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RUNHETC-2002-15; NI 02013-MTH hep-th/0206092

arXiv:hep-th/0206092v1 11 Jun 2002

T-Duality, and the K-Theoretic Partition Function of TypeIIA Superstring Theory

Gregory Moore1 and Natalia Saulina2

1

Department of Physics, Rutgers University Piscataway, NJ 08855-0849, USA

2

Department of Physics, Princeton University Princeton, NJ 08544, USA

We study the partition function of type IIA string theory on 10-manifolds of the form T 2 × X where X is 8-dimensional, compact, and spin. We pay particular attention to the e?ects of the topological phases in the supergravity action implied by the K-theoretic formulation of RR ?elds, and we use these to check the T -duality invariance of the partition function. We ?nd that the partition function is only T -duality invariant when we take into account the T -duality anomalies in the RR sector, the fermionic path integral (including 4-fermi interaction terms), and 1-loop corrections including worldsheet instantons. We comment on applications of our computation to speculations about the role of the Romans mass in M -theory. We also discuss some issues which arise when one attempts to extend these considerations to checking the full U -duality invariance of the theory.

June 10, 2002

1. Introduction & Summary Duality symmetries, such as the U -duality symmetry of toroidally compacti?ed M theory, have been of central importance in the de?nition of string theory and M-theory. Topologically nontrivial e?ects associated with the RR sector have also played a crucial role in de?ning the theory. It is currently believed that RR ?eldstrengths (and their D-brane charge sources) are classi?ed topologically using K-theory [1,2,3,4,5,6,7,8]. Unfortunately, this classi?cation is not U -duality invariant. Finding a U-duality invariant formulation of M-theory which at the same time naturally incorporates the K-theoretic formulation of RR ?elds remains an outstanding open problem. With this problem as motivation, the present paper investigates the interplay between the K-theoretic formulation of RR ?elds and the T-duality group, an important subgroup of the full U-duality group. While T-duality invariance of the theory was one of the guiding principles in the de?nition of the K-theoretic theta function [4][7] we will see that the full implementation of T-duality invariance of the low energy e?ective action of type II string a two-dimensional torus, and X is an 8-dimensional compact spin manifold. We will show theory is in fact surprisingly subtle, even on backgrounds as simple as T 2 × X , where T 2 is

that, in fact, in the RR sector there is a T-duality anomaly. This anomaly is cancelled by a compensating anomaly from fermion determinants together with quantum corrections to the 8D e?ective action. A by-product of our computation is a complete analysis of the 1-loop determinants of IIA supergravity on X × T 2 . As an application of our discussion, we re-examine a proposal of C. Hull [9] for inter-

preting the Romans mass of IIA supergravity in the framework of M-theory. We will show that, while the interpretation cannot hold at the level of classical ?eld theory, it might well hold as a quantum-mechanical equivalence. In section 10 we comment on some of the issues which arise in extending our computation to a fully U-duality invariant partition function. This includes the role of twisted K-theory in formulating the partition sum. This paper is long and technical. Therefore we have attempted to write a readable summary of our results in the remainder of the introduction. 1.1. The e?ective eight-dimensional supergravity, and its partition function Previous studies of the partition function in type II string theory [4][7] considered the limit of a large 10-manifold. One chose a family of Riemannian metrics g = t2 g0 with t → ∞ and g0 ?xed. Simultaneously, one took the string coupling to zero. The focus of 1

these works was on the sum over classical ?eld con?gurations of the RR ?elds. In this paper we consider the limit where only 8 of the dimensions are large. The metric has the form

2 2 ds2 = ds2 T 2 + t dsX

(1.1)

2 2 2ξ where ds2 is constant. T 2 is ?at when pulled back to T . The background dilaton gstring = e

We will work in the limit t→∞ e?2ξ := e?2φ V → ∞ (1.2)

where V is the volume of T 2 and φ is the 10-dimensional dilaton. Finally - and this is ? is identically important -until section 10 we assume the background NSNS 3-form ?ux, H, ? is a globally well-de?ned harmonic form on zero. In particular, the 2-form potential, B, X × T 2. As is well-known the background data for the toroidal compacti?cation (1.1) include a

pair of points (τ, ρ) ∈ H ×H where H is the upper half complex plane. τ is the Teichmuller

parameter of the torus and ρ := B0 + iV , where B0 dσ 8 ∧dσ 9 is an harmonic 2-form on T 2 . While we work in the limit (1.2), within this approximation we work with exact expressions in the geometrical data (τ, ρ). In this way we go beyond [7]. It is extremely well-known that the low energy e?ective 8D supergravity theory obtained by Kaluza-Klein reduction of type II supergravity on T 2 has a “U -duality symmetry” quantum e?ects [10,11,12,13,14]. These are symmetries of the equations of motion and which is classically SL(3, R) × SL(2, R), and is broken to D := SL(3, Z ) × SL(2, Z ) by

not of the action. (The implementation of these symmetries at the level of the action involves a Legendre transformation of the ?elds.) What is perhaps less well-known is that the K-theoretic formulation of RR ?elds leads to an extra term in the supergravity action which is nonvanishing in the presence of nontrivial ?ux con?gurations. Indeed, the proper ? 3 ] = 0, but formulation of this term is unknown for arbitrary ?ux con?gurations with [H for topologically trivial NSNS ?ux the extra term is known [7] and is recalled in equations (1.14) and (1.15) below. This term breaks naive duality invariance of the classical supergravity theory already for the T-duality subgroup of the U-duality group, and makes the discussion of T-duality nontrivial. Let us now summarize the ?elds and T-duality transformation laws in the conventional description of the eight-dimensional e?ective supergravity theory on X. The T-duality group is DT = SL(2, Z )τ × SL(2, Z )ρ . The theory has the following bosonic ?elds. From 2

the NSNS sector there is a scalar t, characterizing the size of X, a unit volume metric gM N , a 2-form potential

1

B(2) , with ?eldstrength H(3) , and a dilaton ξ , all of which are

invariant under DT . In addition, there is a multiplet of 1-form potentials Amα (1) transforming

in the (2, 2) of DT . Finally, the pair of scalars (τ, ρ), transform under (γ1 , γ2 ) ∈ DT as therefore call the factors SL(2, Z )τ , SL(2, Z )ρ, respectively.

(τ, ρ) → (γ1 · τ, γ2 · ρ) where γ · is the action by a fractional linear transformation. We

α The ?eldstrengths from the RR sector include a 0-form and a 2-form, g( p) , p = 0, 2, α =

?eld does not transform locally under T -duality, rather its equation of motion mixes with its Bianchi identity [14]. The fermionic partners are described in section 7 below. The real part of the standard bosonic supergravity action takes the form

3

transforming in the (2′ , 1) of DT . Finally there is a 4-form ?eldstrength g(4) on X . This

1, 2 transforming in the (1, 2) of DT , and a 1-form and 3-form g(p)m , p = 1, 3, m = 8, 9

Re

(8D) Sboson

= SNSNS +

p=0

Sp g(p) + S4 g(4)

(1.3)

In the action (1.3) all of the terms except for the last term are manifestly T-duality invariant. The detailed forms of the actions are: SNSNS = 1 2π 1 e?2ξ t6 R(g ) + 4dξ ∧ ?dξ + 28t?2 dt∧ ? dt + t2 H(3) ∧ ?H(3) 2 X

1 dτ ∧ ?dτ 1 dρ ∧ ?dρ 1 4 nβ + t6 + t6 + t gmn Gαβ Fmα (2) ∧ ? F(2) 2 2 2 (Imτ ) 2 (Imρ) 2 where ? stands for the Hodge dual with the metric gM N , we also denote

mα Fmα (2) = dA(1) ,

(1.4)

1 nβ H(3) = dB(2) ? ?mn Eαβ Amα (1) F(2) , 2

(1.5)

?mn and Eαβ are invariant antisymmetric tensors for SL(2, Z)τ and SL(2, Z)ρ respectively, gmn = M(τ ), and ?nally M(z ) :=

1

g mn = M(τ )?1 , 1 Imz 1 Rez

Gαβ = M(ρ)

(1.6)

Rez |z |2

.

(1.7)

We will always indicate by the subscript (p) the degree p of a di?erential p-form on X

3

The real part of the RR sector action is given by

3

Sp g(p) = π

p=0 X

β α t8 Gαβ g(0) + t6 g mn g(1)m ∧ ? g(1)n + ∧ ? g(0)

(1.8)

β α t4 Gαβ g(2) + t2 g mn g(3)m ∧ ? g(3)n ∧ ? g(2)

together with S4 g(4) = π

X

Im(ρ)g(4) ∧ ?g(4) .

(1.9)

1.2. The semiclassical expansion The vevs of the two ?elds t and e?2ξ (the 8-dimensional length scale of X and the inverse-square 8D string coupling) de?ne semiclassical expansions when they become large. We will expand around a solution of the equations of motion on X . To leading order in our expansion this means X admits a Ricci ?at metric2 gM N . We also have constant scalars t, ξ, τ, ρ, and Fmα (2) = 0, H(3) = 0, so the background action SNSNS is zero. Finally, we expand around a classical ?eld con?guration for the RR ?uxes, and to leading order these ?uxes g(p) are harmonic forms. Nonzero ?uxes contribute terms to the partition function going like O(e?t

8 ?2 p

).

Let us consider the leading order contribution to the partition function. There are

3

several sources of contributions even at leading order, but, since we are interested in questions of T-duality, most of these can be neglected. The volume of X suppresses the contribution of ?uxes g(p) , p = 0, 1, 2, 3, and, to leading order in the t → ∞ expansion

these can be set to their classical values. Note, however, that neither the string coupling, nor the volume of X , suppress the action for g(4) , and thus we must work in a fully quantum mechanical way with this ?eld. This is just as well, since (not coincidentally)

2

Almost nothing in what follows relies on the Ricci ?atness of the metric. We avoid using

this condition since a T -duality anomaly on non-Ricci ?at manifolds would signal an important inconsistency in formulating string theory on manifolds of topology X × T 2 .

3

In particular we are negelecting determinants of KK and string modes, and perturbative

on the NSNS action simply renormalizes V to Vef f , where ρ = B0 + iVef f is the variable on which SL(2, Z)ρ acts by fractional linear transformations.

2 ). These are all T -duality invariant. The backreaction of nonzero RR ?uxes corrections O (gstring

4

this is the term in the action which is not manifestly T-duality invariant. Fortunately, in our approximation, g(4) is a free, nonchiral ?eld and hence quantization is straightforward (after the K -theory subtleties are taken into account). Including subleading terms in the expansion parameter t involves (among other things) summing over the RR ?uxes g(p) , p = 0, 1, 2, 3. Finally, in order to be consistent with our approximation scheme we must allow the possibility of ?at potentials in the background.

4

These contribute nontrivially to the parti-

tion function through important phases and accordingly, we will generalize our background to include these. The real part of the action for the ?at con?gurations vanishes, of course, and hence in the physical partition function one must integrate over these ?at con?gurations. In the RR sector the ?at potentials are thought to be classi?ed by K 1 (X10 ; U (1)) [5]. These contribute no phase to the action and we will henceforth ignore them. space of ?at NSNS potentials is H 2 (X ; U (1)) × (H 1 (X ; U (1)))4. In this paper we will work only with the identity component of this torus. Accordingly, we will identify the space of ?at NSNS potentials with the torus H 2 (X ) 2 (X ) × HZ H 1 (X ) 1 (X ) HZ

4 5

The

(1.10)

normalized harmonic p-forms on X. The ?rst factor is for B(2) and the second factor for the ?elds Amα (1) transforming in the (2, 2) of DT . schematically written as Z (t, gM N , ξ, τ, ρ) =

?at potentials

p (X ) is the lattice of integrally where Hp (X ) is a space of harmonic p-forms on X and HZ

Putting all these ingredients together the partition function we wish to study can be

d??at

RR ?uxes

Det · e?Scl + · · ·

(1.11)

where d??at is a T -duality invariant measure on the ?at potentials, Det is a product of 1-loop determinants and Scl is the classical action. Now, to investigate T -duality it is

4 5

By “?at” we mean the DeRham representative of the relevant ?eldstrength is zero.

If treated as di?erential forms, RR zero modes do contribute to the overall dependence of the ? = te?ξ/3 . See eq.(7.39) below.) In the K-theoretic treatment they also give a partition sum on t

0 factor of |Ktors (X × T 2 )| .

5

convenient to denote by F the collection of all ?elds occuring in (1.11) which transform

locally and linearly under DT . These include the ?at NSNS potentials above as well as integration over the ?at potentials and summation over the ?uxes for p = 0, 1, 2, 3. This measure is T -duality invariant, and we can write Z (t, gM N , ξ, τ, ρ) = [dF ]Z (F ; t, gM N , ξ, τ, ρ). (1.12) the classical ?uxes g(p) , p = 0, . . . , 3. We introduce a measure [dF ] on F which includes

The invariance of (1.12) under the subgroup SL(2, Z )τ of the T-duality group is essentially trivial. The relevant actions and determinants are all based on SL(2, Z )τ -invariant di?erential operators. The invariance of the theory under SL(2, Z )ρ is, however, much of (1.12). Now, checking T -duality invariance is reduced to checking the invariance of Z (F , ρ). This function is constructed from a. The K-theoretic sum over RR ?uxes of g(4) in the presence of F . more nontrivial. Therefore we simplify notation and just write Z (F , ρ) for the integrand

b. The integration over the Fermi zeromodes in the presence of g(4) and F . ?elds and the quantum corrections due to worldsheet instantons.

c. The inclusion of 1-loop determinants, including determinants of the 8D supergravity In the following subsections we sketch how each of these elements enters Z (F , ρ).

function turns out to transform anomalously under T -duality. The integration over the anomalously. The inclusion of 1-loop e?ects, including the string 1-loop e?ects ?nally cancels the anomaly. 1.3. The K-theoretic RR partition function In order to write explicit formulae for the quantities in (1.12) we must turn to the K-theoretic formulation of RR ?elds. In practical terms the K-theoretic formulation alters the standard formulation of supergravity in two ways: First it restricts the allowed ?ux con?gurations through a “Dirac quantization condition” on the ?uxes. Second, it changes the supergravity action by the addition of important topological terms in the action.

6 6

Brie?y, the K-theoretic sum over RR ?uxes g(4) leads to a theta function Θ(F , ρ). This

fermion zeromodes corrects this to a function Θ(F , ρ). This function still transforms

It also alters the overall normalization of the bosonic determinants by changing the nature

of the gauge group for RR potentials, but we will not discuss this in the present paper.

6

In more detail, the K-theoretic Dirac quantization condition states that the DeRham class of the total RR ?eldstrength [G/(2π )] is related to a K-theory class x ∈ K 0 (X10 ) via [ G ? ] = ch(x) A 2π (1.13)

The topological terms in the action can be described as follows. On a general 10-manifold this term involves the mod-two index of a Dirac operator and cannot even be written as a traditional local term in the supergravity action [4,5,7]. In the case of zero NS-NS ?uxes, the general expression for the phase in the supergravity theory is: Im(S10D ) = ?2π Φ, Φ = Φ 1 + Φ2 (1.14)

where e2πiΦ2 is the mod-two index and Φ1 is given by the explicit expression Φ1 =

X10

?

1 15 p1 48

G2 2π G4 2π

5

+

1 6

G2 2π

3

G4 2π

+

p1 G0 1+ 12 8π p1 48

2

(1.15)

?

G2 2π

+

?8 G0 G0 A 1+ + 2 2π 4π

? is expressed in terms of where G2j , j = 0, 1, 2 are RR ?uxes on X10 , p1 = p1 (T X10 ) and A the Pontryagin classes of X10 as ? = 1 ? 1 p1 + 1 7p2 A 1 ? 4p2 . 24 5760 (1.16)

In the case that we reduce to 8 dimensions, taking our manifold to be of the form X × T 2 with the choice of supersymmetric spin structure on T 2 the above considerations simplify and can be made much more concrete. Consider ?rst the Dirac quantization condition. We reduce RR ?eldstrengths as: G0 2 = g(0) 2π G2 1 2 = g(0) dσ 8 ∧dσ 9 + g(1)m ∧dσ m + g(2) 2π G4 1 = g(4) + g(3)m ∧dσ m + g(2) ∧dσ 8 ∧dσ 9 2π

7 7

(1.17)

Beware of notation! The subscript (p) indicates form degree, while the other sub- and

2 superscripts on g(p) indicate DT transformation properties. Thus, for example, g(0) is the second α of 0-forms. component of a doublet g(0)

7

where σ m , m = 8, 9 are coordinates on T 2 . In the K-theoretic formulation of ?ux quantizaα α tion the ?eldstrengths g(4) , g(3)m , g(2) , g(1)m , g(0) are related to certain integral cohomology

classes which we denote as a ∈ H 4 (X, Z), fm ∈ H 3 (X, Z) ? Z2 , nα = eα = n1 n0 e′′ e ∈ H 2 (X, Z) ? Z2 , (1.18)

γm ∈ H 1 (X, Z) ? Z2 ,

∈ H 0 (X, Z) ? Z2

The explicit relation between these classes and the g(p) is somewhat complicated and given in equation (4.3) below. The K-theoretic Dirac quantization condition leaves all integral classes in (1.18) unconstrained except for fm . One ?nds that Sq 3 (fm ) = 0. As explained in section 3.3 and 5.2 “turning on” ?at NSNS potentials corresponds to acting on the K-theory torus by an automorphism changing the holonomies of the ?at connection on the torus. In concrete terms, turning on ?at potentials modi?es the reduction formulae (1.17) according to equations (5.15) to (5.18) below. phase e2πiΦ2 arising from the mod 2 index may be expressed in concrete terms as exp[2πiΦ2 ] = exp iπ

X

Now let us consider the phase. It turns out that on 10-folds of the form X × T 2 the

g(3)8 ∪ Sq 2 (g(3)9 ) + g(3)8 ∪ Sq 2 (g(3)8 ) + g(3)9 ∪ Sq 2 (g(3)9 )

2 g(0)

(1.19)

+iπ

X

2 ? g(0) A8

+

g(4) +

48

p1 ?

