Elliptic algebra, Frenkel–Kac construction and root of unity limit

Itoyama, H.; Oota, Takeshi; Yoshioka, Reiji

doi:10.1088/1751-8121/aa8233

Cited by 3 publications

(7 citation statements)

References 122 publications

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“…34 It is not immediately clear to me how these work out when considering different Lagrangian descriptions of n-punctured tori or of higher genus surfaces. 35 As in various other places in this review there are inaccuracies about the global structure of groups.…”

Section: Linear Quiver Su(n ) Theoriesmentioning

confidence: 77%

“…The flavour symmetry of each group is u(N ) = u(1) × su(N ), so that this split makes su(N ) 2 × u(1) 2 flavour symmetry manifest. In analogy to the N = 2 case we associate each of the four factors to one puncture and write an analogue of (4.11): 35 T su(N ), CP 1 \ 4pt, suitable data = SU(N )…”

Section: Linear Quiver Su(n ) Theoriesmentioning

confidence: 99%

“…This leads to a decomposition of super-Liouville CFT into a Liouville and time-like Liouville pieces [353][354][355][356] The general N, M extension is partially worked out in [20,24,26,29,32] and the coset (8.2) studied further in [357][358][359]. Another perspective is to realize R 4 /Z M by dimensional reduction of R 4 × S 1 , which corresponds to taking q to a root of unity [19,31,[33][34][35] in the q-deformed AGT correspondence we explain later.…”

Section: Coulomb Branch Operatorsmentioning

confidence: 99%

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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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Section: Linear Quiver Su(n ) Theoriesmentioning

confidence: 77%

Section: Linear Quiver Su(n ) Theoriesmentioning

confidence: 99%

Section: Coulomb Branch Operatorsmentioning

confidence: 99%

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A slow review of the AGT correspondence

Floch¹

2020

Preprint

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“…Up to this point we have been working with 4d N = 2 class S theories in Minkowski space. We now turn 37 to Euclidean signature. Our aim in this section and the next is to explain both sides of the AGT relation (1.1) for the case g = su(2) with tame punctures:…”

Section: Localization For 4d Quiversmentioning

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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“…The root of unity limit of 5d gauge theory has been considered [33,34] to explore the instanton counting on A-type ALE space, since the Z k orbifold projection is performed in this limit. See also [35][36][37][38]. From this point of view, it is natural to ask what is the geometric representation theoretical meaning of such a twisted limit for quiver variety.…”

Section: Introductionmentioning

confidence: 99%

Twisted reduction of quiver W-algebras

Kimura,

Pestun

2019

Preprint

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We consider the k-twisted Nekrasov-Shatashvili limit (NS k limit) of 5d (Ktheoretic) and 6d (elliptic) quiver gauge theory, where one of the multiplicative equivariant parameters is taken to be the k-th root of unity. We obtain the extended center of the associated q-deformed quiver W-algebras constructed by our formalism [1-3], which provides gauge theoretic proof of Bouwknegt-Pilch's statement on the relation to the representation ring of quantum affine algebra.

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Elliptic algebra, Frenkel–Kac construction and root of unity limit

Cited by 3 publications

References 122 publications

A slow review of the AGT correspondence

A slow review of the AGT correspondence

A slow review of the AGT correspondence

Twisted reduction of quiver W-algebras

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