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 K¨ahlermanifolds. 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 K¨ahler. If the chiral multiplets are coupled to N = 1 supergravity, then M is further required to be a Hodge K¨ahlermanifold. For rigid N = 2 theories, the target space MV for vector multiplets is restricted to be a rigid special K¨ahlermanifold, whose K¨ahlerpotential is determined by a holomorphic prepotential F; the target space for hypermultiplets MH , on the other hand, is a hyper-K¨ahlermanifold. 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 K¨ahler 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 ®0 ! 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 K¨ahlermanifolds 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 K¨ahlermoduli 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 K¨ahlermanifold M is a Hodge manifold if there exists a line bundle L! M whose first Chern class coincides with the K¨ahlerclass, 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. [!] 2 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 K¨ahlermanifold, 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 K¨ahlermanifold M is rigid special K¨ahler,if there is a section Ω 2 Γ(H;M) such that the K¨ahlerform is given by i J = @@¯hΩjΩi 2¼ together with the condition h@iΩj@jΩi = 0 where the inner product 0 1 0 ¡1 hΩjΩi = iΩy @ A 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 0 1 XA Ω = @ A FB A;B=1;¢¢¢;n The K¨ahlerpotential is therefore ³ ´ A A K(z; z¯) = hΩjΩi = i X F A ¡ X FA The non-degeneracy of the K¨ahlerform guarantees that locally XA’s span a set of coordinates on M, conventionally called the special coordinates. The integrability condition in special coor- dinates, h@AΩj@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 K¨ahlermanifold M is (local) special K¨ahler,if there is section Ω 2 Γ(H;M) such that the K¨ahlerform is given by i J = ¡ @@¯lnhΩjΩi 2¼ and satisfies hΩj@iΩi = 0 Note that the last condition is well-defined on sections of H: since DiΩ = @iΩ + (@iK)Ω and hΩjΩi = 0, we can replace @i by covariant derivative Di. Under a local trivialization, the section Ω takes the form 0 1 ZI Ω = @ A 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 = ¡ihΩj@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.