Pushing around loop equations

Abstract: The term “loop equations” was coined by Migdal in the early 90's in the context of quark confinement to refer to certain sets of equations describing the propagation of test particles in quantum field theory.

They have since appeared many times at the boundary between mathematics and physics. From statistical spectroscopy to enumerative geometry via integrable hierarchies, they encode conformal features of various problems in an analytic way.

The prototypical situation where loop equations are valuable is that of matrix integrals, with applications ranging from all the aforementioned topics to number theory and even artificial neural networks.

A seemingly antipodal approach to matrix integrals is that of orthogonal polynomials, unveiling a subtle relationship between loop equations and the geometry of meromorphic connections. This bridge can be abstracted in terms of $W$-algebra constraints and we shall cross it back and forth.

mathematical physicsalgebraic geometryquantum algebrarings and algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Comments: Hybrid delivery (in person on University of Saskatchewan campus and via Zoom).

PIMS Geometry / Algebra / Physics (GAP) Seminar