Special Geometry (Term Paper for 18.396 Supersymmetric Quantum Field Theories)

Special Geometry (Term paper for 18.396 Supersymmetric Quantum Field Theories)

Xi Yin

Physics Department Harvard University

Abstract

This paper is an elementary survey of special geometry that arises in N = 2 supersymmetric theories. We review the definition of rigid and local special Kählermanifolds. For rigid special geometry, we discuss their connection to N = 2 supersymmetric gauge theories and the Seiberg-Witten solution. For local special geometry, we study their emergence in N = 2, d = 4 supergravity and Calabi-Yau moduli space. In particular, we discuss the connection between string effective action and the metric on Calabi-Yau moduli space, and the non-renormalization theorem of type IIB complex moduli space. Finally, Seiberg-Witten theory is recovered from the rigid limit of type IIB string compactified on certain Calabi-Yau manifolds. 1 Introduction

It is well known that supersymmetry impose strong constraints on the geometry of the scalar manifold M, on which the scalar ﬁelds are viewed as coordinates[1, 2] (for reviews see [3]). In d = 4, N = 1 theories, the scalar manifold for chiral multiplets, M, is restricted to be K¨ahler. If the chiral multiplets are coupled to N = 1 supergravity, then M is further required to be a

Hodge Kählermanifold. For rigid N = 2 theories, the target space MV for vector multiplets is restricted to be a rigid special Kählermanifold, whose Kählerpotential is determined by a holomorphic prepotential F; the target space for hypermultiplets MH , on the other hand, is a hyper-Kählermanifold. When coupled to N = 2 supergravity, MV is restricted to be a

(local) special K¨ahlermanifold, and MH becomes a quarternionic K¨ahlermanifold. For N ≥ 3 supergravity theories (possibly coupled to vector multiplets for N ≤ 4), the scalar manifolds are in fact completely determined as homogeneous coset spaces. This paper is devoted to a study of some of the rich structures in special geometry and N = 2 theories.

Remarkably, the structure of special geometry also emerges in the moduli space of Calabi- Yau manifolds[7]. This is not merely a coincidence. In fact, via the Zamolodchikov metric of (2, 2) superconformal field theories, the prepotential of Calabi-Yau moduli space is closedly related to the prepotential of the low energy effective theories of string compactifications[8]. In type IIA compactifications, vector multiplets are identified with deformations of the Kähler moduli, and hypermultiplets are identified with deformations of the complex moduli. In type IIB theory the identifications are reversed. In general, the Calabi-Yau prepotential captures the classical geometry, which may get quantum corrections in the full string theory, or even at the world-sheet level. It is rather striking that in certain cases quantum corrections are absent. For example, the non-renormalization theorem proved in [9] says that there is no quantum corrections to the complex moduli space of type IIB theory. So in this case the exact coupling of vector multiplets can be computed by classical geometry. These ideas have been applied to obtain exact solutions of various quantum field theories. For example, using the technique of mirror symmetry, Candelas et al [10] were able to solve exactly the superconformal field theory corresponding to the quintic 3-fold (and its mirror).

Rigid special geometry, on the other hand, finds its beautiful application in Seiberg-Witten theory. In this case, the exact prepotential of N = 2, d = 4 SYM is obtained from periods of the Seiberg-Witten (hyper)elliptic curve[5], fibered over the Coulomb branch moduli space. This is in fact closely related to the previous story. Rigid Yang-Mills theories are contained in the α0 → 0 limit of string theory on certain Calabi-Yau geometry. Usually, to produce the desired gauge groups, we take the CY geometry to be K3 (or ALE space) fibration over a base P1.

1 Roughly speaking, the rigid limit amounts to looking very closely at the locus of the singular K3 ﬁber, whose geometry can be shown to reduce to the Seiberg-Witten curve[16].

The paper is organized as follows. In section 2 we define rigid and special Kählermanifolds and review their basic mathematical properties. In section 3 we study the rigid special geometry that arises in N = 2, d = 4 gauge theories, and describe the Seiberg-Witten solution. Section 4 demonstrates the structure of local special geometry in N = 2, d = 4 supergravity coupled to vector multiplets, using the formulation of superconformal tensor calculus. In section 5 we study the geometry of complex and Kählermoduli space on Calabi-Yau manifolds. In section 6, we discuss the connection between Calabi-Yau moduli space and the effective lagrangian of string theories, and an important non-renormalization theorem. This is the key to the whole story. To understand these connections is the original motivation for the author to write this paper. In section 7, we recover Seiberg-Witten curve from the rigid limit of type IIB string theory compactified on certain Calabi-Yau manifolds.

