The geometry of Casimir W-algebras

TL;DR

This paper explores the geometric structure of Casimir W-algebras associated with simply laced Lie algebras, linking conformal field theory, Fuchsian systems, and moduli spaces to understand correlation functions and conformal blocks.

Contribution

It introduces a geometric framework connecting W-algebra correlation functions with Fuchsian differential systems and their moduli, providing new insights into the structure of W-algebra conformal field theories.

Findings

Correlation functions expressed via Fuchsian system solutions

Construction of a bundle encoding conformal blocks

Identification of cycles with moduli parameters

Abstract

Let $\mathfrak{g}$ be a simply laced Lie algebra, $\widehat{\mathfrak{g}}_1$ the corresponding affine Lie algebra at level one, and $\mathcal{W}(\mathfrak{g})$ the corresponding Casimir W-algebra. We consider $\mathcal{W}(\mathfrak{g})$-symmetric conformal field theory on the Riemann sphere. To a number of $\mathcal{W}(\mathfrak{g})$-primary fields, we associate a Fuchsian differential system. We compute correlation functions of $\widehat{\mathfrak{g}}_1$-currents in terms of solutions of that system, and construct the bundle where these objects live. We argue that cycles on that bundle correspond to parameters of the conformal blocks of the W-algebra, equivalently to moduli of the Fuchsian system.