Witten's conjecture and Virasoro conjecture for genus up to two

Abstract

This is an expository paper based on the results in [12] and [16]. The main goal is to prove the following two conjectures for genus up to two. (1) Witten's conjecture on the relations between higher spin curves and Gelfand-Dickey Hierarchies. (2) Virasoro conjecture for conformal semisimple Frobenius manifolds.

submitted to proceedings of conference "Gromov-Witten Theory of Spin Curves and Orbifolds". Final version