2 Special K¨ahlermanifolds

A Kählermanifold M is a Hodge manifold if there exists a line bundle L → M whose first Chern class coincides with the Kählerclass, i.e.

c1(L) = [J] (1)

From the Lefschetz theorem on (1, 1)-classes [4], it suﬃces that [J] lies in the integer cohomology group, i.e. [ω] ∈ H1,1(M) ∩ H2(M, Z)

Then L can be taken as the canonical bundle of the divisor dual to [J]. The hermitian metric on L is the exponential of the K¨ahlerpotential h(z, z¯) = exp(K(z, z¯)), so that (1) is satisﬁed. Now the connection 1-form on L is given by

θ = h−1∂h = ∂K θ = h−1∂h¯ = ∂K¯

Actually, we will refer to Hodge K¨ahlermanifold as those whose K¨ahlerclass [J] is an even element of H2(M, Z). In the context of d = 4 supergravity, this is due to the fact that fermions are sections of the bundle L1/2, as we’ll see in section 4. It has also been proved by Tian in the context of Calabi-Yau moduli space.

2 Let M be an n-dimensional Kählermanifold, L → M be a flat line bundle, SV → M be a flat holomorphic symplectic vector bundle of rank 2n, and consider the tensor bundle H = SV ⊗ L.

Definition A Kählermanifold M is rigid special Kähler,if there is a section Ω ∈ Γ(H,M) such that the Kählerform is given by i J = ∂∂¯hΩ|Ωi 2π together with the condition

h∂iΩ|∂jΩi = 0 where the inner product   0 −1 hΩ|Ωi = iΩ†   Ω 1 0 is given by the standard hermitian metric on the Sp(2n, R)-bundle SV.

Under a local trivialization, the section Ω can be written as   XA Ω =   FB A,B=1,···,n The K¨ahlerpotential is therefore ³ ´ A A K(z, z¯) = hΩ|Ωi = i X F A − X FA

The non-degeneracy of the Kählerform guarantees that locally XA’s span a set of coordinates on M, conventionally called the special coordinates. The integrability condition in special coordinates, h∂AΩ|∂BΩi = i(∂BFA − ∂AFB) = 0, implies that locally ∂ F = F (X) A ∂XA for a function F (X) holomorphic in XA’s, called the prepotential. Note that F (X) is not Sp(2n, R) invariant. It is defined only in the special coordinate. The period matrix is defined by ∂ N = F = ∂ ∂ F AB ∂XB A A B

Now suppose M be an n-dimensional Hodge K¨ahlermanifold, L → M be a line bundle with c1(L) = [J], SV → M be a ﬂat holomorphic simplectic vector bundle of rank 2n + 2, and consider the tensor bundle H = SV ⊗ L.1

1A motivation for this formulation can be seen from section 5.

3 Definition A Hodge Kählermanifold M is (local) special Kähler,if there is section Ω ∈ Γ(H,M) such that the Kählerform is given by i J = − ∂∂¯lnhΩ|Ωi 2π and satisfies

hΩ|∂iΩi = 0

Note that the last condition is well-deﬁned on sections of H: since

DiΩ = ∂iΩ + (∂iK)Ω and hΩ|Ωi = 0, we can replace ∂i by covariant derivative Di.

Under a local trivialization, the section Ω takes the form   ZI Ω =   FJ I,J=0,1,···,n The K¨ahlerpotential is therefore ³ ´ I I K = − ln i Z F I − Z FI

Under K¨ahlertransformation K → K + f + f, the holomorphic section Ω transformation as Ω → e−f Ω. So ZI ’s can be regarded as a set of homogeneous coordinates on M. Suppose

Zi ti = Z0 form a local coordinate system. Then from the condition

I I 0 = −ihΩ|∂iΩi = Z ∂iFI − ∂iZ FI it is not hard to show2 that (locally) up to a symplectic transformation, we can write ∂ F (Z) = F (Z) I ∂ZI where the prepotential F(Z) is holomorphic and of homogeneous degree 2 in ZI ’s.

2 0 i j Write FI = Z FI (t ), then the constraint gives ∂iF0 − Fi + t ∂iFj = 0. Taking its derivative with respect to j 0 k 0 t , we get (a) ∂iFj = ∂j Fi ⇒ locally Fi = Z ∂iF for some F(t); (b) ∂i∂j F0 + t ∂i∂j Fk = 0 ⇒ ∂i∂j (F0 − 2Z F + 0 k 0 k 0 2 i 0 Z t ∂kF) = 0. Up to a symplectic transformation, F0 = Z (2F − t ∂kF). So F (Z) = (Z ) F(Z /Z ) is the desired prepotential. In terms of F(t), the K¨ahlerpotential is given by £ ¤ i i K(t, t) = − ln i 2(F − F) − (∂iF + ∂iF)(t − t )

4 3 Rigid N = 2, d = 4 theories

N = 2 theories of vector multiplets can be conveniently formulated using N = 2 superspace. We introduce fermionic coordinates θα, θ˜α and their conjugates; the corresponding superderivatives generalizes in a straightforward way. We denote d2θd2θ˜ by d4θ, etc.

