On generalized Macdonald polynomials

Миронов, А.; Морозов, А.

doi:10.1007/jhep01(2020)110

Cited by 11 publications

(2 citation statements)

References 75 publications

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“…Symmetries and special polynomials. The renewed interest in refined topological strings in the context of AGT led to developing many families of special polynomials generalizing Jack and Macdonald polynomials, including Macdonald-Kerov functions, generalized Schur functions, elliptic generalized Macdonald polynomials and more: see [598,762,[1007][1008][1009][1010][1011][1012][1013][1014][1015][1016][1017][1018][1019][1020] for recent developments.…”

Section: Discussionmentioning

confidence: 99%

A slow review of the AGT correspondence

Floch¹

2022

J. Phys. A: Math. Theor.

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Starting with a gentle approach to the Alday–Gaiotto–Tachikawa (AGT) correspondence from its 6d origin, these notes provide a wide (albeit shallow) survey of the literature on numerous extensions of the correspondence up to early 2020. This is an extended writeup of the lectures given at the Winter School ‘YRISW 2020’ to appear in a special issue of J. Phys. A. Class S is a wide class of 4d N = 2 supersymmetric gauge theories (ranging from super-QCD (quantum chromodynamics) to non-Lagrangian theories) obtained by twisted compactification of 6d N = ( 2 , 0 ) superconformal theories on a Riemann surface C. This 6d construction yields the Coulomb branch and Seiberg–Witten geometry of class S theories, geometrizes S-duality, and leads to the AGT correspondence, which states that many observables of class S theories are equal to 2d conformal field theory (CFT) correlators. For instance, the four-sphere partition function of a 4d N = 2 S U ( 2 ) superconformal quiver theory is equal to a Liouville CFT correlator of primary operators. Extensions of the AGT correspondence abound: asymptotically-free gauge theories and Argyres–Douglas theories correspond to irregular CFT operators, quivers with higher-rank gauge groups and non-Lagrangian tinkertoys such as T N correspond to Toda CFT correlators, and nonlocal operators (Wilson–’t Hooft loops, surface operators, domain walls) correspond to Verlinde networks, degenerate primary operators, braiding and fusion kernels, and Riemann surfaces with boundaries.

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Section: Discussionmentioning

confidence: 99%

A slow review of the AGT correspondence

Floch¹

2022

J. Phys. A: Math. Theor.

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“…Symmetries and special polynomials. The renewed interest in refined topological strings in the context of AGT led to developping many families of special polynomials generalizing Jack and Macdonald polynomials, including Macdonald-Kerov functions, generalized Schur functions, elliptic generalized Macdonald polynomials and more: see [425,[731][732][733][734][735][736][737][738][739][740][741][742][743][744][745] for recent developments.…”

Section: Discussionmentioning

confidence: 99%

A slow review of the AGT correspondence

Floch¹

2020

Preprint

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Starting with a gentle approach to the Alday-Gaiotto-Tachikawa (AGT) correspondence from its 6d origin, these notes provide a wide survey of the literature on numerous extensions of the correspondence. This is the writeup of the lectures given at the Winter School "YRISW 2020" to appear in a special issue of JPhysA.Class S is a wide class of 4d N = 2 supersymmetric gauge theories (ranging from super-QCD to non-Lagrangian theories) obtained by twisted compactification of 6d N = (2, 0) superconformal theories on a Riemann surface C. This 6d construction yields the Coulomb branch and Seiberg-Witten geometry of class S theories, geometrizes S-duality, and leads to the AGT correspondence, which states that many observables of class S theories are equal to 2d conformal field theory (CFT) correlators. For instance, the four-sphere partition function of 4d N = 2 SU(2) superconformal quiver theories is equal to a Liouville CFT correlator of primary operators.Extensions of the AGT correspondence abound: asymptotically-free gauge theories and Argyres-Douglas (AD) theories correspond to irregular CFT operators, quivers with higher-rank gauge groups and non-Lagrangian tinkertoys such as T N correspond to Toda CFT correlators, and nonlocal operators (Wilson-'t Hooft loops, surface operators, domain walls) correspond to Verlinde networks, degenerate primary operators, braiding and fusion kernels, and Riemann surfaces with boundaries.

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Commutative families in DIM algebra, integrable many-body systems and q, t matrix models

Mironov,

Morozov,

Popolitov

2024

J. High Energ. Phys.

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We extend our consideration of commutative subalgebras (rays) in different representations of the W1+∞ algebra to the elliptic Hall algebra (or, equivalently, to the Ding-Iohara-Miki (DIM) algebra $$ {U}_{q,t}\left({\hat{\hat{\mathfrak{gl}}}}_1\right) $$ U q , t gl ̂ ̂ 1 ). Its advantage is that it possesses the Miki automorphism, which makes all commutative rays equivalent. Integrable systems associated with these rays become finite-difference and, apart from the trigonometric Ruijsenaars system not too much familiar. We concentrate on the simplest many-body and Fock representations, and derive explicit formulas for all generators of the elliptic Hall algebra en,m. In the one-body representation, they differ just by normalization from $$ {z}^n{q}^{m\hat{D}} $$ z n q m D ̂ of the W1+∞ Lie algebra, and, in the N -body case, they are non-trivially generalized to monomials of the Cherednik operators with action restricted to symmetric polynomials. In the Fock representation, the resulting operators are expressed through auxiliary polynomials of n variables, which define weights in the residues formulas. We also discuss q, t-deformation of matrix models associated with constructed commutative subalgebras.

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On generalized Macdonald polynomials

Cited by 11 publications

References 75 publications

A slow review of the AGT correspondence

A slow review of the AGT correspondence

A slow review of the AGT correspondence

Commutative families in DIM algebra, integrable many-body systems and q, t matrix models

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