Orbifold conformal blocks and the stack of pointed \(G\)-covers (Q2496777)

It is by now well-understood that any CFT in two dimensions gives rise to a sheaf with a projectively flat connection (the Knizhnik-Zamolodchikov connection) on the moduli stack \({\mathcal M}_{g,n}\) of smooth pointed stable curves [\textit{E. Frenkel} and \textit{D.~Ben-Zvi}, Vertex algebras and algebraic curves. Mathematical Surveys and Monographs. 88. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0981.17022)]. This procedure can be described as follows: to a vertex algebra \(V\) (corresponding to a choice of CFT model), a collection \({\mathbb M}=(M_1,\dots,M_n)\) of \(V\)-modules and an \(n\)-pointed genus \(g\) curve \((X,{\mathbf p})\) it is associated the vector space of conformal blocks \(C_V(X,{\mathbf p},{\mathbb M})\). As \((X,{\mathbf p})\) varies in \({\mathcal M}_{g,n}\), the spaces of conformal blocks give a sheaf with a projectively flat connection. A convenient and elegant way of describing the \(D\)-module structure on the sheaf of conformal blocks given by the KZ connection is via the Beilinson-Bernstein localization functor associated to the transitive action of the Virasoro algebra on \({\mathcal M}_{g,n}\) (the so-called Virasoro uniformization) [\textit{E. Arbarello, C. De Concini, V. G. Kac} and \textit{C.~Procesi}, Commun. Math. Phys. 117, No. 1, 1--36 (1988; Zbl 0647.17010); \textit{A. A. Beilinson} and \textit{V.~V.~Shekhtman}, Commun. Math. Phys. 118, No. 4, 651--701 (1988; Zbl 0665.17010); \textit{M. L. Kontsevich}, Funct. Anal. Appl. 21, 156--157 (1987); translation from Funkts. Anal. Prilozh. 21, No. 2, 78--79 (1987; Zbl 0647.58012)]. Moreover, in certain cases, when the CFT is rational, the spaces \(C_V(X,{\mathbf p},{\mathbb M})\) are finite dimensional, the sheaf of conformal blocks is a vector bundle which extends to the Deligne-Mumford compactification \(\overline{\mathcal M}_{g,n}\), and the KZ connection extends to a connection with logarithmic singularities along the boundary divisor. In the joint work with \textit{E.~Frenkel} [ Adv. Math. 187, No. 1, 195--227 (2004; Zbl 1049.17024)], the author has extended the notion of conformal blocks to include the twisted modules coming from orbifolding CFTs. On the vertex algebra side one has an action of a finite group \(G\) of automorphisms of the vertex algebra \(V\), and twisted \(V\)-modules \(M^{[g]}\) for each conjugacy class \([g]\) in \(G\). On the smooth projective curves side, one has the notion of pointed \(G\)-cover: an \(n\)-pointed (not necessarily connected) smooth curve \((C,{\mathbf q})\) with an effective action of \(G\) such that the quotient map \(\pi\colon C\to C/G\simeq X\) makes \(C\) a \(G\)-principal bundle over \(X\setminus\{p_i\}\), and \(q_i\in\pi^{-1}(p_i)\). To these data it is attached the space of (twisted) conformal blocks \(C_V^G(C\to X,{\mathbf p},{\mathbf q},{\mathbb M})\); the \(i\)-th module \(M_i\) of the collection \({\mathbb M}\) is twisted by the monodromy generator of \(C\to X\) at the point \(q_i\). The collection \({\mathcal M}_{g,n}^G\) of smooth pointed \(G\)-covers has a structure of smooth Deligne-Mumford stack [\textit{D. Abramovich, A. Corti} and \textit{A. Vistoli}, Commun. Algebra 31, No. 8, 3547--3618 (2003; Zbl 1077.14034)]; [\textit{T. J. Jarvis, R. Kaufmann} and \textit{T. Kimura}, Compos. Math. 141, No. 4, 926--978 (2005; Zbl 1091.14014)], so it is a natural question to ask how the spaces of twisted conformal blocks \(C_V^G(C,X,{\mathbf p},{\mathbf q},{\mathbb M})\) vary as \((C\to X,{\mathbf p},{\mathbf q})\) moves in \({\mathcal M}_{g,n}^G\). This problem is investigated in the present paper. The author shows that, as in the untwisted case, the spaces of twisted conformal blocks define a sheaf with a projectively flat connection on \({\mathcal M}_{g,n}^G\). The key ingredient is the proof of the \(G\)-equivariant Virasoro uniformization of \({\mathcal M}_{g,n}^G\) and the construction of the corresponding Beilinson-Bernstein localization functor. The boundary behaviour of the \(D\)-module of twisted conformal blocks on the compactification \(\overline{\mathcal M}_{g,n}^G\) is not addressed in the paper. Yet, it is conjectured that a logarithmic extension of the KZ connection should be possible for Wess-Zumino-Witten models of positive integer level, when \(V=L_k({\mathfrak g})\), the integrable basic \(\hat{\mathfrak g}\)-module of level \(k\in{\mathbb Z}_+\), and the twisted modules are realized as integrable representations of appropriate twisted affine algebras. In the final part of the paper, an explicit formula is given for the KZ connection on the locus of pointed \(G\)-covers with a fixed based curve, i.e. along the fibers of the projection \({\mathcal M}^G_{g,n}\to {\mathcal M}_{g}\) sending \((C\to X,{\mathbf p},{\mathbf q})\) to \(X\).