For example, an N = 2 chiral superﬁeld Ψ = Ψ(x, θ, θ,˜ θ, θ˜) is a singlet under SU(2)R, and satisﬁes the constraints

Dα˙ Ψ = D˜ α˙ Ψ = 0

We can expand Ψ in N = 1 superspace as √ α 2 Ψ = Φ(˜y, θ) + i 2θ˜ Wα(˜y, θ) + θ˜ G(˜y, θ) where y˜µ = xµ + iθσµθ + iθσ˜ µθ˜

Now Φ is a matter chiral field, Wα is the superfield strength. To get a sensible Lagrangian for the chiral fields, we need to impose additional constraints on Ψ so that G takes the form Z G(˜y, θ) = d2θ Φ(˜yµ − iθσµθ, θ)e2V (˜y−iθσθ,θ,θ)

The most general N = 2 action of an adjoint vector multiplet can be written down in terms of a holomorphic prepotential F(Ψ), ·Z ¸ 1 Im d4xd4θTrF(Ψ) 4π or in N = 1 language, "Z Z # 1 ∂F 1 ∂2F Im d4θΦe2V + d2θ W αW (2) 4π ∂Φ 2 ∂Φ2 α

There cannot possibly be dynamically generated superpotential, as constrained by N = 2 supersymmetry. It is straightforward to generalize the above to more than one vector multiplets, whose explicit form we don’t bother to write down. The eﬀective action is governed by the prepotential F(ΦA), and we see that the target space for ΦA is a rigid special K¨ahlermanifold. The gauge coupling matrix is given by the period matrix

∂2F N = AB ∂ΦA∂ΦB

Now suppose (2) is the low energy effective action of N = 2 pure SU(n) SYM. At a generic point in the Coulomb branch, the gauge group is broken to U(1)n−1. The off-diagonal fields are massive and integrated out. There are only massless fields left in the action. Following

5 the convention of [5], we call aA the VEV of the scalar component of ΦA, and deﬁne the dual A variable aD = ∂F(a)/∂aA, A = 1, ··· , n − 1. One can immediately see that there is a natural

Sp(2n − 2, R) action on the vector (a, aD), which leaves the scalar kinetic term invariant.

For simplicity let’s work with the SU(2) case. Near the classical region where perturbation theory is valid, the prepotential takes the form Ã ! µ ¶ 1 i a2 1 X∞ Λ 4l F(a) = τ a2 + a2 ln + a2 c 2 0 π Λ2 2πi l a l=1 where Λ is the dynamical scale, τ0 is the bare coupling. In the above expression, the second term comes from the 1-loop perturbatively exact NSVZ beta function, the inﬁnite sum comes from instanton corrections. In order to have non-anomalous R-symmetry, we need to assign Λ R-charge 2. On the other hand, l-instanton processes have anomalous R-number 8l by the index theorem, so their contribution must be proportional to Λ4l.

The SL(2, R) action on the symplectic vector is generated by two transformations:

(i) a → a, aD → aD + ba. This eﬀectively shifts the theta angle θ → θ + 2πb. Because of instanton eﬀects, only for integer values of b this would be a symmetry of the full quantum theory.

(ii) a → aD, aD → −a. This should convert the gauge coupling τ = ∂aD/∂a to −1/τ, which is the electric-magnetic duality of the theory. To see this, we can think of the ﬁeld strength W as an independent ﬁeld, and introduce the dual photon VD to impose Bianchi identity ImDW = 0 via coupling Z Z 1 1 Im d4xd4θV DW = − Im d4xd2θW W 4π D 4π D Simply integrating out W does the job.

Interestingly, the structure of rigid special geometry also arises in the moduli space of Rie- A mann surfaces. A genus g Riemann surface has 1-cycles α , βB, A, B = 1, ··· , g, with intersection form

A B α · α = βA · βB = 0 A A A α · βB = −βB · α = δB

This basis is determined up to symplectic transformations. In general there can be monodromies for the 1-cycles in the moduli space M. By Hodge decomposition, there are g linearly independent holomorphic 1-forms ωi, i = 1, ··· , g on the curve. Their periods form a symplectic vector µZ Z ¶ ωi, ωi A α βB

6 We have the basic relations[4] (Riemann’s ﬁrst and second relation) Z P ³R R R R ´ A ω · ω − ω · A ω = ω ∧ ω = 0 A α i βA j βA i α j i j ZΣ P ³R R R R ´ A αA ωi · β ωj − β ωi · αA ωj = ωi ∧ ωj (3) A A Σ

If we deﬁne the period matrix ΠAB by Z X Z ωi = ΠAB ωi B βA B α Then it follows from (3) that

ΠAB = ΠBA and Im Π > 0 (4)

