Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry

TL;DR

This paper introduces a non-commutative deformation of symplectic invariants for algebraic curves, linking algebraic geometry with non-commutative algebra and matrix models, and exploring their topological and enumerative implications.

Contribution

It defines a novel non-commutative deformation of symplectic invariants, extending algebraic geometry concepts into non-commutative settings inspired by matrix models.

Findings

Non-commutative Bergmann kernel satisfies Rauch variational formula.

Deformation reduces to classical symplectic invariants in the commutative limit.

Potential connection to enumeration of non-orientable surfaces.

Abstract

In this article, we define a non-commutative deformation of the "symplectic invariants" of an algebraic hyperelliptical plane curve. The necessary condition for our definition to make sense is a Bethe ansatz. The commutative limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus we define non-commutative deformations of some algebraic geometry quantities. In particular our non-commutative Bergmann kernel satisfies a Rauch variational formula. Those non-commutative invariants are inspired from the large N expansion of formal non-hermitian matrix models. Thus they are expected to be related to the enumeration problem of discrete non-orientable surfaces of arbitrary topologies.