Towards Lax formulation of integrable hierarchies of topological type

TL;DR

This paper develops a Lax formulation for deformed integrable hierarchies associated with topological field theories, using Givental group actions, and compares Hamiltonians to previous results, opening avenues for broader extensions.

Contribution

It introduces an explicit Lax formulation for Givental-deformed hierarchies of topological type and compares the resulting Hamiltonians with existing formulas.

Findings

Deformed hierarchy admits an explicit Lax formulation.

Deformed Hamiltonians agree with previous formulas.

Commentary on extending formulations over the Givental orbit.

Abstract

To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable hierarchies. We consider the Hirota and Lax formulations of the deformation of the hierarchy of N copies of KdV obtained by an infinitesimal action of the Givental group. By first deforming the Hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo-differential operators, we show that such deformed hierarchy admits an explicit Lax formulation. We then compare the deformed Hamiltonians obtained from the Lax equations with the analogous formulas obtained in [1, 2], to find that they agree. We finally comment on the possibility of extending the Hirota and Lax formulation on the whole orbit of the Givental group action.