In the relevant examples, we can ﬁnd a meromorphic 1-form λ (the Seiberg-Witten diﬀerential) which depends holomorphically on the complex moduli ui, i = 1, ··· , g, such that their derivatives with respect to ui give the ωi’s (up to exact forms) Z Z Z Z ∂ ∂ ωi = λ, ωi = λ A A α ∂ui α βB ∂ui βB The relation (4) then implies that locally there exists a prepotential F such that Z Z ∂F(a) aA = λ, = λ A α ∂aA βA

Now to make contact with N = 2, d = 4 SU(n) SYM, aA is to be identified with the VEV of the scalar fields in the low energy effective action, and ∂F/∂aA is to be identified with the electric-magnetic dual variable. The corresponding ui’s are usually chosen as

nX−1 n n−i−1 WAn−1 (x, ui) ≡ x − ui(a)x = hdetn(xI − φ)i i=1

Classically the singularities in the moduli space are located at the discriminant of WAn−1 (x, ui), where W-bosons becomes massless. Quantum mechanically this is no longer true. The argument[5] involves the monodromy around classical region (large u) and the positivity of the gauge coupling. The simplest guess (supported by many evidences but not rigorously proved) is that each classical singularity splits into two singularities in the quantum theory.

How do we know that the prepotential determined in this way is the true prepotential for

N = 2 SYM? In fact, the monodromies of (a, aD) around the singularities, together with the asymptotic behavior and the positivity of the metric on M, unique determines the solution to F. It suﬃces to ﬁnd the SW curve that has the correct monodromy properties. We will not

7 get into this analysis, which can be found in [5]. The answer for SU(n) gauge group is a genus n − 1 (hyper-)elliptic curve, deﬁned by

2 2 2n y = WAn−1 (x, ui) − Λ (5) and correspondingly, the Seiberg-Witten diﬀerential is 1 xdx λ = √ W 0 (6) 2 2π An−1 y The normalization factor is determined from the asymptotic form of F near the classical region.

4 N = 2, d = 4 supergravity

In this section we’ll describe the superconformal tensor calculus formulation of N = 2 supergravity, where local special geometry arises naturally. The idea is to start out with an N = 2 superconformal ﬁeld theory, including extra compensate multiplets, then ﬁx the extra gauge symmetries and break to N = 2 supergravity coupled to vector multiplets. We are following the reference [6].

N = 2, d = 4 superconformal algebra has two ordinary supercharges Qi, and two special supercharges Si,3 where i is the index for extended supersymmetry. In addition to the conformal group generators Mµν,Pµ,Kµ,D which combine to generate SO(4, 2), we also need R-symmetry i i i generators Vj and A of SU(2)R×U(1)R. Q and S transforms as (2, −1) and (2, +1) respectively, under R-symmetry. The full superconformal group is SU(2, 2|2), with its bosonic part being

SU(2, 2) × U(2)R.

First, to build a superconformal gauge theory, we use the Weyl multiplet n o a i i − i eµ, bµ, ψµ,Aµ, Vµj,Tab, χ ,D

They are the gauge ﬁelds corresponding to vierbein, dilatation, gravitino, U(1)R and SU(2)R, together with auxiliary ﬁelds including a ASD anti-symmetric tensor, a doublet of fermions and a real scalar.

We also need n + 1 conformal N = 2 vector multiplets ³ ´ I I I I I X = X , Ωi ,Wµ ,Yij ,I = 0, ··· , n

3They are related to the ordinary supercharges by 1 [K ,Qi] = γ Si, [P ,Si] = γ Qi µ µ µ 2 µ

8 They are a complex scalar, a fermion doublet, a vector gauge ﬁeld and a triplet of auxiliary scalars. The action can be built the same way as the rigid case, where we integrate a holomorphic prepotential F(X) over the N = 2 chiral superspace. Now conformal weight restricts F(X) to be homogeneous of degree 2 in XI .

In order to fix the extra gauge symmetries and break down to super Poincarégroup, we need to introduced an additional compensate multiplet, which can be chosen to be a hypermultiplet, tensor multiplet, or a nonlinear multiplet. We will not discuss the full details of gauge fixing procedure, due to the lack of space, but to describe the main results. Among the fields in the Weyl multiplet, only the vierbein and gravitino survive as physical fields. The hypermultiplet i disappear by gauge fixing SU(2)R and by eliminating auxiliary fields D, χ . For the extra vector multiplet, the complex scalar disappears by gauge fixing the dilatation and U(1)R, the fermions disappear by fixing S-supersymmetry, and the vector becomes the graviphoton. Finally within the original n + 1 vector multiplets, we are left with n complex scalars, n SU(2)R doublets of fermions, and n + 1 vectors.

