Geometrization of the TUY/WZW/KZ connection (Q6571066)

This article provides a generalization of the results of \textit{Y. Laszlo} [J. Differ. Geom. 49, No. 3, 547--576 (1998; Zbl 0987.14027)] to the parabolic setting.\N\NMore concretely, let \(G\) be a simple, simply connected, and complex algebraic group and \(C\) a smooth projective complex curve. Consider the moduli space \(M_G(C)\) of semistable principal \(G\)-bundles over \(C\). A faithful linear representation \(\phi\) of \(G\) provides a natural line bundle on \(M_G(C)\) called the determinant of cohomology line bundle \(\mathrm{Det}\). This line bundle is a generalization of the theta divisor on the Jacobian of \(C\). The space of non-abelian (or generalized) theta functions of level \(\ell\) is the space of sections \(H^0(M_G(C), \mathrm{Det}^{\otimes \ell})\). By letting \(C\) vary on the moduli space of curves \(\mathcal{M}_{g,n}\), these spaces collect together to define the Friedan-Shenker bundle over \(\mathcal{M}_{g,n}\) with fiber \(H^0(M_G(C), \mathrm{Det}^{\otimes \ell})\). This bundle is known to be naturally isomorphic (up to scalars) to the bundle with fiber \(\mathcal{V}^{\dagger}_{0} (\hat{\mathfrak{g}}, \ell)\) of Wess-Zumino-Witten (WZW) conformal blocks of level \(\ell\), an instance of the Chern-Simons/WZW correspondence. Both these bundles carry natural projectively flat connections: Hitchin's connection on the bundle of non-abelian theta functions and the WZW connection (coming from conformal field theory) on the bundle of conformal blocks. Laszlo showed exactly that the natural isomorphism is flat with respect to these connections, i.e. the isomorphism also identifies the flat structures with one another.\N\NIn this article, the authors show that the same is true when one deals with the more general setting of parabolic bundles. In particular, for fixed parabolic data of \((\mathbf{p},\tau)\) of marked points \(\mathbf{p}\) on the curve \(C\) and weights \(\tau\), they consider the analogous construction of the determinant of cohomology line bundle \(\mathrm{Det}_{par,\phi}(\tau)\) over the space of parabolic semistable \(G\)-bundles \(M_{G,\mathbf{\tau}}^{par,ss}(C,\mathbf{p})\) on \(C\). The results in [\textit{I. Biswas} et al., J. Reine Angew. Math. 803, 137--181 (2023; Zbl 1524.14023)] generalize the construction of Hitchin's connection to the bundle of non-abelian parabolic theta functions defined for families of smooth projective curves and whose fiber is \(H^{0}(M_{G,\mathbf{\tau}}^{par,ss}(C,\mathbf{p}),\mathrm{Det}_{par,\phi}^{\otimes a}(\tau))\). As in the non-parabolic case, this Friedan-Shenker bundle can be identified (up to scalars) with a bundle of WZW conformal blocks which carries a natural connection. The main theorem of the paper is the statement that this identification is flat with respect to these connections.\N\NUsing this theorem, the authors also provide an application to the theory of the Knizhnik-Zamolodchikov (KZ) equation. By considering the trivial bundle over the configuration space of points in the projective line, whose typical fiber is the space of invariants of tensor product of representation, and its description as a space of conformal blocks, they use the flatness of the identification to provide a geometric description of the KZ connection.