z2 . From the lattice for K 3/G given in table 12, we that there are two ways of decompactifying to 8 dimensions. Peeling o? a Γ1,1 factor gives us a 7-dimensional theory that attaches to the standard component as in our preceeding discussion. The other limit involves peeling o? a Γ1,1 (2) factor. This leads to a 7-dimensional theory that attaches to

the component of the moduli space containing the CHL string. Note that we also need to worry about the additional signature (1, 1) summand in the lattice for F theory on (K 3 × T 2 )/G. The correct relative normalization for this summand is tricky to determine, although preliminary computations suggest that it is either Γ1,1 or Γ1,1 (2). This is certainly supported from a study of the Z2 × Z2 asymmetric orbifold of the heterotic string. From that approach, it seems clear that there is no new 7-dimensional theory to which the 6dimensional orbifold could decompactify. From this 6-dimensional theory, we therefore arrive at no new 7-dimensional theory. 4.3.2 Automorphisms of del Pezzo surfaces

Let us begin by explaining why del Pezzo surfaces play a natural role in our F theory discussion. Consider an elliptically ?bered K 3 surface Z whose base B is large, and whose complex structure is close to a degenerate one in which the K 3 surface degenerates into a

69

pair of del Pezzo surfaces Z1 , Z2 meeting on an elliptic curve E . This is the limit in which comparison with the E8 × E8 heterotic string becomes possible [75]. (In fact, it corresponds

to both large volume and weak string coupling on the heterotic side of the duality.) It is then natural to expect that automorphisms of del Pezzo surfaces will be classi?ed by the same data used to classify E8 gauge bundles on T 3 . Further, the anomaly cancellation

condition should have a purely geometric realization as a constraint on whether we can “glue” two del Pezzo surfaces with automorphisms into a K 3. So let us begin by considering a rational elliptic surface X in Weierstrass form. This implies that X has an elliptic ?bration π : X → CP1 , and a section σ : CP1 → X contained

which are the identity on E and which stabilize σ (ie map the image of σ to itself). We call these automorphisms of (X, E, σ ) Equivalently, we could consider a degree one del Pezzo surface X obtained by collapsing σ . The elliptic ?bration on X becomes the anti-canonical pencil of elliptic curves (or more generally Weierstrass curves) on this surface. This surface has at worst rational double point singularities. The automorphism group (X, E, σ ) acts naturally on X and is identi?ed with the subgroup of the automorphism group of X ?xing E pointwise. As a ?rst step, let us argue that the automorphism group of (X, E, σ ) is a ?nite cyclic group that preserves the elliptic ?bration structure of X . Along the way, we shall also see that the induced action of the group of automorphisms on the base CP1 of the elliptic ?bration is faithful. The group acts on CP1 ?xing two points, i.e., the automorphism group of (X, E, σ ) stabilizes E and exactly one other ?ber. To see this, let f ? X be a ?ber of the elliptic ?bration and let α : X → X be an automorphism of (X, E, σ ). Then α(f ) is a divisor in X with zero algebraic intersection with E . Hence, its projection to CP1 must be a single point. That is to say α(f ) is contained in a ?ber of the elliptic ?bration. By homological considerations, we see that it is exactly a ?ber of the elliptic ?bration structure. This shows that α preserves the elliptic ?bration structure and hence induces an automorphism α of the base CP1 . If α is trivial, then α stablizes each ?ber of the elliptic ?bration. Since it acts by the identity on E , it acts by the identity on the homology of each smooth ?ber and hence, it is a translation on each smooth ?ber. But, α also stablizes the section σ , and hence it must be the identity on each smooth ?ber. Since the smooth ?bers are dense, it follows that α is the identity. This proves that the automorphism group of (X, E, σ ) acts faithfully on the base CP1 . The elliptic ?bration structure on X has at least two singular ?bers. This means that 70

in the smooth points of X . Since X is rational, σ 2 = ?1, that is to say σ is an exceptional curve. We ?x a ?ber E ? X of π , and we wish to study the group of automorphisms of X

