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 $C\in\overline{M}_g\setminus M_g$?
Best Answer
I don't think that this has been worked out in full generality, but there are some partial results by Ivan Kausz. See for example his paper "A canonical decomposition of generalized theta functions on the moduli stack of Gieseker vector bundles". J. Algebraic Geom. 14 (2005), no. 3, 439–480.