Twisted conformal blocks and their dimension

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.