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 supersym-

metric theories. We review the definition of rigid and local special Kahler manifolds. For

rigid special geometry, we discuss their connection to N = 2 supersymmetric gauge theo-

ries 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 Kahler.

If the chiral multiplets are coupled to N = 1 supergravity, then M is further required to be a

Hodge Kahler manifold. For rigid N = 2 theories, the target space MV for vector multiplets

is restricted to be a rigid special Kahler manifold, whose Kahler potential is determined by

a holomorphic prepotential F ; the target space for hypermultiplets MH , on the other hand,

is a hyper-Kahler manifold. When coupled to N = 2 supergravity, MV is restricted to be a

(local) special Kahler manifold, and MH becomes a quarternionic Kahler manifold. 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 Kahler

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

α′ → 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 Kahler manifolds

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 Kahler moduli 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 Kahler manifolds

A Kahler manifold M is a Hodge manifold if there exists a line bundle L → M whose first Chern

class coincides with the Kahler class, 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 Kahler potential 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 Kahler manifold as those whose Kahler class [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 Kahler manifold, 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 Kahler manifold M is rigid special Kahler, if there is a section Ω ∈ Γ(H,M)

such that the Kahler form is given by

J =i

2π∂∂〈Ω|Ω〉

together with the condition

〈∂iΩ|∂jΩ〉 = 0

where the inner product

〈Ω|Ω〉 = iΩ† 0 −1

1 0

Ω

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

Ω =

XA

FB

A,B=1,···,n

The Kahler potential is therefore

K(z, z) = 〈Ω|Ω〉 = i(XAFA −X

AFA

)

The non-degeneracy of the Kahler form guarantees that locally XA’s span a set of coordinates

on M , conventionally called the special coordinates. The integrability condition in special coor-

dinates, 〈∂AΩ|∂BΩ〉 = i(∂BFA − ∂AFB) = 0, implies that locally

FA =∂

∂XAF (X)

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

NAB =∂

∂XBFA = ∂A∂BF

Now suppose M be an n-dimensional Hodge Kahler manifold, 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 Kahler manifold M is (local) special Kahler, if there is section Ω ∈ Γ(H,M)

such that the Kahler form is given by

J = − i

2π∂∂ ln〈Ω|Ω〉

and satisfies

〈Ω|∂iΩ〉 = 0

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

DiΩ = ∂iΩ + (∂iK)Ω

and 〈Ω|Ω〉 = 0, we can replace ∂i by covariant derivative Di.

Under a local trivialization, the section Ω takes the form

Ω =

ZI

FJ

I,J=0,1,···,n

The Kahler potential is therefore

K = − ln i(ZIF I − Z

IFI

)

Under Kahler transformation K → K + f + f , the holomorphic section Ω transformation as

Ω → e−fΩ. So ZI ’s can be regarded as a set of homogeneous coordinates on M . Suppose

ti =Zi

Z0

form a local coordinate system. Then from the condition

0 = −i〈Ω|∂iΩ〉 = ZI∂iFI − ∂iZIFI

it is not hard to show2 that (locally) up to a symplectic transformation, we can write

FI(Z) =∂

∂ZIF (Z)

where the prepotential F(Z) is holomorphic and of homogeneous degree 2 in ZI ’s.2Write FI = Z0FI(t

i), then the constraint gives ∂iF0 − Fi + tj∂iFj = 0. Taking its derivative with respect to

tj , we get (a) ∂iFj = ∂jFi ⇒ locally Fi = Z0∂iF for some F(t); (b) ∂i∂jF0 + tk∂i∂jFk = 0 ⇒ ∂i∂j(F0 − 2Z0F +

Z0tk∂kF) = 0. Up to a symplectic transformation, F0 = Z0(2F − tk∂kF). So F (Z) = (Z0)2F(Zi/Z0) is the

desired prepotential. In terms of F(t), the Kahler potential is given by

K(t, t) = − ln i[2(F − F)− (∂iF + ∂iF)(ti − ti)

]

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, θα 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

DαΨ = DαΨ = 0

We can expand Ψ in N = 1 superspace as

Ψ = Φ(y, θ) + i√

2θαWα(y, θ) + θ2G(y, θ)

where

yµ = xµ + iθσµθ + iθσµθ

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

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),

14π

Im[∫

d4xd4θTrF(Ψ)]

or in N = 1 language,

14π

Im

[∫d4θΦe2V ∂F

∂Φ+

∫d2θ

12

∂2F∂Φ2

WαWα

](2)

There cannot possibly be dynamically generated superpotential, as constrained by N = 2 su-

persymmetry. 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 Kahler manifold.

The gauge coupling matrix is given by the period matrix

NAB =∂2F

∂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

variable aAD = ∂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

F(a) =12τ0a

2 +i

πa2 ln

(a2

Λ2

)+

12πi

a2∞∑

l=1

cl

(Λa

)4l

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 coupling14π

Im∫

d4xd4θVDDW = − 14π

Im∫

d4xd2θWDW

Simply integrating out W does the job.

Interestingly, the structure of rigid special geometry also arises in the moduli space of Rie-

mann surfaces. A genus g Riemann surface has 1-cycles αA, βB, A,B = 1, · · · , g, with intersection

form

αA · αB = βA · βB = 0

αA · βB = −βB · αA = δAB

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 inde-

pendent holomorphic 1-forms ωi, i = 1, · · · , g on the curve. Their periods form a symplectic

vector (∫

αAωi,

∫

βB

ωi

)

6

We have the basic relations[4] (Riemann’s first and second relation)

∑A

(∫αA ωi ·

∫βA

ωj −∫βA

ωi ·∫αA ωj

)=

∫

Σωi ∧ ωj = 0

∑A

(∫αA ωi ·

∫βA

ωj −∫βA

ωi ·∫αA ωj

)=

∫

Σωi ∧ ωj (3)

If we define the period matrix ΠAB by∫

βA

ωi =∑

B

ΠAB

∫

αBωi

Then it follows from (3) that

ΠAB = ΠBA and ImΠ > 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)∫

αAωi =

∂

∂ui

∫

αAλ,

∫

βB

ωi =∂

∂ui

∫

βB

λ

The relation (4) then implies that locally there exists a prepotential F such that

aA =∫

αAλ,

∂F(a)∂aA

=∫

βA

λ

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

WAn−1(x, ui) ≡ xn −n−1∑

i=1

ui(a)xn−i−1 = 〈detn(xI− φ)〉

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 cou-

pling. 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

y2 = W 2An−1

(x, ui)− Λ2n (5)

and correspondingly, the Seiberg-Witten differential is

λ =1

2√

2πW ′

An−1

xdx

y(6)

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 super-

gravity, 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

generators V ij and A of SU(2)R×U(1)R. Qi and Si 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

eaµ, bµ, ψi

µ, Aµ,Vµij , T

−ab, χ

i, 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

XI =(XI , ΩI

i ,WIµ , Y I

ij

), I = 0, · · · , n

3They are related to the ordinary supercharges by

[Kµ, Qi] = γµSi, [Pµ, Si] =1

2γµQi

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 Poincare group, 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

disappear by gauge fixing SU(2)R and by eliminating auxiliary fields D,χi. 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 transfor-

mations 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

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

−i(XIF I −XIFI)R

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

−i(XIF I −XIFI) = 1

or equivalently |a|2 = eK, where

K(z, z) = − ln i(Z

IFI(Z)− ZIF 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

ZI → e−f(z)ZI , FI → e−f(z)FI

K(z, z) → K(z, z) + f(z) + f(z)

This is nothing but our old friend Kahler transformation. If we know that K is really the Kahler

potential on the scalar manifold M, then clearly M is special Kahler. In particular, (ZI , 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 Kahler form, so M is a Hodge manifold.

However, the scalar component of XI is not the component with smallest nonzero U(1)R

charge. The fermions Ωi has U(1)R charge −1/2, so they are sections of L1/2. This further

requires the Kahler class [J ] = 2c1(L1/2) to be an even element of the integer cohomology

group.

Under the Kahler transformation, XI and FI(X) transform as

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

So V = (XI , FI(X)) are sections of the U(1)-bundle U →M naturally associated to L. Let ∇be the connection on U , i.e.