1 2 g 2 (2)

3

2

1 1 2 g(2) ? g(0) g(2) + g(1)8 g(1)9 2

2

+

p1 2

+

p2 1 2 + g(1)8 g(1)9 g(2) 8

2 ? g(2)

?mn g(1)m g(3)n

This expression is cohomological although it is still unconventional in supergravity theory since it involves the mod-two valued Steenrod squares, denoted Sq 2 (g(3) ), in the ?rst line. The above topological term (1.14) is deduced from the K-theory theta function ΘK de?ned in [4,5,7], and reviewed below. As explained above, it is convenient to ?x the ?elds F . We can de?ne a function Θ(F , ρ) by writing ΘK as a sum ΘK = e?SB (F ) Θ(F , ρ) (1.20)

The sum is over all integral classes except a. That is, we sum over nα , γm , eα , fm subject to the constraint on Sq 3 fm . The action SB (F ) is the manifestly T -duality invariant action 8

for the ?uxes given in (1.8). ΘK is a function of gM N , ρ, τ and the ?at background NSNS ?uxes. As we have mentioned, turning on ?at potentials corresponds, in the K-theoretic automorphisms act naturally on the theta function. We give concrete formulae for this action by showing how the inclusion of nonzero ?at NSNS ?elds B0 , B(2) , Amα (1) modi?es the phase Φ. The explicit formula is in equations (5.20)-(5.24) below. spin 8-folds X it turns out that Θ(F ) is, essentially, a Siegel-Narain theta function for the lattice H 4 (X ; Z). More precisely, there is a quadratic form on H 4 (X ; R) given by Q = Im(ρ)HI ? iRe(ρ)I where H is the action of Hodge ? and I is the integral intersection pairing on H 4 (X, Z). Then α ? Θ(F , ρ) = ei2π ?Φ(F ) Θ ? (Q) β (1.21) Since the K-theoretic constraint Sq 3 a = 0, a ∈ H 4 (X, Z) is automatically satis?ed on interpretation, to acting by automorphisms of the K-theory group K 0 (X ) ? R. These

α ? Here Θ ? (Q) is the Siegel-Narain theta function with characteristics. The characteristics β are written explicitly in equations (5.10), (5.20), and (5.21) below. Finally, the prefactor ?Φ(F ) in (1.21) is de?ned in (5.23) and (5.24) below. 1.4. T-duality transformations One of the more subtle aspects of the K-theoretic formulation of RR ?uxes, is that the very formulation of the action depends crucially on a choice of polarization of the Ktheory lattice K (X10 ) with respect to the quadratic form de?ned by the index. In the above discussion we have chosen the “standard polarization” for IIA theory, i.e Γ2 is the sublattice of K (X10 ) with vanishing G4 , G2 , G0 . Γ1 is then a complementary Lagrangian sublattice such that K (X10 ) = Γ1 + Γ2 . The standard polarization is distinguished for any large

2 10-manifold in the following sense. When the metric of X10 is scaled up g ?M ?M ?N ? → t g ?N ? √ 2 10?4p the action X10 g ?|G2p | of the Type IIA RR 2p-form scales as t . This allows the

sensible approximation of ?rst summing only over G4 , with G2 = G0 = 0, then including G2 with G0 = 0, and ?nally summing over all classical ?uxes G4 , G2 , G0 .

longer distinguished. Various equally good choices are related by the action of the Tduality group DT on ΓK := K (X × T 2 ).

8 8

In the case of X10 = T 2 × X with the metric (1.1) the standard polarization is no

In section 4 we explain how the duality group

There is also a polarization on manifolds of the type S 1 × X9 , (in our case X9 = S 1 × X ) where the measure is purely real and the imaginary part of the action is an integral multiple of iπ (without ?at NSNS potentials). However, this polarization does not lead to a good long-distance approximation scheme.

9

DT acts as a subgroup of symplectic transformations on the K-theory lattice and we give 4.2, since DT acts symplectically, the function Θ(F , ρ) must transform under T -duality

an explicit embeding DT ? Sp(2N, Z), where 2N = rank (ΓK ). As explained in section

modular forms. Nevertheless, this transformation law leaves open the possibility of a

as Θ(γ · F , γ · ρ) = j (γ, ρ)Θ(F , ρ) where j (γ, ρ) is a standard transformation factor for

T-duality anomaly through a multiplier system in j (γ, ρ). In order to investigate this potential anomaly more closely we must choose an explicit duality frame and perform the relevant modular transformations. a nontrivial “multiplier system” under SL(2, Z )ρ . That is, using the standard generators T, S of SL(2, Z )ρ we have: Θ(T · F , ρ + 1) = ?(T )Θ(F , ρ) (1.22) We ?nd that, in fact, the function Θ(F , ρ) does transform as a modular form with

?) 2 b4 Θ(F , ρ) Θ(S · F , ?1/ρ) = ?(S )(?iρ) 2 b4 (iρ

1 +

1 ?

? where T ·F , S ·F denotes the linear action of DT on the ?uxes. Here b+ 4 , b4 is the dimension

of the space of self-dual and anti-self-dual harmonic forms on X and the multiplier system ?(T ) = exp iπ 4 iπ ?(S ) = exp 2 λ2

X

is

(1.23) λ

2

X

where λ is the integral characteristic class of the spin bundle on X. (So, 2λ = p1 ). The multiplier system is indeed nontrivial on certain 8-manifolds. As an example, on all CalabiYau 4-folds we have the relation 1 4 λ2 = 62

X

?8 ? 4 + 1 χ A 12 X

(1.24)

and hence ? is nontrivial if χ is not divisible by 12. In particular, a homogeneous polynomial of degree 6 in P 5 , has χ = 2610. See, e.g. [15]. In more physical language, the “multiplier system” signals a potential T -duality anomaly. Such an anomaly would spell disaster for the theory since the T -duality group should be regarded as a gauge symmetry of M-theory. Accordingly, we turn to the remaining functional integrals in the supergravity theory. We will ?nd that the anomalies cancel, of course, but this cancellation is surprisingly intricate. 10

1.5. Inclusion of 1-loop e?ects We ?rst turn to the 1-loop functional determinants of the quantum ?uctuations of the bosonic ?elds. We show that these are all manifestly T -duality invariant functions of is given in equation (6.20) below. The net e?ect of inlcuding the bosonic determinants is thus to replace e?SB (F ) Θ(F , ρ) → ZB (F , ρ) := DetB e?SB (F ) Θ(F , ρ) (1.25) F except for the quantum ?uctuations of g(4) . The full bosonic 1-loop determinant DetB

1 weight ( 1 4 (χ + σ ), 4 (χ ? σ )), in close analogy to the theory of abelian gauge potentials on

Inclusion of this determinant alters the modular weight so that ZB (F , ρ) transforms with a 4-manifold [16,17]. Here χ, σ are the Euler character and signature of the 8-fold X . The

multiplier system (1.23) is left unchanged. Now let us consider modi?cations from the fermionic path integral. Recall that we may always regard a modular form as a section of a line bundle over the modular curve H/SL(2, Z )ρ . On general grounds, we expect the fermionic path integral to provide a trivializing line bundle. The gravitino and dilatino in the 8d theory transform as modular forms under the T-duality group DT with half-integral weights and consequently they too are subject to potential T -duality anomalies. The inclusion of the fermions modi?es the bosonic partition function in two ways: through zeromodes and through determinants. The fermion action in the 8D supergravity has the form SFermi = Skinetic + Sfermi??ux + S4?fermi

(8)

(1.26)

where kinetic terms Skinetic as well as fermion-?ux couplings Sfermi??ux are quadratic in fermions and S4?fermi denotes the four-fermion coupling. Skinetic is T-duality invariant but Sfermi??ux and S4?fermi contain some non-invariant terms. The non-invariant fermion zeromode couplings are collected together in the form S (zm)ninv =

X

4π Imρ g(4) ∧ ? Y(4) + 2π Imρ Y(4) ∧ ? Y(4)

(1.27)

where the harmonic 4-form Y(4) is bilinear in the fermion zeromodes. The explicit expression for Y(4) can be found in equations (7.21) and (7.41) below. 11

The inclusion of the integral over the fermionic zeromodes of Skinetic modi?es the partition function by replacing the expression Θ(F , ρ) in (1.21) by Θ(F , ρ) = Here Θ d?F

(zm)

ei2π ?Φ(F ) Θ

α (Q) β

(1.28)

α (Q) β

is a supertheta function for a superabelian variety based on the K-theory theta function. ? di?er from α, ? ? β (This is explained in Appendix F.) In particular, the characteristics α, ? β by expressions bilinear in the fermion zeromodes. Similarly, the prefactor ?Φ di?ers from ?Φ by an expression quartic in the fermion zeromodes. Finally, d?F includes the T -duality invariant term e?S at

′ ?SB (F ) ZB+F (F , ρ) := Det′ Θ(F , ρ) B DetF e

(zm)inv

(zm)

is a T -duality

invariant measure for the ?nite dimensional integral over fermion and ghost zeromodes. It from the action. Including the one-loop fermionic determinants of the non-zero modes we ?nally arrive (1.29)

The formula we derive for (1.29) allows a relatively straightforward check of the Tduality transformation laws and we ?nd: ZB+F (T · F , ρ + 1) = ?(T )ZB+F (F , ρ) ZB+F (S · F , ?1/ρ) = (?iρ) 4

1 1 χ+ 8 X

(p2 ?λ2 )

(iρ ?) 4

1

1 χ? 8

X

(p2 ?λ2 )

(1.30) ZB+F (F , ρ)

Perhaps surprisingly, the fermion determinants have not completely trivialized the RR contribution to the path integral measure. However, there is one ?nal ingredient we must take into account: In the low energy supergravity there are quantum corrections which contribute to leading order in the t → ∞ and ξ → ?∞ limit. From the string worldsheet instanton corrections. From the M -theory viewpoint we must include the oneloop correction C3 X8 in M -theory together with the e?ect [18] of membrane instantons. 1 1 χ? 2 4 The net e?ect is to modify the action by the quantum correction Squant = 1 1 χ+ 2 4

X

worldsheet viewpoint these consist of a 1-loop term in the α′ expansion together with

(p2 ? λ2 ) log [η (ρ)] +

X

(p2 ? λ2 ) log [η (?ρ ?)]

(1.31)

Where η (ρ) is the Dedekind function. The ?nal combination Z (F , ρ) = e?Squant ZB+F (F , ρ) is the fully T-duality invariant low energy partition function. 12 (1.32)

1.6. Applictions As a by-product of the above results we will make some comments on the open problem of the relation of M-theory to massive IIA string theory. In [9] C. Hull made an interesting suggestion for an 11-dimensional interpretation of certain backgrounds in the Romans equivalent to M -theory on a certain 3-manifold, the nilmanifold. theory. One version of Hull’s proposal states that massive IIA string theory on T 2 × X is In section 9 we review Hull’s proposal. For reasons explained there we are motivated to introduce a modi?cation of Hull’s proposal, in which one does not try to set up a 1-1 correspondence between M-theory geometries and massive IIA geometries, but nevertheless, the physical partition function Z (F , ρ) of the massive IIA theory can be identi?ed with a certain sum over M-theory geometries involving the nilmanifold. The detailed proposal can be found in section 9.3. 1.7. U -duality and M -theory In the ?nal section of the paper we comment on some of the issues which arise in trying to extend these considerations to writing the fully U -duality-invariant partition function. SL(2, Z )ρ duality invariance, and we make some preliminary remarks on how one can see K-theory theta functions for twisted K-theory from the M theory formulation. We summarize brie?y the M -theory partition function on X × T 3 , we comment on the

2. Review of T-duality invariance in the standard formulation of type IIA supergravity We start by reviewing bosonic part of the standard 10D IIA supergravity action [19]. Fermions will be incorporated into the discussion in section 7. 2.1. Bosonic action of the standard 10D IIA supergravity ?2 and string frame metric The 10D NSNS ?elds are the dilaton φ, 2-form potential B ? ? ? , where M , N = 0, . . . 9. The 10D RR ?eldstrenghts are the 4-form G4 , 2-form G2 and We measure all dimensionful ?elds in units of 11D Planck length lp and set k11 = π, so 13

g ?M ?N

0-form G0 .

Sbos =

(10)

1 2π

e?2φ

X10

√

1? ?3 g10 R(? ?H g ) + 4dφ ∧ ? ?dφ + H 3∧? 2 √ g10 G2 0

(2.1)

1 + 4π

X10

? 4 ∧? ? 4 + iB ? ∧G ? 4 ∧G ?4 + G ? 2 ∧? ?2 + G ?G ?G

where ? ? stands for the 10D Hodge duality operator. The ?elds in (2.1) are de?ned as ? 2 = G2 + B ?2 G0 , G ?2 B ?2 G0 , ? 4 = G4 + B ?2 G2 + 1 B G 2 ? 3 = dB ?2 . H

We explain the relation between our ?elds and those of [19] in Appendix(B). 2.2. Reduction of IIA supergravity on a torus We now recall some basic facts about the reduction of the bosonic part of the 10D action on T 2 . Let us consider X10 = T 2 × X and split coordinates as X M = (xM , σ m ), where M = 0, . . . , 7, m = 8, 9. The standard ansatz for the reduction of the 10d metric has the form:

2 M N m n ds2 10 = t gM N dx dx + V gmn ω ? ω ?

(2.2)

where gmn is de?ned in (1.6), t2 gM N is 8D metric, detgM N = 1. V is the volume of T 2 and ω m = dσ m + Am (1) . The other bosonic ?elds of the 8D theory are listed below.

α α 1 . g(0) , g(2) ,

α = 1, 2 g(1)m , g(3)m

m = 8, 9

and g(4) are de?ned from9

G0 2 = g(0) 2π ?2 G 1 1 2 2 = g(0) ?mn ω m ω n + g(1)m ω m + g(2) + g(0) B0 2π 2 ?4 1 G 2 1 = g(4) + g(3)m ω m + B0 g(2) ?mn ω m ω n + g(2) 2π 2 2 . The 8D dilaton ξ is de?ned by e?2ξ = e?2φ V

(2.3)

(2.4)

3 . B(2) , B(1)m , B0 are obtained from the KK reduction of the NSNS 2-form potential in the following way

9

?89 = 1,

?89 = 1

14

?2 = 1 B0 ?mn ω m ω n + B(1)m ω m + B(2) + 1 Am B(1)m B 2 2 (1) Now, the real part of the 8D bosonic action obtained by the above reduction is

3

(2.5)

Re Sboson = SNS +

p=0

(8D)

Sp g(p) + S4 g(4)

(2.6)

where SNS = 1 2π 1 e?2ξ t6 R(g ) + 4dξ ∧ ?dξ + 28t?2 dt∧ ? dt + t2 H(3) ∧ ?H(3) 2

1 6 dρ ∧ ?dρ 1 4 1 dτ ∧ ?dτ + t + t gmn Gαβ Fmα ∧ ? Fnβ + t6 2 (Imτ )2 2 (Imρ)2 2

(2.7)

of

where Gαβ is de?ned in (1.6) and Am (1) and B(1)m are combined into 1-form as a collection Amα (1) = Also, we denote10 1 H(3) = dB(2) ? ?mn Eαβ A(1)mα Fnβ (2) 2

3

?mn B(1)n Am (1)

(2.8)

(2.9)

Sp g(p) = π

p=0 X

β α t8 Gαβ g(0) + t6 g mn g(1)m ∧ ? g(1)n + ∧ ? g(0)

(2.10)

β α t4 Gαβ g(2) + t2 g mn g(3)m ∧ ? g(3)n ∧ ? g(2)

Finally we have S4 g(4) = π

X

Im(ρ)g(4) ∧ ?g(4)

3 p=0

(2.11) Sp g(p) for the value of the

actions evaluated on a background ?ux ?eld con?guration. SB (F ) will enter the partition sum ZB+F (F , τ, ρ) in equation (8.1) below.

10

It is convenient to introduce the notation SB (F ) =

E12 = 1,

E21 = ?1

15

2.3. T-duality action on 8D bosonic ?elds The T-duality group of the 8D e?ective theory obtained by reduction on T 2 is known which acts on τ to be DT = SL(2, Z)τ × SL(2, Z)ρ, where the ?rst factor is mapping class group of T 2 τ→ and the second factor acts on ρ = B0 + iV ρ→ Let us denote generators of SL(2, Z)ρ by S : ρ → ?1/ρ, and generators of SL(2, Z)τ by ? : τ → ?1/τ, S ? :τ →τ +1 T T :ρ→ ρ+1 αρ + β γρ + δ (2.13) aτ + b cτ + d (2.12)

We now recall how T-duality acts on the remaining bosonic ?elds of the 8D theory mentioned in the introduction. These transform linearly under T -duality. They include the NS potential B(2) , which is T-duality invariant, as well as Amα (1) , which transform in

α α the (2, 2). The other components of F are the RR ?eldstrengths g(0) , g(2) , α = 1, 2 which

[14]. First, ξ, t, gM N are T -duality invariant. Next, there is the collection of ?elds F

DT .

transform in the (1, 2) of DT and g(1)m , g(3)m , m = 8, 9 which transform in the (2′ , 1) of Finally, the ?eld g(4) is singled out among all the other ?elds since according to the

conventional supergravity [14] SL(2, Z)ρ mixes g(4) with its Hodge dual ?g(4) and hence g(4) does not have a local transformation. More concretely, ?Reρg(4) + iImρ ? g(4) g(4)

(8D)

(2.14)

8D action Sboson is not manifestly invariant under SL(2, Z)ρ.

transforms in the (1, 2) of DT . Due to this non-trivial transformation the classical bosonic

3. Review of the K-theory theta function In this section we review the basic ?ux quantization law of RR ?elds and the de?nition of the K-theory theta function. We follow closely the treatment in [4,5,7]. 16

3.1. K-theoretic formulation of RR ?uxes As found in [1]-[4] RR ?elds in IIA superstring theory are classi?ed topologically by ? 2 = 0 is an element x ∈ K 0 (X10 ). The relation for B G = 2π

10

? Achx,

G=

j =0

Gj

(3.1)

? is expressed in terms of the Pontryagin classes where ch is the total Chern character and A as ? = 1 ? 1 p1 + 1 7p2 ? 4p2 A 1 24 5760 (3.2)

In (3.1), the right hand side refers to the harmonic di?erential form in the speci?ed real cohomology class. The quantization of the RR background ?uxes is understood in the sense that they are derived from an element of K 0 (X10 ). 3.2. De?nition of the K-theory theta function Let us recall the general construction of a K-theory theta function, which serves as the RR partition function in Type IIA. One starts with the lattice ΓK = K 0 (X10 )/K 0 (X10 )tors . This lattice is endowed with an integer-valued unimodular antisymmetric form by the formula ω (x, y ) = I (x ? y ?), where for any z ∈ K 0 (X10 ), I (z ) is the index of the Dirac operator with values in z. Given a metric on X10 , one can de?ne a metric on ΓK g (x, y ) =

X10

(3.3)

?G(y ) G(x) ? ∧ 2π 2π

(3.4)

where ? ? is the 10D Hodge duality operator.

terpreted as a symplectic form and a metric, respectively, on T. To turn T into a Kahler manifold one de?nes the complex structure J on T as g (x, y ) = ω (Jx, y )

Let us consider the torus T = (ΓK ?Z R) /ΓK . The quantities ω and g can be in-

(3.5)

Now, if it is possible to ?nd a complex line bundle L over T with c1 (L) = ω, then T 17

becomes a “principally polarized abelian variety.” L has, up to a constant multiple, a

unique11 holomorphic section which is the contribution of the sum over ?uxes to the RR partition function. vature ω are in one-one correspondence with U(1)-valued functions ? on ΓK such that ?(x + y ) = ?(x)?(y )(?1)ω(x,y). As explained in detail in [20], holomorphic line bundles L over T with constant cur-

(3.6)

For weakly coupled Type II superstrings one can take ? to be valued in Z2 . Motivated by T-duality, and the requirements of anomaly cancellation on D-branes [5], Witten proposed that the natural Z2 ? valued function ? for the RR partition function is given by a mod X10 , and for any v ∈ KO (X10 ), there is a well-de?ned mod 2 index q (v ) [21]. We take ?(x) = (?1)j (x) where j (x) = q (x ? x ?). two index [4]. For any x ∈ K 0 (X10 ), x ? x ? ∈ KO (X10 ) lies in the real K-theory group on

(3.7)

As explained in [4,5,7] there is an anomaly in the theory unless ?(x) is identically 1 on

the torsion subgroup of K (X10 ). In the absence of this anomaly it descends to a function RR partition function. on ΓK = K 0 (X10 )/K 0 (X10 )tors and can be used to de?ne a line bundle L and hence the To de?ne the theta function one must choose a decomposition of ΓK as a sum Γ1 ⊕ Γ2 ,

where Γ1 and Γ2 are “maximal Lagrangian” sublattices. ω establishes a duality between Γ1 and Γ2 , and therefore there exists θK ∈ Γ1 /2Γ1 such that ?(y ) = (?1)ω(θK ,y) , ?y ∈ Γ2

(3.8)

Following [7] we choose the standard polarization:the sublattice Γstd is de?ned as the 2 set of x with vanishing G0 , G2 , G4 . This choice implies that G0 , G2 , G4 are considered as independent variables. This is a distinguished choice for every large 10-manifold in the sense that it allows for a good large volume semiclassical approximation scheme on any 10-manifold ( see sec.5).

