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

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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 Γ-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 Γ-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.

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Authors and Affiliations

  1. Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia

    Anton A. Gerasimov & Dimitri R. Lebedev

  2. School of Mathematics, Trinity College Dublin, Dublin 2, Ireland

    Anton A. Gerasimov

  3. Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland

    Anton A. Gerasimov

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  1. Anton A. Gerasimov
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  2. Dimitri R. Lebedev
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Correspondence to Anton A. Gerasimov.

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ArXiv ePrint: 1011.0403

Extended version of a talk given by the first author at Quantum field theory and representation theory, October, 2010, Moscow, Russia.

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Gerasimov, A.A., Lebedev, D.R. On topological field theory representation of higher analogs of classical special functions. J. High Energ. Phys. 2011, 76 (2011). https://doi.org/10.1007/JHEP09(2011)076

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