Baxter Operator Formalism for Macdonald Polynomials

We propose integral representations of the Whittaker functions for the classical Lie algebras sp 2ℓ , so 2ℓ and so 2ℓ+1 . These integral representations generalize the integral representation of gl ℓ+1 -Whittaker functions first introduced by Givental. One of the salient features of the Givental representation is its recursive structure with respect to the rank ℓ of the Lie algebra gl ℓ+1 . The proposed generalization of the Givental representation to the classical Lie algebras retains this property. It was shown elsewhere that the integral recursion operator for gl ℓ+1 -Whittaker function in the Givental representation coincides with a degeneration of the Baxter Q-operator for gl ℓ+1 -Toda chains. We construct Q-operator for affine Lie algebras so 2ℓ , so 2ℓ+1 and a twisted form of gl 2ℓ . We demonstrate that the relation between recursion integral operators of the generalized Givental representation and degenerate Q-operators remains valid for all classical Lie algebras.

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“…Finally, note that some of the results presented in this paper was announced previously in [GLO], [GLO1].…”

Section: Introductionsupporting

confidence: 61%

New integral representations of Whittaker functions for classical Lie groups

Gerasimov¹,

Лебедев²,

Oblezin³

2012

Russ. Math. Surv.

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“…x × f where f ⊂ R 2 79 . From the point of view of the 3d boundary theory on R 2 01 × S 1 x this defect is two-dimensional.…”

Section: Type Iib String Theory Constructionmentioning

confidence: 99%

“…given by an eigenvalue of the Baxter operatorQ(x) acting on space of symmetric polynomials in variables {e −m i } constructed in [79]. The eigenfunctions are given by Macdonald polynomials P λ (e −m i ; q, t) [80].…”

Section: Trigonometric Ruijsenaars-schneider Systemmentioning

confidence: 99%

Walls, lines, and spectral dualities in 3d gauge theories

Gadde

Gukov

Putrov

2014

J. High Energ. Phys.

128

242

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Abstract:In this paper we analyze various half-BPS defects in a general three dimensional N = 2 supersymmetric gauge theory T . They correspond to closed paths in SUSY parameter space and their tension is computed by evaluating period integrals along these paths. In addition to such defects, we also study wall defects that interpolate between T and its SL(2, Z) transform by coupling the 3d theory to a 4d theory with S-duality wall. We propose a novel spectral duality between 3d gauge theories and integrable systems. This duality complements a similar duality discovered by Nekrasov and Shatashvili. As another application, for 3d N = 2 theories associated with knots and 3-manifolds we compute periods of (super) A-polynomial curves and relate the results with the spectrum of domain walls and line operators.

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“…In this section we use the Baxter operators to define the b -Whittaker functions by means of a Givental-type integral formula, and establish some of their basic properties. A similar approach has been taken in [GKLO08] in the case of undeformed Whittaker functions, and in [GKLO14] for Macdonald polynomials and their degenerations, the q-Whittaker polynomials. We also explain the relation of our Givental formula to the Mellin-Barnes integral representation of the b -Whittaker functions that was first discovered in [KLS02].…”

Section: The B -Whittaker Functionsmentioning

confidence: 99%

On $b$-Whittaker functions

Schrader,

Shapiro

2018

Preprint

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The b -Whittaker functions are eigenfunctions of the modular q -deformed gl n open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a Mellin-Barnes integral representation for these eigenfunctions. In the present paper, we develop the analytic theory of the b -Whittaker functions from the perspective of quantum cluster algebras. We obtain a formula for the modular open Toda system's Baxter operator as a sequence of quantum cluster transformations, and thereby derive a new modular b -analog of Givental's integral formula for the undeformed Whittaker function. We also show that the b -Whittaker functions are eigenvectors of the Dehn twist operator from quantum higher Teichmüller theory, and obtain b -analogs of various integral identities satisfied by the undeformed Whittaker functions, including the continuous Cauchy-Littlewood identity of Stade and Corwin-O'Connell-Seppäläinen-Zygouras. Using these results, we prove the unitarity of the b -Whittaker transform, thereby completing the analytic part of the proof of the conjecture of Frenkel and Ip on tensor products of positive representations of Uq(sln), as well as the main step in the modular functor conjecture of Fock and Goncharov. We conclude by explaining how the theory of b -Whittaker functions can be used to derive certain hyperbolic hypergeometric integral evaluations found by Rains.

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Baxter Operator Formalism for Macdonald Polynomials

Cited by 10 publications

References 13 publications

New integral representations of Whittaker functions for classical Lie groups

New integral representations of Whittaker functions for classical Lie groups

Walls, lines, and spectral dualities in 3d gauge theories

On $b$-Whittaker functions

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