Abstract
Let \(\Gamma \) be a finite group acting on a simple Lie algebra \({\mathfrak {g}}\) and acting on a s-pointed projective curve \((\Sigma , \vec {p}=\{p_1, \ldots , p_s\})\) faithfully (for \(s\ge 1\)). 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.
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Acknowledgements
We thank Constantin Teleman for some helpful correspondences and conversations. The first author was partially supported by the NSF grant DMS-2001365 and the second author was partially supported by the NSF Grant DMS-1802328.
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Hong, J., Kumar, S. Twisted conformal blocks and their dimension. Math. Z. 306, 76 (2024). https://doi.org/10.1007/s00209-024-03461-4
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DOI: https://doi.org/10.1007/s00209-024-03461-4