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" 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.

Latex, 59 pages

References in corpus (4)

• Remodeling the B-model
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity
• Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants

Cited by in corpus (9)

• Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems
• Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach
• Topological expansion of the Bethe ansatz, and quantum algebraic geometry
• Uncertainty Quantification in Lattice QCD Calculations for Nuclear Physics
• Quantum Hitchin Systems via beta-deformed Matrix Models
• Generalized matrix models and AGT correspondence at all genera
• Large N limit of beta-ensembles and deformed Seiberg-Witten relations
• Seiberg-Witten equations and non-commutative spectral curves in Liouville theory
• Elements of proof for conjectures of Witte and Forrester about the combinatorial structure of Gaussian Beta Ensembles