Singularity theory and integrable hierarchies

A remarkable conjecture, suggested by E. Witten [W1] and proved by M. Kontsevich [Ko], is that the topology of moduli spaces of curves is governed by the KdV integrable hierarchy. More generally, let X be a compact Kahler manifold. Is there a deep relation between integrable hierarchies and the topology of the moduli spaces of stable maps from compact complex curves to X?; The question posed above inspired new approaches to the theory of integrable systems. One of them was developed by B. Dubrovin and Y. Zhang [DZ1], and is based on the theory of Frobenius manifolds. In particular, when applied to the Frobenius manifold corresponding to the quantum cohomology of a compact Kahler manifold X (with semi-simple quantum cup product), Dubrovin and Zhang's method should yield an hierarchy closely related to the Gromov-Witten theory of X. The only manifold for which the theory was applied is CP1 [DZ2]. In this way Dubrovin and Zhang obtained a proof of the so called Toda conjecture: the Gromov-Witten invariants of CP1 are governed by the flows of the ETH. However their proof relies also on two additional results: existence of tau-functions of the Extended Toda Hierarchy (ETH shortly) [CDZ] and Virasoro constraints for CP1 (proved in [G3]).; The Toda conjecture was formulated by T. Eguchi and S.-K. Yang [EY]. The first proof, modulo the Toda equation (proved by A. Okounkov and R. Pandharipande [OP1]) and the Virasoro constraints, was given by E. Getzler. Another approach to Toda conjecture is based on taking the non-equivariant limit of its equivariant version. The equivariant version was formulated and proved by A. Okounkov and R. Pandharipande [OP2]. However, taking the non-equivariant limit is not an easy task (and it is not written anywhere). One needs to use Virasoro constraints, otherwise half of the equations will be lost. Thus CP1 and the ETH are a test example for any theory which attempts to describe a relation between integrable hierarchies and Gromov-Witten theory.; The main motivation of this thesis is a recent work of A. Givental who, by introducing a certain vertex operators construction, obtained a new interpretation of the n-KdV hierarchy in terms of period mappings in singularity theory. Our goal is to test the ideas from singularity theory to the mirror model of CP1 . In particular we obtain a new proof of the Toda Conjecture. Our proof does not depend on Virasoro constraints and also it yields naturally Hirota Quadratic Equations for the ETH, which were unknown until now. The ideas involved in our argument are parallel to the ones in [G1] and [GM]. The only new feature, which might be useful in the future, is that we use vertex operators with coefficients in the algebra of differential operators on the affine line.