Symplectic invariants, Virasoro constraints and Givental decomposition

TL;DR

This paper demonstrates that symplectic invariants derived from spectral curves satisfy Virasoro constraints and can be decomposed into Givental formulas, revealing dualities between matrix models and connections to KP tau functions.

Contribution

It establishes Virasoro constraints for symplectic invariants and links their decompositions to Givental formulas, unifying different matrix model representations.

Findings

Symplectic invariants satisfy Virasoro constraints at poles and zeros of differential forms.

Duality between matrix models is explained via symplectic invariants and Givental decompositions.

Decomposition of symplectic invariants as Kontsevich integrals and hermitian matrix integrals.

Abstract

Following the works of Alexandrov, Mironov and Morozov, we show that the symplectic invariants of \cite{EOinvariants} built from a given spectral curve satisfy a set of Virasoro constraints associated to each pole of the differential form $ydx$ and each zero of $dx$ . We then show that they satisfy the same constraints as the partition function of the Matrix M-theory defined by Alexandrov, Mironov and Morozov. The duality between the different matrix models of this theory is made clear as a special case of dualities between symplectic invariants. Indeed, a symplectic invariant admits two decomposition: as a product of Kontsevich integrals on the one hand, and as a product of 1 hermitian matrix integral on the other hand. These two decompositions can be though of as Givental formulae for the KP tau functions.