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

Bertrand Eynard (1) , Olivier Marchal (1, 2)

1 SPhT - Service de Physique Théorique
2 CRM - Centre de Recherches Mathématiques [Montréal]

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.

Mots clés: random matrices; algebraic geometry; Bethe ansatz; integrable

Domaines: Physique mathématique [math-ph]; Physique des Hautes Energies - Théorie [hep-th]

Bertrand Eynard, Olivier Marchal. Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry. 2008. ⟨hal-00323064⟩