The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups
The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups
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发表时间：
2000-01
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通讯作者：
M. Kapranov
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作者：
M. Kapranov
建立了Kac-Moody群s对偶猜想中出现的生成函数与几何爱森斯坦级数之间的关系。对于由曲面和曲线组成的一对，我们考虑一个精细的几何函数E（涉及沿曲线具有抛物线结构的g束），它既依赖于椭圆变量，也依赖于模变量。我们证明了E关于仿射Weyl群的一个泛函方程，从而建立了E的椭圆性。当曲线为P^1时，我们显式地计算了爱森斯坦-卡克-穆迪级数，结果表明它是一个不可约的卡克-穆迪特征的某种变形，更准确地说，是仿射根系统的Hall-Littlewood多项式的类比。得到了任意单连通结构群的泛爆破函数的显式表达式。
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.