The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups

By M. Kapranov. 2000

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We establish a relation between the generating functions appearing in the S-duality conjecture of Vafa and Witten and geometric Eisenstein series for Kac-Moody groups. For a pair consisting of a surface and a curve on it, we consider a refined geometric function E (involving G-bundles with parabolic structures along the curve) which depends both on elliptic and modular variables. We prove a functional equation for E with respect to the affine Weyl group, thus establishing the elliptic behavior. When the curve is P^1, we calculate the Eisenstein-Kac-Moody series explicitly and it turns out to be a certain deformation of an irreducible Kac-Moody character, more precisely, an analog of the Hall-Littlewood polynomial for the affine root system. We also get an explicit formula for the universal blowup function for any simply connected structure group.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cf3f64ac1c8b0334091a7822bfad13eae3854f81

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Citations in the corpus

On the Motive of the Hilbert scheme of points on a surface (2000, background): “ng out, all negative powers of [A1] vanish. In [L-Q1], [L-Q2] a blowup formula was proven for the Euler numbers and the virtual Hodge numbers of moduli spaces of stable rank 2 sheaves on surfaces. In [Ka] a blowup formula is proven for principal bundles. We can show that the formula of [L-Q1], [L-Q2”
Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points (2004, background): “M. The series �[M](t) is the Kapranov zeta function of the variety M: [ 7 ].”
On the motives of moduli of chains and Higgs bundles (2011, methodology): “ CHAINS AND HIGGS BUNDLES 5 1.2. Zeta functions and their relation with Hodge polynomials. To compare our formulas with more classical formulas, we need to recall several facts on zeta functions from [25] and [19]. (1) For any variety X its zeta function is the formal power series Z(X,t) = P [X(n)]t”
Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures) (2017, background): “Using Kapranov’s results [10] on the motivic zeta function of a curve one can show that there is a ring homomorphism evC : Rg → Mot which sends q to L and when extended to Rg[[z]] → Mot[[z]] sends the formal zeta function to the motivic zeta function: 2 For some interesting counter-examples in this ”
Motivic measures and stable birational geometry (2001, methodology): “ in part by the NSF grant DMS-0100537. The second author was supported in part by the NSA grant MDA904-01-1-0020and the CRDF grant RM1-2089. 1 2 MICHAEL LARSEN AND VALERY A. LUNTS we deﬁne (following [Ka]) the motivic zeta-function of X as the formal power series ζµ(X,t) = 1+ X∞ n=1 µ([X(n)])tn ∈ A[”
Rationality criteria for motivic zeta functions (2002, background): “-cycles on X of degree n (where by convention, S0X = Spec F). A celebrated theorem of B. Dwork asserts that the zeta-function ZX(t) = X∞ i=0 |Symn(X)(F)|tn is a rational function in t. Kapranov asked [Ka] whether this rationality lifts to the Grothendieck ring of varieties over F. Explicitly, let K ”
Non-Archimedean analytic geometry as relative algebraic geometry (2013, unknown): “ith trivial valuation, our formalism will be useful for studying G-bundles over an algebraic surface or Kac-Moody bundles over an algebraic curve with an eye towards representation theory as in [11], [30], [21]. This is based on the relationship between G-bundles on a punctured formal neighborhood o”
The spherical Hecke algebra for affine Kac-Moody groups I (2008, background): “ we have →a ·F∈e im(q) (here by → a ·Fe we mean the action of → a on Fe as an element of the Zr-torsor Bung G(S0)). (2) q has ﬁnite ﬁbers. 22 Remark. When X is smooth this is exactly Theorem 2.2.1 of [5] (since in this case our condition is equivalent to the fact that X has negative self-intersectio”
Refined curve counting on complex surfaces (2012, background): “ropical Severi degrees on toric surface 2 Invariants of a single curve 2.1 Reﬁned invariants Let M denote the Grothendieck ring of varieties; let L = [A1] denote the class of the afﬁne line. Kapranov [Kap] introduced the motivic zeta function of a variety: X(q) = X1 n=0 SymnX qn 2MJqK: When X is a ”
Chow motives versus non-commutative motives (2011, background): “Recall from Kapranov [28] that the motivic zeta function of a smooth algebraic kvariety X with respect to a motivic measure is the following formal power series”