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 last Lecture we provide a functional integral representation of the Archimedean L-factors in terms of a type B S¹ equivariant topological sigma model on a disk. This representation leads naturally to the classical Euler integral representation of the ¡-functions. These two integral representations of L-factors in terms of A and B topological sigma models are related by a mirror map and we provide explicit derivation of the 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. We extend our approach to the case of class one Whittaker functions.

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];

2. New integral representations of Whittaker functions for classical Lie groups. To be published in Uspehi Math. Nauk; [math.RT/0705.2886];