Geometrization of the TUY/WZW/KZ connection

Abstract

Given a simple, simply connected, complex algebraic group G, a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over any family of smooth projective curves with marked points was constructed by the authors in an earlier paper. Here, it is shown that the identification between the bundle of nonabelian theta functions and the bundle of WZNW conformal blocks is flat with respect to this connection and the one constructed by Tsuchiya-Ueno-Yamada. As an application, we give a geometric construction of the Knizhnik-Zamolodchikov connection on 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 representations. BISWAS, MUKHOPADHYAY, AND WENTWORTH 7.1. The line bundle on the universal moduli stack 21 7.2. The relative uniformization and parabolic theta functions 22 8. Sugawara tensor and the parabolic Hitchin heat operator 22 8.1. Twisted D-modules via quasi-section of Drinfeld-Simpson 23 8.2. Projective heat operator from Sugawara 24 8.3. The parabolic duality pairing and the Hitchin symbol 25 8.4. Proof of the Main theorem (Theorem 1.1) 27 9. Proof of Theorem 6.3 28 9.1. Reduction to the SL r case 28 9.2. Reduction to the SL r with full flag 28 9.3. Reduction to abelian varieties 31 9.4. Abelianization and determinant of cohomology 33 10. Geometrization of the KZ equation on invariants 34 10.1. KZ connection 34 10.2. Invariants as global sections 34 10.3. Differential operators 35 References 36