Instanton counting via affine Lie algebras II: From Whittaker vectors to the Seiberg-Witten prepotential

Summary

Let G be a simple simply connected algebraic group over ℂ with Lie algebra $\mathfrak{g}$ . Given a parabolic subgroup P ⊂ G, in tikya[1] the first author introduced a certain generating function Z ^aff _G,P . Roughly speaking, these functions count (in a certain sense) framed G-bundles on ℙ² together with a P-structure on a fixed (horizontal) line in ℙ². When P = B is a Borel subgroup, the function Z ^aff _G,B was identified in tikya[1] with the Whittaker matrix coefficient in the universal Verma module over the affine Lie algebra $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathfrak{g}} _{aff}$ (here we denote by $\mathfrak{g}_{aff}$ the affinization of $\mathfrak{g}$ and by $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathfrak{g}} _{aff}$ the Lie algebra whose root system is dual to that of $\mathfrak{g}_{aff}$ ).

For P = G (in this case we shall write Z ^aff _G instead of Z ^aff _{G, P} and G = SL(n) the above generating function was introduced by Nekrasov (see tikya[7]) and studied thoroughly in tikya[5] and tikya[8]. In particular, it is shown in loc. cit. that the leading term of certain asymptotic of Z ^aff _G is given by the (instanton part of the) Seiberg-Witten prepotential (for G = SL(n)). The prepotential is defined using the geometry of the (classical) periodic Toda integrable system. This result was conjectured in tikya[7].

The purpose of this paper is to extend these results to arbitrary G. Namely, we use the above description of the function Z ^aff _G,B to show that the leading term of its asymptotic (similar to the one studied in tikya[7] for P = G) is given by the instanton part of the prepotential constructed via the Toda system attached to the Lie algebra $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathfrak{g}} _{aff}$ . This part is completely algebraic and does not use the original algebro-geometric definition of Z ^aff _G,B . We then show that for fixed G these asymptotic are the same for all functions Z ^aff _G,P .