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OU-HET 220 July, 1995

arXiv:hep-th/9508001v2 2 Aug 1995

Calabi-Yau Compacti?cation of Type-IIB String and a Mass Formula of the Extreme Black Holes

Hisao Suzuki?

Department of Physics, Osaka University Toyonaka, Osaka 560, Japan

Abstract Recently proposed mechanism of the black hole condensation at conifold singularity in type II string is an interesting idea from which we can interpret the phase of the universal moduli space of the string vacua. It might also be expected that the true physics is on the conifold singularity after supersymmetry breaking. We derive a mass formula for the extreme black holes caused by the self-dual 5-form ?eld strength, which is stable and supersymmetric. It is shown that the formula can be written by the moduli parameters of Calabi-Yau manifold and can be calculated explicitly.

?

e-mail address: suzuki@phys.wani.osaka-u.ac.jp

There are huge number of consistent string theories in four dimensions. Uni?cation of the string vacua is required to recover the predictive power for physics. Recently, an interesting observation was given by Strominger[1] who argued that the conifold singularity may be the key to understand the physics. Motivated by the argument given by Seiberg and Witten[2], he interpreted the conifold singularity as a point where black holes become massless and probably the physics choose the point after supersymmetry breaking. The Wilsonian e?ective action acquires the logarithmic term at conifold singularity and induces such terms in the period matrix which coincides with the classical calculation[1]. On the other hand, it is known by mathematicians that we can glue together di?erent Calabi-Yau space at conifold singularity. In Ref.[3], physical understanding of such phase transition through conifold singularity was advocated. At present, the true physical process has not fully understood but t! he argument is very interesting. Motivated by these fascinating idea, we construct a mass formula of the classical extreme black holes, which should be an fundamental formula for Calabi-Yau compacti?cation of the type II string. In type IIB string such black holes are constructed by the self-dual 5 form ?eld strength in ten dimensions[4]. After compacti?cation, the 5-form turns to the ?eld strength of the U(1) Yang-Mills ?elds and the black holes constructed by these gauge ?eld are stable and supersymmetric when other ?elds are neutral with respect to these charges[5]. The masses of such black holes are expected to be the function of the moduli[1]. The mass formula will turn out to be di?erent from the generalized BPS mass formula for non-aberian dyon solutions[6]. More generally, the d-form ?eld strength in 2d dimensions contains the information of the internal space in the mass formula of the black holes. Therefore, it will be convenient to treat the example in dimensions lower that ten. To begin with, we are going to consider torus compacti?cation of the self-dual 3-form ?eld strength in 6 dimensions. We denote the ?eld strength by H = H?νρ dx? ∧ dxν ∧ dxρ .

By the T 2 compacti?cation, the internal space can be identi?ed by x = x + R1 , y = y + R2 , .

(1)

1

and corresponding cycle can be called A-cycle and B-cycle respectively. We take the one-form basis α, β such that α = 1, β = ?1, α ∧ β = 1. (2)

A

B

T2

The explicit form of the one forms are α= dx dx ,β = ? . R1 R2 (3)

The holomorphic (1, 0) form ? = dx + idy can be written by these basis as ? = R1 α ? iR2 β, and satis?es ? ≡ Z = R1 , ? ≡ G = ?iR2 . (5) (4)

A

B

We expand the antisymmetric tensor ?elds as H = F ∧ α + G ∧ α, (6)

where F and G are the two form ?eld strength with respect to space-time and we have omitted the scalar component. Then the action can be written as H ?H = F ∧ ?F α ∧ ?α + 1 2 R1

M4

T2 ×M 4

G∧?∧β∧?∧ 1 2 R2

M4

= V ol(T 2 )[

F ∧ ?F +

G ∧ ?G],

(7)

where V ol(T 2 ) = R1 R2 and this volume factor of the internal space is common to all terms of the action so that we can omit it. The relation (4) implies that the coupling constant depends explicitly on the moduli parameter of the internal space. The limit R1 → ∞ can be regarded as the strong coupling limit. As a classical solution, We require the magnetic charge to be quantized; H = nCg , H = mCg , (8)

α×S 2

β×S 2

2

where the elemental charge Cg can not be determined by our argument. The quantum number m and n may be regarded as winding numbers with respect to the internal space. For the metric, we take Reisner-Nordstrom solution. The condition (8) implies that magnetic component of the antisymmetric tensor can be written as Fm = Cg n sin θdθ ∧ d?, Gm = ?Cg m sin θdθ ∧ d?. (9)

