Combinatorics of vertex operators and deformed $W$-algebra of type D$(2,1;α)$

Feigin, Boris; Jimbo, Michio; Mukhin, E.

doi:10.48550/arxiv.2103.15247

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…• Although we focused on the 5d AGT correspondence, it is possible to study 2d/3d [55,102], 4d [62], and 6d generalizations [61,110,111] following previous studies of the intertwiner formalism. Moreover, we can change the space time which the theory is defined on or introduce defects to the system by changing the base quantum toroidal algebra to other quantum toroidal algebras such as quantum toroidal gl n [63,86], gl m | n [112,113], D(2, 1; α) [93,114], and toroidal quiver BPS algebras [87,89,90,92,93].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

5d AGT correspondence of supergroup gauge theories from quantum toroidal $\mathfrak{gl}_{1}$

Noshita¹

2022

Preprint

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We discuss the 5d AGT correspondence of supergroup gauge theories with A-type supergroups. We introduce two intertwiners called positive and negative intertwiners to compute the instanton partition function. The positive intertwiner is the ordinary Awata-Feigin-Shiraishi intertwiner while the negative intertwiner is an intertwiner obtained by using central charges with negative levels. We show that composition of them gives the basic Nekrasov factors appearing in supergroup partition functions. We explicitly derive the instanton partition functions of supergroup gauge theories with A and D-type quiver structures. Using the intertwiners, we briefly study the Gaiotto state, qq-characters and the relation with quiver W-algebra. Furthermore, we show that the negative intertwiner corresponds to the anti-refined topological vertex recently defined by Kimura and Sugimoto. We also discuss how superquiver theories should appear in our formalism if they exist. The existence of the AGT correspondence of the theories we study in this paper implies that there is a broader 2d/4d (5d/q-algebra) correspondence, or more generally the BPS/CFT correspondence, where new non-unitary theories play important roles.

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Section: Conclusion and Discussionmentioning

confidence: 99%

5d AGT correspondence of supergroup gauge theories from quantum toroidal $\mathfrak{gl}_{1}$

Noshita¹

2022

Preprint

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“…Algebro-geometrically, this moduli space admits a description and virtual cycle in the style of Oh-Thomas [OT20] 1 . Combinatorially, (the original) qq-characters may be constructed by recursive expansion [FJM21] in a similar fashion as for q-characters [FM01]. This is a great approach for explicit computation, especially for A Γ -modules which are not geometrically realizable like in (2), but it is not the direction we will take in this paper.…”

Section: 13mentioning

confidence: 99%

A representation-theoretic approach to $qq$-characters

Liu¹

2022

Preprint

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We raise the question of whether (a slightly generalized notion of) qq-characters can be constructed purely representation-theoretically. In the main example of the quantum toroidal gl 1 algebra, geometric engineering of adjoint matter produces an explicit vertex operator RR which computes certain qq-characters, namely Hirzebruch χ y -genera, completely analogously to how the R-matrix R computes q-characters. We prove the independence of preferred direction for the refined vertex in this and more general non-toric settings.

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“…We suppose that this is the quantum toroidal D(2, 1; α), which is yet to be studied in detail. See [47,48] for recent developments. For the Drinfeld second realization of the quantum affine superalgebras of D(2, 1; α), see [49].…”

Section: Some Issues On the Asymmetric Quivermentioning

confidence: 99%

A Note on Quiver Quantum Toroidal Algebra

Noshita,

Watanabe

2021

Preprint

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Recently, Li and Yamazaki proposed a new class of infinite-dimensional algebras, quiver Yangian, which generalizes the affine Yangian. The characteristic feature of the algebra is the action on BPS states for non-compact toric Calabi-Yau threefolds, which are in one-to-one correspondence with the crystal melting models. These algebras can be bootstrapped from the action on the crystals and have various truncations.In this paper, we propose a q-deformed version of the quiver Yangian, referred to as the quiver quantum toroidal algebra (QQTA). We examine some of the consistency conditions of the algebra. In particular, we show that QQTA is a Hopf superalgebra with a formal super coproduct, like known quantum toroidal algebras. QQTA contains an extra central charge. When it is trivial, QQTA has the three-dimensional representation acting on the three-dimensional crystals, like Li-Yamazaki's quiver Yangian. When Calabi-Yau threefolds have compact 4-cycles, the action seems to have some inconsistency, whose nature is not clear at this moment.

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Combinatorics of vertex operators and deformed $W$-algebra of type D$(2,1;α)$

Cited by 4 publications

References 14 publications

5d AGT correspondence of supergroup gauge theories from quantum toroidal $\mathfrak{gl}_{1}$

5d AGT correspondence of supergroup gauge theories from quantum toroidal $\mathfrak{gl}_{1}$

A representation-theoretic approach to $qq$-characters

A Note on Quiver Quantum Toroidal Algebra

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