  Looking for a quantum field theory model of Archimedean algebraic geometry a
class of infinite-dimensional integral representations of classical special
functions was introduced. Precisely the special functions such as Whittaker
functions and Gamma-function were identified with correlation functions in
topological field theories on a two-dimensional disk. Mirror symmetry of the
underlying topological field theory leads to a dual finite-dimensional integral
representations reproducing classical integral representations for the
corresponding special functions. The mirror symmetry interchanging infinite-
and finite-dimensional integral representations provides an incarnation of the
local Archimedean Langlands duality on the level of classical special
functions. In this note we provide some directions to higher-dimensional
generalizations of our previous results. In the first part we consider
topological field theory representations of multiple local L-factors introduced
by Kurokawa and expressed through multiple Barnes's Gamma-functions. In the
second part we are dealing with generalizations based on consideration of
topological Yang-Mills theories on non-compact four-dimensional manifolds.
Presumably, in analogy with the mirror duality in two-dimensions, S-dual
description should be instrumental for deriving integral representations for a
particular class of quantum field theory correlation functions and thus
providing a new interesting class of special functions supplied with canonical
integral representations.