∇iV =(

∂i +12∂iK

)V, ∇iV =

(∂i −

12∂iK

)V

Now the period matrix is defined by

FI(X) = NIJXJ , ∇iF I(X) = NIJ∇iXJ

The reader should be aware that: we haven’t shown why the scalar fields can be described

by a sigma model with Kahler potential 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

−NIJDµXIDµXJ, where DµXI = (∂µ + iAµ)XI

Aµ is the auxiliary gauge field for U(1)R. It is eliminated by the equation of motion

Aµ =i

2NIJ

(XI∂µX

J − ∂µXIXJ)

Since we have fixed dilatation, there is identity

NIJXIXJ = −1,

10

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

−gIJ∂µZI∂µZJ, gIJ = ∂I∂JK = NIJeK + ∂IK∂JK

The terms in the Lagrangian that contributes to the vector kinetic term are

i

4FIJF−I

ab F−Jab +i

8(F I − FIJX

J)F−Iab T−ab − 1

64NIJX

IX

JT−abT

−ab + c.c. (7)

where F+ab and F−

ab are the self-dual and anti-self-dual part of Fab, related by complex conjugation.

The auxiliary anti-self-dual tensor T−ab is eliminated by its equation of motion

T−ab = 4NIJX

JF−I

ab

XK

NKLXL

(8)

Using the identity

NIJ = F IJ + iNIKZKNJLZL

ZMNMNZN,

and plug (8) into (7), we arrive at the gauge kinetic term

i

4N IJF−I

ab F−Jab − i

4NIJF+I

ab F+Jab

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

5 Calabi-Yau Moduli Space as a Special Kahler Manifold

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.

The Hodge bundle H → M has fiber H3(X) of complex dimension b3 = 2(h2,1 + 1), where

h2,1 = dimC H2,1(X). A canonical hermitian metric on H is given by

〈α|β〉 = i

∫

Xα ∧ β

Poincare duality implies that H3(X) is self-dual, and there is a real basis Aa, Bb unique up to

symplectic transformations such that

〈Aa|Ab〉 = 〈Ba|Bb〉 = 0

〈Aa|Bb〉 = −〈Bb|Aa〉 = iδab

So we learned thatH 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 Kahler potential by

K(z, z) = − ln〈Ω|Ω〉

Under transformation Ω → ef(z)Ω, the Kahler potential 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 Kahler class with the first Chern class c1(L). It has

been proved by Tian that the Kahler class is an even element in the cohomology group.

Let ta be a set of coordinates onM. 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

〈Ω|∂aΩ〉 = 0

From section 2 we see that M is special Kahler. It is easy to determine Ka

Ka = −〈∂aΩ|Ω〉〈Ω|Ω〉 = ∂aK

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

Ga = DaΩ

The metric on M is thus given by

gab = ∂a∂bK = −eK〈Ga|Gb〉

This is positive definite by Riemann’s bilinear relations. The prepotential is determined by

Za =∫

AaΩ, Fa ≡ ∂F

∂Za=

∫

Ba

Ω

Similarly, the complexified Kahler structure moduli space is also special Kahler. There is

Kahler potential

K = − ln∫

XJ ∧ J ∧ J

12

where J is the Kahler form. 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 Kahler moduli. Let

eA, A = 1, · · · , h1,1 be a basis for H2(X,Z), and write the complexified Kahler form as

B + iJ = wAeA

Then the prepotential for Kahler moduli can be expressed as

F = − 13!

wAwBwC

w0

∫

XeA ∧ eB ∧ eC

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 Kahler moduli. In the context of type IIA theory the identifications are reversed. The

scalar manifold for the vector multiplets is special Kahler, 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 Kahler, 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 Kahler moduli, 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

12Ωij

lδglkdxi ∧ dxj ∧ dxk

to be harmonic4.4It is also equivalent to that δgij satisfies the Lichnerowicz equation

∇k∇kδgij + 2Rik

jlδgkl = 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,

ds2 =1V

∫

Md6x

√det g 2gikgjl

[δgijδgkl + (δgilδgjk + δBilδBjk)

](9)

Let ta, 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

Gaijk = ∂agklΩlij

Or inversely,

δgij =1

2||Ω||2 Ωkli Gakljδt

a

where ||Ω||2 = 13!ΩijkΩ

ijk is constant since Ω is harmonic. Plugging this into (9), after a

straightforward calculation, we obtain the expression for the metric on MC ,

Gab =∫M Ga ∧Gb∫M Ω ∧ Ω

This is precisely the Weil-Peterson metric we’ve seen in the last section. Similar conclusion also

holds for the Kahler moduli.