11

The uniqueness follows from the index theorem on T using unimodularity of ω and the fact i > 0.

that for any complex line bundle M over T with positive curvature we have H i (T; M ) = 0,

18

It was demonstrated in [7] that Γstd in the standard polarization consists of K-theory 1 classes of the form x = n0 1 + x(c1 , c2 ). 1 is a trivial complex line bundle and x(c1 , c2 ) is de?ned for c1 ∈ H 2 (X10 , Z) and c2 ∈ H 4 (X10 , Z) with Sq 3 c2 = 0, as 1 ) + .... ch(x(c1 , c2 )) = c1 + (?c2 + c2 2 1 (3.9)

The higher Chern classes indicated by . . . are such that x(c1 , c2 ) is in a maximal Lagrangian sublattice Γstd complementary to Γstd 1 2 . Then, θK for the standard polarization can be chosen to satisfy ch0 (θK ) = 0, where ch1 (θK ) = 0, ch2 (θK ) = ?λ + 2? a0 , I (θK ) = 0 (3.10)

1 λ= 2 p1 and a ?0 is a ?xed element of H 4 (X10 , Z) such that

?c ? ∈ L′ where L′ =

f (? c) =

X10

c ? ∪ Sq 2 a ?0 Sq 3 (? c) = 0

(3.11) and f (? a) stands for the

with the characteristic class a ? ∈ H 4 (X10 , Z) and supersymmetric spin structure on the S 1 . (We will show in section 5.1 below that for X10 = X × T 2 in fact a ?0 = 0.) The K-theory theta function in the standard polarization is ΘK = eiu

x∈Γ1

mod 2 index of the Dirac operator coupled to an E8 bundle on the 11D manifold X10 × S 1

4 4 c ? ∈ Htors (X10 , Z)/2Htors (X10 , Z),

eiπτK (x+ 2 θK ) ?(x)

1

(3.12)

by

where u = ? π 4

X10

ch2 (θK )ch3 (θK ) and the explicit form of the period matrix τK is given (G0 G10 ? G2 G8 + G4 G6 ) G2p ∧? ?G2p (3.13)

1 1 ReτK (x + θK ) = 2 (2π )2

X10 2

1 1 ImτK (x + θK ) = 2 (2π )2 p=0 The RR ?elds which enter (3.13),(3.14) are: 1 G0 (x + 2π 1 G2 (x + 2π 1 G4 (x + 2π

(3.14)

X10

1 θK ) = n0 2 1 θK ) = e ? 2 1 1 2 1 θK ) = a ?+ e ? ? (1 + n0 /12)λ 2 2 2 19

(3.15)

where we denote e ? = c1 (x),

From (3.12) and (3.13),(3.14) the following topological term was found in [7] to be the K-theoretic corrections to the 10D IIA supergravity action.

?,a ?) e2πiΦ(n0 ,e = exp ?2πin0

a ? = ?c2 (x) + a ?0 .

e ?

X10

? A

8

(?(1))

n0

e,a ?) e2πiΦ(?

(3.16)

e,a ?) a0 ) a) e2πiΦ(? = (?1)f (? (?1)f (? exp 2πi

X10

e ?5 e ?3 a ? 11? e3 λ e ?a ?λ e ?λ2 1 ? + ? ? + ? e ?A8 60 6 144 24 48 2 (3.17)

? 3] = 0 3.3. Turning on the NSNS 2-form ?ux with [H In the presence of an H -?ux we expect K -theory to be replaced by twisted K -theory KH classifying bundles of algebras with nontrivial Dixmier-Douady class. The Morita equivalence class of the relevant algebras only depends on the cohomology class of H , but this does not mean that the choice of “connection” that is, the choice of B ?eld is ? ] = 0, the choice of irrelevant to formulating the K -theory theta function. Indeed, when [H ? in H ? = dB ? changes the action in supergravity and “turning on” this ?eld trivialization B in supergravity corresponds to acting with an automorphism on the K-theory torus. In this section we describe this change explicitly. See [22][23] for recent mathematical results relevant to this issue. ?2 ∈ H 2 (X10 , R). We normalize B ?2 so that it is de?ned mod H 2 (X10 , Z) Let us turn on B under global tensor?eld gauge transformation. By Morita equivalence, the RR ?elds are still classi?ed topologically by x ∈ K 0 (X10 ). The standard coupling to the D-branes implies ? ( x) G ? = eB2 ch(x) 2π Let us de?ne ? ( x) G ? := e?B2 ch(? x) 2π ? A (3.19)

that the cohomology class of the RR ?eld is

? A

(3.18)

The bilinear form on ΓK = K 0 (X10 )/K 0 (X10 )tors is still given by the index: ω (x, y ) = 1 (2π )2 ? (x)∧G ? (y ) = I (x ? y G ?) 20 (3.20)

X10

while the metric on ΓK is modi?ed to be g ?(x, y ) = 1 (2π )2 ? (x)∧? ? (y ) G ?G (3.21)

X10

and the Z2 valued function ?(x) is unchanged. If we continue to use the standard polarization then θK ∈ Γ1 /2Γ1 is unchanged as well. τK The net e?ect to modify (3.12) is that the period matrix τK should be substituted for ? ). = τ K (G → G ?2 = eiu ΘK B eiπ τK (x+ 2 θK ) ?(x)

1

(3.22)

x∈Γ1

Note, that the constant phase eiu in front of the sum remains the same as in (3.12) The imaginary part of the 10D Type IIA supergravity action now becomes Im(S10D ) = ?2π Φ, where Φ=Φ+ 1 ? 2 1 ?5 2 1 ?3 2 1 ?4 2 ?2 B G , B2 G4 + B G2 G4 + B 2 G2 + G0 G4 + B2 G0 G2 + 2 8π 3 4 20 2 0 (3.23)

1 Φ is de?ned in (3.16),(3.17) and G2p (x + 2 θK ),

p = 0, 1, 2 are given in (3.15). ?2 coincide with the imaginary From (3.23) we ?nd that corrections to Φ depending on B

part of the standard supergravity action (see, for example [12].) ? de?ned in (3.19) is a gauge invariant ?eld if the global tensor?eld gauge Note, that G transformation ?2 → B ? 2 + f2 , B also acts on K 0 (X10 ) as: x → L(?f2 ) ? x, x ∈ K 0 (X10 ) (3.25) f2 ∈ H 2 (X10 , Z) (3.24)

where the line bundle L(?f2 ) has c1 (L(?f2 )) = ?f2 .

Thus, according to (3.25) a tensor?eld gauge transformation acts as an automorphism

of ΓK , preserving the symplectic form ω. (3.25) acts on theta function (3.22) by multiplication by a constant phase: ?2 + f2 = ei 4 ΘK B

π X10

f2 (λ?2? a0 )2

?2 ΘK B

(3.26)

4.

Action of T-duality in K-theory In this section we consider X10 = T 2 × X and describe the action of T-duality on the 21

K-theory variables.

As we have mentioned, the standard polarization is distinguished for any large 102 manifold in the following sense. When the metric of X10 is scaled up g ?M ?M ?N ? → t g ?N ? √ 2 10?4p the action X10 g ?|G2p | of the Type IIA RR 2p-form scales as t . This allows the

successive approximation of keeping only G4 whose periods have the smallest action, then

including G2 and ?nally keeping all G4 , G2 , G0 . no longer distinguished. Various equally good choices are related by the action of the

0 T-duality group DT on ΓK = K 0 (T 2 × X )/Ktors (T 2 × X ).

In the case of X10 = T 2 × X with the metric (1.1), the standard polarization is

denotes the complex dimension of the K-theory torus T = K 0 (T 2 × X ) ?Z R/ΓK and Sp(2N, Z) stands for the group of symplectic transformations of the lattice ΓK . 4.1. Background RR ?uxes in terms of integral classes on X. of integral classes on X. Let us start from the standard polarization 12 and write a general element of Γstd as 1 x = n0 1 + L(n1 e0 + e + γm dσ m ) ? 1 + x(e0 e′ + a + hm dσ m ) + ? where e0 = dσ 8 ∧dσ 9 , so that

T2

We argue below that DT can be considered as a subgroup of Sp(2N, Z), where N

To describe the action of DT on K-theory variables, we will write RR ?elds in terms

(4.1)

1 is a trivial line bundle, and for any a ? ∈ H 4 (X10 ; Z ), x(? a) is a K -theory lift (if it exists). In (4.1) ? puts x into the Lagrangian lattice Γstd and we also introduce the notations: 1 a ∈ H 4 (X ; Z), e, e′ ∈ H 2 (X ; Z), hm ∈ H 3 (X, Z), γm ∈ H 1 (X ; Z) m = 8, 9 (4.2)

e0 = 1. L(? e) is a line bundle with c1 (L) = e ? ∈ H 2 (X10 ; Z),

The RR ?elds entering (3.13),(3.14) are given by 1 1 G0 (x + θK ) = n0 , 2π 2 1 1 G2 (x + θK ) = n1 e0 + e + γm dσ m , 2π 2 1 1 1 1 G4 (x + θK ) = a + e2 + e0 e′′ + fm dσ m ? (1 + n0 /12)λ 2π 2 2 2 where e′′ = n1 e + e′ ? γ1 γ2 , fm = hm + am + eγm

(4.3)

(4.4)

Note that (3.13) is in fact only a function of these variables, by the Lagrangian property. ? 3 ] = 0, we ?nd the From the 10D constraint Sq 3 a ? = Sq 3 a ?0 , valid in the case [H constraints on the integral cohomology classes: Sq 3 fm = Sq 3 am , that actually Sq fm = 0,

12 3

m = 8, 9. We will show

m = 8, 9 ( see comment below 5.8).

Γstd and Γstd are de?ned on page 19. 1 2

22

4.2. The embedding DT ? Sp(2N, Z) From the transformation rules of the RR ?elds under the T-duality group [24] we ?nd of n0 , n1 and e, e′′ in the following way: nα = that fm and γm transform in the (2′ , 1) of DT and we can form a representation (1, 2) out n1 n0 eα = e′′ e

,

(4.5)

We would like to reformulate the transformation rules for RR ?elds in terms of the action on ΓK . 13 The action of SL(2, Z)τ on ΓK is via standard pullback under topologically nontrivial di?eomorphisms. The action of SL(2, Z)ρ is more novel. We will explain the action of the two generators S, T of SL(2, Z)ρ separately. To begin, the action of T on ΓK is a particular case of the global gauge transformation (3.24),(3.25)

std polarization since it maps Γstd 2 → Γ2 :

with f2 = e0 and for this reason T ∈ Sp(2N, Z). The action of T preserves the standard G2p (y ? L(?e0 )) = 0, ?y ∈ Γstd 2 p = 0, 1, 2 (4.6)

The action of the generator S on ΓK is more interesting. By the Kunneth theorem we can decompose K 0 (X × T 2 ) = K 0 (X ) ? K 0 (T 2 ) ⊕ K 1 (X ) ? K 1 (T 2 ) (4.7)

for K 1 (T 2 ) we denote the basis as ζ m , m = 8, 9. We now have a Lagrangian decomposition of ΓK = Γ1 ⊕ Γ2 : Γ1 = K 0 (X ) ? 1 ⊕ K 1 (X ) ? ζ 8 , Γ2 = K 0 (X ) ? (L(e0 ) ? 1) ⊕ K 1 (X ) ? ζ 9 (4.8)

as the standard symplectic operator iσ2 . For K 0 (T 2 ) we choose basis 1 and L(e0 ) ? 1, and

Both K 0 (T 2 ) = Z ⊕ Z and K 1 (T 2 ) = Z ⊕ Z have natural symplectic bases on which S acts

on which the T -duality generator S acts simply. However, the decomposition (4.8) is not compatible with the standard polarization, and hence the action of S in the standard polarization appears complicated. We now give an explicit description of the action of S in the standard polarization. Let us write a generic element y ∈ Γstd as 2 y = x(? a) ? L(e0 ) ? 1 + z1 + z2 + z3 ? L(e0 ) ? 1 ,

13

a ? ∈ H 4 (X, Z)

(4.9)

Some discussion of T -duality in the K -theoretic context can be found in [25].

23

In (4.9) z1 , z2 , z3 are such that G (z1 ) = jm dσ m , 2π where jm ∈ H 5 (X, R) ⊕ H 7 (X, R), G (z2 ) = k, 2π G (z3 ) = k ′ 2π (4.10)

transformation rules of RR ?elds [24] S acts on y as S : y → y′,

k, k ′ ∈ H 6 (X, R) ⊕ H 8 (X, R) According to the

y ′ = x(? a) + z1 + z3 ? z2 ? L(e0 ) ? 1

14

(4.11)

std std From (4.11) we ?nd that the image Γ′ 2 := S (Γ2 ) di?ers from Γ2 .

K-theory theta function ΘK . This follows from the fact that ΘK is an holomorphic section symplectic transformations, and has a one-dimensional space of holomorphic sections, it follows that under T-duality transformatons ΘK can at most be multiplied by a constant. Nevertheless, this leaves open the possibility of a T-duality anomaly, as indeed takes place. To conclude this section we show how the multiplier system of (1.22)(1.23) is related to the standard 8th roots of unity appearing in theta function transformation laws. Let us recall the general transformation rule under Sp(2N, Z) for the theta function θ [m] (τ ) of m′ ∈ R2N are the a principally polarized lattice Λ = Λ1 + Λ2 of rank 2N. Here m = m′′ characterstics and the period matrix τ ∈ MN (C), τ T = τ is a quadratic form on Λ1 . It was found in [26] that under symplectic transformations σ·τ = the general θ [m](τ ) transforms as ?[σ · m] (σ · τ ) = κ(σ )e2πiφ(m,σ)det(Cτ + D)1/2 ?[m](τ ) where σ · m = mσ ?1 +

14

de?ned transformation laws under DT of the function Θ(F , ρ), related by (1.20) to the

Since we have an embedding DT ? Sp(2N, Z), we can deduce the existence of well-

of the the line bundle L over the K-theory torus with c1 (L) = ω . Since L is not a?ected by

Aτ + B , Cτ + D

σ ∈ Sp(2N, Z)

(4.12)

(4.13)

1 2

CT D AT B

d d

? In following [24] we have actually combined the transformation S with the transformation S

from SL(2, Z)τ . This is a more convenient basis for checking the invariance of the theory.

24

φ(m, σ ) = ?

′ ′ ′′ 1 m T DB T m′ ? 2m T BC T m′′ + m T CAT m′′ + 2

+

′ ′′ 1 m TD ? m TC 2

AT B

d

where (A)d denotes a vector constructed out of diagonal elements of matrix A. The factor κ(σ ) in (4.13) has quite nontrivial properties [26]. In particular κ2 (σ ) is a character of Γ(1, 2) ? Sp(2N, Z), where σ ∈ Γ(1, 2) if f AT B

d

∈ 2Z,

CT D

d

∈ 2Z

(4.14)

σ (S ) and σ (T ) in Sp(2N, Z) . We give σ (S ) and σ (T ) in Appendix(A). (3.13),(3.14) we ?nd that in (4.13) det(C (S )τK + D(S ))1/2 = ei 4 b4 (?iρ) 2 b4 (iρ ?) 2 b4 , det(C (T )τK + D(T ))1/2 = 1,

π 1 + 1 ?

