Archimedean local L-factors were introduced to simplify functional equations of global L-functions. From the point of view of arithmetic geometry these factors complete the Euler product representation of global L-factors. A known construction of non-Archimedean local L-factors is rather transparent and uses characteristic polynomials of the image of the Frobenius homomorphism in finite-dimensional representations of the local Weil-Deligne group closely related to the local Galois group. On the other hand, Archimedean L-factors are expressed through products of Gamma-functions and thus are analytic objects avoiding simple algebraic interpretation. In the series of lectures we approach the problem of proper interpretation of Archimedean L-factors in terms of topological field theory.

In this second Lecture we propose a functional integral representation for local Archimedean L-factors given by products of the Gamma-functions. In particular we derive a representation of the Gamma-function as a properly regularized equivariant symplectic volume of an infinite-dimensional space. The corresponding functional integral arises in the description of S¹ x U_n equivariant topological linear sigma model of a type A on a disk. We extend our approach to the construction of parabolic GL_n/P, class one, Whittaker function as a correlation function in a S¹ x U_n topological sigma model of a type A. (Here we shall assume P is maximal parabolic subgroup).

The lecture is based on common papers with A.Gerasimov and S.Oblezin:

1. Archimedean L-factors and Topological Field Theories I, II in Communications in Number Theory and Physics, v 5, no 1, 2011 [math.NT/0906.1065];[hep-th/0909.2016];

3. Parabolic Whittaker Functions and Topological Field Theories I, Communications in Number Theory and Physics, v 5, no 1, 2011 [arXiv:1002.2622].