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We propose that the Virasoro algebra controls quantum cohomologies of general Fano manifolds M (c 1 (M) ? 0) and determines their partition functions at all genera. We construct Virasoro operators in the case of complex projective spaces and show that they reproduce the results of Kontsevich-Manin, Getzler etc. on the genus-0,1 instanton numbers. We also construct Virasoro operators for a wider class of Fano varieties. The central charge of the algebra is equal to Ø(M ), the Euler characteristic of the manifold M . As is well known, the quantum cohomology of a (symplectic) manifold M is described by the topological oe-model with M being its target space [1]. The partition function of the topological oe-model is given by the sum over holomorphic maps from a Riemann surface \Sigma g to M . When the degree d of the map is zero (constant map), correlation functions of the oe-model reproduces the classical intersection numbers among homology cycles of M . When the degree is non-zero, h...

This paper outlines the construction of invariants of Hamiltonian group actions on symplectic manifolds. The invariants are derived from the solutions of a nonlinear rst order elliptic partial dierential equation involving the Cauchy-Riemann operator, the curvature, and the moment map (see (17) below). They are related to the Gromov invariants of the reduced spaces. Our motivation arises from the proof of the Atiyah-Floer conjecture in [17, 18, 19] which deals with the relation between holomorphic curves ! M S in the moduli space M S of at connections over a Riemann surface S and anti-self-dual instantons over the 4-manifold S. In [3] Atiyah and Bott interpret the space M S as a symplectic quotient of the space A S of connections on S by the action of the group G S of gauge transformations. A moment's thought shows that the various terms in the anti-self-duality equations over S (see equation (64) below) can be interpreted symplectically. Hence they should give rise to meaningful equations in a context where the space A S is replaced by a nite dimensional symplectic manifold M and the gauge group G S by a compact Lie group G with a Hamiltonian action on M . In this paper 2 we show how the resulting equations give rise to invariants of Hamiltonian group actions. The same adiabatic limit argument as in [19] then leads to a correspondence between these invariants and the Gromov{Witten invariants of the quotient M==G (Conjecture 3.6). This correspondence is the subject of the PhD thesis [27] of the second author. In Section 2 we review the relevant background material about Hamiltonian group actions, gauge theory, equivariant cohomology, and holomorphic curves in symplectic quotients. The heart of this paper is Section 3, where we discuss the equations and the...

We discuss local mirror symmetry for higher-genus curves. Specifically, we consider the topological string partition function from higher genus curves contained in a Fano surface within a Calabi-Yau. Our main example is the local P2 case. The Kodaira-Spencer theory of gravity, tailored to this local geometry, can be solved to compute this partition function. Then, using the results of Gopakumar and Vafa [1] and the local mirror map [2], the partition function can be rewritten in terms of expansion coefficients, which are found to be integers. We verify, through localization calculations in the A-model, many of these Gromov-Witten predictions. The integrality is a mystery, mathematically speaking. The asymptotic growth (with degree) of the invariants is analyzed. Some suggestions are made towards an enumerative interpretation, following the BPS-state description of Gopakumar and Vafa.

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Abstract. We prove that the algebraic Witten’s “top Chern class ” constructed in [9] satisfies the axioms for the spin virtual class formulated in [5]. This paper is a sequel to [9]. Its goal is to verify that the virtual top Chern class c1/r in the Chow group of the moduli space of higher spin curves M 1/r g,n, constructed in [9], satisfies all the axioms of spin virtual class formulated in [5]. Hence, according to [5], it gives rise to a cohomological field theory in the sense of Kontsevich-Manin [7]. As was observed in [9], the only non-trivial axioms that have to be checked for the class c1/r are two axioms that we call Vanishing axiom and Ramond factorization axiom. The first of them requires c 1/r to vanish on all the components of the moduli space M 1/r g,n, where one of the markings is equal to r − 1. The second demands vanishing of the push-forward of c 1/r restricted to the components of the moduli space corresponding to the so called Ramond sector, under some natural finite maps. Recall that the virtual top Chern class is a crucial ingredient in the generalized Witten’s conjecture formulated in [10], [11]. The original index-theoretic construction of this

We introduce a precise notion, in terms of some Schlessinger’s type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessinger and obstruction theories. The inverse mapping theorem holds for natural transformations of extended deformation functors and all such functors with finite dimensional tangent space are prorepresentable in the homotopy category. Finally we prove that the primary obstruction map induces a structure of graded Lie algebra on the tangent space. Mathematics Subject Classification (1991): 13D10, 14B10, 14D15.

There is exactly one straight line passing through any two given distinct points; there is exactly one quadratic curve on the complex projective plane

We discuss how the theory of quantum cohomology may be generalized to “gravitational quantum cohomology ” by studying topological σ-models coupled to twodimensional gravity. We first consider σ-models defined on a general Fano manifold M (manifold with a positive first Chern class) and derive new recursion relations for its two point functions. We then derive bi-Hamiltonian structures of the theories and show that they are completely integrable at least at the level of genus 0. We next consider the subspace of the phase space where only a marginal perturbation (with a parameter t) is turned on and construct Lax operators (superpotentials) L whose residue integrals reproduce correlation functions. In the case of M = CP N the Lax operator is given by L = Z1 +Z2 + · · ·+ZN +etZ −1 1 Z−1 2 · · · Z −1 N and agrees with the potential of the affine Toda theory of the AN type. We also obtain Lax

We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant version of the mirror theorem allows us to write down a hypergeometric series, which together with a mirror map allows one to compute the invariants to all orders, similar to the closed string model or the physics approach via mirror symmetry. In the noncompact example where the Calabi-Yau is K P 2, our results agree with physics predictions at genus zero obtained using mirror symmetry for open strings. At higher genera, our results satisfy strong integrality checks conjectured from physics.