One can easily check that SL(2, Z )ρ ? Γ(1, 2) by writing out explicit representations Using the explicit expressions for σ (S ) and σ (T ) as well as the de?nition of τK

φ(m, σ (S )) = 0

(4.15) (4.16)

φ(m, σ (T )) = 0

Now comparing (4.13) and the explicit formulae (5.31) for the transformation laws of Θ(F , ρ) derived in the next section we ?nd the relation between κ(σ ) and the multiplier system ?(S ), ?(T ) κ(S )ei 4 b4 = ?(S ),

π

κ(T ) = ?(T )

(4.17)

5. Θ(F , ρ) as a modular form the K-theory theta function ΘK and we check that Θ(F , ρ) transforms under the T-duality group DT as a modular form. 5.1. Zero NSNS ?elds We ?rst assume that all NSNS background ?elds are zero. In this case Θ(F , ρ), de?ned

?π Im(ρ)g(4) ∧?g(4)

In this section we derive an explicit expression for Θ(F , ρ) using its relation (1.20) to

in (1.20) is given by an expression of the form Θ(F , ρ) =

ei2π Φ(a,F ) e

a∈H 4 (X,Z)

X

(5.1)

25

substitute

where the imaginary part of the 8D e?ective action 2π Φ(a, F ) is derived as follows. We a ? = a + e0 e′ + hm dσ m ,

?,a ?) into the de?nition (3.16) of ei2π Φ(n0 ,e .

e ? = e + n1 e0 + γm dσ m

(5.2)

We need to evaluate f (a + e0 e′ + hm dσ m ). We use the bilinear identity from [7] f ( u + v ) = f ( u) + f ( v ) +

X10

uSq 2 v,

?u, v ∈ H 4 (X10 ; Z)

(5.3)

to ?nd f (a + e0 e′ + hm dσ m ) = f (a + e0 e′ ) + f (hm dσ m ). Let us consider f (hm dσ m ) ?rst. Again using the bilinear identity we obtain: f (hm dσ m ) = f (h8 dσ 8 ) + f (h9 dσ 9 ) +

X

(5.4)

h8 Sq 2 (h9 )

(5.5)

over, from the di?eomorphism invariance of the mod two index we see that f (hdσ 8 ) = f (hdσ 8 + ?hdσ 9 ), for any integer ? and, using the bilinear identity once more we ?nd that f (hdσ m ) = r (h), m = 8, 9 where r (h) =

X

From (5.3) it follows that f (hdσ m ), m = 8, 9 are linear functions of h ∈ H 3 (X, Z). More-

hSq 2 h,

h ∈ H 3 (X, Z)

(5.6)

is a spin-cobordism invariant Z2 -valued function. In fact, r (h) is a nontrivial invariant since for X = SU (3) and h = x3 the generator of H 3 (SU (3), Z) we have r (h) = 1. In conclusion: f (hm dσ m ) =

X

h8 Sq 2 h8 + h9 Sq 2 h9 + h8 Sq 2 (h9 )

(5.7)

Now we consider f (a + e0 e′ ): f (e0 e′ + a) = f (a) + f (e0 e′ ) +

X10 2

e0 e′ Sq 2 a 1 aλ + (e ) (a ? λ) 2 X

′ 2

1 = (a) ? (e′ )2 λ + (e′ )2 a = 2 X

(5.8)

This uses the bilinear identity (5.3), the reduction of the mod two index along T 2 , and the formula eq.(8.40) for f (u ∪ v ) from [7]. We can now evaluate a ?0 de?ned in (3.11). The kernel of Sq 3 is given by those elements

a + e0 e′ + hm dσ m such that h8 ∪ h8 = h9 ∪ h9 = 0. If we add the condition that the element 26

is a torsion class then f (a + e0 e′ ) = 0 and we need only evaluate (5.7). Now, since condition that hm is torsion we ?nd that the right hand side of (5.7) is zero. It follows that a ?0 = 0. We can now evaluate the phase. Using (4.4) we reexpress (5.5) as f (hm dσ m ) =

X

Sq 3 (hm ) = hm ∪ hm = 0 it follows that Sq 2 (hm ) has an integral lift. Using again the

f8 Sq 2 (f9 ) + f8 Sq 2 (f8 ) + f9 Sq 2 (f9 ) + e2 (γ9 f8 ? γ8 f9 ) + e3 γ8 γ9

(5.9)

Taking into account (5.9) and (5.8) we ?nd the total phase Φ(a, F ) in (5.1) is given by: Φ(a, F ) = ?Φ + where the characteristics are de?ned as: α= 1 1 1 2 (e) + (1 ? n0 /12) λ + (e′′ + e) ?mn γm γn 2 2 2 1 ′′ 2 1 1 β = (e ) + (1 ? n1 /12) λ + (e′′ ? e) ?mn γm γn 2 2 2 (a + α)β,

X

(5.10)

(5.11)

shift of the summation variable in (5.1) a → a + λ + The prefactor ?Φ is given by

1 mn ? γm γn . Note that for convenience we have made a and we recall that e′′ = n1 e + e′ ? 2 1 2

(e + e′′ ) ?mn γm γn .

exp[2πi?Φ] = exp πi

X

f8 Sq 2 (f9 ) + f8 Sq 2 (f8 ) + f9 Sq 2 (f9 )

(5.12)

exp 2πi

X

1 1 1 1 1 1 2 2 ? (e′′ e) ? e′′ eλ + e3 e′′ ? e2 λ + n0 λ (e′′ ) + (1 + n0 /12)λ2 + 4 24 6 4 48 4 1 ?8 ? 1 n0 n1 A ?8 + + (n0 ? n1 )A 2 2 + λ 24

2

+

λ mn ? γ m fn + 24

1 n0 (e′′ ? e)λ ? 12e2 e′′ ? 4eλ ? 4e3 ?mn γm γn 48

In deriving ?Φ we have used ? A = 1 ? A8 ? 2 27 λ 24

2

8

Also, in bringing ?Φ to the form (5.12) we have used the congruences 1 1 1 3 2 (e′′ ) e + e′′ e3 + (e′′ ) e2 ? λe′′ e ∈ Z 6 4 12 (e′′ e) ∈ 2Z,

2

(5.13) (5.14)

X

X

X

e′′ eλ ∈ 2Z.

which follow from the index theorem on X : 1 4 1 e ? λe2 ∈ Z, 24 24 ?e ∈ H 2 (X, Z). (5.15)

X

5.2. Including ?at NSNS potentials Let us now take into account globally de?ned NSNS ?elds: ?2 = 1 B0 ?mn ω m ω n + B(1)m ω m + B(2) + 1 Am B(1)m , B 2 2 (1) Am (1)

and recall that Am (1) and B(1)m are combined into the (2, 2) of DT as in (2.8). ?2 ? = eB G as in (3.18) where G are given in We de?ne a gauge invariant ?eldstrength G 1 ? x + θK as (4.3) and we expand G

2

?0 G 2π ?2 G 2π ?4 G 2π

1 x + θK 2 1 x + θK 2 1 x + θK 2

2 = g(0) 1 2 = g(0) + g(0) B0

1 2 ?mn ω m ω n + g(1)m ω m + g(2) 2 1 ?mn ω m ω n 2

(5.16)

2 1 = g(4) + g(3)m ω m + B0 g(2) + g(2)

α α ?elds g(0) , g(1)m , g(2) , g(3)m are now linear combinations of the integral classes γm , fm , eα , nα

The ?rst e?ect of including ?at NSNS ?elds is to modify the ?elds which enter SB (F ). These

de?ned in (4.2),(4.4) with coe?cients constructed from Amα (1) and B(2) : n1 n0 1 α α γm + ξ(1)m + B(2) g(0) g(2) = eα + Amα (5.17) (1) 2 (5.18)

α g(0) =

,

g(1)m = γm + ξ(1)m ,

1 1 nβ g(3)m = fm + B(2) g(1)m + λ(3)m + k(3)m + ?mn Eαβ Apα (1) ξ(1)p A(1) 2 6 where we denote

α ξ(1)m = ?mn Eαβ g(0) Anβ (1) ,

λ(3)m = ?mn Eαβ eα Anβ (1) , 28

nβ k(3)m = ?mn Eαβ Apα (1) γp A(1) (5.19)

The other e?ect of including ?at NSNS ?elds is to shift the characteristics and the prefactor of Θ(F , ρ). Now Θ(F , ρ) has the form: Θ(F , ρ) = e2πi?Φ exp

a∈H 4 (X,Z) X

? ?πIm(ρ)g(4)∧ ? g(4) + iπRe(ρ)g(4)∧g(4) + 2πig(4) β (5.20)

where [g(4) ] = a + α, ?

? are a ∈ H 4 (X, Z), and the shifted characteristics α, ? β α ? = α + ?2 , ? = β + ?1 β (5.21)

where α, β are de?ned in terms of integral classes n0 , n1 , γm , eα in (5.11), while ?α transform in the (1, 2) of DT . Explicitly, 1 1 fm + λ(3)m + k(3)m ?α = Amα (1) 2 6 1 γm + ξ(1)m + B(2) eα + Amα (1) 2 + (5.22)

where ξ(1)m , λ(3)m , k(3)m ζ(4) =

1 α α + B(2) B(2) g(0) ? ζ(4) g(0) 2 are given in (5.19) and we also denote (5.23)

1 m2 β4 n2 β3 m1 β2 n1 β1 A(1) ?m1 m2 A(1) ?n1 n2 A(1) Eβ1 β2 Eβ3 β4 A(1) 64

The shifted prefactor ?Φ in (5.20) is given by ?Φ = ?Φ ? 1 β ∧?2 + ?1 ∧?2 + (?Φ)inv 2 (5.24)

X

where ?Φ is de?ned in terms of integral classes n0 , n1 γm , eα , fm in (5.12) and (?Φ)inv Explicitly, is the part of the phase which is manifestly invariant under the T-duality group DT . (?Φ)inv =

3 B(2)

X

1 1 1 1 α β Eαβ g(0) e ? ?mn γm γn ? ?mn ξ(1)m γn ? ?mn ξ(1)m ξ(1)n + (5.25) 12 6 4 8

1 1 3 1 2 B(2) ? ?mn ξ(1)m fn ? ?mn λ(3)m γn ? ?mn λ(3)m ξ(1)n ? ?mn k(3)m ξ(1)n + 4 2 8 24 X 1 1 1 1 B(2) ? ?mn fm fn ? ?mn λ(3)m fn ? ?mn λ(3)m λ(3)n ? ?mn λ(3)m k(3)n + 2 2 4 6 X + 1 1 β m + ζ(4) ?mn γm γn + ξ(1)m q(5) + ζ(4) Eαβ eα g(0) 12 2

X

1 m λ(3)m q(5) + ζ(4) ?mn γm fn 12

mβ m where q(5) = Eαβ Apα (1) fp A(1)

29

5.3. Derivation of T-duality transformations. Let us study transformations of Θ(F , ρ) de?ned in (5.20) under DT . First, we note

that Θ(F , ρ) is invariant under SL(2, Z )τ . Next, we consider the action of the generator S. For any function h(F ) of ?uxes F , we denote

S h(F ) := h(S · F ) and δS [h] := S [h] ? h where S ·F denotes the linear action on ?uxes. To check the transformation under S we need law is: ? θ ?φ (0| ? 1/τ ) = (?iτ )1/2 e2πiθφ ? (0|τ ) φ θ to do a Poisson resummation on the self-dual lattice H 4 (X, Z). The basic transformation (5.26)

and its generalization to self-dual lattices (4.13). ?nd that Θ(F , ρ) transforms under S as Θ(S · F , ?1/ρ) = e

2πi

X

After the Poisson resummation and a shift of summation variable a → a + e2 + λ we

? +δS ?Φ S α ? S β

(?iρ) 2 b4 (iρ ?) 2 b4 Θ(F , ρ)

1 +

1 ?

(5.27)

? (5.21),(5.22) and ?Φ (5.24) as well as the transformation Now using the de?nitions of α, ? β rules for F , we ?nd after some tedious algebra δS ?Φ = ? ? + S α ? S β

X X

λ2 +Z 4

(5.28)

We conclude that the generator S acts as Θ(S · F , ?1/ρ) = e

iπ

X

λ2 /2

(?iρ) 2 b4 (iρ ?) 2 b4 Θ(F , ρ)

1 +

1 ?

(5.29)

f2 = e0 . In this way we ?nd from (3.26) that

the K-theory theta function ΘK as well as the transformation of ΘK under global gauge ?2 → B ?2 + f2 (3.26) where the action of the generator T corresponds to transformation B

iπ λ2 /4

To check how Θ(F , ρ) transforms under the generator T we use its relation (1.20) to

Θ(T · F , ρ + 1) = e 30

X

Θ(F , ρ)

(5.30)

5.4. Summary of T-duality transformation laws Below we summarize the transformation laws of the function Θ(F , ρ) under the gen-

erators of T-duality group DT .

Θ(F , ρ) is invariant under SL(2, Z )τ : ? · F , ρ) = Θ(F , ρ) Θ(T ? · F , ρ) = Θ(F , ρ) Θ(S (5.31)

SL(2, Z )ρ . That is, using the standard generators T, S of SL(2, Z )ρ we have: Θ(T · F , ρ + 1) = ?(T )Θ(F , ρ)

Θ(F , ρ) transforms as a modular form with a nontrivial “multiplier system” under

?) 2 b4 Θ(F , ρ) Θ(S · F , ?1/ρ) = ?(S )(?iρ) 2 b4 (iρ

1 +

1 ?

(5.32)

? where T ·F , S ·F denotes the linear action of DT on the ?uxes. Here b+ 4 , b4 is the dimension

of the space of self-dual and anti-self-dual harmonic forms on X and the multiplier system

is ?(T ) = exp iπ 4 iπ ?(S ) = exp 2 λ2

X

(5.33) λ

2

X

where p1 = p1 (T X ). These de?ne the “T-duality anomaly of RR ?elds.”

6. The bosonic determinants In this section we compute bosonic quantum determinants around the background speci?ed in section 2. Let us factorize bosonic quantum determinants as: DRR (DNS ) denotes the contribution from RR (NSNS) ?elds. 6.1. Quantum determinants DRR for RR ?elds Quantum determinants DRR for RR ?elds have the form

4

DetB = DRR DNS , where

DRR =

ZRR,p

p=1

(6.1)

31

where ZRR,p is the quantum determinant for g(p) . First, we present the contribution ZRR,4 arising from the ?uctuation dC(3) of g(4) . From (2.11) we ?nd the kinetic term for C(3) S3,cl = πIm(ρ) dC(3) , dC(3) (6.2)

where ( , ) denotes the standard inner product on the space of p-forms on X , constructed with the background metric gM N . We use the standard procedure [27,28] for path-integration over p-forms, which can be summarized as follows. Starting from the classical action for the p-form Sp,cl = α dC(p) , dC(p) one constructs the quantum action as15 :

p

Sp,qu = α C(p) , ?p C(p) +

m=1

α

1 m+1

m+1 k=1 k uk (p?m) , ?p?m u(p?m)

(6.3)

is the Laplacian acting on p-forms and constructed with gM N 16 . To compute ZRR,4 we apply (6.3) for p = 3, normalized as [DCp ]e?(Cp ,Cp ) = 1:

′ ′ ′ ′ ?1 2 (B3 ?B2 +B1 ?B0 )

example, uk (p?1) ,

where uk (p?m) ,

k = 1, . . . m + 1, m = 1, . . . p are ghosts of alternating statistics. For k = 1, 2 are fermions, uk (p?2) , k = 1, 2, 3 are bosons, etc. In (6.3) ?p α = π Im(ρ) and use the measure [DCp ]

ZRR,4 = (α)

det′ ?3 V3

?1 2

det′ ?2 V2

det′ ?1 V1

?3/2

det′ ?0 V0

2

(6.4)

where det′ ?p is the determinant of nonzero modes of the Laplacian acting on p-forms.

p p the dimension and the determinant of the metric of the harmonic torus Tharm = Hp /HZ . ′ Bp = Bp ? bp , where Bp denotes the (in?nite ) number of eigen-p-forms and bp and Vp are

The appearance of Vp in (6.4) is due to the appropriate treatment of zeromodes and is The determinants det′ ?p together with the in?nite powers depending on Bp , here and below, require regularization and renormalization, of course. These can be handled using, for example, the techniques of [29]. In particular the expression q (Imρ) := (Imρ)? 2 (B3 ?B2 +B1 ?B0 )

15

1 1

explained in Appendix(E).

(6.5)

Factors α m+1 should be understood as a mnemonic rule to keep track of the dependence on ? = dd? + d? d

α which follows from the analysis of various cancellations between ghosts and gauge-?xing ?elds

16

32

is a local counterterm of the form e the 1-loop action: S1?loop = π Imρ

?π Imρ

X

(uλ2 +vp2 )

, where the numbers u, v depend on

X

the regularization. From now on we will assume that πImρ

(uλ2 + vp2 ) is included into

(uλ2 + vp2 ) +

X

iπ Reρ 24

X

p2 ? λ2

(6.6)

the ?uctuations for g(3)m , g(1)m respectively. Let us also make ?eld rede?nition ?(0)m , m = 8, 9 to ?elds C(0)m , m = 8, 9 which have well de?ned of the quantum ?elds C transformation properties under the full U-duality group17 C(0)8 = √ ?(0)8 , τ2 eξ C 1 ?(0)9 C(0)9 = √ eξ C τ2

In section 8 we will show that T-duality invariance determines u and v uniquely. ?(0)m which are Next, we consider the contributions to DRR from dC(2)m , dC α , dC

(1) α g(2) ,

(6.7)

From (2.10) we ?nd classical action quadratic in the above ?uctuations: ?6 g ′mn C(0)m , d? dC(0)n , S0,cl = π t

β α S1,cl = πt4 Gαβ C(1) , d? dC(1)

S2,cl = πt2 g mn C(2)m , d? dC(2)n ? = te?ξ/3 is U-duality invariant, and g ′88 = 1 g 88 , g ′99 = τ2 g 99 , g ′89 = g 89 . where t τ2 6 ′mn 4 2 mn ? Now, using (6.3) with a = π t g , πt Gαβ , πt g and p = 0, 1, 2 correspondingly we ?nd: ZRR,1 ?6 = πt

′ ′ B0 ?B1 ′ ?B0

det′ ?0 V0

?1

?1

(6.8)

2

ZRR,2 = πt4 ZRR,3 = πt

2

det′ ?1 V1

?1

det′ ?0 V0

2

(6.9)

?3

′ ′ ′ ?B2 +B1 ?B0

det′ ?2 V2

det′ ?1 V1

det′ ?0 V0

(6.10)

In computing (6.8)-(6.10) we also used that detm,n g mn = 1, detα,β Gαβ = 1.

detm,n g ′mn = 1 and

Collecting together (6.4) and (6.8)-(6.10) we ?nd that DRR has the form: DRR det′ ?3 = rRR (t, ρ) V3

?1 2

det′ ?1 V1

?1 2

(6.11)

17

For some discussion of U-duality see sec.10

33

where rRR (t, ρ) = (e )

′ ξ 2B0

Imρ

1 2 (b3 ?b2 +b1 ?b0 )

t?2B2 ?2B1 ?4B0 (π )? 2 (B0 +B1 +B2 +B3 )

′

′

′

1

′

′

′

′

and we recall that q (Imρ) was included into S1?loop .

ential forms. It would be more natural if these determinants had a K-theoretic formulation. This might be an interesting application to physics of di?erential K-theory. 6.2. Quantum determinants for NSNS ?elds

mα Let us ?rst consider ?uctuations damα (1) and db(2) of the NSNS ?eld F(2) and H(3) .

We have computed the quantum determinants DRR treating RR ?uctuations as di?er-

From (2.7) we ?nd the quadratic action for ?uctuations: Scl = 1 ?2ξ 4 nβ 2 ? b(2) , d? db(2) t gmn Gαβ amα e (1) , d da(1) + t 4π (6.12)

Now, again using (6.3) we ?nd ZNS,2 = and ZNS,3 t2 ?2ξ = e 4π t4 ?2ξ e 4π

′ ′ 2(B0 ?B1 )

det′ ?1 V1

1 ?2

?2

det′ ?0 V0

4

(6.13)

′ ′ ′ 1 2 (B1 ?B2 ?B0 )

det′ ?2 V2

det′ ?1 V1

det′ ?0 V0

?3/2

(6.14)

Let us now consider ?uctuations of scalars: δξ, δτ, δρ. From (2.7) we write the action quadratic in these ?uctuations: Sscal = β

X

8? M δξ?M δξ +

1 1 ? M δτ ?M δ τ ?+ ? M δρ?M δ ρ ? 2 (τ 2 ) (ρ 2 )2

(6.15)

where β =

1 ?2ξ 6 t . 4π e

Now using the scalar measures de?ned as

?

δρ∧?δ ρ ? X (Imρ)2

[Dδρ][Dδ ρ ?]e

= 1,

?8

[Dδτ ][Dδ τ ?]e

δξ∧?δξ

?