the automorphism group of (X, E, σ ) is faithfully represented as a group of automorphisms of CP1 ?xing a point of CP1 and permuting a ?nite set of points of cardinality at least two. All such groups are ?nite cyclic and ?x two points of CP1 . We use E ′ to denote the ?ber other than E stabilized by the automorphism group of (X, E, σ ). Suppose that an automorphism α is the identity on E ′ . It is also the identity on E . We consider the degree one del Pezzo model X where the section has been collapsed. The image of σ is a smooth point of this surface, and α descends to an automorphism α of X ?xing the point of intersection x of E and E ′ and acting by the identity on E and on E ′ . If follows that the di?erential of α at x is the identity, and hence that the restriction of α to the exceptional curve in X obtained from blowing up x is the identity. It follows that α stabilizes each ?ber of the elliptic ?bration. But we have already seen that this implies that α is the identity. The action of the automorphism group of (X, E, σ ) on E ′ is therefore faithful. Let us examine the automorphism groups of the various types of Weierstrass curves ?xing a given smooth point of the curve. The automorphism group of a generic elliptic curve ?xing a point is Z2 acting by ?1 and ?xing 4 points. In the case of special elliptic

curves the automorphism group is either Z4 acting with two ?xed points and two points with stabilizers Z2 , or Z6 acting with one ?xed point, 3 points with stabilizer Z2 and two points with stabilizer Z3 . The automorphism group of an ordinary double point ?ber ?xing a smooth point is Z2 ?xing the singular point and the other point. The automorphism group of a Weierstrass cusp ?xing a smooth point is C? acting so as to ?x only the singular point and the given smooth point. Let Y be an elliptic surface and let f ? Y be a ?ber. Let p ∈ CP1 be the image of f

under projection mapping. Let Y → Y be the minimal resolution of Y and let p : Y → CP1 be the induced projection mapping. Let f ? Y be the full preimage of f . Fix a disk

? ? CP1 centered at p su?ciently small so that the preimage of ? ? {p} contains the preimage of no singular point of Y or of the projection mapping. We de?ne the local contribution of f to the Euler characteristic of the smooth model of Y to be the Euler characteristic of p?1 (?). It is easy to see that f is a smooth ?ber if and only if its local contribution to the Euler characteristic is zero and otherwise that the local contribution to the Euler characteristic is positive. Also, the local contribution to the Euler characteristic is 1 if and only if f misses the singularities of X and contains an ordinary double point. Lastly, if the elliptic surface in question is rational, then the sum over all ?bers of the local contributions to the Euler characteristic is 12. The quotient of X by the automorphism group is a surface Y with an induced elliptic 71

?bration and with rational double point singularities. In the singular model, the ?ber which contains the image of E ′ has multiplicity equal to the order of the automorphism group of X . On the other hand, in the minimal resolution, the strict transform of this curve is one of the components of the set of rational curves indexed by the nodes of an extended Dynkin diagram of type A, D , or E and which intersect each other as indicated by the bonds of the extended Dynkin diagram. Furthermore, the multiplicities of the various components in the ?ber in the smooth model are given by the coroot integers on the corresponding nodes of this diagram. These numbers are all at most 6, and hence every component has multiplicity at most 6 in the ?ber. It follows in the singular model, that the ?ber containing the image of E ′ has multiplicity at most 6, and hence the order of the automorphism group is at most 6. (Notice that the cases of smooth and nodal ?bers E ′ follow directly from the classi?cation of the automorphism groups of these ?bers given above.) The automorphism group of (X, E, σ ) is therefore a cyclic group of order at most 6. Now we simply list the possibilities: the automorphism group is Z2 and the ?ber E ′ stabilized by the action is smooth. The quotient surface has four A1 singularities and the image of E ′ passes through all of these. The local contribution of this ?ber to the Euler characteristic of the smooth model is 6. The local contribution to the Euler characteristic of the smooth model of X from the singular ?bers of the elliptic ?bration is 12, and since the automorphism group acts freely on these ?bers, the local contribution of the images of these ?bers to the Euler characteristic of the smooth model of the quotient is 12/2 = 6. In the minimal resolution of the quotient surface, the preimage of this ?ber is a tree of rational curves intersecting according to the extended Dynkin diagram of D4 . The strict transform of E ′ has multiplicity 2 in the ?ber, i.e., it corresponds to the central node in the extended Dynkin diagram, the one with coe?cient two in the dominant root. The automorphism group is Z3 and the ?ber E ′ stabilized by the action is smooth. The quotient surface has three A2 singularities and the image of E ′ passes through all of them. In the minimal resolution of the quotient surface, the preimage of this ?ber is a tree of rational curves intersecting according to the extended Dynkin diagram of E6 and this ?ber contributes 8 to the Euler characteristic of the smooth model. The singular ?bers contribute 12 to the Euler characteristic of the smooth model of X and hence their images in the quotient contribute 12/3 = 4 to the Euler characteristic of its smooth model. The strict transform of E ′ is the curve of multiplicity three in the ?ber, i.e., it corresponds to the central node in the extended Dynkin diagram, the one with coe?cient three in the dominant root. The local contribution of this ?ber to the Euler characteristic of the smooth model is 8. 72

