Plane partitions with a “pit”: generating functions and representation theory

Recently, Gaiotto and Rapčák (GR) proposed a new family of the vertex operator algebra (VOA) as the symmetry appearing at an intersection of five-branes to which they refer as Y algebra. Procházka and Rapčák, then proposed to interpret Y algebra as a truncation of affine Yangian whose module is directly connected to plane partitions (PP). They also developed GR's idea to generate a new VOA by connecting plane partitions through an infinite leg shared by them and referred it as the web of W-algebra (WoW). In this paper, we demonstrate that double truncation of PP gives the minimal models of such VOAs. For a single PP, it generates all the minimal model irreducible representations of W -algebra. We find that the rule connecting two PPs is more involved than those in the literature when the U (1) charge connecting two PPs is negative. For the simplest nontrivial WoW, N = 2 superconformal algebra, we demonstrate that the improved rule precisely reproduces the known character of the minimal models.After the success of two-dimensional conformal field theory (CFT), the construction of the new chiral algebras and the study of their representation theory such as their minimal models had been an essential issue in string theory during the 80s. The first examples were the superconformal algebra (SCA) and the W-algebras [1].In the process of proving Alday-Gaiotto-Tachikawa (AGT) duality [2], there was a dramatical change in the analysis of the chiral algebra [3]. Instead of using the module generated by applying the chiral algebra generators, one introduces the orthogonal basis labeled by the Young diagrams, which also distinguish the fixed points in the instanton moduli space of the supersymmetric Yang-Mills theory. Such basis describes a representation of affine Yangian [3,4,5,6] 1 . The equivalence between the W n -algebra with U(1) current and the affine Yangian was essential in the proof of AGT conjecture. Indeed, the affine Yangian is equivalent to W 1+∞ [µ] [7,8] which contains W n algebra as its truncation.Recently, Gaiotto and Rapčák [9] introduced a vertex operator algebra (VOA) through the intersection of D5, NS5 and (−1, −1) 5-brane and putting various numbers of D3-branes between these 5-branes. The algebra is called Y L,M,N [Ψ] where L, M, N are the non-negative integers which represent the number of D3-branes and Ψ ∈ C is the parameters of the algebra. By the analysis of gauge theory with the interfaces, they showed that the chiral algebra (or VOA) thus constructed can be described as the BRST reduction of various super Lie algebras. At the same time, the use of the trivalent vertex implies that it would be natural to connect them in the form of the Feynman diagram to generate new VOAs.Procházka and Rapčák [10] developed this idea further. They used a realization of Y L,M,N algebra through the affine Yangian. They used the fact that the plane partition (PP) with three asymptotes written by Young diagrams gave a natural representation of the affine Yangian and identified Y L,M,N with a degenerate representation...

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“…The distinction between the different representation is described by the modification of the leg factor in x 1 direction. 13…”

Section: N = 1 Casementioning

confidence: 99%

Plane partition realization of (web of) $$ \mathcal{W} $$-algebra minimal models

Harada

Matsuo

2019

J. High Energ. Phys.

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“…It was observed in [27] (based on results of [25,[28][29][30]) that W 1+∞ algebra actually contains a three-parameter family of truncations Y N 1 ,N 2 ,N 3 parametrized by non-negative integers N i . These more general truncations can be furthermore identified with VOAs from [31] defined in terms of the quantum Hamiltonian reduction.…”

Section: Jhep01(2020)042mentioning

confidence: 99%

On extensions of $$ \mathfrak{gl}\widehat{\left(\left.m\right|n\right)} $$ Kac-Moody algebras and Calabi-Yau singularities

Rapčák

2020

J. High Energ. Phys.

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We discuss a class of vertex operator algebras W m|n×∞ generated by a supermatrix of fields for each integral spin 1, 2, 3,. .. . The algebras admit a large family of truncations that are in correspondence with holomorphic functions on the Calabi-Yau singularity given by solutions to xy = z m w n. We propose a free-field realization of such truncations generalizing the Miura transformation for W N algebras. Relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations. The discussion provides a concrete example of a non-trivial interplay between vertex operator algebras, algebraic geometry and gauge theory.

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“…The variety A M r (n) is an affine bundle on M r (n) A , whose rank is equal to the codimension of A M r (n) in M r (n). 7 When c (1) , c (2) = 0, in [93], the Heisenberg subalgebra of SH c is generated by {b −l , b l , b 0 , E 0 | l ≥ 0}. To compare with the notation in the current paper, we have…”

Section: Hyperbolic Localizationmentioning

confidence: 99%

“…We define W r1,r2,r3 as a subalgebra of a tensor product of m = r 1 +r 2 +r 3 Heisenberg algebras. The algebra W r1,r2,r3 can be defined according to [7,64,89] as an intersection of kernels of vertex operators of the form…”

Section: Hyperbolic Localizationmentioning

confidence: 99%

Cohomological Hall Algebras, Vertex Algebras and Instantons

Rapčák

Soibelman

Yang

et al. 2019

Commun. Math. Phys.

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We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of gl(1). Based on that we derive the vertex algebra at the corner Wr 1 ,r 2 ,r 3 of Gaiotto and Rapčák. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau 3-folds.

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Plane partitions with a “pit”: generating functions and representation theory

Cited by 59 publications

References 49 publications

Plane partition realization of (web of) $$ \mathcal{W} $$-algebra minimal models

Plane partition realization of (web of) $$ \mathcal{W} $$-algebra minimal models

On extensions of $$ \mathfrak{gl}\widehat{\left(\left.m\right|n\right)} $$ Kac-Moody algebras and Calabi-Yau singularities

Cohomological Hall Algebras, Vertex Algebras and Instantons

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