Title: The Combinatorics, Algebra, and Geometry of Conformal Blocks

 

Abstract: For any choice of smooth, marked projective curve and some representation-theoretic data, the  Wess-Zumino-Novikov-Witten (WZNW) model of conformal field theory produces a finite dimensional vector space called a space of conformal blocks.  As the marked curve is varied, the conformal blocks form a vector bundle over the space of curves; this gives rise to the so-called "fusion rules" of the WZNW theory.  I will explain how these rules allow us to count the dimensions of the space of conformal blocks using Ehrhart theory in a case associated to the Lie algebra sl_2.  Hiding behind this surprising connection are deep algebraic properties of the total coordinate ring of a closely related moduli space associated to a curve: it's spaces of parabolic SL_2 principal bundles.  An honorary appearance will be made by the chain of loops.