Hecke modifications of conformal blocks outside the critical level

TL;DR

This work develops a non-critical-level framework for Hecke modifications of conformal blocks by coupling Hecke operations on $G$-bundles with insertions of twisted vacuum modules. A central result is the canonical isomorphism $H_\lambda(\Delta_{\underline{K}}(\underline{M})) \cong \Delta_{(\underline{K},G(\mathcal{O})_\lambda)}(\underline{M},V_{G(\mathcal{O}),k}^\lambda)$, realized as twisted $\mathcal{D}$-modules via Beilinson–Bernstein–Jantzen theory. The authors then describe the $N\to N+1$ transition through coordinate transformations, both for $X=\mathbb{P}^1$ and in general, and provide explicit calculations for $G=\mathrm{PGL}_2(\mathbb{C})$ including Ward identities and the Knizhnik–Zamolodchikov equations. The results extend the geometric Langlands perspective to non-critical levels, linking conformal blocks, twisted vacuum modules, and affine Grassmannian geometry with concrete parabolic and coordinate-transform techniques, thereby enabling explicit computations of conformal blocks and their differential equations in this broader setting.

Abstract

We define Hecke modifications of conformal blocks over affine Lie algebras at non-critical level by using the Hecke modifications of the underlying $G$-bundles. We show that this procedure is equivalent to the insertion of a twisted vacuum module at an additional marked point and provide an explicit description using coordinate transformations.