hep-th/0607100 Topological Strings and 6 (Almost) Modular Forms 0 0 2 l u J Mina Aganagic,1 Vincent Bouchard,2 Albrecht Klemm,3 7 1 1 1 University of California, Berkeley, CA 94720, USA v 0 2 Mathematical Sciences Research Institute, Berkeley, CA 94720, USA 0 1 3 University of Wisconsin, Madison, WI 53706, USA 7 0 6 0 / h t - p e Abstract h : v The B-model topological string theory on a Calabi-Yau threefold X has a symmetry i X groupΓ, generated by monodromiesoftheperiodsofX. Thisactsonthetopologicalstring r a wave function in a natural way, governed by the quantum mechanics of the phase space H3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi- Yaumanifolds giving rise to Seiberg-Witten gauge theories infour dimensions and localIP 2 and IP IP . Asa byproduct, wealso obtaina simple way ofrelating the topologicalstring 1 1 × amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifoldC3/ZZ . 3 July 2006 1. Introduction Topological string theory has led to many insights in both physics and mathematics. Physically, it computes non-perturbative F-terms of effective supersymmetric gauge and gravitytheoriesinstringcompactifications. Moreover, many dualitiesofsuperstring theory are better understood in terms of topological strings. Mathematically, the A-model ex- ploresthe symplecticgeometry andcan bewritteninterms ofGromov-Witten, Donaldson- Thomas or Gopakumar-Vafa invariants, while the mirror B-model depends on the complex structure deformations and usually provides a more effective tool for calculations. The topological string is well understood for non-compact toric Calabi-Yau manifolds. For example, the B-model on all non-compact toric Calabi-Yau manifolds was solved to all genera in [1] using the W symmetries of the theory. Geometrically, the W symmetries ∞ ∞ are the ω-preserving diffeomorphisms of the Calabi-Yau manifold, where ω is the (3,0) holomorphic volume form. By contrast, for compact Calabi-Yau manifolds the genus ex- pansion of the topological string is much harder to compute and so far only known up to genus four in certain cases, for instance for the quintic Calabi-Yau threefold. It is natural to think that understanding quantum symmetries of the theory may hold the key in the compact case as well. In this paper, we will not deal with the full diffeomorphism group, but we will ask how does the finite subgroup Γ of large, ω-preserving diffeomorphisms, constrain the am- plitudes. In other words, we ask: what can we learn from the study of the group of symmetries Γ generated by monodromies of the periods of the Calabi-Yau? To answer this question we need to know how Γ acts in the quantum theory. This is simple for the following reason. One of the most remarkable facts about the toplogical string partition function Z = exp( g2g−2 ) of a Calabi-Yau manifold X is that it is a wave function g s Fg of a quantum mechanical system where H (X) acts as a phase space and g2 plays the P 3 s role of h¯. Classically, Γ acts on H (X) as a discrete subgroup of the group Sp(2n,ZZ) of 3 symmetries that preserve the symplectic form, where n = 1b (X). This has a natural lift 2 3 to the quantum theory. The answer to the question turns out to be both beautiful and natural. Namely, the ’s turn out to be (almost) modular forms of Γ. By “(almost) modular form” we mean g F one of two things: a form which is holomorphic, but quasi-modular (i.e. it transforms with shifts), or a form which is modular, but not quite holomorphic. By studying monodromy transformations of the topological string partition function in “real polarization”, where 1 Z is a holomorphic function on the moduli space, we find that it is a quasi-modular form of Γ of weight 0. Moreover, the symmetry transformations under Γ imply that the genus g partition function is fixed recursively in terms of lower genus data, up to g F the addition of a holomorphic modular form. The latter is in turn fixed (at least in principle) by its behavior at the boundaries on the moduli space. On the other hand, if we consider the topological string partition function in “holomorphic polarization”, which is not holomorphic on the moduli space, this turns out to be a modular form of weight 0. While it fails to be holomorphic, it turns out to be “almost holomorphic” in a