On the irreducibility of associated varieties of W-algebras

Arakawa, Tomoyuki; Moreau, Anne

doi:10.1016/j.jalgebra.2017.06.007

Cited by 13 publications

(12 citation statements)

References 42 publications

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“…• First consider the theory constructed from using a trivial nilpotent orbit f , so that the 4d theory has a flavor symmetry group G. The corresponding VOA is the nonadmissible AKM V −h ∨ + n u (g), and we assume that its associated variety O g,n,u is irreducible (see the discussion in [99]) and is the closure of a nilpotent orbit O g,n,u .…”

Section: Adding Regular Singularitymentioning

confidence: 99%

4d $$ \mathcal{N} $$ = 2 SCFTs and lisse W-algebras

Xie

Yan

2021

J. High Energ. Phys.

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We continue our studies of the correspondence between 4d $$ \mathcal{N} $$ N = 2 SCFTs and 2d W-algebras. The purpose of this paper is to study the relationship between 2d lisse W-algebras and their 4d SCFT partners. The lisse W-algebra is the W-algebra whose associated Zhu’s C2 algebra is finite dimensional. As the associated variety of Zhu’s C2 algebra is identified with the Higgs branch in the 4d/2d correspondence, the lisse condition is equivalent to the absence of the Higgs branch on the 4d side. We classify 4d $$ \mathcal{N} $$ N = 2 SCFTs which do not admit Higgs branch, then these theories would give lisse W-algebras through the 4d/2d correspondence. In particular, we predict the existence of a large class of new non-admissible lisse W-algebras, which have not been studied before. The 4d theories corresponding to lisse W-algebra can appear in the Higgs branches of generic 4d $$ \mathcal{N} $$ N = 2 SCFTs, therefore they are crucial to understand the Higgs branches of $$ \mathcal{N} $$ N = 2 SCFTs.

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Section: Adding Regular Singularitymentioning

confidence: 99%

4d $$ \mathcal{N} $$ = 2 SCFTs and lisse W-algebras

Xie

Yan

2021

J. High Energ. Phys.

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“…The Higgs branch of the above quiver gauge theory is identical to the Higgs branch of the (A N N −1 [−N + 2], F ) theory. [63,64], and the corresponding associated variety is exactly the minimal nilpotent orbit.…”

Section: Examplesmentioning

confidence: 99%

Vertex operator algebras of Argyres-Douglas theories from M5-branes

Song

Xie

Yan

2017

J. High Energ. Phys.

157

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We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N = (2, 0) theory of type J on a punctured sphere. We denote the AD theories as (and Y represent an irregular and a regular singularity respectively. We restrict to the 'minimal' case where J b [k] has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the Walgebra W k 2d (J, Y ), where k 2d = −h + b b+k with h being the dual Coxeter number of J. We verify this conjecture by showing that the Schur index of the AD theory is identical to the vacuum character of the corresponding VOA, and the Hall-Littlewood index computes the Hilbert series of the Higgs branch. We also find that the Schur and Hall-Littlewood index for the AD theory can be written in a simple closed form for b = h. We also test the conjecture that the associated variety of such VOA is identical to the Higgs branch. The M5-brane construction of these theories and the corresponding TQFT structure of the index play a crucial role in our computations.

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“…At some point in the stabilization process, we have to resort to the normality of certain classical nilpotent orbits. Mysteriously, these orbits showed up as well in the study of W -algebras [AM18] which suggests intricate relations between symmetric pairs and W -algebras deserving to be further explored.…”

Section: Introductionmentioning

confidence: 93%

Quasi-split symmetric pairs of 𝑈(𝔰𝔩_{𝔫}) and Steinberg varieties of classical type

2021

Represent. Theory

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We provide a Lagrangian construction for the fixed-point subalgebra, together with its idempotent form, in a quasi-split symmetric pair of type A n − 1 A_{n-1} . This is obtained inside the limit of a projective system of Borel-Moore homologies of the Steinberg varieties of n n -step isotropic flag varieties. Arising from the construction are a basis of homological origin for the idempotent form and a geometric realization of rational modules.

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On the irreducibility of associated varieties of W-algebras

Abstract: Abstract. We investigate the irreducibility of the nilpotent Slodowy slices that appear as the associated variety of W -algebras. Furthermore, we provide new examples of vertex algebras whose associated variety has finitely many symplectic leaves.

Cited by 13 publications

References 42 publications

4d $$ \mathcal{N} $$ = 2 SCFTs and lisse W-algebras

4d $$ \mathcal{N} $$ = 2 SCFTs and lisse W-algebras

Vertex operator algebras of Argyres-Douglas theories from M5-branes

Quasi-split symmetric pairs of 𝑈(𝔰𝔩_{𝔫}) and Steinberg varieties of classical type

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