Gromov--Witten Theory of CP¹ and Integrable Hierarchies

الأصل · EN

The ancestor Gromov--Witten invariants of a compact manifold X can be organized in a generating function called the total ancestor potential of X. In this paper, we construct Hirota Quadratic Equations (HQE shortly) for the total ancestor potential of P¹. The idea is to adopt the formalism developed in G1,GM to the mirror model of P¹. We hope that the ideas presented here can be generalized to other manifolds as well. As a corollary, using the twisted loop group formalism from G3, we obtain a new proof of the following version of the Toda conjecture: the total descendant potential of P¹ (known also as the partition function of the P¹ topological sigma model) is a tau-function of the Extended Toda Hierarchy.