δτ ∧?δ τ ? X (Imτ )2

=1

(6.16) (6.17)

[Dδξ ]e

X

=1

we ?nd the quantum determinants for the NSNS scalars ZNS,0 : ZNS,0 = β ? 2 B0 34

5 ′

det′ ?0 V0

?5 2

(6.18)

Finally, we consider the ?uctuation hM N of the metric t2 gM N . Recall that we work in the limit e?ξ → ∞ so that in computing the quantum determinant for the metric we drop couplings to RR background ?uxes. From (2.7) we ?nd the quadratic action: Smetr = β

X

(DN hM P )P M P QS DN hQS + hM P RM NP Q hNQ 1 ? DM hM N ? DN h 2

2

(6.19)

where h = g M N hM N and P M P QS = 1 M Q P S 1 M P QS g g ? g g 2 4

The covariant derivative DM is performed with the background metric, and indices are raised and lowered with this metric. Following standard procedure [30,31] we ?rst insert the gauge ?xing condition into

1 the path-integral δ κN ? (DM hM N ? 2 DN h) Then, we insert the unit

In (6.19) RM NP Q is the Riemann tensor of the Ricci-?at18 background metric gM N .

1=

det β 11

Dκ(1) e?β (κ(1) ,κ(1) )

(6.20)

and integrate over κ(1) in the path-integral. This procedure brings the kinetic term for the ?uctuation hM N to the form β

X QS hQS , hM P P M P NR KNR Q S QS δR DL DL + 2RN Q RS = ?δN KNR

(6.21)

Gauge ?xing also introduces fermionic ghosts k(1) , l(1) with the action Sgh = β 1/2 l(1) , ?1 k(1) Using the measure [DhM N ]e

?

X

(6.22)

hM N P M N P Q hP Q

= 1 we obtain the result for the quantum

determinant Zmetr of the metric: Zmetr = (β )

18

′ ′ 1 ?2 ?B1 ) (NK

det K

′

1 ?2

det′ ?1 V1

(6.23)

If the background metric is not Ricci-?at there are terms involving the Ricci-tensor in (6.19)

as well as in (6.22) below.

35

where det′ K is a regularized determinant of nonzero modes of the operator K de?ned in the second rank symmetric tensors and nK is the number of zeromodes of the operator K. We will explain how we regularize det′ K shortly. Combining all NSNS determinants together we ?nd: DNS = rNS (t, ξ ) det′ K where rNS (t, ξ ) = (4π ) 2

1 ′ ′ ′ ′ 1 NK +B0 +B1 +2 B2

′ (6.21) and NK = NK ? nK , where NK stands for the dimension (in?nite ) of the space of

?1 2

det′ ?2 V2

1 ?2

(6.24)

eξ

′ ′ ′ ′ ′ ′ ′ ′ NK +B2 +2B1 +2B0 ?3NK ?B2 ?4B1 ?8B0

t

(6.25)

Finally, from (6.11) and (6.24) we ?nd the full expression for bosonic determinants DetB = Q(t, gM N ) Imρ

1 2 (b3 ?b2 +b1 ?b0 )

(6.26)

where Q is a function only of the T-duality invariant variables gM N , t and ξ. Explicitly, Q(t, gM N ) = rtot det K NK so that

′

1 ?2

det′ ?3 V3

?1 2

det′ ?2 V2

1 ?2

det′ ?1 V1

1 ?2

(6.27)

where we regularized det′ K in a way that eliminates dependence on i?nite numbers Bp and ?)3(nK +b2 +2b1 +4b0 ) rtot = (t (6.28)

? = te?ξ/3 . where we recall t that DetB is manifestly invariant under all generators of DT except generator S. Using, Im(?1/ρ) = we ?nd that under S , DetB transforms as DetB (?1/ρ) = sB DetB (ρ), 7. Inclusion of the fermion determinants In this section we include the e?ects of the fermionic path integral. We recall the fermion content in the 10-dimensional and 8-dimensional supergravity theories and derive their actions. In the presence of nontrivial ?uxes these fermionic path integrals are nonvanishing, even for the supersymmetric spin structure on T 2 . 36 sB = ρ ρ ?

1 2 (b0 ?b1 +b2 ?b3 )

Now, let us check the transformation laws of DetB under DT . From (6.26) it is obvious Im(ρ) ρρ ?

(6.29)

(6.30)

7.1. Fermions in 8D theory and their T-duality transformations. Let us begin by listing the fermionic content in the 8-dimensional supergravity theory (this content will be derived from the 10-dimensional theory below.) The fermions in the 8D theory include two gravitinos ψ A , η A , A = 0, . . . , 7 and spinors Σ, Λ, l, ?, ? l, ? ?. 19 The relation of these ?elds to the 10D ?elds is explained in (7.13),(7.14) below. There are also bosonic spinor ghosts b1 , c1 , Υ2 and b2 , c2 , Υ1 which accompany ψ A and η A respectively. The fermions and ghosts transform under T-duality generators as follows. The gen?, S ? act trivially on fermions and ghosts while the under the generator S they erators T, T transform as ψ A → eiαΓ ψ A , l → e2iαΓ l, and ghosts transform as Υ1 → Υ1 ,

? ? ?

ηA → ηA ,

Λ → e?iαΓ Λ, ? → eiαΓ ?,

? ?

?

Σ→Σ

?

(7.1) (7.2)

? ? l → e?2iαΓ ? l,

? ? → eiαΓ ? ?

Υ2 → e?iαΓ Υ2

(7.3) (7.4)

{c1 , b1 } → eiαΓ {c1 , b1 } {c2 , b2 } → {c2 , b2 } where α is de?ned by 1 α = ν + π, 2 ? is the 8D chirality matrix. and Γ iρ ? = eiν |ρ|

(7.5)

The above transformation rules for space-time fermions follow from the transformation rules for the appropriate vertex operators on the world-sheet (as discussed for example in [11]). The only generator of DT acting non-trivially on fermions is S . The components are invariant. This follows since S does not act on the left-moving components of vertex operators. In this way we ?nd the transformation rules for η A , b2 , c2 , Σ, Υ1 , l, ? l, which origwe recall that these ?elds originate from N S ? R sector and that the right-moving R vertex S : VR → eiαΓ VR .

? a A VNS , a = 8, 9 of the right-moving NS vertex are rotated by 2α, while the components VNS

inate from R ? N S sector. To account for the transformation rules for ψ A , b1 , c1 , Λ, Υ2 , ?, ? ?

VR transforms under S as

19

(7.6)

These ?elds are MW in Lorentzian signature. We supress 16 component spinor indices below

37

7.2. 10D fermion action We start from the part of the 10D IIA supergravity action quadratic in fermions[19]. We work in the string frame. Sf erm = 1 16 √

(10) 20

√

?g10 e?2φ

1 1? ? ?N ?B ? ? ? ? ?N ? ? ?? + 1Λ ? ?Γ ? ?Γ ?N ?A ? ?A DN DN ψ ? Λ ? √ ( ?N ? ψB ? ? φ)ΛΓ Γ ψA A 2 2 2

+

? E 3 ? 5? ?C ? ? ? 11 ?F ? ?C ? ? 11 ? ? ?A ?C ? ? 11 ? ? ? Γ ? ?? ψ ? ?Γ ?A ? ?A ? ?E ?g10 e?φ G ΓF Γ Λ Γ ψE ? ] Γ ψ + √ ΛΓ Γ ? + ΛΓ AC [E 4 2

+ 1 192 + √

√

?g10 e?φ G0

5 ? 21 ? 1? ?B ? ? ?? ? ?Γ ?Λ ? + ?A ? ?A Λ ψB ψ ? + √ ΛΓ ψA ?? A 8 32 8 2

(7.7)

+

? E 3? 1 ? ?B ?C ?D ?? ?B ?C ?D ? ? ? ?A ? ?B ?C ?D ?? ? ? Γ ?F ? ?A ? ???? ψ ? ?E ? ?Γ ?A Λ ?g10 e?φ G ψE ΓF ? + ΛΓ ? ] ψ + √ ΛΓ Γ AB C D [E 4 2

1 48

√

? √ ? E E ?B ?C ? ? ? 11 ?F ? ?B ?C ? ? 11 ? ? ? ? ?Γ ?A ? ? ? ?A ?g10 e?2φ HA ψ Γ ΓF Γ ψE ?B ?C ? ? ] Γ ψ + 2ΛΓ Γ ? [E

? ?A ? and ψ are the Majorana dilatino and gravitino and covariant derivatives act on where Λ

them as

1 ?C ? ?A ? ? ? ? ?B ? B A ?A ?A ψ DN ?B ?C ?Γ ? ψ = ?N ? ψ + ωN ? B ? ψ + ωN 4 ?C ?? ? ? 1 ? ? ? ΓB DN Λ ? Λ = ?N ? Λ + ωN BC 4

There are also terms quartic in fermions in the action. It turns out that it is important to take them into account to check the T-duality invariance of partition sum. We recall the 4-fermionic terms in Appendix(C). 7.3. Reduction on T 2 . To carry out the reduction of the fermionic action to 8D we choose the gauge for the 10D veilbein as

? ?A E ? = M A tEM 0

2 a Am M V em 1 V 2 ea m 1

,

(7.8)

? ?A (recall a = 8, 9 and A = 0, ..., 7) and use the following basis of 10D 32 × 32 matrices Γ ,

? A = σ2 ? ΓA Γ

20

A = 0, . . . 7,

? 8 = σ1 ? 116 , Γ

? 9 = σ2 ? Γ ?, Γ

? = Γ0 . . . Γ7 Γ

(7.9)

We explain the relation between our conventions and those of [19] in Appendix(B).

38

Here ΓA are symmetric 8D Dirac matrices, which in Euclidean signature can be all chosen ? 11 and charge to be real, and σ1,2,3 are Pauli matrices. In this basis the 10D chirality Γ conjugation matrices C (10) have the form ? 11 = σ3 ? 116 , Γ C (10) = iσ2 ? 116 . (7.10)

? ?A ? in the following and Λ The 8D fermions listed in section 7.1 are related to 10D ?elds ψ

way

21

: ψA ηA ?a , ?A + 1 Γ ?AΓ ?aψ =ψ 6 √ l a ? ? 2Λ ?, ?aψ =Γ ? 2 Σ Λ ? ? ? l

√ 2 ? ?a 3? Γa ψ , = Λ+ 4 4 ?8 ? Γ ?9 ? 89 ψ =ψ

(7.11)

(7.12)

7.4. 8D fermion action Now we present the 8D action Squad = Skin + Sf ermi?f lux quadratic in fermionic

(8)

?uctuations 22 over the 8D background speci?ed in section 2.2. The kinetic term is standard

Skin =

X

e?2ξ t7

1 ? AM B 1 2 2 ψA Γ DM ψB + η ?A ΓAM B DM ηB + ΣΓM DM Σ + ΛΓM DM Λ 2 2 3 3 (7.13) 1? M 1 M 1? M 1 M lΓ DM ? l+ ? lΓ DM l + ? ?Γ DM ? + ? ?Γ DM ? ? + ? 4 4 4 4

The coupling of ?uxes to fermion bilinears is: Sf ermi?f lux = π 4 e?ξ t8

X

n0 ρ ? + n1 ? n0 ρ + n1 m √ X(0) ? √ + (7.14) X(0) + t7 g(1)m ∧ ? X(1) Imρ Imρ

+t

6

2 1 2 1 g(2) ρ + g(2) g(2) ρ ? + g(2) ? (2) + t5 g(3)m ∧ ? X m √ ∧ ? X(2) ? √ ∧?X (3) Imρ Imρ

+t4

21 22

? (4) Imρg(4) ∧ ? X(4) + X

?a are mixed to give the 8D “dilatino”, the superpartner of e?2ξ = e?2φ V . ? and Γ ?aψ Λ ?A = ψ ? Γ0 . In Euclidean signature ψ ?A and ψA are treated as In Minkowski signature ψ A

independent ?elds.

39

where the harmonic ?uxes g(p) , p = 0, . . . , 4 were de?ned in (5.16). These harmonic ?elds ? (p) constructed out of fermi bilinears. We now give couple to di?erential p-forms X(p) , X explicit formulae for X(p) : √ (+) (?) (?) (+) (?) (?) X(0) = ?ΨA ΓAB WB ? WB ΓAB ΨA + i 2Λ ΓA WA √ (?) √ (+) √ (?) (?) ?i 2WA ΓA Λ(+) + i 2Σ ΓA ΨA ? i 2ΨA ΓA Σ(+) i ?(?) A (+) i (+) A (?) i (?) A (+) i (+) A (?) + l Γ ΨA ? ΨA Γ l ? Γ WA ? WA Γ ? + ? 2 2 2 2 1 ?(+) (+) 1 ? (+) ?(+) (+) (+) l ? ? + ? ? l +4Σ Λ(+) ? 4Λ Σ(+) ? ? 2 2 = ΨA Γ[A ΓM N ΓB] WB + WA Γ[A ΓM N ΓB] ΨB + √ √ (+) √ (+) (?) (?) (?) i 2Λ ΓM N ΓA WA ? i 2WA ΓA ΓM N Λ(+) + i 2Σ ΓM N ΓA ΨA X(2)

MN (?) (?) (?) (?)

(7.15)

(7.16)

√ (?) i (+) i (?) A (+) +i 2ΨA ΓA ΓM N Σ(+) + ? l Γ ΓM N ΨA + ΨA ΓM N ΓA l(?) 2 2 i (+) i (?) A (+) (+) ? Γ ΓM N WA ? WA ΓM N ΓA ?(?) + 4Σ ΓM N Λ(+) ? ? 2 2 1?(+) 1 ? (+) (+) +4Λ ΓM N Σ(+) ? ? l ΓM N ? ?(+) ? ? ? ΓM N ? l(+) 2 2

(±)

where ψA

=

1 2

? ψA ,etc. and we use the combinations of 8D ?elds 116 ± Γ 2 ΓA Λ, ΨA = ψA + i 3 √ WA = ηA ? i √ 2 ΓA Σ 3

to make the expressions for X(0) , X(2) have nicer coe?cients. ? (0) , X ? (2) can be obtained from X(0) , X(2) by exchange of 8D chiralities The forms X (?) ? (+). Under the T-duality generator S the above forms transform as X(0) , X(2) → e?iα X(0) , X(2) , so that the combinations √ 1

Imρ

? (0) , X ? (2) ? (0) , X ? (2) → eiα X X √1

Imρ

(7.17)

(n0 ρ + n1 )X(p) ,

? (p) for p = 0, 2 which (n0 ρ ? + n1 )X

appear in the action (7.13) are invariant under S. Also we have de?ned the 1-form

m X(1) M

[A B] [A B] = em + ΨA Γ ΓM Γ WB ? W A Γ ΓM Γ ΨB

(?)

(+)

(+)

(?)

(7.18)

40

√ (+) √ √ (?) (+) (+) (?) ?i 2Λ ΓM ΓA WA + i 2WA ΓA ΓM Λ(+) ? i 2Σ ΓM ΓA ΨA √ (?) i ?(+) A i (+) (+) l(+) l Γ ΓM ΨA + ΨA ΓM ΓA ? +i 2ΨA ΓA ΓM Σ(?) ? ? 2 2 i ? (?) A i (?) (?) (?) ? ? ?(?) ? 4Σ ΓM Λ(+) ? Γ ΓM WA + WA ΓM ΓA ? 2 2 1 (?) 1 (+) (+) +4Λ ΓM Σ(?) ? ? l ΓM ?(+) + em ? ΓM l(?) + ? ? (+) ? (?) 2 2

and the 3-form

m X(3) M NP [A B] [A B] = em + ?ΨA Γ ΓM NP Γ WB ? WA Γ ΓM NP Γ ΨB (?) (+) (+) (?)

(7.19)

√ (+) √ √ (?) (+) (+) (?) +i 2Λ ΓM NP ΓA WA + i 2WA ΓA ΓM NP Λ(+) ? i 2Σ ΓM NP ΓA ΨA √ (?) i ?(+) A i (+) (+) l(+) ?i 2ΨA ΓA ΓM NP Σ(?) ? ? l Γ ΓM NP ΨA ? ΨA ΓM NP ΓA ? 2 2 i ? (?) A i (?) (?) (?) + ? ?(?) ? 4Σ ΓM NP Λ(+) ? Γ ΓM NP WA + WA ΓM NP ΓA ? 2 2 1 (?) 1 (+) (+) ?4Λ ΓM NP Σ(?) + ? ? ΓM NP l(?) + ? l ΓM NP ?(+) + em ? (+) ? (?) 2 2

m m where we denote em ± = e8 ? ie9 .

m m The forms X(1) and X(3) transform in the 2 of SL(2, Z)τ . Also from (7.18),(7.19) we

m m ?nd that X(1) and X(3) are invariant under SL(2, Z)ρ if we accompany the action of the

generator S by the U(1) rotation of em a

±iα m em e± ± → e

(7.20)

Since there are no local Lorentz anomalies, we can make this transformation. The most important objects in (7.14) are the self-dual23 form X(4) and the anti-self? (4) which couple to the ?ux g(4) . X(4) is de?ned by dual form X X(4)

M NP Q

= ?iΨA Γ[A ΓM NP Q ΓB] WB + iWA Γ[A ΓM NP Q ΓB] ΨB

(+)

(+)

(+)

(+)

(7.21)

√ √ (?) (+) (+) ? 2Λ ΓM NP Q ΓA WA + 2WA ΓA ΓM NP Q Λ(?) √ (+) √ (?) 1 (+) A (?) (+) l Γ ΓM NP Q ΨA ? 2Σ ΓM NP Q ΓA ΨA + 2ΨA ΓA ΓM NP Q Σ(?) ? ? 2 1 (?) 1 (+) A 1 (?) (?) + ΨA ΓM NP Q ΓA l(+) ? ? ? Γ ΓM NP Q WA + WA ΓM NP Q ΓA ?(+) 2 2 2

23

1 ? ?A1 A2 A3 A4 B1 B2 B3 B4 ΓB1 B2 B3 B4 Γ In our conventions ΓA1 A2 A3 A4 = ? 4!

41

+4iΣ

(?)

ΓM NP Q Λ(?) ? 4iΛ

(?)

i ? (?) i ?(?) l ΓM NP Q ? ?(?) + ? ? ΓM NP Q ? l(?) ΓM NP Q Σ(?) ? ? 2 2

? (4) can be obtained from X(4) by the exchange of 8D chiralities (+) ? (?). and X Under the T-duality generator S these forms transform as X(4) → eiα X(4) , ? 4 → e?iα X ? (4) X

(7.22)

We have also checked using Appendix(C) that the 4-fermion terms in the 8D action can be written as

′ ′′ S4?f erm = S4 ?f erm + S4?f erm , (8D) ′ S4 ?f erm =

π 128

X

? (4) ∧ ? X ? (4) e?2ξ t8 X(4) ∧ ? X(4) + X (7.23)

′ term S4 ?f erm is required for T-duality invariance of the total partition sum Z (F , ρ) of

′′ While S4 ?f erm is manifestly invariant under T-duality, we will see that the non-invariant

(1.32).

