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

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

2008-09-19

arxiv.org/abs/0809.3367v1

JHEP 0903:094,2009

Physics

Mathematical Physics

Latex, 59 pages

Scientific paper

10.1088/1126-6708/2009/03/094

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.

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