Some Algebraic Perspectives on Conformal Field Theory

There will be very little actual physics in this talk. Rather, I'll discuss a not-atypical quest to extract some interesting mathematical ideas from the structure of what physicists may or may not want to do. From our limited vantage point, part of conformal field theory, as encoded within the theory of vertex operator algebras, gives a tool for constructing vector spaces associated with algebraic curves (i.e., Riemann surfaces) with marked points. As these curves and marking vary, these vector spaces also move in very well-behaved families, giving us, in fact, tools to study various moduli spaces (i.e., well-behaved parameter spaces) of marked curves.

In this talk, I'll discuss some perspectives from recent joint work with Chiara Damiolini and Angela Gibney, where we try to understand various aspects of this picture as our curves degenerate and acquire mild singularities. While this doesn't come up in the original physical picture, we find that various aspects of the structure seem to carry through in certain situations and seem to give new perspectives on the mathematical applications (and perhaps the physics?).