SH c realization of minimal model CFT: triality, poset and Burge condition

122

Abstract:The relation between the bosonic higher spin W ∞ [λ] algebra, the affine Yangian of gl 1 , and the SH c algebra is established in detail. For generic λ we find explicit expressions for the low-lying W ∞ [λ] modes in terms of the affine Yangian generators, and deduce from this the precise identification between λ and the parameters of the affine Yangian. Furthermore, for the free field cases corresponding to λ = 0 and λ = 1 we give closed-form expressions for the affine Yangian generators in terms of the free fields. Interestingly, the relation between the W ∞ modes and those of the affine Yangian is a non-local one, in general. We also establish the explicit dictionary between the affine Yangian and the SH c generators. Given that Yangian algebras are the hallmark of integrability, these identifications should pave the way towards uncovering the relation between the integrable and the higher spin symmetries.

“…The triality symmetry of the affine Yangian, which we studied in section 3.4, has also an incarnation for the case of SH c ; this was already studied in [37].…”

Section: Jhep04(2017)152mentioning

confidence: 96%

Higher spins and Yangian symmetries

Gaberdiel

Gopakumar

et al. 2017

122

“…Here we provide an alternative approach to the refined character: the partially ordered set (POSET) [27] approach. The POSET method was developed in [28,29] to compute the character of W-algebraic models with singular vectors.…”

Section: Song's Prescription For Chiral Algebramentioning

confidence: 99%

Testing Macdonald index as a refined character of chiral algebra

Watanabe

Zhu

2020

We test in (A n−1 , A m−1 ) Argyres-Douglas theories with gcd(n, m) = 1 the proposal of Song's in [1] that the Macdonald index gives a refined character of the dual chiral algebra. In particular, we extend the analysis to higher rank theories and Macdonald indices with surface operator, via the TQFT picture and Gaiotto-Rastelli-Razamat's Higgsing method. We establish the prescription for refined characters in higher rank minimal models from the dual (A n−1 , A m−1 ) theories in the large m limit, and then provide evidence for Song's proposal to hold (at least) in some simple modules (including the vacuum module) at finite m. We also discuss some observed mismatch in our approach.

“…An affine Yangian description of the minimal model was studied in [36]. Here we give an alternative picture by PP.…”

Section: Analytic Continuation To Negative Weightmentioning

confidence: 99%

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

Harada

Matsuo

2019

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...