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, Qi, i = 1, · · · , h. The Kahler

potential is some general function K(Qi, Qi, Φa, c.c.). Denote the scalar components of Φa, Qi, Qi

by φa, qi, qi, the gaugino by λ, and ψq the hypermultiplet fermion. Then the kinetic terms for

the scalars involves a cross term

∂i∂aK ∂µqi∂µφa† + c.c.

N = 2 supersymmetry consequently requires a term

∂i∂aK ψqi/∂λa + c.c.

Now one directly observes that to cancel its N = 1 supersymmetry variation, one needs a term

involving two derivatives, Aaµ, and scalars, which cannot possibly be Lorentz invariant. This

implies that ∂i∂aK = 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 Kahler moduli. 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 Kahler modulous is effectively the sigma model coupling, which governs world-

sheet corrections, and the dilaton eΦ ∼ 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 Kahler moduli or the dilaton, hence doesn’t received any quantum corrections.

In other words, the tree level result

K = ln(

i

∫

MΩ ∧ Ω

)

is exact in the full string theory!

We can also say something about the Kahler moduli in type IIA theory5. Since they cannot

be coupled to the dilaton, they do not received string loop corrections. Let TA = wA/w0 be

the Kahler moduli. In world-sheet perturbation theory, there is Peccei-Quinn (PQ) symmetry,

which is an analog of the shift in theta angle,

TA → TA + iεA

As long as the image of the world-sheet in X has trivial homology class, the PQ symmetry holds.

The Kahler potential will be invariant provided

δF = cABXAXB

for real coefficients cAB’s. This determines the prepotential F to be of the form

F =dABCwAwBwC

w0+ iλ(w0)2

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

order exp(−nATA/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

Calabi-Yau 3-folds X3 (or its mirror X3). If X3 is a K3 fibration over P1, 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.

It is determined by the periods of the holomorphic 3-form Ω on a symplectic basis (αI , βJ) of

3-cycles,

XI =∫

αI

Ω, FJ =∫

βJΩ

If X3 is defined by a polynomial WX3in a weighted projective space, the holomorphic 3-form

can be explicitly written as

Ω =∫

γ

1WX3

5∑

i=1

(−1)ixidx1 ∧ · · · dxi · · · ∧ dx5

where γ is a curve winding around the hypersurface WX3= 0. The integrand is a closed form,

so this integral is well-defined regardless the choice of γ.

The charged objects under XI , FJ are D3-branes wrapped on 3-cycle ν = MIαI + NJβJ ,

with central charge

Z =∫

νΩ

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 strings6For 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

WX3(xi, z; uk) = ε

[z +

Λ2n

z+ 2WALE

ADE(xi, uk)

]+O(ε2)

where WALEADE are the defining equations of ALE spaces, or “simple singularities”, associated with

ADE gauge groups. They are of the form

WALEAn−1

= WAn−1(x1, uk) + x22 + x2

3

WALED,E = WD,E(x1, x2, uk) + x2

3

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,

X1 : z +Λ2n

z+ 2WAn−1(x1, uk) = 0 (10)

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

[z−i , z+i ]. 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

Ω =dz

z∧ dx1 ∧ dx2

∂W/∂x3

We want to integrate Ω over the 3-cycles to obtain the periods. It is easy to integrate out x2,

since∫ dx2

∂W/∂x3doesn’t depend on the size of the sphere (therefore is a constant). Up to constant

factors, ∫

x2

Ω =dx1dz

z= d

(x1dz

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 [z−i , z+i ] 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

heterotic picture, where the size of the P1 is determined by the heterotic string dilaton ys ∼ e−s, as proposed in

[15]. From the leading order running of gauge coupling constant, ys is traded to the dynamically generated scale

by ys = (α′Λ2)n (for SU(n) gauge group). The rigid limit amounts to taking ε = (α′)n/2 → 0 and keep Λ fixed.

For a more detailed discussion see [16].8Although W ALE

D,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 [z−i , z+i ] (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.

x1dz/z from z−i to z+i on one sheet, and from z+

i back to z−i on the other sheet9 (with different

values of x1). This path lifts to a 1-cycle in the Seiberg-Witten curve. The reduced form

λ =x1dz

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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20