Toric degenerations of the algebra of conformal blocks and compactifications of character varieties

Abstract: The algebra of conformal blocks is a commutative ring built from the spaces of conformal blocks of the Wess-Zumino-Novikov-Witten model of conformal field theory attached to a smooth projective curve and a simple Lie algebra. For algebraic geometers, these rings emerge naturally as the total coordinate rings of the moduli of principal bundles on the curve. I'll describe a construction which realizes the algebras of conformal blocks for curves of fixed genus as the fibers of a flat family over the Deligne-Mumford compactification of the moduli of smooth curves. The algebras which appear over the singular curves in this construction then have a novel interpretation as the projective coordinate rings of certain Fano compactifications of another moduli space: the free group character variety. In the $SL_2$ case, I'll explain how the construction of so-called Newton-Okounkov bodies for free group character varieties can then be used to produce toric degenerations of algebras of conformal blocks.