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

Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry - Mathematical Physics - Download this document for free, or read online. Document in PDF available to download.

Abstract: In this article, we define a non-commutative deformation of the -symplecticinvariants- of an algebraic hyperelliptical plane curve. The necessarycondition for our definition to make sense is a Bethe ansatz. The commutativelimit reduces to the symplectic invariants, i.e. algebraic geometry, and thuswe define non-commutative deformations of some algebraic geometry quantities.In particular our non-commutative Bergmann kernel satisfies a Rauch variationalformula. Those non-commutative invariants are inspired from the large Nexpansion of formal non-hermitian matrix models. Thus they are expected to berelated to the enumeration problem of discrete non-orientable surfaces ofarbitrary topologies.

Author: ** Bertrand Eynard SPhT, Olivier Marchal SPhT, CRM**

Source: https://arxiv.org/