The automorphism group is Z4 and the ?ber E ′ stabilized by the action is smooth. The quotient surface has two A3 singularities and an A1 singularity. The image of E ′ passes through all these singularities. In the minimal resolution of the quotient surface the preimage of this ?ber is a tree of rational curves intersecting according to the extended Dynkin diagram of E7 and the local contribution of this ?ber to the Euler characteristic of the smooth model is 9. The singular ?bers contribute 12 to the Euler characteristic of the smooth model of X , and hence the images in the quotient of the singular ?bers of X contribute 12/4 = 3 to the Euler characteristic of the smooth model of the quotient. The strict transform of E ′ is the curve of multiplicity 4 in the ?ber, i.e., corresponds to the node with coe?cient 4 in the dominant root. The automorphism group is Z5 and the ?ber E ′ is a Weierstrass cusp. The quotient surface has two A4 singularities and the image of E ′ passes through both of them. The preimage of this ?ber in the minimal resolution of the quotient is a tree of rational curves intersecting according to the extended Dynkin diagram of E8 and contributes 10 to the Euler characteristic of the smooth model of the quotient. The other singular ?bers of X contribute 10 to the Euler characteristic of the smooth model of X and their images in the quotient contribute 10/5 = 2 to the Euler characteristic of its smooth model. The strict transform of E ′ is the curve in this con?guration with multiplicity 5 in the divisor representing the ?ber. That is to say it corresponds to the node with coe?cient 5 in the dominant root. The local contribution of this ?ber to the Euler characteristic of the smooth model is 10. Notice that since E ′ is a cusp, the sum of the contributions of the all other ?bers to the Euler characteristic of the smooth model of X is 10. Thus, in the quotient the sum of the local contributions of the ?bers besides the image of E ′ to the Euler characteristic of the smooth model is 2, giving us a total of 12 as required. The automorphism group is Z6 and the ?ber E ′ is smooth. The quotient surface has three singularities—of types A5 , A2 and A1 , respectively, re?ecting the three singular orbits of the action of this cyclic group on a smooth elliptic curve. The image of E ′ passes through all these singularities and its preimage in the minimal resolution is a tree of rational curves intersecting according to the extended Dynkin diagram of E8 . The strict transform of E ′ has multiplicity 6 in the ?ber and hence corresponds to the trivalent node in the extended Dynkin diagram, the one with coe?cient 6 in the dominant root. The local contribution of this ?ber to the Euler characteristic of the smooth model is 10. The singular ?bers of X contribute 12 to the Euler characteristic of its smooth model, and the images of these ?bers in the quotient contribute 12/6 = 2 to the Euler characteristic of its smooth model. This completes the list of possibilities. Notice how the extended Dynkin diagram of E8 73

predicts the automorphism groups and the singularities of the quotient surfaces. First of all there will be an automorphism group of order k if and only if one of the coe?cients on the extended Dynkin diagram of E8 is divisible by k . Given an integer k with this property, the singularity in the quotient surface when the automorphism group is cyclic of order k is determined as follows: One takes the extended Dynkin diagram of E8 with the usual coe?cients and removes all nodes whose coe?cients are divisible by k . There remains a collection of Ani diagrams. These label the singularities of the quotient surface. Furthermore, we can connect all of these diagrams to a central node and form an extended Dynkin diagram of some subgroup of E8 . In that new diagram the coe?cient of the central node that we added will be exactly k . This is the ?ber in the minimal resolution of the quotient. The strict transform of the image of E ′ is the component corresponding to the central node that we added. Note that the list of possibilities matches perfectly with the gauge theory picture of commuting pairs in E8 and their centralisers. Of course, this is not an accident: one can prove by abstract methods that equivalence between the group theory and del Pezzo surfaces which goes through the Looijenga space of E ? Λ(E8 )/W (E8 ) is categorical and

