Representation Theory over Tropical Semifield and Langlands Duality

Gerasimov, A.; Лебедев, Д. В.

doi:10.1007/s00220-013-1705-2

Cited by 2 publications

(3 citation statements)

References 41 publications

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“…Tropical geometry is an emerging tool in algebraic geometry that can transform certain questions into combinatorial problems by replacing a variety with a polyhedral object called a tropical variety. It has had striking applications to a range of subjects, such as enumerative geometry [Mik05, FM10, GM08, AB13], classical geometry [CDPR12,Bak08], intersection theory [Kat09, GM12,OP13], moduli spaces and compactifications [Tev07, HKT09, ACP15, RSS14], mirror symmetry [Gro10, GPS10,Gro11], abelian varieties [Gub07,CV10], representation theory [FZ02,GL13], algebraic statistics and mathematical biology [PS04,Man11] (and many more papers by many more authors). Since its inception, it has been tempting to look for algebraic foundations of tropical geometry, e.g., to view tropical varieties as varieties in a more literal sense and to understand tropicalization as a degeneration taking place in one common algebro-geometric world.…”

Section: Introductionmentioning

confidence: 99%

Equations of tropical varieties

Giansiracusa¹,

Giansiracusa²

2016

Duke Math. J.

121

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We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.Comment: 36 pages, final version to appear in Duk

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Section: Introductionmentioning

confidence: 99%

Equations of tropical varieties

Giansiracusa¹,

Giansiracusa²

2016

Duke Math. J.

121

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“…Let us first recall the basic constructions of the hierarchy of Γ-functions [Ba]. The simplest Γfunction (called elementary Γ-function in [GL1]) is given by Γ 0 (s) = 1 s .…”

Section: Multiple Gamma-functionsmentioning

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]. This note is a very preliminary discussion of a large project of higher dimensional generalizations of [GLO2], [GLO3], [GLO4].…”

Section: Introductionmentioning

confidence: 99%

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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Representation Theory over Tropical Semifield and Langlands Duality

Cited by 2 publications

References 41 publications

Equations of tropical varieties

Equations of tropical varieties

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

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