Twisted conformal blocks and their dimension

TL;DR

This paper studies twisted conformal blocks associated with finite group actions on Lie algebras and curves, establishing isomorphisms, dimension reduction techniques, and formulas generalizing Verlinde and Kac-Walton results.

Contribution

It introduces a framework for analyzing twisted conformal blocks under group actions, deriving dimension formulas and extending classical results to twisted settings.

Findings

Isomorphism between twisted conformal blocks and quotient group actions

Dimension calculation reduces to simpler cases with 3 marked points

Derived Verlinde and Kac-Walton formulas for twisted conformal blocks

Abstract

Let $\Gamma$ be a finite group acting on a simple Lie algebra $\mathfrak{g}$ and acting on a $s$-pointed projective curve $(\Sigma, \vec{p}=\{p_1, \dots, p_s\})$ faithfully (for $s\geq 1$). Also, let an integrable highest weight module $\mathscr{H}_c(\lambda_i)$ of an appropriate twisted affine Lie algebra determined by the ramification at $p_i$ with a fixed central charge $c$ is attached to each $p_i$. We prove that the space of twisted conformal blocks attached to this data is isomorphic to the space associated to a quotient group of $\Gamma$ acting on $\mathfrak{g}$ by diagram automorphisms and acting on a quotient of $\Sigma$. Under some mild conditions on ramification types, we prove that calculating the dimension of twisted conformal blocks can be reduced to the situation when $\Gamma$ acts on $\mathfrak{g}$ by diagram automorphisms and covers of $\mathbb{P}^1$ with 3 marked points. Assuming a twisted analogue of Teleman's vanishing theorem of Lie algebra homology, we derive an analogue of the Kac-Walton formula and the Verlinde formula for general $\Gamma$-curves (with mild restrictions on ramification types). In particular, if the Lie algebra $\mathfrak{g}$ is not of type $D_4$, there are no restrictions on ramification types.