The Virasoro-like Algebra of a Frobenius Manifold

TL;DR

This paper introduces a new infinite-dimensional Lie algebra called the Virasoro-like algebra for Frobenius manifolds, leading to quadratic PDEs and Virasoro constraints for related partition functions.

Contribution

It constructs the Virasoro-like algebra for arbitrary Frobenius manifolds and derives associated PDEs and Virasoro constraints, extending the algebraic framework of Frobenius manifolds.

Findings

Defined the Virasoro-like algebra as a deformation of the Virasoro algebra.

Established quadratic PDEs satisfied by the genus-zero free energy.

Derived Virasoro constraints for the Hodge partition function under semisimplicity.

Abstract

For an arbitrary calibrated Frobenius manifold, we construct an infinite dimensional Lie algebra, called the Virasoro-like algebra, which is a deformation of the Virasoro algebra of the Frobenius manifold. By using the Virasoro-like algebra we give a family of quadratic PDEs that are satisfied by the genus-zero free energy of the Frobenius manifold. We also derive, under the semisimplicity assumption, the Virasoro constraints for the corresponding abstract Hodge partition function.