Now we look more carefully at the gauge ﬁxing of vector multiplets. There are n+1 XI ’s that will be modded out by dilatation and U(1)R symmetry, so the scalar manifold is n-dimensional complex manifold M. Let zA be a set of local coordinates. The parameters of gauge transformations are holomorphic in z: XI → eΛD(z)−iΛA(z)XI

It is convenient to split XI to XI = aZI and introduce new gauge symmetry

a → eΛ(z)a, ZI → e−Λ(z)ZI

I Now we let dilatation and U(1)R act only on a. The scalar curvature is coupled to X ’s via

I I −i(X F I − X FI )R

To work in the Einstein frame is equivalent to ﬁxing the dilatation gauge:

I I −i(X F I − X FI ) = 1 or equivalently |a|2 = eK, where ³ ´ I I K(z, z¯) = − ln i Z FI (Z) − Z F I (Z)

In above we used the fact that FI (X) is homogeneous of degree 1 in X, so FI (X) = aFI (Z). To

ﬁx U(1)R, we can take the phase of a to be

a = eK/2

9 There is a remaining ambiguity parameterized by Λ(z) = f(z), acting as

I −f(z) I −f(z) Z → e Z ,FI → e FI K(z, z¯) → K(z, z¯) + f(z) + f¯(z)

This is nothing but our old friend Kählertransformation. If we know that K is really the Kähler I potential on the scalar manifold M, then clearly M is special Kähler. In particular, (Z ,FI ) lives on a line bundle L over M (more precisely, its tensor with an Sp(2n + 2, R) bundle, as in section 2), and h(z, z¯) = eK(z,z¯) is the hermitian metric on the fiber. The curvature of L is precisely the Kählerform, so M is a Hodge manifold.

I However, the scalar component of X is not the component with smallest nonzero U(1)R 1/2 charge. The fermions Ωi has U(1)R charge −1/2, so they are sections of L . This further 1/2 requires the K¨ahlerclass [J] = 2c1(L ) to be an even element of the integer cohomology group.

I Under the K¨ahlertransformation, X and FI (X) transform as

I I X → exp(−iImf)X ,FI (X) → exp(−iImf)FI (X)

I So V = (X ,FI (X)) are sections of the U(1)-bundle U → M naturally associated to L. Let ∇ be the connection on U, i.e. µ ¶ µ ¶ 1 1 ∇ V = ∂ + ∂ K V, ∇¯V = ∂¯ − ∂¯K V i i 2 i i i 2 i Now the period matrix is deﬁned by

J J FI (X) = NIJ X , ∇¯iF I (X) = NIJ ∇¯iX

The reader should be aware that: we haven’t shown why the scalar ﬁelds can be described by a sigma model with K¨ahlerpotential K, and why NIJ is the gauge coupling. The rest of the section will demonstrate this fact.

Let us ﬁrst deﬁne NIJ = 2ImFIJ , so the action for the scalar is

I µ J I I −NIJ DµX D X , where DµX = (∂µ + iAµ)X

Aµ is the auxiliary gauge ﬁeld for U(1)R. It is eliminated by the equation of motion i ³ ´ A = N XI ∂ XJ − ∂ XI XJ µ 2 IJ µ µ Since we have ﬁxed dilatation, there is identity

I J NIJ X X = −1,

10 it is not hard to verify directly that the scalar kinetic term is

I µ J K −gIJ¯∂µZ ∂ Z , gIJ¯ = ∂I ∂J K = NIJ e + ∂I K∂J K

The terms in the Lagrangian that contributes to the vector kinetic term are i i 1 F F −I F −Jab + (F − F XJ )F −I T −ab − N XI XJ T −T −ab + c.c. (7) 4 IJ ab 8 I IJ ab 64 IJ ab + − where Fab and Fab are the self-dual and anti-self-dual part of Fab, related by complex conjugation. − The auxiliary anti-self-dual tensor Tab is eliminated by its equation of motion

J −I − NIJ X Fab Tab = 4 K L (8) X NKLX Using the identity K L NIK Z NJLZ NIJ = F IJ + i M N , Z NMN Z and plug (8) into (7), we arrive at the gauge kinetic term i i N F −I F −Jab − N F +I F +Jab 4 IJ ab 4 IJ ab

This shows that the period matrix NIJ is indeed the gauge coupling matrix.

5 Calabi-Yau Moduli Space as a Special K¨ahlerManifold

Let’s ﬁrst consider the moduli space M of complex structures on a Calabi-Yau 3-folds X. There is a unique (up to rescaling) holomorphic 3-form preserved by the SU(3) holonomy. 3 The Hodge bundle H → M has ﬁber H (X) of complex dimension b3 = 2(h2,1 + 1), where 2,1 h2,1 = dimC H (X). A canonical hermitian metric on H is given by Z hα|βi = i α ∧ β X

3 a Poincar´eduality implies that H (X) is self-dual, and there is a real basis A ,Bb unique up to symplectic transformations such that

hAa|Abi = hBa|Bbi = 0

a a a hA |Bbi = −hBb|A i = iδb

So we learned that H is a ﬂat, holomorphic Sp(2h2,1+2, R) vector bundle. This natural structure is the motivation for our deﬁnition of special geometry in section 2.

11 Now we turn to the holomorphic 3-form Ω. It is a holomorphic section of the projectivization of H. As our notation in section 2 indicated, we can deﬁne a K¨ahlerpotential by

K(z, z¯) = − lnhΩ|Ωi

Under transformation Ω → ef(z)Ω, the K¨ahlerpotential transforms as

K(z, z¯) → K(z, z¯) − f(z) − f(z)

So we can think of Ω as a section of the bundle H ⊗ L for some line bundle L, and eK is the hermitian metric on L. This identifies the Kählerclass with the first Chern class c1(L). It has been proved by Tian that the Kählerclass is an even element in the cohomology group.

Let ta be a set of coordinates on M. Under infinitesimal deformation of the complex structure on X, a holomorphic differential dωi on X is mixed with anti-holomorphic differentials. So Ω will mix only with a (2, 1)-form Ga,

∂aΩ = Ga − KaΩ

Simply by looking at the degrees of the forms, there is relation

hΩ|∂aΩi = 0

From section 2 we see that M is special K¨ahler.It is easy to determine Ka

h∂aΩ|Ωi Ka = − = ∂aK hΩ|Ωi

The connection on L is Da = ∂a + ∂aK, so

Ga = DaΩ

The metric on M is thus given by

K ga¯b = ∂a∂¯bK = −e hGa|Gbi

This is positive deﬁnite by Riemann’s bilinear relations. The prepotential is determined by Z Z a ∂F Z = Ω,Fa ≡ = Ω a a A ∂Z Ba

Similarly, the complexified Kählerstructure moduli space is also special Kähler. There is

K¨ahlerpotential Z K = − ln J ∧ J ∧ J X

12 where J is the Kählerform. This can be derived from (9) in the next section. Since we are mainly concerned with complex moduli, we’ll just state the results for Kählermoduli. Let 2 eA,A = 1, ··· , h1,1 be a basis for H (X, Z), and write the complexified Kählerform as

A B + iJ = w eA

Then the prepotential for K¨ahlermoduli can be expressed as Z 1 wAwBwC F = − 0 eA ∧ eB ∧ eC 3! w X where w0 is introduced to make F homogeneous of degree 2. (Actually, this prepotential diﬀers from the usual convention by a factor 3/4.)

6 String Eﬀective Action and a Non-renormalization Theorem

Consider type IIB string theory compactified on a Calabi-Yau 3-fold X. The low energy effective theory is N = 2 supergravity coupled to h2,1 vector multiplets and h1,1 +1 hypermultiplets. The vector multiplets are deformations of the complex moduli, and hypermultiplets are deformations of the Kählermoduli. In the context of type IIA theory the identifications are reversed. The scalar manifold for the vector multiplets is special Kähler,as we’ve seen in section 4. On the other hand, we have the Weil-Peterson metric on the moduli space of complex structures on X, which is also special Kähler,as shown in the previous section. Now the crucial question is, are they the same metric? The answer to type IIB case is YES; and the answer for type IIA case is no, where the full metric gets corrections from world-sheet instantons.

Let first work at classical level. Let δgij, δBij be the deformation of the metric and B-field associated to the complex and Kählermoduli, corresponding to massless fields in the low energy effective theory. The latter fact requires δB and δJ to be harmonic forms. For Calabi-Yau manifold X of full SU(3) holonomy, h2,0 = 0, which restricts δB of type (1, 1). Requiring that δJ is harmonic is equivalent to requiring the (2, 1)-form

1 ¯l i j k Ω δg¯¯dx ∧ dx ∧ dx 2 ij lk to be harmonic4.

4 It is also equivalent to that δgij satisﬁes the Lichnerowicz equation

k k l ∇ ∇kδgij + 2Ri j δgkl = 0.

13 Now we make a Kluza-Klein reduction. Take the 6-dimensional Einstein-Hilbert action plus B-ﬁeld kinetic term on X, and vary g, B. This gives the tree level metric on the scalar manifold M, Z p h i 2 1 6 ik¯ j¯l ds = d x det g 2g g δgijδgk¯¯l + (δgi¯lδgjk¯ + δBi¯lδBjk¯) (9) V M a Let t , a = 1, ··· , h2,1 be the coordinates on the complex moduli space, the (2, 1) deformations

Ga in the previous section can be written explicitly as

¯l Gaijk¯ = ∂agk¯¯lΩ ij

Or inversely, 1 kl a δg¯¯ = Ω¯ G ¯δt ij 2||Ω||2 i aklj

2 1 ijk where ||Ω|| = 3! ΩijkΩ is constant since Ω is harmonic. Plugging this into (9), after a straightforward calculation, we obtain the expression for the metric on MC , R M Ga ∧ G¯b Ga¯b = R M Ω ∧ Ω This is precisely the Weil-Peterson metric we’ve seen in the last section. Similar conclusion also holds for the K¨ahlermoduli.