Because of the self duality of H, the electric component can be obtained by adding hodge dual of the tensor; H = Fm ∧ α + Gm ∧ β + ?Fm ∧ ?α + ?Gm ∧ ?β. (10)

In the case of the extreme black hole, the mass can be written by the sum of the electric and magnetic charge as

2 M 2 = 2Cg (

m2 n2 + 2 ), 2 R1 R2

(11)

where the factor 2 comes from the fact that the contribution of the electric ?elds and the magnetic ?eld are identical due to the electro-magnetic duality. When we denote the √ undetermined constant as Cg = V ol(T 2 )v = R1 R2 v/ 2, the mass of the extreme hole can be written as

M 2 = v 2 |Zm ? Gn|2 ,

(12)

which is exactly the BPS formula[2, 6] This formula is basically for dyon solutions in NonAbelian theories. You can easily generalize the construction to the torus compacti?cation of the 5-form ?eld strength in ten dimensions. But you will ?nd that the formula di?ers from that obtained in Ref.[6] by the special geometry. This fact implies that the mass formula is not unique, which may be caused by the fact that the the elementary quantity is not M but M 2 which can contain tensor with respect to Sp(n, Z). Before discussing Calabi-Yau compacti?cation of the 5-form ?eld strength in ten dimensions, we will derive some formula of Calabi-Yau manifolds. 3

On the Calabi-Yau manifold M, we take a canonical homology basis for H3 (M; Z) as Aa , Bb , a, b = 0, 1, ..., b2,1 and let (αa , β b) be the dual cohomology basis such that[7] αb =

b αa ∧ β b = δa ,

Aa

M

Ba

βb =

M

b β b ∧ αa = ?δa .

(13)

Note that holomorphic 3-form ? is also the element of H 3 (M, Z). We de?ne the period of ? as za = Therefore, we can express ? as ? = z a αa ? Ga (z)β a . (15) ?, Ga = ?. (14)

Aa

Ba

In Ref.[8], it is shown that the complex structure of M can be completely determined by the choice of z a , so that Ga can be regarded as a function of z a . For the application to the algebraic geometry, it will be convenient to take z a as a function of other moduli parameters ta [8], but here for simplicity, we take z a as elemental variables. Note that the holomorphic 3-form ? is de?ned up to constant, the same must be required for z a and Ga ; Ga (λz) = λGa (z), from which we have Ga = z b ?b Ga . (17) (16)

By Kodaira’s theorem[8], the in?nitesimal variation of the holomorphic 3-form with respect to the moduli parameters can be expanded by the (3, 0)?form and (2, 1)?forms. Therefore, ? ∧ ?a ? = 0, (18)

M

which implies 2Ga = ?a (z b Gb ). Therefore, G can be written as Ga = ?a G. 4 (19)

The vector space H (3,0) ⊕ H (2,1) can be spaned by ωa = ?a ? = αa ? ?a ?b Gβ b . (20)

We hereafter denote the matrix ?a ?b G as Σab ≡ Gab + iBab . The holomorphic 3-form ? can be written by this basis as ? = z a ωa . (21)

Since any vector of H (3,0) ⊕ H (2,1) can be written in the form C = C a ωa , a natural inner product of these basis can be de?ned as < C|D >= i 2 ? ? C ∧ D = C a Bab D b , (22)

M

In other words, a natural metric of the vectors are imaginary part of the period matrix Σab . These inner products are the basic set-up for the special geometry whereas the basis (αa , βa ) can be called integral basis[9]. The most important ingredient of our discussion is Hodge ? operator. Hodge ? operation maps the 3-forms to 3-forms. We write the e?ect on the basis as

b ? αa = ?fa αb + eab β b ,

? β a = ?g ab αb + fba β b , so that we have αa ∧ ?αb = eab , αa ∧ ?β b = β a ∧ ?β b = g ab ,

(23)

M

M

M

b β b ∧ ?αa = fa .

(24)

Since the relation ? ? γ = ?γ should hold for any 3-form γ, these coe?cients satisfy the following relations;

c b eac g cb ? fa fcb = δa , c fa eab = fbc eac

,

fcb g ac = fca g cb .

