On topological field theory representation of higher analogs of classical special functions
- 75 Downloads
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.
KeywordsTopological Field Theories Integrable Field Theories
- D.M. Austin and P.J. Braam, Morse-Bott theory and equivariant cohomology, in The Floer Memorial Volume, H. Hofer et al. eds., Springer, U.S.A. (1995).Google Scholar
- E.W. Barnes, On the theory of the multiple gamma functions, Trans. Cambridge Philos. Soc. 19 (1904) 374.Google Scholar
- A. Braverman and P. Etingof, Instanton counting via affine Lie algebras. II. From Whittaker vectors to the Seiberg-Witten prepotential, math/0409441.
- D. Bump, Automorphic forms and representations, Cambridge University Press, Cambridge U.K. (1998).Google Scholar
- R.L. Cohen, J.D.S. Jones and G.B. Segal, Floer’s infinite dimensional Morse theory and homotopy theory, in The Floer Memorial Volume, H. Hofer et al. eds., Springer, U.S.A. (1995).Google Scholar
- A. Gerasimov, A quantum field theory model of archimedean geometry, talk given at Rencontres Itzykson 2010: New trends in quantum integrability, June 21–23, IPhT Saclay, France (2010).Google Scholar
- A.A. Gerasimov and D.R. Lebedev, Representation theory over tropical semifield and Langlands correspondence, arXiv:1011.2462.
- J. Bernstein and S. Gelbart, An introduction to the Langlands program, lectures presented at the Hebrew University of Jerusalem, Jerusalem, March 12–16 (2001), Springer, U.S.A. (2003).Google Scholar
- H. Nakajima and K. Yoshioka, Lectures on instanton counting, math/0311058.