# Conformal blocks vector bundles on $\overline{M}_{g}$ in terms of generalized theta functions?

Conformal field theory uses representation theory to produce various vector bundles on the Deligne-Mumford compactified moduli spaces $\overline{M}_{g}$ and $\overline{M}_{g,n}$, known as bundles of conformal blocks or covacua. Several authors (Beauville, Laszlo, Sorger, Pauly, Faltings, Kumar, Narasimhan, Ramanathan) proved in the late 80s and early 90s that the fiber over a point in the interior (i.e., a smooth curve) is naturally identified with a certain space of generalized theta functions. At the time this was proven not much had been said about moduli of vector bundles over singular curves, so I wonder if any of the more recent treatments of moduli of G-bundles allow for an extension of this theorem to the Deligne-Mumford boundary.

For instance, the conformal blocks bundle on $\overline{M}_g$ corresponding to $\mathfrak{sl}_r$, level $l$, has fiber over $C\in M_g$ equal to $H^0(SU_C(r),\mathcal{L}^l)$, where $SU_C(r)$ is the moduli space of semistable rank $r$ vector bundles with trivial determinant on $C$ and $\mathcal{L}$ is the determinant line bundle. Is there a similar interpretation for fibers over $C\in\overline{M}_g\setminus M_g$?

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