(25)

5

b Let us determine the coe?cients eab , fa , g ab. These coe?cients can be completely

determined by z a . Note that the e?ect of ?-operator on (3, 0) form ? is ? ? ? = ?i?, whereas on (2, 1) form C the operation acts as ? ? C = iC. (27) (26)

This di?erence is su?ce to determine the coe?cients. The basis of (2, 1) form should be perpendicular to ? with respect to the inner product de?ned in (22). These are written as ωa ? Bab z b ? ?, < z|z > (28)

function of the moduli as complex conjugation to get real number for the norm1 . By

where < z|z >= Bab z a z b as de?ned in (22). We require that ?-operator acts on the ?

acting the star operator on (23), and by separation the real and imaginary part, we obtain eab =

b fa

g ab

1 ? ? (Ga Gb + Ga Gb ) ? Gac (B ?1 )cd Gdb ? Bab , < z|z > 1 ? = (Ga z b + Ga z b ) ? Gac (B ?1 )cb , ? < z|z > za zb + zazb ? ? = ? (B ?1 )ab . < z|z >

(29)

After preparing the mathematical formula, we are now going to derive the mass formula of the black hole coming from the compacti?cation of the self-dual 5-form ?eld strength. We expand the 5-form F by F = F a αa + Ga β a , (30)

where F a and Ga are 2-form ?eld strength with respect to our space-time. Then the action can be written as

M×M 4

1

F ?F =

M4

b [eab F a ∧ ?F b + 2fa F a ∧ ?Gb + g ab Ga ∧ ?Gb ].

(31)

We should say that another expression of the operation can be obtained without this requirement.

I am not convinced that this this is the proper choice at present

6

b We can therefore regard the parameters eab , fa and g ab as coupling constants. These cou-

pling constants are functions of the muduli parameters as was shown in (28). Equations of motions are satis?ed when dF a = dGa = d ? F a = d ? Ga = 0. We consider ReisnerNordstrom solution and impose the quantization conditions on the magnetic component as

Aa ×S 2 Ba ×S 2

F = Cg na , F = Cg ma . (32)

The corresponding ?elds may be regarded as the ones having an elemental charge Cg and the winding number (na , mb ) with respect to the internal manifold. The magnetic component can be solved by taking (Fm )a = Cg na sin θdθ ∧d?, Then the solution of the 5-form F can be given by F = (Fm )a ∧ αa + (Gm )a ∧ β a + ?(Fm )a ∧ ?αa + ?(Gm )a ∧ ?β a . (33) (Gm )a = ?Cg ma sin θdθ ∧d?

We can now derive the mass formula of the extreme black hole. The mass is the sum of the contributions of the electric charges and those of magnetic charges. These contribution should be identical by electromagnetic duality.. As a matter of fact, When we denote the electric charge and the magnetic charge of F a and Gb by Qa , P a and Qb , P b respectively, the solution (31)implies the electric charge can be written as P a = fba Qb ? g ab Qb ,

b Pa = eab Qb + fa Qb .

(34)

Using this relation, we ?nd the following identity;

b eab Qa Qb + 2fa Qa Qb + g ab Qa Qb b = eab P a P b + 2fa P a Pb + g ab Pa Pb .

(35)

The ?nal formula of the extreme black hole is given by

2 2Cg b [eab na nb ? 2fa na mb + g ab ma mb ], M = V olM 2

(36)

where V olM is the volume of the Calabi-Yau space coming from the fact that the Einstein action aquires this factor. 7

Note that this formula is di?erent from an generalization of BPS mass formula[6] M = v|z a ma ? Ga na |

b where corresponding eab , fa and g ab are identi?ed as

(37)

? ? eab = Ga Gb + Ga Gb ,

b ? fa = (z a Gb + z a Gb ), ?

g ab = z a z b + z a z b . ? ?

(38)

At present, the derived formula (36) is merely classical. When we proceed further, we should consider the quantum behavior of the coupling constants. It might be that the classical result contain the information of the loop correction.[1, 3] It seems also instructive to consider models where the e?ective coupling constants can be calculated explicitly.

8

References

[1] A. Strominger, ” Massless Black Holes and Conifolds in String Theory,” hep-th 9504090. [2] N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19. [3] B. R. Greene, D. R. Morrison and A. Strominger, ”Black Hole Condensation and the Uni?cation of String Vacua,” CLNS-95/1335, hep-th 9504145. [4] G. Horowits and A. Strominger, Nucl. Phys. B360 (1991) 197. [5] G. W. Gibbons and C. M. Hull, Phys. Lett. 109B (1982) 190. [6] A. Ceresole, R. D’Auria, S. Ferrara and A. Van Proeyen, ”Duality Transformations in Supersymmetric Yang-Mills Theory Coupled to Supergravity,” hep-th 9502072. [7] P. Candelas, P.S. Green and H. H¨ bsch, Nucl. Phys. B330 (1990) 49. u [8] R. Bryant and P. Gri?ths, Progress in Mathematics 36, pp.77, (Birk¨user, Boston, o 1983) [9] A. Strominger, Comm. Math. Phys. 133 (1990) 163.

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