Rank reduction of conformal blocks

TL;DR

This paper demonstrates that under certain conditions, the space of conformal blocks for a simple, simply-connected group can be decomposed into products of conformal blocks for lower-rank groups, revealing a rank reduction phenomenon.

Contribution

It establishes a geometric framework for rank reduction of conformal blocks when weights lie on specific faces or walls of the eigenvalue polytope, using moduli stacks and theta functions.

Findings

Conformal blocks decompose into products over lower-rank groups on certain faces.

Explicit isomorphisms of conformal blocks bundles are constructed.

The approach uses geometric methods and moduli stacks of parahoric bundles.

Abstract

Let $X$ be a smooth, pointed Riemann surface of genus zero, and $G$ a simple, simply-connected complex algebraic group. Associated to a finite number of weights of $G$ and a level is a vector space called the space of conformal blocks, and a vector bundle of conformal blocks over $\bar{\text{M}}_{0,n}$. We show that, assuming the weights are on a face of the multiplicative eigenvalue polytope, the space of conformal blocks is isomorphic to a product of conformal blocks over groups of lower rank. If the weights are on a degree zero wall, then we also show that there is an isomorphism of conformal blocks bundles, giving an explicit relation between the associated nef divisors. The methods of the proof are geometric, and use the identification of conformal blocks with spaces of generalized theta functions, and the moduli stacks of parahoric bundles recently studied by Balaji and Seshadri.