To prove the non-renormalization theorem, we ﬁrst show that the scalar manifold factorizes as MV × MH , at least locally. This in particular means that the scalar of the hypermultiplets cannot enter the vector multiplet metric, and vice versa. We will follow the argument of [11], in the case of global supersymmetry. The more general case for matter ﬁelds coupled to N = 2 supergravity is worked out in [1], which will not be presented here.

Consider a system of v Abelian vector multiplets and h neutral hypermultiplets. In N = 1 language, they contain chiral multiplets Φa, a = 1, ··· , v and Qi, Q˜i, i = 1, ··· , h. The K¨ahler potential is some general function K(Qi, Q˜i, Φa, c.c.). Denote the scalar components of Φa,Qi, Q˜i a i i by φ , q , q˜ , the gaugino by λ, and ψq˜ the hypermultiplet fermion. Then the kinetic terms for the scalars involves a cross term

i µ a† ∂i∂a¯K ∂µq ∂ φ + c.c.

N = 2 supersymmetry consequently requires a term

a ∂i∂a¯K ψq˜i /∂λ + c.c.

Now one directly observes that to cancel its N = 1 supersymmetry variation, one needs a term a involving two derivatives, Aµ, and scalars, which cannot possibly be Lorentz invariant. This implies that ∂i∂a¯K = 0, the metric on the moduli space factorizes.

14 On the other hand, in the context of (2, 2) superconformal ﬁeld theory, by computing four point amplitude one can show that[12] the moduli space is the direct product of the complex moduli and K¨ahlermoduli. This is very much in the same spirit as the factorization of vector and hypermultiplet moduli space.

The above facts are extremely powerful in constraining quantum corrections to type IIB compactification. In this case the vector multiplets are deformations of the complex moduli. Note that the Kählermodulous is effectively the sigma model coupling, which governs world- Φ sheet corrections, and the dilaton e ∼ gs lies in the hypermultiplet, which governs string loop corrections. By the factorization property, the prepotential for MV cannot possibly be depen- dent on either Kählermoduli or the dilaton, hence doesn’t received any quantum corrections. In other words, the tree level result µ Z ¶ K = ln i Ω ∧ Ω M is exact in the full string theory!

We can also say something about the K¨ahlermoduli in type IIA theory5. Since they cannot be coupled to the dilaton, they do not received string loop corrections. Let T A = wA/w0 be the K¨ahlermoduli. In world-sheet perturbation theory, there is Peccei-Quinn (PQ) symmetry, which is an analog of the shift in theta angle,

T A → T A + i²A

As long as the image of the world-sheet in X has trivial homology class, the PQ symmetry holds. The K¨ahlerpotential will be invariant provided

A B δF = cABX X for real coeﬃcients cAB’s. This determines the prepotential F to be of the form

d wAwBwC F = ABC + iλ(w0)2 w0 dABC is the intersection number we saw in section 5. Since n-loop world-sheet corrections are proportional to T 3−n(w0)2, the last piece corresponds to a possible 3-loop correction, which in general does appear. Non-perturbatively, F also receives world-sheet instanton corrections of A 0 order exp(−nAT /2πα ).

5An enlightening discussion can be found in J. Polchinski, String Theory, Volume II.

15 7 Seiberg-Witten Curve Revisited

As an application of local special geometry and the non-renormalization theorem demonstrated in the previous section, we will now recover the Seiberg-Witten solution from considerations of Calabi-Yau compactiﬁcations.

N = 2 supersymmetric gauge theories arise in the rigid limits of type IIA (IIB) string on 1 Calabi-Yau 3-folds X3 (or its mirror X˜3). If X3 is a K3 ﬁbration over P , type IIA theory on X3 is believed to be dual to heterotic string on K3×T 2, possibly with nontrivial compactiﬁcation of the gauge bundle6. This dual description will be useful in understanding the gauge symmetries and the role of the dilaton.

We’ve seen in the previous section that in type IIB theory, the tree level prepotential is exact. J It is determined by the periods of the holomorphic 3-form Ω on a symplectic basis (αI , β ) of

3-cycles, Z Z I X = Ω,FJ = Ω J αI β If X˜ is deﬁned by a polynomial W in a weighted projective space, the holomorphic 3-form 3 X˜3 can be explicitly written as Z 1 X5 Ω = (−1)ixidx1 ∧ · · · dxdi · · · ∧ dx5 γ W X˜3 i=1 where γ is a curve winding around the hypersurface W = 0. The integrand is a closed form, X˜3 so this integral is well-deﬁned regardless the choice of γ.