precise sense. Moreover, it is again entirely fixed recursively, up to an holomorphic modular form. Thus, the price to pay for insisting on holomorphicity is that the ’s fail to be precisely g F modular, and the price of modularity is failure of holomorphicity! The recursive relations we obtain contain exactly the same information as what was extracted in [6] from the holomorphic anomaly equation. In [6], through a beautiful study of topological sigma models coupled to gravity, the authors extracted a set of equations that the partition functions satisfy, expressing an anomaly in holomorphicity of . g g F F The equations turn out to fix in terms of lower genus data, up to an holomorphic func- g F tion with a finite set of undetermined coefficients. Bringing forth the underlying symmetry of the theory makes certain key aspects of [6], in particular the laborious construction of propagators, transparent and appealing. Namely, the propagators are simply the “gener- ators” of (almost) modular forms, that is the analogues of the second Eisenstein series of SL(2,ZZ) and its non-holomorphic counterpart! That a reinterpretation of [6] in the lan- guage of (almost) modular forms should exist was anticipated by R. Dijkgraaf in [13]. For local Calabi-Yau manifolds, the relevant modular forms are Siegel modular forms. In the compact Calabi-Yau manifold case, our formalism seems to predict the existence of a new theory of modular forms of (subgroups of) Sp(2n,ZZ), defined on spaces with Lorentzian signature (instead of the usual Siegel upper half-space). The paper is structured as follows. In section 2, we describe the B-model topological string theory, from a wave function perspective, for both compact and non-compact target spaces. Insection3, wetakeafirst lookathowthetopologicalstringwavefunctionbehaves under the symmetry group Γ generated by the monodromies. Then, we give a more precise analysis of the resulting constraints on the wave function in section 4. We also explain the close relationship between the topological string amplitudes and (almost) modular forms in this section. In the remaining sections we give examples of our formalism: in section 5 we study SU(N) Seiberg-Witten theory, in section 6 local IP2 — where we also use the 2 wave function formalism to extract the Gromov-Witten invariants of the orbifold C3/ZZ , 3 and in section 7 local IP1 IP1. To conclude our work, in section 8 we present some open × questions, speculations and ideas for future research. Finally, in Appendix A we derive the local holomorphic anomaly equation for the wave function with non-compact target spaces, and Appendix B and C are devoted to a review of essential facts and conventions about modular forms, quasi-modular forms and Siegel modular forms. 2. B-model and the Quantum Geometry of H3(X,C) The B-model topological string on a Calabi-Yau manifold X is a theory of variations of complex structure of X. When X has a mirrorY, this is dual to the A-model topological string, which is the Gromov-Witten theory of Y. As is often the case, many properties of the theory become transparent when the moduli of X and Y are allowed to vary, and the global structure of the fibration of the theory over its moduli space is considered. This is quite hard to do in the A-model directly, but the mirror B-model is ideally suited for these types of questions. 2.1. Real Polarization Let us first recall the classical geometry of H3(X,C) = H3(X,ZZ) C. In the fol- ⊗ lowing, we will assume that X is a compact Calabi-Yau manifold, and later explain the modifications that ensue in the non-compact, local case. Choose a complex structure on X by picking a 3-form ω. Given a symplectic basis of H (X,ZZ), 3 AI B = δI, ∩ J J where I,J = 1,...n, andn = 1b (X), we canparameterize thedifferent choices ofcomplex 2 3 structures by the periods xI = ω, p = ω. I ZAI ZBI The periods are not independent, but satisfy the special geometry relation: ∂ p (x) = (x). (2.1) I ∂xIF0 As is well known, turns out to be the classical, genus zero, free energy of the topological 0 F strings on X. 3 In the above, we picked a symplectic basis of H . Different choices of symplectic basis 3 differ by Sp(2n,ZZ) transformations: p˜ = A Jp +B xJ I I J IJ (2.2) x˜I = CIJp +DI xJ J J where A B M = Sp(2n,ZZ). C D ∈ (cid:18) (cid:19) For future reference, note that the period matrix τ, defined by ∂ τ = p IJ ∂xJ I transforms as τ˜ = (Aτ +B)(Cτ +D)−1. (2.3) For a discrete subroup Γ Sp(2n,ZZ), the changes of basis can be undone by picking a ⊂ different 3-form ω. Conversely, we should identify the choices of complex structure that are related by changes of basis of H (X,ZZ). 