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Looking for a quantum field theory model of Archimedean algebraic geometry aclass of infinite-dimensional integral representations of classical specialfunctions was introduced. Precisely the special functions such as Whittakerfunctions and Gamma-function were identified with correlation functions intopological field theories on a two-dimensional disk. Mirror symmetry of theunderlying topological field theory leads to a dual finite-dimensional integralrepresentations reproducing classical integral representations for thecorresponding special functions. The mirror symmetry interchanging infinite-and finite-dimensional integral representations provides an incarnation of thelocal Archimedean Langlands duality on the level of classical specialfunctions. In this note we provide some directions to higher-dimensionalgeneralizations of our previous results. In the first part we considertopological field theory representations of multiple local L-factors introducedby Kurokawa and expressed through multiple Barnes's Gamma-functions. In thesecond part we are dealing with generalizations based on consideration oftopological Yang-Mills theories on non-compact four-dimensional manifolds.Presumably, in analogy with the mirror duality in two-dimensions, S-dualdescription should be instrumental for deriving integral representations for aparticular class of quantum field theory correlation functions and thusproviding a new interesting class of special functions supplied with canonicalintegral representations.

On topological field theory representation of higher analogs of classical special functions