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

Abstract

In this article, we define a non-commutative deformation of the ''symplectic invariants'' (introduced in [13]) of an algebraic hyperelliptic 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 Bergman 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. © 2009 SISSA.