Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdVr equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank r− 1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity Ar−1. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of Gromov-Witten invariants and quantum cohomology. 0. Introduction The moduli space of stable curves of genus g with n marked points Mg,n is a fascinating object. Mumford [37] introduced tautological cohomology classes associated to the universal curve Cg,n Mg,n, and Witten conjectured [42] and Kontsevich [28] proved that the intersection numbers of certain tautological cohomology classes on the moduli space of stable curves Mg,n have a generating function which satisfies the equations of the Korteweg-de Vries hierarchy (more precisely that it is a τ function of the KdV hierarchy satisfying some additional equations). This remarkable result provided an unexpected link between the algebraic geometry of these moduli spaces and integrable systems. The spaces Mg,n can be generalized in two ways. The first is by choosing a smooth projective variety V and considering the moduli space of stable maps into V of curves of genus g with n marked points, Mg,n(V ). When V is a point, Mg,n(V ) reduces to Mg,n. The second is Date: March 6, 2008. Research of the first author was partially supported by NSA grant MDA904-991-0039. Research of the second author was partially supported by NSF grant number DMS-9803427. 1 2 T. J. JARVIS, T. KIMURA, AND A. VAINTROB by considering moduli space of r-spin curves M g,n , introduced in [21, 22], which, upon forgetting the r-spin structure, reduces to Mg,n. It is natural to ask if Kontsevich’s theorem admits a generalization to either of these cases. The case of Mg,n(V ) remains mysterious. It gives the GromovWitten invariants of V (and their so-called gravitational descendants), which assemble into a generating function whose exponential is an analog of a τ function. In the case where V is a point, one recovers the τ function of the KdV hierarchy by Kontsevich’s theorem. More generally, there is a conjecture of Eguchi-Hori-Xiong [12] and of S.Katz which essentially states that this generating function is a highest weight vector for a particular representation of the Virasoro algebra. Presumably, there is some analog of an integrable system which gives rise to this Virasoro algebra action, should the conjecture hold. On the other hand, the KdV hierarchy is known to be merely the first in a series of integrable hierarchies known as the KdVr (or r-th Gelfand-Dickey) hierarchies, where r = 2, 3, 4, . . . . In the case r = 2 this becomes the usual KdV hierarchy. Each of these hierarchies has a formal solution corresponding to the unique τ function which satisfies an equation known as the string (or puncture) equation. In [40, 41] Witten formulated a generalization of his original conjecture, which states that for each r ≥ 2 there should exist moduli spaces and cohomology classes on them whose intersection numbers assemble into the τ function of the KdVr hierarchy. In this paper, we are motivated by analogy with the construction of Gromov-Witten invariants from the moduli space of stable maps to introduce axioms which must be satisfied by a cohomology class c (called the virtual class) on the moduli space of r-spin curves M g,n in order to obtain a cohomological field theory (CohFT) of rank (r−1) in the sense of Kontsevich-Manin [29]. This virtual class on M g,n should be regarded as an analog of the Gromov-Witten classes of a variety V , i.e. the pullbacks via the evaluation maps of elements in H•(V ). We realize this virtual class in genus zero as the top Chern class of a tautological bundle over M 0,n associated to the r-spin structure. This yields a Frobenius manifold structure [10, 20, 33] on the state space of the CohFT which is isomorphic to the Frobenius manifold associated to the versal deformation of the Ar−1 singularity [10]. This is an indication of the existence of a kind of “mirror symmetry” between the moduli space of r-spin curves and singularities. According to Manin [34] “isomorphisms of Frobenius manifolds of different HIGHER SPIN CURVES AND INTEGRABLE HIERARCHIES 3 classes remain the most direct expression of various mirror phenomena.” Proving the generalized Witten conjecture for all genera would provide further evidence of this relationship. As in the case of Gromov-Witten invariants, one can construct a potential function from the integrals of c on different components of M g,n to form the small phase space of the theory. The large phase space is constructed by introducing the tautological classes ψ associated to the universal curve C g,n M g,n . It can be regarded as a parameter space for a family of CohFTs. A very large phase space (see [25, 26, 13, 35]) parametrizing an even larger space of CohFTs is obtained by considering tautological classes λ, associated to the Hodge bundles, and tautological classes μ, associated to the universal spin structure. We show that the corresponding potential function satisfies analogs of the puncture and dilaton equations and also a new differential equation obtained from a universal relation involving the class μ1. These relations hold in all genera. Topological recursion relations are also obtained from presentations of these classes in terms of boundary classes in low genera. Finally, using the new relation on μ1, we show that the genus zero part of the large phase space potential Φ0(t) is completely determined by the geometry, and this potential agrees with the generalized Witten conjecture in genus zero. Some of our constructions were foreshadowed by Witten, who formulated his conjecture even before the relevant moduli spaces and cohomology classes had been constructed, just as he had done in the case of the topological sigma model and quantum