7.5. T -duality invariance of the ghost interactions The classical 8D action obtained from the reduction of 10D IIA supergravity on T 2 is invariant under local supersymmetry (all 32 components survive the reduction ). To construct the quantum action we have to impose a gauge ?xing condition on the gravitino ?(8D) := ψA and include ghosts. Since the susy transformation laws involve ?uxes, ψ ηA there is a potential T-duality anomaly from the ghost sector. In fact no such anomaly will occur as we now demonstrate. There are two generic properties of supergravity theories: 1 .) In addition to a pair of Faddeev-Popov ghosts associated to the local susy gauge ?(8D) → ψ ?A + δ? ?A transformation ψ ?ψ(8D) a “third ghost,” the Nielsen-Kallosh ghost, (8D) appears [32]. 2 .) Terms quartic in Faddeev-Popov ghosts are required [33]. Let us recall ?rst how the “third ghost” appears. Following the standard procedure ?A ?Aψ we ?x the local susy gauge by inserting δ f ? Γ (8D) into the path integral. Then we also insert the unit24 1= det

24

1

1 ?2ξ 7 ? e t D 2

?f f

[df ]e 2

i

X

?Df ? e?2ξ t7 f

,

? = iΓ ? N DN D

(7.24)

We use the measure

[df ]e

i

X

= 1.

42

? has zeromodes this expression is formally 0/0, but (7.27) and integrate over [df ]. (If D below still makes sense.) As a result we ?rst ?nd that the gravitino kinetic term gets modi?ed to ? i 2 ?A MAB ψ B + η e?2ξ t7 ψ ?A MAB η B

25

(7.25)

X

where the operator MAB acts on sections of the bundle

Spin(X ) ? T X as (7.26)

MAB = δAB iΓM DM ? 2iΓA DB

M where DA = EA DM . The determinant in (7.24) is expressed as the partition function for ? with action the “third ghost”Υ

SΥ ? =?

i 2

? ?D ?Υ ? e?2ξ t7 Υ

X

(7.27)

? is a bosonic 32 component spinor, which we decompose into 16 component spinors Υ as ? = Υ Υ1 Υ2

Now we come to the most interesting part of quantum action which involves FaddeevPopov ghosts ? b, c ?. Sbc = Sbc + Sbc where Sbc

(2) (2) (4)

(7.28)

Sbc

(4)

denotes the parts of the action quadratic (quartic) in FP ghosts. Let us

discuss the quadratic part ?rst. According to the standard FP procedure we have Sbc =

(2)

X8

?? ?A t7 e?2ξ ? bΓA δc ?ψ(8D)

(7.29)

We decompose bosonic 32 component spinors ? b, c ? as c ?= c1 c2 , ? b= b1 b2 .

We can write the action as a sum of two pieces Sbc = Sbc

25 (2) (2)0

+ Sbc

(2)2

Spin(X ) and T X are spinor and tangent bundles on X

43

Here Sbc

(2)0

does not contain fermionic matter ?elds while Sbc

(2)0 Sbc

(2)2

is quadratic in fermions.

We now present

and put

(2)2 Sbc

in Appendix(D). ? + n1 ? gh 2 8 n0 ρ + n1 gh n0 ρ √ X(0) ? √ X(0) t 3 Imρ Imρ

Sbc

(2)0

=

X8

? ? )? t7 e?ξ ? b(?iD c ? πe?ξ

(7.30)

2 1 2 1 g(2) ρ ? + g(2) g(2) ρ + g(2) 1 1 gh gh m ? gh √ ? √ + t7 g(1)m ∧ ? X(1) ∧ ? X(2) ∧?X + t6 Im (2) 2 3 Imρ Imρ

1 gh m + t5 g(3)m ∧ ? X(3) 8 where we de?ne forms bilinear in FP ghosts as

gh = X(0)

1 (?) (?) (?) (?) (?) (?) b2 c1 ? c1 (?) b2 ? b1 c2 + c2 (?) b1 2

(7.31)

gh X(2)

MN

=

1 (?) (?) (?) (?) (?) (?) b2 ΓM N c1 + c1 (?) ΓM N b2 + b1 ΓM N c2 + c2 (?) ΓM N b1 2

(7.32)

1 m (+) (?) (?) (+) (+) (?) e+ b2 ΓM c1 ? c1 (?) ΓM b2 ? b1 ΓM c2 + c2 (+) ΓM b1 (7.33) M 2 1 (+) ? (?) + em 2 ? 1 (?) (+) (+) (+) (?) gh m b2 ΓM NP c1 + c1 (?) ΓM NP b2 + b1 ΓM NP c2 (7.34) = em X(3) + M NP 2 1 (?) +c2 (+) ΓM NP b1 (+) ? (?) + em 2 ? ? gh can be obtained from X gh , X gh by exchange of 8D chiralities (?) ? ? gh , X The forms X (2) (0) (2) (0) (+). Note, that ? b, c ? do not couple to the ?ux g(4) .

gh m X(1)

=

Let us now present the part of the quantum 8D action which is quartic in ghosts (as obtained by following the procedure of [33]): Sbc = e?2ξ t8

(4)

1 ? ? ? ABC c bΓ ? 84

1 ? ?? A ? ? ? ABC c bΓ c ? bΓ ? + 3

? ? ?Ac bΓ ?

(7.35)

The presence of this quartic action is due to the fact that gauge symmetry algebra is open ?A contains a term proportional to the equation of motion of in supergravity: [δ? ? , δ? ? ]ψ

1 1

(8D)

?A . ψ (8D) that Sbc

(2)2

The T-duality invariance of Sbc , Sbc

(4)

(2)0

and SΥ ? is manifest and we have also checked

is T-duality invariant, so we conclude that the part of the 8D quantum action

which contains ghosts is T-duality invariant. 44

7.6. Computation of the determinants We can now compute the fermionic quantum determinants including ghosts. Let us expand the ?elds Λ, Σ, l, ? l, ?, ? ?, b1 , b2 , c1 , c2 , Υ1 , Υ2 and ψA , ηA in the full orthonormal basis (7.26). Note that since we are assuming that background ?uxes are harmonic, fermionic ? = iΓN DN and M respectively, where the operator M was de?ned in of the operators D

non-zero modes do not couple to them. Moreover,we can rescale non-zero modes by a factor of e?ξ t7/2 so that kinetic terms appear without any dependence on ξ and t, but four-fermionic terms are supressed as e2ξ t?6 with respect to the kinetic terms. Since kinetic terms are manifestly T-duality invariant the integration over nonzero modes will just give a factor Det′ F depending only on the Ricci ?at metric gM N and the constants t and ξ, all of which are T-duality invariant. Det′ F has the form

′ Det′ F = rF (ξ, t)det M

(7.36)

where det′ M is determinant of the operator M de?ned in (7.26) regularized in a way that rF (ξ, t) = const e?2ξ t7 where nM denotes the number of zero modes of M.

?nM

(7.37)

Note, that determinants of nonzero modes of the fermions Σ, Λ, l, ?, ? l, ? ? and bosons The situation is quite di?erent for zero-modes: the kinetic terms are zero but there is

Υ1 , Υ2 , b1 , b2 , c1 , c2 cancel each other and do not contribute to Det′ F. nonzero coupling to harmonic ?uxes, so that if we rescale fermion zeromodes by e? 2 ξ t2 we make both the fermion coupling to g(4) and the fermion quartic terms independent of ξ and t. We will also rescale ghost zeromodes by e? 2 ξ t2 and include the factor e?ξ t4 we de?ne new rF :

new rF (ξ, t) := rF (ξ, t) e?ξ t4 nM

1 1

nM

which

comes from the rescaling of fermion and ghost zeromodes into the de?nition of Det′ F , i.e.

= const(t)?3nM (eξ )nM

(7.38)

From (6.28) and (7.38) we ?nd that the full quantum determinants depend on t and ξ in the following way ??3 )nM ?nK ?b2 ?2b1 ?4b0 (t 45 (7.39)

? = te?ξ/3 is the U-duality invariant combination.26 Note that the where we recall that t ? in (7.39) comes entirely from the volume of the space of zero modes. dependence on t fermion zero modes is shrinking. Since (7.39) is an overall factor in the partition sum, it is ? → ∞, but the volume of The volume of bosonic zero modes is blowing up in the limit t

a question of a net balance between fermion and boson zero modes whether the partition ? → ∞. sum blows up or vanishes in the limit t 7.7. Integration over the space of fermion zeromodes We can split the action of the rescaled fermion and ghost zeromodes as S (zm) = S (zm)inv + S (zm)ninv . Here the part S (zm)inv is invariant under T-duality and includes all the ghost zeromode interactions, the coupling of the fermion zeromodes to all RR ?uxes except for g(4) and the invariant part of the 4-fermion zeromode couplings, denoted S4?f erm . recast in the following way: S (zm)ninv =

X (zm)′′

S (zm)ninv transforms non-trivially under the generator S of T-duality and can be

4π Imρg(4) ∧ ? Y(4) + 2π ImρY(4) ∧ ? Y(4)

(7.40)

where we de?ne the harmonic 4-form Y(4) as Y(4) = This object transforms under S as S · Y(4) = ?ReρY(4) + iImρ ? Y(4) . We now expand the harmonic 4-forms in the basis ωi of H 4 (X, Z) g(4) = (ni + α ? i ) ωi , Y(4) = y i ωi , ?=β ?i ωi β (7.42) 1 1 (zm) ? (zm) . √ +X X (4) 16 Imρ (4) (7.41)

? are given in (5.21). Next, we de?ne where the chracteristics α ?, β Θ(F , ρ) =

26

d?F

(zm) ? i2π ?Φ

he

Θ

α (Q) β

(7.43)

For any Ricci-?at spin 8-manifold the numbers nM and nK can be expressed in terms of

topological invariants.

46

where the shifted characterstics are de?ned as αi = α ?i + yi, d?F

(zm)

which depends on τ, ρ, t, gM N as well as fermion and ghost zeromodes. The dependence on τ, ρ, t, gM N comes entirely from the coupling of the rescaled zeromodes (of fermions and ghosts) to the ?uxes g(p) , p = 0, 1, 2, 3. Finally, we have also de?ned 1 ?Φ(F , ρ, y) := ?Φ ? yIS · y ? yI β 2 where ?Φ was de?ned in (5.24). Θ(F , ρ) is invariant under SL(2, Z)τ and transforms under SL(2, Z)ρ as ?) 2 b4 Θ(F , ρ) Θ(S · F , ?1/ρ) = sF ?(S )(?iρ) 2 b4 (iρ Θ(T · F , ρ + 1) = ?(T )Θ(F , ρ) mation27 of d?zm F sF = eiα

I M

1 + 1 ?

denotes the measure of the rescaled fermion and ghost zeromodes. Recall that ? = e?S (zm)inv is a T-duality invariant expression Q(ρ) = [H Imρ ? iReρ]I. In (7.43) h

?i + S · y i , and βi = β

(7.44)

(7.45) (7.46)

We do Poisson ressumation to ?nd (7.45) and the extra phase sF is due to the transfor-

= (i)I

M

(?iρ)? 2 I

1

M

(iρ ?) 2 I

1

M

(7.47)

putation of the chiral anomaly [34], only the zeromodes contribute to the transformation of fermionic measure. Indeed, the contribution of the bosonic ghosts c1 , b1 , Υ2 to the transformation of the measure cancels that of the contribution of the fermions ?, ? ?, Λ, l, ? l.

where I M is the index of the operator M de?ned in (7.26). As in the standard com-

8. T-duality invariance 8.1. Transformation laws for ZB+F (F , τ, ρ) Now we study the transformation laws for

?SB (F ) ZB+F (F , τ, ρ) = DetB Det′ Θ(F , ρ) Fe

(8.1)

where Θ(F , ρ) is de?ned in (7.43), while DetB and Det′ F are de?ned in (6.26) and evaluated on the background ?eld con?guration.

27

(7.36),(7.38) respectively. We also recall that SB (F ) is the real part of the classical action

Here we use the fact that the 10D fermions are Majorana fermions in Minkowski signature.

47

ZB+F (F , τ, ρ) transforms under SL(2, Z )ρ by using the transformation rules of DetB (6.30) and Θ(F , ρ) (7.45),(7.46). We ?nd: ZB+F (S · F , τ, ?1/ρ) = sB sF ?(S )(?iρ) 2 b4 (iρ ?) 2 b4 ZB+F (F , τ, ρ) ZB+F (T · F , τ, ρ + 1) = ?(T )ZB+F (F , τ, ρ) where sB is taken from the transformation of DB . Now, using the de?nition of χ and σ 1 1 (b0 ? b1 + b2 ? b3 + b± 4 ) = (χ ± σ ), 2 4 as well as the index theorem: I M +

1 1 χ+ 8 1 + 1 ?

First, we note that ZB+F (F , τ, ρ) is invariant under SL(2, Z )τ . Second, we learn how

(8.2) (8.3)

? σ = b+ 4 ? b4

(8.4)

λ2 =

X X

?8 248A

we obtain the ?nal result for the transformation under the generator S ZB+F (S · F , τ, ?1/ρ) = (?iρ) 4

X

(p2 ?λ2 )

(iρ ?) 4

1

χ? 1 8

X

(p2 ?λ2 )

ZB+F (F , τ, ρ)

(8.5)

From (8.3) and (8.5) we ?nd that there is a T-duality anomaly. Let us note in passing that the transformations (8.3),(8.5) are consistent for any 8dimensional spin manifold. This can be seen by computing ZB+F (ST )6 · F , τ, ρ) = e

iπ 4

X

28

(7λ2 ?p2 )

ZB+F F , τ, ρ)

(8.6)

ZB+F S 4 · F , τ, ρ) = ZB+F F , τ, ρ) and then noting that the index theorem for 8-dimensional spin manifolds implies

X

(7λ2 ? p2 ) ∈ 1440Z.

(8.7)

Euler characteristic is given by [35]:

Incidentally, when X admits a nowhere-vanishing Majorana spinor of ± chirality the χ=± 1 2 (p2 ? λ2 )

1

(8.8)

X

and the transformation rule (8.5) simpli?es to: ZB+F (S · F , τ, ?1/ρ) = (?iρ) 2 χ ZB+F (F , τ, ρ) ZB+F (S · F , τ, ?1/ρ) = (iρ ?) 2 χ ZB+F (F , τ, ρ) for positive and negative chirality, respectively.

28

1

(8.9) (8.10)

FR

The branches for the 8 ? th roots of unity are chosen in such a way that S 2 = (?)FR , where is a space-time fermion number in right-moving sector of type IIA string

48

8.2. Including quantum corrections Now we recall that there is a 1-loop correction to the e?ective 8D action: S1?loop = π Imρ where we recall that π Imρ

X

(uλ2 + vp2 ) +

X

iπ Reρ 24

X

p2 ? λ2

(8.11)

(uλ2 + vp2 ) comes from the regularization of q (Imρ) in(6.5)

and the numbers u and v depend on the regularization. We now demonstrate that to construct a T-duality invariant partition function this term should be replaced with Squant = 1 1 χ+ 2 4

1 24

X

(p2 ? λ2 ) log [η (ρ)] +

1 1 χ? 2 4

X

(p2 ? λ2 ) log [η (?ρ ?)]

(8.12)

1 u = ? 24 and v =

where η (ρ) is Dedekind function. Taking the limit Imρ → ∞ one can uniquely determine in (8.11). η has the following transformation laws: η (?1/ρ) = (?iρ) 2 η (ρ), so that e?Squant transforms as e?Squant (?1/ρ) = (?iρ)

1 χ? 1 ?4 8 X 1

η (ρ + 1) = e 12 η (ρ)

πi

(8.13)

(p2 ?λ2 )

π ?i 24

(iρ ?)

X

1 1 χ+ 8 ?4

X

(p2 ?λ2 ) ?Squant

e

(ρ )

(8.14) (8.15)

e?Squant (ρ + 1) = e

(p2 ?λ2 ) ?Squant

e

(ρ )

Finally, we ?nd that the total partition function Z (F , ρ) := e?Squant ZB+F (F , ρ) is invariant: Z T · F, ρ + 1 = Z F, ρ , Z S · F , ?1/ρ = Z F , τ, ρ . This is our main result. As a consistency check consider( for simplicity) the case when X admits a noweherevanishing spinor of positive chirality and take the limit Imρ = V → ∞ Squant → iπ ρ+ 12 1 2πinmρ χ. e m (8.19) (8.17) (8.18) (8.16)

n≥1 m≥1

We recognize the multiple cover formula for world-sheet instantons on T 2 from [18]. 49

9. Application: Hull’s proposal for interpreting the Romans mass in the framework of M -theory As a by-product of the above results we will make some comments on an interesting open problem concerning the relation of M-theory to IIA string theory. It is well known that IIA supergravity admits a massive deformation, leading to the Romans theory. The proper interpretation of this massive deformation in 11-dimensional terms is an intriguing open problem. In [9] C. Hull suggested an 11-dimensional interpretation of certain backgrounds in the Romans theory. His interpretation involved T-duality in an essential way, and in the light of the above discussion we will make some comments on his proposal. ( For a quite di?erent proposal for interpreting this massive deformation see [36]. ) 9.1. Review of the relation of M-theory to IIA supergravity Naive Kaluza-Klein reduction says that for an appropriate transformation of ?elds gM ?theory , CM ?theory → gIIA , HIIA , φIIA , CIIA we have SM ?theory = SIIA (9.1)

One of the main points of [7] was that, in the presence of topologically nontrivial ?uxes equation (9.1) is not true! Indeed, given our current understanding of these ?elds, there is not even a 1-1 correspondence between classical M-theory ?eld con?gurations and classical IIA ?eld con?gurations. Rather, certain sums of IIA-theoretic ?eld con?gurations were asserted to be equal to certain sums of M-theoretic ?eld con?gurations. In this sense, the equivalence of type IIA string theory to M -theory on a circle ?bration is a quantum equivalence. sum over K -theory lifts x(? a) of a class a ? ∈ H 4 (X10 ; Z) is proportional to the sum over

a0 ) N (?)Arf(q )+f (? √ N2 NK

To be more precise, in [7] it was shown that for product manifolds Y = X10 × S 1 , the

torsion shifts of the M-theory 4-form of Y . We have:

x(? a)

e?SIIA = exp ?||GM ?theory (? a)||2

4 (X10 ,Z) c ?∈Htors

a+? c) (?1)f (?

(9.2)

2π a ?? 1 λ and the equivalence class of a ? is de?ned to contain M -theory ?eld con?gurations 2 with ?xed kinetic energy ||GM ?theory (? a)||2 = 1 4π GM ?theory (? a)∧? ?GM ?theory (? a), 50

The above formula is the main technical result of [7]. We recall that [GM ?theory (? a)] =

X10

from which follows that these ?elds are characterized by a ?′ = a ?+c ?,

0 4 Also, in (9.2) NK and N is the order of Ktors (X10 ) and Htors (X10 ; Z) respectively,

4 c ? ∈ Htors (X10 , Z).