I I J The charged objects under X ,FJ are D3-branes wrapped on 3-cycle ν = MI α + N βJ , with central charge Z Z = Ω ν Near the points in the moduli space where ν degenerates, new massless hypermultiplets emerge in the theory. This is very similar to what happens in Seiberg-Witten theory, where monopoles and dyons become massless near the singularities.

We will restrict our attention to CY 3-folds as K3 fibration, or ALE fibration over a P1. The ADE singularities on the fiber give rise to enhanced gauge symmetries[14]. It has three different descriptions: i) In type IIA theory, D2-branes wrapped on vanishing rational curves become massless. They give new massless vector multiplets, resulting in enhanced gauge symmetry. ii) In type IIB theory, D3-branes wrapped on vanishing cycles, giving rise to tensionless strings

6For a detailed review, see P.S. Aspinwall [13].

16 in 6-dimensions. iii) In heterotic theory, the ADE root lattice coincides with the Narain lattice (think of K3 as elliptic ﬁbration over P1), giving rise to enhanced gauge symmetry.

Near the locus of singular K3 on the P1, the CY looks like7 " # 2n Λ ALE 2 W ˜ (xi, z; uk) = ² z + + 2W (xi, uk) + O(² ) X3 z ADE

ALE where WADE are the deﬁning equations of ALE spaces, or “simple singularities”, associated with ADE gauge groups. They are of the form

ALE 2 2 WAn−1 = WAn−1 (x1, uk) + x2 + x3 ALE 2 WD,E = WD,E(x1, x2, uk) + x3

In particular WAn−1 coincide with the characteristic polynomials of SU(n) gauge groups. The Seiberg-Witten curve (5) can be rewritten as a ﬁbration of weight diagrams over a P1, Λ2n X : z + + 2W (x , u ) = 0 (10) 1 z An−1 1 k

The local geometry of CY looks very much alike this, except extra pieces x2, x3. These extra coordinates can be integrated out to reproduce the SW curve8.

The Seiberg-Witten curve projects to a two-sheet cover of the z-plane branched along − + [zi , zi ]. Each K3 fiber of the CY contains n − 1 independent 2-spheres, with north and south poles defined by (10). The 3-cycles in CY can be thought of as fibrations of homology 2-spheres over the branch cuts, as in figure 1. The holomorphic 3-form in this case is dz dx ∧ dx Ω = ∧ 1 2 z ∂W /∂x3

We want to integrate Ω over the 3-cycles to obtain the periods. It is easy to integrate out x2, R since dx2 doesn’t depend on the size of the sphere (therefore is a constant). Up to constant ∂W/∂x3 factors, Z µ ¶ dx dz x dz Ω = 1 = d 1 x2 z z

If we further integrate out x1, we are left with the diﬀerence of x1dz/z for a pair of roots of (10). − + Finally we integrate along [zi , zi ] and obtain the periods of Ω. The total eﬀect is to integrate 7Taking the rigid limit on the CY geometry is slightly tricky. The situation is more clear from the dual 1 −s heterotic picture, where the size of the P is determined by the heterotic string dilaton ys ∼ e , as proposed in

[15]. From the leading order running of gauge coupling constant, ys is traded to the dynamically generated scale 0 2 n 0 n/2 by ys = (α Λ ) (for SU(n) gauge group). The rigid limit amounts to taking ² = (α ) → 0 and keep Λ ﬁxed. For a more detailed discussion see [16]. 8 ALE Although WD,E are diﬀerent from the characteristic polynomials, it can be shown that they lead to physically equivalent answers.

17 − + Figure 1: The SW curve projects to a double-sheet cover of the z-plane, branched along [zi , zi ] (with one branch cut displayed). A 3-cycle in the CY is a ﬁbration of homology 2-spheres over the z-plane. The spheres degenerates at the branch points z−, z+, corresponding to singular ALE ﬁbers. The period of the holomorphic 3-form Ω over the 3-cycle can be reduced to the integral of a 1-form λ along the branch cut: from z− to z+ on the upper sheet, and from z+ back to z− on the lower sheet.

− + + − 9 x1dz/z from zi to zi on one sheet, and from zi back to zi on the other sheet (with different values of x1). This path lifts to a 1-cycle in the Seiberg-Witten curve. The reduced form x dz λ = 1 z is indeed the Seiberg-Witten differential! It differs from (6) by an exact form. In fact, one can go ahead and think of the SW curve as the geometry where tensionless strings live on. This is extremely useful[16] in studying the BPS spectrum of the theory.

Acknowledgements

The author thanks Dan Jaﬀeris, Joe Marsano and Andy Neitzke for valuable conversations.

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