3 Thus, the x’s can be viewed as coordinates on the Teichmuller space of X, on which T Γ acts as the mapping class group. Consequently, the space of truly inequivalent complex structures is = /Γ. M T Generically, the moduli space has singularities in complex codimension one, and Γ is M generated by monodromies around the singular loci. Since ω is defined up to multiplication by a non-zero complex number, this makes x’s and p’s sections of a line bundle over the L moduli space. Thus, the x’s are really projective coordinates on . T It is natural to think of H3(X,ZZ) as a classical phase space, with symplectic form dxI dp . I ∧ In the quantum theory xI and p become canonically conjugate operators J [p ,xJ] = g2δJ (2.4) I s I acting on the quantum wave function Z , where g2 plays the role of h¯. Then, | i s xI Z = Z(xI) h | i 4 describes the topological string partition function in the real polarization1 of H3(X). This has perturbative expansion ∞ Z(xI) = exp[ g2g−2 (xI)] s Fg g=0 X where is the genus g free energy of the topological string. Acting on Z(x), g F ∂ ∂ p Z(x) = g2 Z(x) ( ) Z(x). I s∂xI ∼ ∂xIF0 Note that due to (2.4), g is a section of , so that is a section of 2−2g. s g L F L The partition function Z implicitely depends on the choice of symplectic basis. Clas- sically, changes of basis (p,x) (p˜,x˜) which preserve the symplectic form are canonical → transformations of the phase space. For the transformation in (2.2), the corresponding generating function S(x,x˜) that satisfies dS = p dxI p˜ dx˜I I I − is given by 1 1 S(x,x˜) = (C−1D) xJxK +(C−1) xJx˜K (AC−1) x˜Jx˜K. (2.5) JK JK JK −2 − 2 This has an unambiguous lift to the quantum theory, with the wave function transforming as Z˜(x˜) = dx e−S(x,x˜)/gs2 Z(x). (2.6) Z More precisely, we should specify the contour used to define (2.6); however, as long as we work with the perturbative g2 expansion of Z(x), the choice of contour does not enter. To s make sense of (2.6) then, consider the saddle point expansion of the integral. Given x˜I, the saddle point of the integral xI = xI solves the classical special geometry cl relations that follow from (2.2) : ∂S = p (x ). ∂xI|xcl I cl Expanding around the saddle point, and putting xI = xI +yI, cl 1 For us, ω naturally lives in the complexification H3(X,C) =C⊗H3(X,IR), so “real” polar- ization is a bit of a misnomer. 5 we can compute the integral over y by summing Feynman diagrams where ∆ = (τ +C−1D) (2.7) IJ IJ − is the inverse propagator, and derivatives of , g F ∂ ...∂ (x ), (2.8) I1 InFg cl the vertices. As a short hand we summarize the saddle point expansion by ˜ = +Γ (∆IJ, ∂ ...∂ (x )) Fg Fg g I1 InFr<g cl where Γ (∆IJ, ∂ ...∂ (x )) is a functional that is determined by the Feynman g I1 InFr<g cl rules in terms of the lower genus vertices ∂ ...∂ (x ) for r < g and the propagator I1 InFr cl ∆IJ. The latter is related to the inverse propagator ∆ in (2.7) by ∆IJ∆ = δI . For IJ JK K example, at genus 1 the functional is simply 1 Γ (∆IJ) = logdet( ∆), 1 2 − where by ∆ we mean the propagator ∆IJ in matrix form. At genus two one has 1 1 Γ (∆IJ, ∂ ...∂ ) = ∆IJ ( ∂ ∂ + ∂ ∂ ) 2 I1 InFr<2 2 I JF1 2 IF1 JF1 1 1 + ∆IJ∆KL( ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ) I 1 J K L 0 I J K L 0 2 F F 8 F (2.9) 1 + ∆IJ∆KL∆MN( ∂ ∂ ∂ ∂ ∂ ∂ I J K 0 L M N 0 8 F F 1 + ∂ ∂ ∂ ∂ ∂ ∂ ), I K M 0 J L N 0 12 F F where we suppressed the argument x for clarity. cl It is easy to see from the path integral that this describes all possible degenerations of a Riemann surface of genus g to “stable” curves of lower genera, with ∆IJ being the corresponding contact term, as shown in the figure below. Stable here means that the conformal Killing vectors were removed by adding punctures, so that every genus zero component has at least three punctures, and every genus one curve, one puncture. 