Twisted conformal blocks and their dimension

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