Title: Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors Show Chinese title

Title: 通过仿射李代数进行瞬子计数 第一卷：(仿射)旗流形的等变J函数和惠特aker向量

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. Show Chinese abstract

Abstract: 对于半单的单连通代数群G，我们引入了一些抛物线类比的Nekrasov分划函数（由Nekrasov引入，并最近由Nekrasov-Okounkov和Nakajima-Yoshioka对G=SL(n)进行了研究）。这些函数大致上计数在射影平面上具有无穷远处平凡化以及在水平线上具有抛物结构的主G丛。当上述抛物子群是一个Borel子群时，我们证明相应的分划函数基本上等于某个仿射李代数的Whittaker矩阵系数——即其根系与Lie(G)的仿射化的根系对偶的仿射李代数。我们解释了如何将这一结果视为Givental和Kim关于旗流形的Gromov-Witten不变量（更准确地说，是等变J函数）结果的仿射类比。因此，本文的主要结果可以看作是通过相应的“Langlands对偶”仿射李代数来计算与G相关的仿射旗流形的等变J函数（特别是，我们重新证明了关于普通旗流形的相应结果）。作为主要工具，我们使用了由Finkelberg、Gaitsgory和作者引入的Uhlenbeck空间的代数几何版本。这些结果与Seiberg-Witten预势的联系将在后续出版物中进行讨论。