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

TL;DR

This paper explores topological field theory representations of higher analogs of classical special functions, extending previous two-dimensional models to higher dimensions and connecting them with Langlands duality and S-duality concepts.

Contribution

It introduces higher-dimensional topological field theory models for special functions, generalizing previous two-dimensional results and linking them to Langlands duality and S-duality.

Findings

Representation of multiple L-factors via Barnes's Gamma-functions

Extension of topological field theory models to four-dimensional Yang-Mills theories

Proposed dualities for deriving new integral representations of special functions

Abstract

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.