Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors
Alexander Braverman
TL;DR
The paper develops a unified, algebraic framework for instanton counting by introducing parabolic (and affine) Nekrasov-type partition functions via Bun_G and its Uhlenbeck compactifications, and shows these partition functions can be realized as Whittaker matrix coefficients in (affine) Verma modules of the Langlands dual algebra. It proves a precise link between these partition functions and equivariant J-functions of affine flag varieties, establishing that the finite- and affine-parabolic cases satisfy quantum-Toda eigenfunction relations and provide a representation-theoretic interpretation of affine Gromov–Witten-type invariants. The work constructs and analyzes the action of the dual Lie algebras on equivariant intersection cohomology (IH) of zastava spaces and related moduli, using graph spaces and localization to prove the central identities (j-versus-z) and to derive explicit SL(2) examples. Overall, it provides a conceptual bridge between geometric representation theory, equivariant cohomology, and integrable systems in the context of affine Lie algebras and their Whittaker models, with further Seiberg–Witten connections to be addressed in subsequent work.
Abstract
For a semi-simple simply connected algebraic group G we introduce certain parabolic analogues of the Nekrasov partition function (introduced by Nekrasov and studied recently by Nekrasov-Okounkov and Nakajima-Yoshioka for G=SL(n)). These functions count (roughly speaking) principal G-bundles on the projective plane with a trivialization at infinity and with a parabolic structure at the horizontal line. When the above parabolic subgroup is a Borel subgroup we show that the corresponding partition function is basically equal to the Whittaker matrix coefficient in the universal Verma module over certain affine Lie algebra - namely, the one whose root system is dual to that of the affinization of Lie(G). We explain how one can think about this result as the affine analogue of the results of Givental and Kim about Gromov-Witten invariants (more precisely, equivariant J-functions) of flag manifolds. Thus the main result of the paper may considered as the computation of the equivariant J-function of the affine flag manifold associated with G (in particular, we reprove the corresponding results for the usual flag manifolds) via the corresponding "Langlands dual" affine Lie algebra. As the main tool we use the algebro-geometric version of the Uhlenbeck space introduced by Finkelberg, Gaitsgory and the author. The connection of these results with the Seiberg-Witten prepotential will be treated in a subsequent publication.