Instanton counting via affine Lie algebras I Equivariant J-functions of (affine) flag manif

d UK
a rational function of a ∈ k and (ε1 , ε2 ) ∈ Consider now the generating function
∞
R2 .
Z=
d=0
Qd
d UK
1d .
It can (and should) be thought of as a function of the variables q and a, ε1 , ε2 as before. In [23] it was conjectured that the first term of the asymptotic in the limit lim ln Z
INSTANTON COUNTING VIA AFFINE LIE ALGEBRAS I: EQUIVARIANT J-FUNCTIONS OF (AFFINE) FLAG MANIFOLDS AND WHITTAKER VECTORS
arXiv:math/0401409v2 [math.AG] 15 Oct 2004
ε1 ,ε2 →0
is closely related to Seiberg-Witten prepotential of K . For K = SU (n) this conjecture has been proved in [24] and [22]. Also in [23] an explicit combinatorial expression for Z has been found. 1.2. Algebraic version. In this paper we want to generalize the definition of the function Z in several directions and compute it in some of these new cases. First of all, it will be convenient for us to make the whole situation completely algebraic. Namely, let G be a complex semi-simple algebraic group whose maximal compact subgroup is isomorphic to K . We shall denote by g its Lie algebra. Let also S = P2 and denote by D∞ ⊂ S the ”straight line at ∞”; thus S\D∞ = A2 . It is well-known d that Md K is isomorphic to the moduli space BunG (S, D∞ ) of principal G-bundles on S endowed with a trivialization on D∞ of second Chern class −d. When it does not lead to a confusion we shall write BunG instead of BunG (S, D∞ ). The algebraic analog of d has been constructed in [3]; we denote this algebraic variety by U d . This variety UK G is endowed with a natural action on G × (C∗ )2 . 1.3. Parabolic generalization of the partition function. Let C ⊂ S denote the ”horizontal line”. Choose a parabolic subgroup P ⊂ G. Let BunG,P denote the moduli space of the following objects: 1) A principal G-bundle FG on S; 2) A trivialization of FG on D∞ ⊂ S; 3) A reduction of FG to P on C compatible with the trivialization of FG on C. Let us describe the connected components of BunG,P . Let M be the Levi group ˇ the Langlands dual group of M and let Z (M ˇ ) be its center. We of P . Denote by M aff ˇ ). Let also Λ denote by ΛG,P the lattice of characters of Z (M G,P = ΛG,P × Z be the ∗ aff ˇ lattice of characters of Z (M ) × C . Note that ΛG,G = Z.
d UK
This research has been partially supported by t来自e NSF.12
ALEXANDER BRAVERMAN
of the unit K × (S 1 )2 -equivariant cohomology class (which we denote by 1d ) over d ; the integral takes values in the field K which is the field of fractions of the algebra UK 1 ∗ A = HK ×(S 1 )2 (pt) . Note that A is canonically isomorphic to the algebra of polynomial functions on k × R2 (here k denotes the Lie algebra of K ) which are invariant with 1d may naturally be regarded as respect to the adjoint action of K on k. Thus each
ALEXANDER BRAVERMAN Abstract. Let g be a simple complex Lie algebra, G - the corresponding simply connected group; let also gaff be the corresponding untwisted affine Lie algebra. aff For a parabolic subgroup P ⊂ G we introduce a generating function ZG,P which 2 roughly speaking counts framed G-bundles on P endowed with a P -structure on the horizontal line (the formal definition uses the corresponding Uhlenbeck type aff compactifications studied in [3]). In the case P = G the function ZG,P coincides with Nekrasov’s partition function introduced in [23] and studied thoroughly in [24] and [22] for G = SL(n). In the ”opposite case” when P is a Borel subgroup of G aff we show that ZG,P is equal (roughly speaking) to the Whittaker matrix coefficient in the universal Verma module for the Lie algebra ˇ gaff – the Langlands dual Lie algebra of gaff . This clarifies somewhat the connection between certain asymptotic aff of ZG,P (studied in loc. cit. for P = G) and the classical affine Toda system. We also explain why the above result gives rise to a calculation of (not yet rigorously defined) equivariant quantum cohomology ring of the affine flag manifold associated with G. In particular, we reprove the results of [13] and [18] about quantum cohomology of ordinary flag manifolds using methods which are totally different from loc. cit. We shall show in a subsequent publication how this allows one to connect certain aff asymptotic of the function ZG,P with the Seiberg-Witten prepotential (cf. [2], thus proving the main conjecture of [23] for an arbitrary gauge group G (for G = SL(n) it has been proved in [24] and [22] by other methods.