TITLE: Conformal blocks vector bundles on $\overline{M}_{g}$ in terms of generalized theta functions?
QUESTION [27 upvotes]: 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 $C\in\overline{M}_g\setminus M_g$?

REPLY [3 votes]: Under a certain natural compatibility assumption, there are counterexamples due to Belkale-Gibney-Kazanova. 
The assumption essentially says that at a point $[\mathrm{C}]$ of the moduli space $\overline{\mathscr{M}}_{g,n}$, an algebra obtained from the fiber at $[\mathrm{C}]$ of the conformal blocks bundle is isomorphic to the algebra of sections of a line bundle $\mathscr{L}$ over some variety. (This holds whenever $[\mathrm{C}]$ is in $\mathscr{M}_{g,n}$ if one takes $\mathscr{L}$ to be the theta line bundle over some moduli space of sheaves on $\mathrm{C}$, all determined by the input data (level, type of Lie algebra, etc.) defining the conformal blocks.) 
Consider the classical case $\overline{\mathscr{M}}_2$, $\mathfrak{sl}_2$, $\ell=1$. In case $\mathrm{C}$ is smooth, the moduli space of sheaves on $\mathrm{C}$ with trivial determinant is nothing but $\mathbf{P}^3$, and the theta line bundle is $\mathscr{O}(1)$ (theorem of Narasimhan and Ramanan). Belkale-Gibney-Kazanova show (example 4.2 in their paper) that if one takes $\mathrm{C}$ to be a curve with a separating node, one arrives at a counterexample by considering the first Chern class of the bundle of conformal blocks. 
A formula for the Chern character of the bundle of conformal blocks can be found in a paper of Marian-Oprea-Pandharipande-Pixton-Zvonkine.