N2 stands for the number of elements in the quotient L′′ = L/L′ , where L =

4 4 Htors (X10 ; Z)/2Htors (X10 ; Z) and L′ =

invariant of the quadratic form q (? c) = f (? c) +

c ? ∈ L,

Sq 3 c ? = 0 . Finally, Arf(q ) is the Arf

X10

extends to the case where Y is a nontrivial circle bundle over X10 [7].

c ? ∪ Sq 2 a ?0 on L′′ . The identity (9.2)

As we have mentioned, we interpret the fact that we must sum over ?eld con?gurations only quantum-equivalent. This point might seem somewhat tenuous, relying, as it does, on the fact that the torsion groups in cohomology and K-theory are generally di?erent. Nevertheless, as we will now show, a precise version of Hull’s proposal again requires equating sums over IIA and M-theory ?eld con?gurations. In this case, however, the sums are over non-torsion cohomology classes, and in this sense the claim that IIA-theory and M-theory are only quantum equivalent becomes somewhat more dramatic. 9.2. Review of Hull’s proposal One version of Hull’s proposal states that massive IIA string theory on T 2 × X is in (9.2) as a statement that IIA-theory on X10 and M-theory on Y = X10 × S 1 are really

equivalent to M -theory on a certain 3-manifold which is a nontrivial circle bundle over a torus. The proposal is based on T-duality invariance, which allows one to transform away

G0 at the expense of introducing G2 along the torus, combined with the interpretation of G2 ?ux as the ?rst Chern class of a nontrivial M-theory circle bundle [7]. We now describe this in more detail. Hull’s proposal is based on the result [12] that dimensional reduction of massive IIA

1 supergravity with mass m on a circle of radius R, (denoted SR ), gives the same theory as 1 Scherk-Schwarz reduction of IIB supergravity on S1 /R . The IIB ?elds are twisted by

g (θ ) =

1 where the coordinate on S1 /R is z = 2π R θ,

1 0

mθ 1

(9.3)

θ ∈ [0, 1] and the monodromy is 1 m 0 1 ∈ SL(2, Z) (9.4)

g (1)g (0)?1 = Schematically: IIAm 1 ×X = SR 9

IIB 1 S1 /R × X9 51

(9.5)

g (θ)

where X9 is an arbitrary 9-manifold. Note, in particular, that the twist acts on the IIB axiodil τB = C0 + ie?φB as τB (θ ) = τB (0) + mθ which implies that the IIB RR ?eld G1 has a nonzero period. Let us also recall the duality between IIB on a circle and M-theory on T 2 :

1 SR ′

(9.6)

IIB M = 2 1 1 × S1/R × X T (τM , AM ) × S1 /R × X

(9.7)

where the T 2 (τM , AM ) on the M-theory side has complex structure τM = τB (0) and area AM = e

φB 3

(R ′ )? 3 .

4

Now, invoking the adiabatic argument we have: IIB 1 × SR ′ × X =

g (θ)

1 S1 /R

M B (m; R′ , R) × X

(9.8)

where B (m; R′ , R) is a 3-manifold with metric: ds =

2

2π R

2

(dθ )2 + AM

1 dx + (ReτM + mθ )dy ImτM

29

2

+ ImτM dy 2

(9.9)

where x, y are periodic x ? x + 1 and y ? y + 1.

Combining (9.5) with (9.8) we get the basic statement of Hull’s proposal:

1 SR

IIAm M = 1 × SR′ × X B (m; R′ , R) × X

(9.10)

We can now see the connection between Hull’s proposal and T-duality. A duality transformation exchanges G0 for a ?ux of G2 through the torus. Then we can interpret the nontrivial ?ux G2 as the ?rst chern class of a line bundle in the M -theory setting. 9.3. A modi?ed proposal In view of what we have discussed in the present paper, the equivalence of classical actions - when proper account is taken of the various phases of the supergravity action cannot be true. This is re?ected, for example, in the asymmetry of the phase (5.12) in

29

It is not entirely obvious that the invocation is justi?ed, since for a large M -theory torus the

twist is carried out over a small radius on the IIB side.

52

exchanging n0 for n1 . However, we follow the lead of (9.2) and therefore modify Hull’s proposal by identifying sums over certain geometries on the IIA and M-theory side.

30

A modi?ed proposal identi?es Z (F , ρ, τ ) de?ned in (8.1),(8.16) with a sum over Mtheory geometries as follows. Recall ?rst that in the 8D theory there is a doublet of p α zeroforms g(0) , arising from G0 and G2 . Next, let us factor g(0) = ? where p, q are q relatively prime integers and ? is an integer. Then we take a matrix N ∈ SL(2, Z)ρ N = such that N g(0) = ? 0 (9.12) r ?q ?s p rp ? sq = 1 (9.11)

This is the T-duality transformation that eliminates Romans ?ux. Now, thanks to the invariance of Z (F , τ, ρ) under T-duality transformations (see (8.17)(8.18) above) we ?nd: Z (F , ρ ) = Z N · F , pρ + s qρ + r (9.13)

By the results of [7] the right hand side of (9.13), having G0 = 0, does have an interpretation as a sum over M-theory geometries. The M-theory geometry is indeed a circle bundle over T 2 × X de?ned by c1 = ?e0 + pe ? qe′′ + γm dσ m (as in Hull’s proposal), but in addition it is necessary to sum over E8 bundles on the 11-manifold B × X . While it is essential to sum over g(4) , all other ?uxes F may be treated as classical - that is, they may be ?xed and it is not necessary to sum over them. Both sides of (9.13) should be regarded as wavefunctions in the quantization of selfdual ?elds. For this reason we propose that there is only an intrinsically quantum mechanical equivalence between IIA theory and M-theory in the presence of G0 .

30

In making these statements we are including the K-theoretic phase as part of the “classical”

action. Since the phase is formally at 1-loop order it is possible that one could associate it with a 1-loop e?ect in such a way that classical equivalence does hold.

53

10. Comments on the U-duality invariant partition function The present paper has been based on weakly coupled string theory. However, our motivation was understanding the relationship between K-theory and U-duality. In generalizing our considerations to the full U-duality group D = SL(3, Z) ×SL(2, Z)ρ of toroidally compacti?ed IIA theory it is necessary to go beyond the weak coupling expansion. Thus, it is appropriate to start with the M -theory formulation. In the present section we make a few remarks on the U -duality of the M -theory partition function and its relation to the K -theory partition functions of type IIA strings. In particular, we will address the following points: a.) The invariance of the M -theory partition function under the nongeometrical SL(2, Z )ρ is not obvious and appears to require surprising properties of η invariants. In section 10.2 we state this open problem in precise terms. b.) We will sketch how one can recover “twisted K -theory theta functions,” at weak coupling cusps when the H -?ux is nonzero in section 10.3. We believe that one can clarify the relation between K-theory and U-duality by studying the behavior of the M-theory partition function at di?erent cusps of the M-theory moduli space. At a given cusp the summation over ?uxes is supported on ?uxes which can be related to K -theory. (See, for example, (9.2).) A U-duality invariant formulation of the theory must map the equations de?ning the support at one cusp to those at any other cusp. This should de?ne the U -duality invariant extension of the K -theory constraints. 10.1. The M -theory partition function Let us consider the contribution to the M-theory partition function from a background Y which is a T 3 ?bration over X.

?3 ? ?mn θ m θ n t2 gM N dxM dxN + V 3 g ds2 11 = V

1 2

(10.1)

g ?mn and V are the shape and the volume of the T 3 ?ber. We denote world indices on T 3 by m = (m, 11), m = 8, 9 and M = 0, . . . , 7 as before.

m ?2 gM N is an 8D Einstein metric with detgM N = 1. where θ m = dxm + Am ∈ [0, 1]. t (1) and x

Topologically, one can specify the T 3 ?bration over X by a triplet of line bundles Lm which transform in the representation 3 of SL(3, Z) and have ?rst Chern classes These are characterized by a homomorphism π1 (X ) → SL(3, Z). 54

m m c1 (Lm ) = F(2) , where F(2) = dθ m . Such a speci?cation is valid up to possible monodromies.

On a manifold Y of the type (10.1) we reduce the M-theory 4-form GM ?theory as 1 GM ?theory k θm θn F(2)mn + εmnk B0 F(2) = G(4) + G(3)m θ m + 2π 2 We also include the ?at potential c(0) = 1 B0 εmnk θ m θ n θ k 6 (10.3) (10.2)

in the Kaluza-Klein reduction.31 (We will list the full set of ?at potentials in this background below.) From the Bianchi identity dGM ?theory = 0 we have

m dG(4) = F(2) G(3)m , n dG(3)m = F(2) F(2)mn m dF(2)mn = 0 dF(2) =0

(10.4)

which implies that ?uxes G(4) and G(3)m are in general not closed forms.32 Let us recall how the various ?elds transform under D = SL(3, Z) × SL(2, Z)ρ [14]. ?, gM N are U-duality invariant. ? t ? ? SL(3, Z) acts on the scalars g ?mn parametrizing SL(3, R)/SO (3) via the mapping Fmα (2) = G(3)m

m F(2) 1 mnk m transform in the (3, 2) of D , where F(2) := 2 ε F(2)nk . m F(2) transform in the (3′ , 1) of D

SL(2, Z)ρ acts on ρ = B0 + iV ∈ H by fractional linear transformations.

class group of T 3 . ? ? ?

G(4) is singled out among all the other ?elds since according to conventional su-

pergravity [14] SL(2, Z)ρ mixes G(4) with its Hodge dual ?G(4) . More concretely, ?Reρ G(4) + iImρ ? G(4) G(4) (10.5)

transforms in the (1, 2) of D . Due to this non-trivial transformation the classical bosonic 8D action is not manifestly invariant under SL(2, Z)ρ . In detail, the action has real part: Re(S8D ) = π

X 31 32 nβ α ?2 g ?4 g ImρG(4) ∧? G(4) + t ?mn G(3)m ∧? G(3)n + t ?mn Gαβ Fm (2) ∧? F(2)

(10.6)

ε11,8,9 = ε11,8,9 = 1.

n = 0, n = 8, 9, so that all background In IIA at weak coupling we assumed G(3)11 = 0 and F(2)

?uxes are closed forms.

55

+

1 2π

1 mn kl ??2 ?M t ??M t ? + 1 ?M ρ? M ρ ?6 R + 28t t ?+ g ? g ? ?M g ?mk ? M g ?nl 2 2ρ 2 3 X

where Gαβ is de?ned in (1.6), g ?kl is inverse of g ?mk and R is the Ricci-scalar of the metric gM N . The imaginary part of the 8D bosonic action follows from the reduction of the Mtheory phase ?M (C ). This phase is subtle to de?ne in topologically nontrivial ?eld con?gurations of the G-?eld. It may be formulated in two ways. The ?rst formulation was given in [37]. It uses Stong’s result that the spin-cobordism group ?11 (K (Z, 4)) = 0 [38].

G That is, given a spin 11-manifold Y and a 4-form ?ux 2 one can always ?nd a bounding π ? of the the ?ux to Z . In these terms the M-theory spin 12-manifold Z and an extension G

phase ?M (C ) is given as: ?M (C ) = ? exp 2πi 6 ? 3 ? 2πi G 48 Z ? (p2 ? λ2 ) G (10.7)

Z

Here ? is the sign of the Rarita-Schwinger determinant. The phase does not depend on the choice of bounding manifold Z , but does depend on the “trivializing” C -?eld at the boundary Y . A second formulation [7,39,40] proceeds from the observation of [37] that the integrand of (10.7) may be identi?ed as the index density for a Dirac operator coupled to an E8 vector bundle. The M-theory 4-form can be formulated in the following terms [7,39,40]. We set: GM ?theory ? + dc =G 2π ?= where G

F2 1 T r + 321 T rR2 , 248 60 8π 2 π2

(10.8)

F is the curvature of a connection A on an E8 bundle V

3 form, and c ∈ ?3 (Y, R)/?3 Z (Y ), where ?Z (Y ) are 3-forms with integral periods. The pair

on Y and R is the curvature of the metric connection on T Y. GM ?theory is a real di?erential (A, c) is subject to an equivalence relation. In these terms the M-theory phase is expressed as: ?M (C ) = exp 2πi η (DV ) + h (DV ) η (DRS ) + h (DRS ) + 4 8 ω (c)

(10.9)

where DV is the Dirac operator coupled to the connection A, DRS is the Rarita-Schwinger operator, h(D) is the number of zeromodes of the operator D on Y , and η (D) is the η invariant of Atiyah-Patodi-Singer. The phase ω (c) is given by ω (c) = exp πi

Y

? 2 + X8 ) + cdcG ? + 1 c(dc)2 c(G 3 56

(10.10)

10.2. The semiclassical expansion ? there is a well-de?ned semiclassical expansion of the M-theory partition For large t function, which follows from the appearance of kinetic terms in the action (10.6) scaling as ?2k for k = 0, 1, 2, 3. In the leading approximation we can ?x all the ?elds except G(4) , but t this last ?eld must be treated quantum mechanically. Note that this semiclassical expansion can di?er from that described in the previous sections because we do not necessarily require weak string coupling. In the second approximation we treat G(4) and G(3)n as quantum ?elds, and so on. In the leading approximation in addition to the sum over ?uxes G(4) we must integrate coming from the KK reduction of c

3 over the ?at potentials. These include ?at connection Am (1) of the T ?bration and potentials

1 ′ ′ m m n c = C(3) + C(2) m θ + C(1)mn θ θ + c(0) 2

(10.11)

(3, 2) by writing

combine the ?at potentials C(1)mn and Am (1) in a U -duality multiplet of D transforming as

α Am (1) = 1 mnk ε C(1)nk 2 Am (1)

in (10.3). C(3) is invariant under U-duality, C(2)m transforms in the (3, 1) of D . We can

p 1 ′ ′ ′ m where C(2) m = C(2)m ? 2 C(1)pm A(1) and C(3) = C(3) ? C(2)m A(1) , and c(0) is de?ned

.

(10.12)

The duality invariance in the leading approximation is straightforward to check. We

4 keep only G(4) . The ?ux is quantized by [G(4) ] = a ? 1 2 λ, where a ∈ H (X, Z) is the

characteristic class of the E8 bundle and λ is the characteristic class of the spin bundle.

invariant. The imaginary part of the 8D e?ective action in this case takes a simple form which can be found from (10.9): Im(S8D ) = ?π 1 a ∪ λ + B0 a ? λ 2

2

We sum over a ∈ H 4 (X, Z). The 8D action, including the imaginary part is SL(3, Z)

(10.13)

X

The invariance under SL(2, Z)ρ then follows in the same way as in our discussion in the weak string coupling regime. Let us now try to go beyond the ?rst approximation. In the second approximation ?eldstrengths F(2) and F(2) . We thus have a family of tori with ?at connections. Already in 57

1 [G(4) ] = a ? 2 λ + [A m (1) G(3)m ]. We allow nonzero ?uxes G(3)m , but still set to zero the

the second approximation, when we switch on nonzero ?uxes G(3)m there does not appear to be a simple expression for the M-theory phase. Nevertheless, one can get some information about the M-theory phase from the requirement of U-duality invariance. We know that SL(3, Z) invariance is again manifest from the de?nition of ?M (C ) and Re(S8D ). But the expected SL(2, Z)ρ invariance is nontrivial. We would simply like to state this precisely. To do that we write M-theory partition function in the second approximation as ZM ?theory (? gmn , ρ) := d?f lat

G(3)m

ZM ?theory (? gmn , G(3)m , ρ)

(10.14)

where ZM ?theory (? gmn , G(3)m , ρ) is the partition function with ?xed, but nonzero, ?ux, G(3)m , d?f lat stands for the integration over H 3 (X ) × 3 HZ (X ) H 2 (X ) 2 HZ (X )

3

×

H 1 (X ) 1 HZ (X )

6

,

(10.15)

normalized harmonic p-forms on X. The ?rst factor is for C(3) , the second factor for C(2)m measure d?f lat is U-duality invariant.

p (X ) is the lattice of integrally where Hp (X ) is a space of harmonic p-forms on X and HZ

α and the third factor is for the ?elds Am (1) transforming in the (3, 2) of D . The integration

The summand in (10.14) with ?xed G(3)m is given by ZM ?theory (? gmn , G(3)m , ρ) = where

?π ?2 g Im(ρ)G(4) ∧?G(4) +t ?mn G(3)m ∧?G(3)n

Det(G(4) , G(3)m )e?Squant e?Scl

a∈H 4 (X,Z)

(10.16)

e

?Scl

X

= ?M G(4) , G(3)m , B0 e

and Det(G(4) , G(3)m ) denotes 1-loop determinants. These depend implicitly on the scalars ? as well as on the metric gM N . We include 1-loop corrections in Squant (see below). ρ, g ?mn, t The M-theory phase ?M in (10.16) depends on the ?eldstrenths G(4) , G(3)m and the ?at potentials, but it is metric-independent, and hence should be a topological invariant. The dependence of ?M on ?at potentials is explicit from (10.10) for c as in (10.11). For example dependence of ?M on B0 has the form e

iπ

X

B0 G(4) G(4)

(10.17)

58

It is conveneient to include 1-loop corrections comes from η (DV ) + h(DV ).

X

B0 X8 together with e?ect of membrane

instantons in Squant . The nontrivial question is dependence on G(4) and G(3)m which also The independence of ?M on the metric on Y = X × T 3 (in the second approximation)

follows from the standard variation formula for η -invariant. To show this let us ?x the connection on the E8 bundle V with curvature F and consider the family of veilbeins e(s) on Y = X × T 3 parametrized by s ∈ [0, 1] such that the metric on T 3 remains ?at and independent of the coordinates on X. The corresponding family of Rieman tensors R(s) ?(s) which is a pullback from X × [0, 1]. Now we can write the gives an A-roof genus A ?(s) ch(V )A

Y ×[0,1]

iF

standard formula for the change in η -invariant under the variation of veilbein [41]: η (e(1)) ? η (e(0)) = j +

(10.18)

1 where integer j is a topological invariant of Y × [0, 1] and ch(V ) := 30 [T r248 e 2π ]. In the ? = G(4) + G(3)m dxm so that neither ch2 (V ) = second approximation we only switch on G

? + 1 λ) nor ch4 (V ) = 1 (G ? + 1 λ)2 have a piece ? dx8 dx9 dx11 and integral in (10.18) ?2(G 2 5 2 vanishes. Now we come to the main point. The requirement of the invariance under the standard generators S, T of SL(2, Z)ρ ZM ?theory (? gmn , ?1/ρ) = ZM ?theory (? gmn , ρ) ZM ?theory (? gmn , ρ + 1) = ZM ?theory (? gmn , ρ) (10.19) (10.20)

gives a nontrivial statement about the properties of the function ?M (G(4) , G(3)m , B0 ). classes which satisfy a system of SL(3, Z) invariant constraints. These constraints can in principle be determined by summing over torsion classes once the phase ?M is known in su?cently explicit terms. In the simple case when G(3)m are all 2-torsion classes, one can act by the generators of SL(3, Z) on the constraint Sq 3 (G(3)9 ) + Sq 3 (G(3)11 ) + G(3)9 ∪ G(3)11 = 0 system of constraints then we ?nd G(3)m ∪ G(3)n = 0, 59 m, n = 8, 9, 11 (10.22) (10.21) The sum over ?uxes G(3)m ∈ H 3 (X, Z) in (10.14) might be entirely supported by

which follows from [7]. If we assume that this constraint is part of SL(3, Z) invariant

10.3. Comment on the connection with twisted K-theory In this section we discuss the behavior of the partition function near a weak-coupling cusp. There is a twisted version of K-theory which is thought to be related to the classi?cation of D-brane charges in the presence of nonzero NSNS H -?ux [2,42,43,44]. It is natural ?uxes with nonzero H(3) := G(3)11 ∈ H 3 (X, Z) are related, in the weak string-coupling cusp, to some kind of twisted K-theory theta function. to ask if the contributions to the M -theory partition function ZM ?theory (? gmn , ρ) from

The weak-coupling cusp may be described by relating the ?elds in (10.1) to the ?elds ? is related to the expansion parameter used in our previous in IIA theory. First, the scale t ?2 = e? 2 3 ξ t2 . Next, we parametrize the shape of T 3 as g sections by t ?mn = eam ebn δab where ? eam = ? as R × R2 × SL(2, R)/SO (2) the modular parameter τ of the IIA torus. As far as we know, nobody has precisely de?ned what should be meant by the “ KH theta function.” Since the Chern character has recently been formulated in [22][23], this should be possible. Nevertheless, even without a precise de?nition we do expect it cohomology, this should be a “maximal Lagrangian” sublattice of ker d3 /Imd3 where d3 : H ? (X10 , Z) → H ? (X10 , Z) is the di?erential d3 (ω ) = ω ∧ [H(3) ]. Using the ?ltration

? this implied by the semiclassical expansion, and working to the approximation of e?t

2

?