cohomology. We prove that his conjecture has a precise algebro-geometric foundation, just as in the case of Gromov-Witten theory. Witten also outlined a formal argument to justify his conjecture in genus zero. Our work shows that the formulas that he ultimately obtained for the large phase space potential function in genus zero are indeed correct, provided that the geometric objects involved are suitably interpreted. This is nontrivial even in genus zero because of the stacky nature of the underlying moduli spaces. We proceed further to prove relations between various tautological classes associated to the r-spin structures and derive differential equations for the potential function associated to them. Notice also that one can introduce moduli spaces M g,n(V ) of stable r-spin maps into a variety V , where one combines the data of both the stable maps and the r-spin structures. The analogous construction 4 T. J. JARVIS, T. KIMURA, AND A. VAINTROB on these spaces yields a Frobenius manifold which combines GromovWitten invariants (and quantum cohomology) with the KdVr hierarchies. Work in this direction is in progress [24]. In the first section of this paper, we review the moduli space of genus g, stable r-spin curves, M g,n , which was introduced in [21, 22]. We also discuss the stratification of the boundary of M g,n . The boundary strata fall into two distinct categories—the so-called Neveu-Schwarz and Ramond types. Only the former has the factorization properties analogous to properties of the moduli space of stable maps. In the second section, we introduce canonical morphisms, tautological bundles, tautological cohomology classes, and cohomology classes associated to the boundary strata of M g,n and derive a new relation involving the μ1 class. In the third section, we define a cohomological field theory (CohFT) in the sense of Kontsevich-Manin, its small phase space potential function, and the associativity (WDVV) equation. We then review the construction of Gromov-Witten invariants for the moduli space of stable maps and define the large and very large phase spaces in the Gromov-Witten theory. Motivated by this example, we explain how to construct a CohFT and the various potential functions from analogous intersection numbers on M g,n , assuming that the virtual class c exists. In the fourth section, we state axioms which c must satisfy in order to obtain a CohFT. We show that these axioms give a complete CohFT with a flat identity, and we construct the class c in genus zero as well as in the case r = 2. In the fifth section, we obtain analogs of the string and dilaton equations (which essentially yield the L−1 and L0 Virasoro generators) for this r-spin CohFT, and a find a new equation based on the relation involving the μ1 class. We also prove the analog of topological recursion relations in genus zero. Finally in the sixth section, we determine the large phase space potential of this theory in genus zero and show that it yields a solution to (the semiclassical limit of) the KdVr hierarchy, thereby proving the genus zero generalized Witten conjecture. We conclude with our own W -algebra conjecture, a KdVr analog of a refinement of the Virasoro conjecture [12]. Acknowledgments . We would like to thank D. Abramovich, M. Adler, B. Dubrovin, J. Figueroa-O’Farrill, E. Getzler, A. Kabanov, Yu. Manin, P. van Moerbeke, A. Polishchuk, M. Rosellen, and J. Stasheff for useful exchanges. We would also like to thank Heidi Jarvis for help with typesetting this paper. A.V. thanks the Max-Planck-Institut für Mathematik in Bonn for hospitality and financial support. HIGHER SPIN CURVES AND INTEGRABLE HIERARCHIES 5 1. The Moduli Space of r-spin Curves 1.1. Definitions. Definition 1.1. Let (X, p1, . . . , pn) be a nodal, n-pointed algebraic curve, and let K be a rank-one, torsion-free sheaf on X. A d-th root of K of type m = (m1, . . . , mn) is a pair (E , b) of a rank-one, torsion-free sheaf E , and an OX -module homomorphism b : E⊗d K ⊗OX(− ∑ mipi) with the following properties: • d · deg E = degK −mi • b is an isomorphism on the locus of X where E is locally free • for every point p ∈ X where E is not free, the length of the cokernel of b at p is d− 1. For any d-th root (E , b) of type m, and for any m′ congruent to m mod d, we can construct a unique d-th root (E ′, b′) of type m′ simply by taking E ′ = E ⊗O(1/d(mi−m′i)pi). Consequently, the moduli of curves with d-th roots of a bundle K of typem is canonically isomorphic to the moduli of curves with d-th roots of type m′. Therefore, unless otherwise stated, we will always assume the type m of a d-th root has the property that 0 ≤ mi < d for all i. Unfortunately, the moduli space of curves with d-th roots of a fixed sheaf K is not smooth when d is not prime, and so we must consider not just roots of a bundle, but rather coherent nets of roots [21]. This additional structure suffices to make the moduli space of curves with a coherent net of roots smooth. Definition 1.2. Let K be a rank-one, torsion-free sheaf on a nodal n-pointed curve (X, p1, . . . , pn). A coherent net of r-th roots of K of type m = (m1, . . . , mn) consists of the following data: • For every divisor d of r, a rank-one torsion-free sheaf Ed on X; • For every pair of divisors d′, d of r, such that d′ divides d, an OX-module homomorphism cd,d′ : E ′ d Ed′ . These data are subject to the following restrictions. 1. E1 = K and c1,1 = 1 2. For each divisor d of r and each divisor d′ of d, letm′′ = (m′′ 1, . . . , m ′′ n) be such that m′′ i is the unique non-negative integer less than d/d ′, and congruent tomi mod d. Then the homomorphism cd,d′ makes (Ed, cd,d′) into a d/d′ root of Ed′ of type m′′. 6 T. J. JARVIS, T. KIMURA, AND A. VAINTROB 3. The homomorphisms {cd,d′} are compatible. That is, the diagram (E′ d )⊗d /d (cd,d′) ⊗d′/d′′ E′′′ d Ed′′ ? cd′,d′′ c d,d ′′