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

Bertrand Eynard (1) , Olivier Marchal (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681

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.

Domaines: Physique mathématique [math-ph]

Bertrand Eynard, Olivier Marchal. Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry. Journal of High Energy Physics, 2009, pp.JHEP03(2009)094. ⟨10.1088/1126-6708/2009/03/094⟩. ⟨hal-00863583⟩