Journal of High Energy Physics

, 2011:76 | Cite as

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

  • Anton A. Gerasimov
  • Dimitri R. Lebedev


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.


Topological Field Theories Integrable Field Theories 


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Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia
  2. 2.School of MathematicsTrinity College DublinDublin 2Ireland
  3. 3.Hamilton Mathematics InstituteTrinity College DublinDublin 2Ireland

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