6 Fig. 1. Pictorial representation of the Feynman expansion at genus 2 in terms of degenerations of Riemann surfaces. Mirror symmetry and Gromov-Witten theory picks out the real polarization which is natural at large radius where instanton corrections are suppressed, and where the classical geometry makes sense. However, also by mirror symmetry, there is a larger family of topological A-model theories which exist, though they may not have an interpretation as counting curves. Foragenericelement M ofSp(2n,ZZ), (2.6)simplytakesonepolarizationintoanother. However, forM inthemapping classgroup Γ Sp(2n,ZZ), thetransformation(2.6)should ⊂ translate into a constraint on , since Γ is a group of symmetries of the theory. We will g F explore the consequences of this in the rest of this paper. 2.2. Holomorphic Polarization Instead of picking a symplectic basis of H (X) to parameterize the variations of com- 3 plex structure on X, we can choose a fixed background complex structure Ω H3(X,C), ∈ and use it to define the Hodge decomposition of H3(X,C): H3 = H3,0 H2,1 H1,2 H0,3. ⊕ ⊕ ⊕ Here Ω is the unique H3,0 form and the D Ω’s span the space of H2,1 forms, where i D = ∂ ∂ K and K is the K¨ahler potential K = log[i Ω Ω¯]. This implies that: i i − i X ∧ R ω = ϕΩ+ziD Ω+z¯iD¯ Ω¯ +ϕ¯Ω¯, (2.10) i i 7 where (ϕ,zi), and (ϕ¯,z¯i) become coordinates on the phase space.2 Correspondingly we can express Z as a wave function in holomorphic polarization | i zi,ϕ Z = Z(zi,ϕ). h | i The topologicalstring partitionfunction Z(zi,ϕ) depends on thechoice of background Ω, and this dependence is not holomorphic. This is the holomorphic anomaly of [6]. One way to see this is through geometric quantization of H3(X) in this polarization [36]. We will take a different route, and exhibit this by exploring the canonical transformation from real to holomorphic polarizations. Using special geometry relations it is easy to see that xI = ω = zI + c.c ZAI p = ω = τ zJ + c.c I IJ ZBI where we defined zI = ϕXI +ziD XI i in terms of XI = Ω, P = Ω, I ZAI ZBI and where ∂ τ = P . IJ ∂XI J From this it easily follows that dp dxI = (τ τ¯) dzI dz¯J I IJ ∧ − ∧ and hence the canonical transformation from (xI,p ) to (zI,z¯I) is generated by I dSˆ(x,z) = p dxI +(τ τ¯) z¯IdzJ I IJ − where 1 1 Sˆ(x,z) = τ¯ xIxJ +xI(τ τ¯) zJ zI(τ τ¯) zJ. IJ IJ IJ 2 − − 2 − 2 Since ω for us does not live in H3(X,IR), but rather in H3(X,C), ϕ¯ and z¯i are not honest complex conjugates of ϕ, zi. 8 In the quantum theory, this implies that the topological string partition function in the holomorphic polarization is related to that in real polarization by: Zˆ(z;t,t¯) = dx e−Sˆ(x,z)/gs2 Z(x) (2.11) Z where ti are local coordinates on the moduli space, parameterizing the choice of back- ground, i.e. XI = XI(t). Note that all the background dependence of Zˆ(z) comes from the kernel of Sˆ.3 From the above, it is easy to show that the dependence of Zˆ on the background can be captured by the holomorphic anomaly equation of [6]. Namely, differ- entiating the left and the right hand side of the above with respect to t¯we get ∂ 1 ∂2 ∂ Zˆ = [ g2C¯ jk +G zj ] Zˆ ∂t¯¯i 2 s ¯i ∂zi∂zj ¯ij ∂ϕ which is the equation (6.11) of [6].4 In the above equation, C is the amplitude at ijk genus zero with three punctures, G is the K¨ahler metric, G = ∂¯∂ K and C¯ jk = ¯ij ¯ij ¯i j ¯i e2KC¯ G¯jjGk¯k. ¯i¯jk¯ Without knowing the choice of contour implicit in defining the integral (2.11) above, we can make sense of the expression in perturbative expansion about a saddle point. For simplicity, let us pick ϕ = 1, zi = 0, so that zI = XI. The saddle point equation, which can be written as5 (τ¯(X) τ(x )) xJ +(τ(X) τ¯(X¯)) zJ = 0, − cl IJ cl − IJ has then a simple solution, xI = XI. cl Expanding around this solution, we can compute the integral by summing Feynman dia- grams where (τ(X) τ¯(X¯)) (2.12) IJ − − 3 In what follows, we will use hats to label quantities which are not holomorphic. 4 In fact there is an important subtelty here. Namely, the definition of Zˆ here and in [6] differ slightly,byafactorof(eϕ+zj∂jK)χ/24−1 innormalization,whereχistheEulercharacteristicofthe Calabi-Yau. Asaconsequence,ouroneloopholomorphicanomalyequationis∂¯i∂jFˆ1 = 12C¯¯ikℓCjkℓ while their equation has an extra factor (χ/24−1)G¯ij on the right hand side. 5 We used here the special geometry relation p = τ xJ. I IJ 9