Archimedean L-factors and Topological Field Theories II

Gerasimov, A.; Лебедев, Д. В.; Oblezin, Sergey

doi:10.48550/arxiv.0909.2016

Cited by 2 publications

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“…In [GLO2], [GLO3], [GLO4], [GLO5], [G] a topological field theory framework for a description of the Archimedean algebraic geometry was proposed. As a first step [GLO2], [GLO3] local Archimedean L-factors were interpreted as correlation functions in two-dimensional equivariant topological field theories on a disk. It was demonstrated that the local Archimedean Langlands correspondence between various constructions of L-factors (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…It was demonstrated that the local Archimedean Langlands correspondence between various constructions of L-factors (see e.g. [B], [L]) is realized as mirror symmetry on the level of underlying topological field theories [GLO3]. These results were generalized to a class of Whittaker functions in [GLO4].…”

Section: Introductionmentioning

confidence: 99%

“…Presumably this picture holds in full generality and provides a realization of the Archimedean Langlands duality for a generic Whittaker function. Moreover, one can expect that the approach of [GLO2], [GLO3], [GLO4], [GLO5], [G] will lead to an interpretation of all basic constructions of Archimedean algebraic geometry in terms of twodimensional topological field theories thus providing a clue to a natural formulation of the geometry over Archimedean fields.…”

Section: Introductionmentioning

confidence: 99%

“…Given the proposed connection of the Archimedean geometry with two-dimensional topological field theories it is natural to ask what is a special role of two dimensions in these considerations and are there any signs of possible generalizations of [GLO2], [GLO3], [GLO4] to other dimensions. The case of zero dimension was considered in [GL1].…”

Section: Introductionmentioning

confidence: 99%

“…The case of zero dimension was considered in [GL1]. This note is a very preliminary discussion of a large project of higher dimensional generalizations of [GLO2], [GLO3], [GLO4]. In the first part we propose a description of higher analogs of local Archimedean L-factors introduced by Kurokawa [Ku1], [Ku2] (see also [Ma]) in terms of equivariant topological field theories with quadratic actions.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Gerasimov

Лебедев

2011

J. High Energ. Phys.

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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. * 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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Gerasimov

Лебедев

2011

J. High Energ. Phys.

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Representation Theory over Tropical Semifield and Langlands Duality

Gerasimov

Лебедев

2013

Commun. Math. Phys.

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Recently we propose a class of infinite-dimensional integral representations of classical gl ℓ+1 -Whittaker functions and local Archimedean local L-factors using two-dimensional topological field theory framework. The local Archimedean Langlands duality was identified in this setting with the mirror symmetry of the underlying topological field theories. In this note we introduce elementary analogs of the Whittaker functions and the Archimedean L-factors given by U ℓ+1 -equivariant symplectic volumes of appropriate Kähler U ℓ+1 -manifolds. We demonstrate that the functions thus defined have a dual description as matrix elements of representations of monoids GL ℓ+1 (R), R being the tropical semifield. We also show that the elementary Whittaker functions can be obtained from the non-Archimedean Whittaker functions over Q p by taking the formal limit p → 1. Hence the elementary special functions constructed in this way might be considered as functions over the mysterious field Q 1 . The existence of two representations for the elementary Whittaker functions, one as an equivariant volume and the other as a matrix element, should be considered as a manifestation of a hypothetical elementary analog of the local Langlands duality for number fields. We would like to note that the elementary local L-factors coincide with L-factors introduced previously by Kurokawa.2 Whittaker functions in classical setting

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Archimedean L-factors and Topological Field Theories II

Cited by 2 publications

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

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

Representation Theory over Tropical Semifield and Langlands Duality

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