hence that the automorphism groups of the three classes of objects—commuting pairs in E8 , E ? Λ(E8 )/W (E8 ) (here, Λ(E8 ) is the coroot lattice of E8 and W (E8 ) is its Weyl group) and del Pezzo surfaces must be the same.18 One last point is worth remarking on: given two rational elliptic surfaces with smooth ?bers and sections (X1 , E1 , σ1 ) and (X2 , E2 , σ2 ) and given an isomorphism from E1 to E2

matching up the intersections with the sections, we can glue X1 and X2 together to form a singular surface X with a normal crossing along E = E1 = E2 . This surface ?bers over CP1 ∪ CP1 with a section σ . Of course, it has a marked ?ber E . It is a singular model of an elliptically ?bered K 3 surface, and in fact represents a point in a divisor at in?nity in a compacti?cation of period space for these surfaces.19 Given automorphisms αi of (Xi , Ei , σi ) we can glue them together to determine an automorphism of (X, E, σ ). To smooth the singular surface X to an elliptically ?bered K 3 we need a trivialization of the tensor product of the normal bundles of E in X1 and X2 . To carry along the group action requires then trivializing the action of this tensor product. This is possible if and only if the actions of αi on the disks in the base CP1 ’s centered at the image points of Ei are inverses of each other. In particular, the orders of α1 and α2 must be the same and these two automorphisms must be in inverse components. This corresponds in the gauge theory
18 19

We wish to thank Bob Friedman for pointing this out to us. On the heterotic side, this represents the in?nite-volume, zero-coupling limit.

74

language to the fact that the Chern–Simons invariant of the commuting triples must be inverses of each other. As promised, we therefore recover our anomaly matching constraint from this gluing condition. 4.4 4.4.1 Type IIA compacti?cations with RR one-form ?ux Equivariant ?at line bundles on T 4

We now take a di?erent tack and consider compacti?cations with ?ux. This is a quite di?erent class of models from the purely geometric compacti?cations just discussed. Among compacti?cations of this kind, we shall ?nd new dual descriptions for the perturbative asymmetric orbifolds of section 2 both in 6 and 7 dimensions. Let us ?rst consider ?at RR 1-form ?elds in a type IIA string compacti?cation on some (possibly singular) manifold X . Such a 1-form ?eld A can be seen as a connection in a principal U (1) bundle P → X . Of course the interpretation in M theory of such a RR 1form ?eld con?guration will be as a compacti?cation on the manifold P . In our preceeding discussion, P took the form of (X × S 1 )/G. As we shall see, it is natural from this purely geometric M theory picture to treat A via equivariant cohomology. In section 4.5, we revisit this treatment from the perspective of K-theory where we ?nd a group of 1-form ?uxes in agreement with the results from equivariant cohomology. K-theory, however, will give us new physics for compacti?cations with more general combinations of ?uxes. U (1) principal bundles, or equivalently complex Hermitian line bundles, over a smooth manifold X are classi?ed topologically by their ?rst Chern class c1 , which is an element of the cohomology group H 2 (X, Z). This Chern class equals [F/2π ] in real cohomology, where F = dA is the curvature two-form. Flat line bundles, i.e., bundles that satis?es F = 0, are classi?ed by the cohomology group H 1 (X, U (1)). One can think of this group of homomorphisms of H1 (X, Z) into U (1) as the holonomies of the ?at connection around the non-trivial homology 1-cycles of X . Note that because F vanishes, in general such a ?at bundle can have only have a torsion ?rst Chern class c1 ∈ Tor H 2 (X, Z). (75)

If there is no torsion in H 2 (X ) then a ?at line bundle is necessarily topologically trivial. It can stil


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