?ξ/3 e√ τ2

0 0

√ e?ξ/3 τ2 0

0

1 ?? 0 ? 0 0 e2ξ/3

0

??

τ1 1 0

? C(0)8 C(0)9 ? 1

(10.23)

We denote frame indices by a = (a, 11), a = 8, 9. The weak coupling cusp may be written (10.24)

where the ?rst factor is for the dilaton ξ , the second for C(0)8 , C(0)9 ,33 and the third for

to be a sum over a “Lagrangian” sublattice of KH (X × T 2 ). At the level of DeRham

means that we should ?rst de?ne a sublattice of the cohomology lattice by the set of integral cohomology classes (a, G(3)8 , G(3)9 ) such that (G(4) , G(3)8 , G(3)9 ) are in the kernel of d3 : H(3) ∧ G(4) = 0,

33

H(3) ∧ G(3)m = 0,

m = 8, 9

(10.25)

?(0)m transforming in the 2′ of SL(2, Z)τ as C(0)8 = These are related to the RR potentials C √ ? ?(0)9 eξ τ 2 C C(0)9 = eξ √1 C (0)8 ,

τ2

60

Then the theta function should be a sum over the quotient lattice obtained by modding out by the image of d3 G(3)8 ? G(3)8 ? pH(3) , G(3)9 ? G(3)9 ? sH(3) , G(4) ? G(4) ? ω(1) H(3) . (10.26)

quotient lattice emerges from (10.14). follow from (10.10): e

2πi

X

Here p, s ∈ Z and ω(1) ∈ H 1 (X, Z). Thus, our exercise is to describe how a sum over this Let us consider the couplings of ?at potentials C(1)89 and C(2)m to the ?uxes which

C(1)89 H(3) G(4) 2πi

e

X

?mn C(2)m G(3)n H(3)

(10.27)

Integrating over C(1)89 and C(2)m gives H(3) ∧G(4) = 0 and ?mn H(3) ∧G(3)n = 0 respectively. Next, we note that, due to the SL(3, Z) invariance of the M-theory action we have (suppressing many irrelevant variables) ZM ?theory (C(0)m , G(3)m ? pm H(3) , A11 (1) , G(4) ? ω(1) H(3) ) = ZM ?theory (C(0)m + pm , G(3)m , A11 (1) + ω(1) , G(4) ) Now we use (10.28) to write the sum over all ?uxes G(4) , G(3)m , kernel of d3 as ZH =

d3 ?kernel

(10.28)

m = 8, 9 in the

ZM ?theory (C(0)m , G(3)m , A11 (1) , G(4) ) =

W

Mf und

(10.29)

the kernel of d3 and W =

where Mf und stands for the ?uxes in the fundamental domain for the image of d3 within ZM ?theory (C(0)m + pm , G(3)m , A11 (1) + ω(1) , G(4) ) (10.30)

pm ∈Z2 ω(1) ∈H 1 (X,Z)

Now, we can recognize that ZH descends naturally to the quotient of the weak-coupling cusp.

2 Γ′ ∞ \ R × R × SL(2, R)/SO (2)

(10.31)

form

? 2 where Γ′ ∞ = Z is the subgroup of the parabolic group Γ∞ consisting of elements of the Lmn 1 = ?0 0 ? ? 0 p 1 s?, 0 1 61 p, s ∈ Z (10.32)

lattice. (Recall that we are working in the DeRham theory, with the ?ltration appropriate to the second approximation.) The interesting point that we learn from this exercise is that in formulating the KH theta function, the weighting factor for the contribution of a class in KH should be given like exp[?Q(pm , ω(1) )] where Q is quadratic form. Therefore W is itself already a theta real part of the classical action (10.6), since, as we have shown, the phase is independent ?2 g t ?mn G(3)m ∧ ? G(3)n of the metric on X × T 3 . The dependence on C(0)m comes from

X 1 a? 2 λ + [A m (1) G(3)m ].

Written this way, ZH is clearly a sum over a Lagrangian sublattice of the KH (X × T 2 )

by (10.30). The dependence of the action on the integers pm and ω(1) ∈ H 1 (X, Z) behaves

function. This follows because the dependence on C(0)m and A11 (1) comes entirely from the

and the dependence on A11 (1) from

ImρG(4) ∧ ? G(4) , where we recall that [G(4) ] = X

It would be very interesting to see if the function ZH de?ned in (10.29) is in accord

with a mathematically natural de?nition of a theta function for twisted K-theory. But we will leave this for future work. As an example, let us consider X = SU (3). Let x3 generate H 3 (X, Z). Then ?xing H(3) = kx3 we ?nd that the fundamental domain of the image of d3 within the kernel of d3 is given by G(3)8 = rx3 , G(3)9 = px3 , 0 ≤ r, p ≤ k ? 1 (10.33)

so that the sum over RR ?uxes in (10.29) is ?nite and it is in this sense that RR ?uxes are k-torsion. This example of X = SU (3) is especially interesting since it is well known [45,46,44] that at weak string coupling D-brane charges on SU (3) in the presence of H(3) = kx3 are classi?ed by twisted K-theory groups of SU (3), and these groups are k -torsion. As argued in [5], from Gauss’s law it is then natural to expect that RR ?uxes are also k -torsion. This is indeed what we ?nd in (10.33).

34

On the other hand, the M -theory

sum is indeed a full sum over all ?uxes. This is in harmony with the result of [47] for brane charges. Clearly, there is much more to understand here.

Acknowledgements:

34

In fact, from [44] we know the order of the torsion group is actually k or k/2, according to

the parity of k . However, given the crude level at which we are working we do not expect to see that distinction. We expect that a more accurate account of the phases in the partition function will reproduce this result.

62

GM would like to thank E. Diaconescu and E. Witten for many important discussions and correspondence related to these matters both during and since the collaboration leading to [5][7]. He would also like to thank B. Acharya, C. Hull, N. Lambert, and J. Raben for useful discussions and correspondence, and C. Hull for some remarks on the manuscript. GM would also like to thank L. Baulieu and B. Pioline at LTPHE, Paris and the Isaac Newton Institute for hospitality during the completion of this manuscript. G.M. is supported in part by DOE grant # DE-FG02-96ER40949.

Appendix A. Duality transformations as symplectic transformations Here we give the explicit expressions for representations of S and T in Sp(2N, Z). Let us choose the following basis of the lattice Γ x = x1 , x2 (A.1)

x1 = yl 1, yl ? (L(e0 ) ? 1) , (L(es ) ? 1) , (L(es ) ? 1) ? (L(e0 ) ? 1) , (L(γr dσ m ) ? 1) , x (fk dσ m ) , x (ωi ) x2 = x(ωi ) ? (L(e0 ) ? 1) , x (dk dσ m ) , x (wr dσ m ) , x(us ), x(us ) ? (L(e0 ) ? 1) , x(hl ), x(hl ) ? (L(e0 ) ? 1) where we introduce yl ∈ H 0 (X, Z), γr ∈ H 1 (X, Z), es ∈ H 2 (X, Z), fk ∈ H 3 (X, Z), hl ∈ H 8 (X, Z), wr ∈ H 7 (X, Z), l = 1, . . . , b0 r = 1, . . . b1

(A.2)

(A.3)

us ∈ H 6 (X, Z) s = 1, . . . , b2 , ωi ∈ H 4 (X, Z), i = 1, . . . , b4 ,

dk ∈ H 5 (X, Z), k = 1, . . . , b3 ,

where bp is the rank of H p (X, Z) and b3 is the rank of the sublattice of H 3 (X, Z) which is span by classes f such that Sq 3 f = 0. In the above basis the generators S and T are represented by σ (S ) = A(S ) B (S ) C (S ) D (S ) , 63 σ (T ) = A(T ) C (T ) B (T ) D (T ) (A.4)

? ?1b0 ? ? ? A(S ) = ? ? ? ? ? ? ? ? C (S ) = ? ? ? ? ? ?1b4

?

1b0 1b2 ?1b2 12b1 12b3 0b4 ? ? ? ? ? ? ? ? ? 1b0 1b2 ?1b2 1b2 12b1 ? ? ? ? D (S ) = ? ? ? ? ? 0b4

? ? ? ? ? ? ? ? ?

? ? ? ? B (S ) = ? ? ? ?

?

? ? ? ? ? ? ? ? ?

12b3 12b1 1b2 ?1b2 ? ? ? ? ? ? ? ? ? ?1b0

1b4 (A.5) ? ? ? ? ? ? ? ? ?

1b0

1b0 ? ?1b0 ? ? ? A(T ) = ? ? ? ?

?

(A.6)

B (T ) = 0N

(A.7)

12b3 1b4 ? 1b4 12b3 12b1 1b2 ?1b2 1b2

C (T ) = 0N ,

? ? ? ? D (T ) = ? ? ? ?

? ? ? ? ? ? ? ? ? (A.8)

1b0 ?1b0

1b0

Appendix B. Supergravity conventions The 10D ?elds that we use are related to the ?elds in [19]as: ? 3φ G Rom √ 4 = e? 4 F4 , 2π

9φ G Rom √ 2 = ?e? 4 F2 , 2π

? 3φ B Rom √ 2 = ?e 2 B2 , 2π

m = G0 e

15φ 4

,

(Rom) ? ? = e? φ 8 ψ , ψ ? A A

We also remind that we set k11

1 φ (Rom) ? = e? φ 8 λ(Rom) , g Λ ?N ? = e 2 gM ?N ? M √ = π while in [19] k11 = 2π was assumed.

64

Appendix C. 4-Fermion terms Below we collect 4-fermionic terms in D=10 IIA supergravity action which are obtained from circle reduction of the D=11 action of [48]. S4?f erm =

(10)

π 2 +

√

?g10 e?2φ ?

1 ? ABCDEF χF + 12χ ? BC χD] χ ? BC χD] (C.1) χ ?E Γ ?[ A Γ ? [A Γ 64

1 ? ABCEF χF χ ? B χC + 1 χ ? A χC χ ? B χC χ ?E Γ ?A Γ ?A Γ ?B Γ 32 4 1 ? B χC χ ? A χC ? 1 χ ? B χC χ ? B χC ? χ ?A Γ ?B Γ ?A Γ ?A Γ 8 16 1 ? ? ?? + √ Γ ?Λ χA ? = ψA 6 2 A √ 2 2 ? 11 ? χ11 = ? Γ Λ 3

(11)

where

? 11). and A = (A, Recall that the graviton EA M and the gravitino ψA to 10D ?elds as [48]:

? 3 ?A EA EM ? =e ?, M ?

φ

of 11D supergravity are related

?

E11 11 = e

2φ 3

,

E11 ? = e M

2φ 3

CM ?

ψA

(11)

1 φ = √ e 6 χA 2π

Appendix B. Quartic couplings of ghosts and fermions Below we collect terms in the 8D quantum action which are bilinear in FP ghosts and bilinear in fermions: Sbc

(2)2

=

π 2

t8 e?2ξ

X

1 ? A χC + 2χ ? B χC χ ?B Γ ?A Γ 8

? ? ?AΓ ? BC c bΓ ? +

(B.1)

1 ? ? ? BC ? ?a ? ?a χ ?B Γ ?a bΓ Γ c ? ? χC + 2χ ? ΓB χC 6 2 1 ? ?BCD ? ? ? BC χD ? ? ? ABCD c ?a χ ?A Γ χ ?a bΓ ? + bΓ c ? + ? ΓBC χD 6 9 1 ? ? ? ABCDE 4 ? ? a ? BCDE ? CD χE ? ? c ? χ ?B Γ b ΓA Γ + Γa ?Γ 48 3 65

? B χA ? ? c ?Γ

? ? ? B χa ? A χB ? 4 ? c ?Γ bΓ ? 3

1 ? ? 11 ? ? ? ?a c ?Γ χ11 bΓ χB + 4

? ? ? ? ? A χA + 4 ? ?a bΓ bΓ χa ? 3

4 1 4 ?a ? ? ? A ? DE ? ? ? ? DE ? D ?? A a ?? a + La Γ χa + LDE? b Γ Γ χA + Γ +LAa ? ? D bΓ χ ? bΓ χ 3 4 3 where we now split indices as A = (A, ? a) , components of LDE are given by: ? A χd L Ad c ?Γ ? = ?? ?, Sbc ψA

(11) (2)2

A = 0, . . . 7,

a ? = (a, 11),

a = 8, 9. Nonzero

? a χ11 La11 = ?? c ?Γ

?A ( gauge parameter ? is obtained by relating 8D gauge ?eld ψ ?) to 11D gravitino (8D)

(gauge parameter ?(11) ) as ?A = ψ (8D) √ 2πe? 6 ψA

φ

(11)

1 ? ?a ? (11) + Γ , A Γ ψa ? 6

?= ?

√

2πe 6 ?(11)

φ

Let us also remind a standard fact that to keep the gauge EA 11 = 0,

?

EA m =0

used in reduction from 11D one has to accompany supersymmetry transformations of [48] with ?eld dependent Lorentz transformations. The last line in the action Sbc To write out ?A := ψ (8D)

(2)2 Sbc (2)2

originates from such Lorentz transformations.

in terms of 8D ?elds , ? (8D) := Λ Σ Λ , ?(8D) := θ l ? , ν ?(8D) := ? l ? ?

ψA ηA

one should substitute √ 1 ?A + Γ ?(8D) + 2 Γ ?Aθ ?AΛ ? (8D) , χA = ψ (8D) 12 6 χ8 = A = 0, . . . , 7

√ √ 1 ?9 ? 1 ?8 ? 1 ? 89 ν ? (8D) , χ9 = 1 Γ ? (8D) θ(8D) + 2Λ ?(8D) + Γ θ(8D) + 2Λ ν ?(8D) + Γ 2 3 2 3 √ √ 2? 2 2 ? 11 ? χ11 = ? Γ Λ(8D) ? θ(8D) 3 4

(2)2

We do not present the ?nal expression but we have checked that Sbc invariant. 66

is T-duality

Appendix E. Measures for path integrals Here we explain why det′ ?p are divided by Vp in (6.4) This is related to the integration over zeromodes.

p p Introducing a basis ai (p) , i = 1, . . . , b in HZ let us denote

Vpij =

X

j ai (p) ∧ ? a(p) ,

Vp = deti,j Vpij

(E.1)

p . Note, that Vp is invariant under the choice of basis in HZ

To explain integration over fermionic zero modes let us consider the following path? ?

integral over fermionic p-forms u and v.

bp bp

DuDv ? In (E.2) we have inserted fermion zero modes. of eigen p-forms of ?p .

u

γi j =1 γj

i=1

v ? e?

v,?p u

(E.2)

where γi , i = 1, . . . bp is a basis of Hp (X, Z).

bp i=1 γi

u

bp j =1 γj

v, to get non-zero answer, i.e. to saturate

To perform the integration in (E.2) we expand u and v in an orthonormal basis {ψn }

u=

n

un ψ n ,

v=

n

vn ψn ,

ψn , ψm = δn,m

(E.3)

Hp (X, Z), i.e

p p Let us choose the basis ai (p) , i = 1, . . . , b of the lattice HZ , dual to the basis γi ∈

γi

aj (p) = δij

Then, orthonormal zero-modes are expressed as

?1 i = aj ψzm (p) Wp i j ij

(E.4) .

p T ?1 : (Vp )ij = Wp Wp where Wp is the inverse of the vielbein for the metric on HZ

Now, we integrate (E.2) and obtain

det′ ?p Vp 67

(E.5)

In the case of bosonic p-forms u and v we do not need to insert anything to get a non-zero answer: DuDve?

u,?p v

=

det′ ?p Vp

?1

(E.6)

where (E.6) the integration over bosonic zero-modes was performed

bp bp

Dui zm

i=1 j =1

j Dvzm =

1 deti,j ψj γi zm

2

= Vp

(E.7)

Appendix F. Super-K -theory theta function Here we explain why Θ(F , ρ) de?ned in (1.28) is a supertheta function for a family of

principally polarized superabelian varieties. To show this we use the results of [49], where supertheta functions were studied.

A generic complex supertorus is de?ned as a quotient of the a?ne superspace with even coordinates zi , i = 1, . . . , Neven and odd coordinates ξa , a = 1, . . . , Nodd by the action of the abelian group generated by {λi , λi+Neven } λi : zj → zj + δij , λi+Neven : zj → zj + (?even )ij , ξa → ξa ξa → ξa + (?odd )ia (F.1) (F.2)

We will restrict to the special case (?odd )ia = 0 relevant for our discussion. Let us also assume that the reduced torus (obtained from the supertorus by forgetting all odd coordinates) has a structure of a principally polarized abelian variety and denote its Kahler form by ω. It follows from the results of [49], that a complex line bundle L on the supertorus with c1 (L) = ω has a unique section (up to constant multiple) i? ?T even = ?even together with the positivity of the imaginary part of the reduced matrix. This section is a supertheta function. Now we can ?nd a family of principally polarized superabelian varieties relevant to our case simply by setting Neven = N and Nodd = Nf erm.zm and by de?ning symmetric ?even as Re(?even )ij = ReτK (xi , xj ), Im(?even )ij = ImτK (xi , xj )+ 68 (F.3) (F.4)

2 p=0 X10

G2p (xi ) + G2p (xj ) ∧? ?J2p (zm) + δij F (zm)

a bilinear expression in fermion(and ghosts) zeromodes and F (zm) is a functional quartic in the 10D fermion action (7.10),(14.1) as well as from the ghost action (7.35),(7.40),(15.1). of the imaginary part of the period matrix described in (F.4). It would be very nice if one could formulate this superabelian variety in a more natural way, without reference to a Lagrangian splitting of ΓK . The modi?ed characteristics α, β and prefactor ?Φ(F ) in (1.28) all originate from the shift fermion( and ghosts) zeromodes, both J2p (zm) and F (zm) can in principle be found from

where xi , i = 1, . . . , N is a basis of Γ1 . In (F.3) J2p (zm) is a 2p-form on X10 constructed as

69

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- The partition function of 2d string theory

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