# 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 $C\in\overline{M}_g\setminus M_g$?

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.

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.