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Семинар по арифметической алгебраической геометрии
10 декабря 2009 г. 15:00, г. Москва, МИАН, комн. 540 (ул. Губкина, 8)
 


Архимедовы $L$-множители и топологические теории поля

Д. Р. Лебедев

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Аннотация: 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.
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