TY - CHAP

T1 - Instanton counting via affine Lie algebras II

T2 - From Whittaker vectors to the Seiberg–Witten prepotential

AU - Braverman, A.

AU - Etingof, P.

PY - 2006

Y1 - 2006

N2 - Let G be a simple simply connected algebraic group over ℂ with Lie algebra g. Given a parabolic subgroup P ⊂ G, in [1] the first author introduced a certain generating function ZaffG, 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 ZaffG,B was identified in [1] with the Whittaker matrix coefficient in the universal Verma module over the affine Lie algebra ğaff (here we denote by gaff the affinization of g and by ğaff the Lie algebra whose root system is dual to that of gaff). For P = G (in this case we shall write ZaffG instead of ZaffG,P) and G = SL(n) the above generating function was introduced by Nekrasov (see [7]) and studied thoroughly in [5] and [8]. In particular, it is shown in loc. cit. that the leading term of certain asymptotic of ZaffG 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 [7]. The purpose of this paper is to extend these results to arbitrary G. Namely, we use the above description of the function ZaffG,B to show that the leading term of its asymptotic (similar to the one studied in [7] for P = G) is given by the instanton part of the prepotential constructed via the Toda system attached to the Lie algebra ğaff. This part is completely algebraic and does not use the original algebro-geometric definition of ZaffG,B. We then show that for fixed G these asymptotic are the same for all functions ZaffG,P.

AB - Let G be a simple simply connected algebraic group over ℂ with Lie algebra g. Given a parabolic subgroup P ⊂ G, in [1] the first author introduced a certain generating function ZaffG, 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 ZaffG,B was identified in [1] with the Whittaker matrix coefficient in the universal Verma module over the affine Lie algebra ğaff (here we denote by gaff the affinization of g and by ğaff the Lie algebra whose root system is dual to that of gaff). For P = G (in this case we shall write ZaffG instead of ZaffG,P) and G = SL(n) the above generating function was introduced by Nekrasov (see [7]) and studied thoroughly in [5] and [8]. In particular, it is shown in loc. cit. that the leading term of certain asymptotic of ZaffG 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 [7]. The purpose of this paper is to extend these results to arbitrary G. Namely, we use the above description of the function ZaffG,B to show that the leading term of its asymptotic (similar to the one studied in [7] for P = G) is given by the instanton part of the prepotential constructed via the Toda system attached to the Lie algebra ğaff. This part is completely algebraic and does not use the original algebro-geometric definition of ZaffG,B. We then show that for fixed G these asymptotic are the same for all functions ZaffG,P.

UR - http://www.scopus.com/inward/record.url?scp=85014151040&partnerID=8YFLogxK

U2 - 10.1007/0-8176-4478-4_5

DO - 10.1007/0-8176-4478-4_5

M3 - Chapter

AN - SCOPUS:85014151040

T3 - Progress in Mathematics

SP - 61

EP - 78

BT - Progress in Mathematics

PB - Springer Basel

ER -