Архимедовы $L$-множители и топологические теории поля

In the first part of this talk we recall shortly two integral representation of Whittaker function (the Mellin–Barnes and the Givental ones). Then we identify eigenvalues of the Baxter $Q$-operator acting on Whittaker functions with local Archimedean $L$-factors. The Baxter $Q$-operator is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra $H(G(R),K)$, $K$ being a maximal compact subgroup of $G$. In the second, main part, of this talk 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 a type A equivariant topological linear sigma model on a disk. Then we provide another functional integral representation of the Archimedean $L$-factors in terms of a type B topological sigma model on a disk. This representation leads naturally to the classical Euler integral representation of the Gamma-functions. These two integral representations of $L$-factors in terms of A and B topological sigma models are related by a mirror map. The mirror symmetry in our setting should be considered as a local Archimedean Langlands correspondence between two constructions of local Archimedean $L$-factors.
The talk is based on papers:
Gerasimov, Lebedev, Oblezin: CMP 284:3 (2008) 867–896;
arXiv: 0906.1065v2;
arXiv: 0909.2106v2.