Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation

TL;DR

This paper calculates the one-loop genus one correction to the integrable hierarchy in 2D topological field theory, revealing its bihamiltonian structure and connecting it to elliptic Gromov-Witten invariants.

Contribution

It introduces the computation of genus one corrections and the central charge of the underlying W-algebra in the context of 2D topological field theories.

Findings

Bihamiltonian structure is described by a classical W-algebra.

Central charge of the W-algebra is computed.

Generating function of elliptic Gromov-Witten invariants is expressed via tau-function.

Abstract

We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov - Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity.