Rationality of the exceptional W-algebras $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$ associated with subregular nilpotent elements of $\mathfrak{sp}_4$

Fasquel, Justine

doi:10.48550/arxiv.2009.09513

Cited by 3 publications

(5 citation statements)

References 28 publications

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“…The most-studied nilpotent elements in a simple Lie algebra are probably those in the principal, subregular, and minimal nilpotent orbits. Thus Main Theorem 4 finishes the proof that all exceptional W -algebras associated to these three nilpotent orbits are strongly rational; many cases (including all principal W -algebras) were already known thanks to [Wan,Ara2,Ara4,AvE,CL1,CL2,CL3,Fa].…”

Section: Introductionmentioning

confidence: 58%

On rationality for $C_2$-cofinite vertex operator algebras

McRae¹

2021

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Let V be an N-graded, simple, self-contragredient, C2-cofinite vertex operator algebra. We show that if the S-transformation of the character of V is a linear combination of characters of V -modules, then the category C of grading-restricted generalized V -modules is a rigid tensor category. We further show, without any assumption on the character of V but assuming that C is rigid, that C is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of V is semisimple, then C is semisimple and thus V is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated to V .We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that C2-cofinite affine W -algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such W -algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the "coset rationality problem" to the problem of C2-cofiniteness for the coset. That is, given a vertex operator algebra inclusion U ⊗ V ֒→ A with A, U strongly rational and U , V a pair of mutual commutant subalgebras in A, we show that V is also strongly rational provided it is C2-cofinite. ROBERT MCRAE 6. Applications 61 6.1. Rationality for C 2 -cofinite W -algebras 61 6.2. Rationality for C 2 -cofinite cosets 66 Appendix A. Differential equations 69 Appendix B. The assumption in Theorem 6.4 for odd nilpotent orbits 74 B.1. Classical types 74 B.2. Exceptional types 76 References 82

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Section: Introductionmentioning

confidence: 58%

On rationality for $C_2$-cofinite vertex operator algebras

McRae¹

2021

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“…W k (g, f ) is rational if k is admissible and f ∈ O k . Conjecture 4.6 has been proved in the cases where f is principal [9], g is of type A [13], g is of type ADE and f is subregular [13], and g is of type B 2 and f is subregular [45].…”

Section: Asymptotic Data Of W -Algebrasmentioning

confidence: 99%

Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras

Arakawa,

van Ekeren,

Moreau

2021

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We apply results from the geometry of nilpotent orbits and nilpotent Slodowy slices, together with modularity and asymptotic analysis of characters, to prove many new isomorphisms between affine W -algebras and affine Kac-Moody vertex algebras and their finite extensions at specific admissible levels. In particular we identify many new collapsing levels for W -algebras.Argyres-Douglas theories in the 4D/2D duality mentioned above ([99, 108, 104]). Collapsing levels which are boundary principal admissible levels have been studied by Xie and Yan [106] in view of the 4D/2D duality, and our results confirm their conjectures [106, 3.4].We also find some cases whereHere, we say a vertex algebra V is a finite extension of a vertex algebra W if W is a conformal vertex subalgebra of V and V is a finite direct sum of simple W -modules.Note that the validity of Conjecture 1.2 in particular implies the widely-believed fact that if V is a finite extension of W and V is lisse, then W is lisse as well.We find examples where W k (g, f ) is a finite extension of L k ♮ (g ♮ ), X W k (g,f ) is not isomorphic to X L k ♮ (g ♮ ) but is birationally equivalent to X L k ♮ (g ♮ ) , see Remarks 10.

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“…The most studied, non-type , non-super example is the type 2 ( = 4 ≃ 5 ) subregular W-algebra whose operator product expansions are listed in Section 5 of [18]. The representation theory of W k ( 4 , sub ) at certain levels has been explored [18], and work on the inverse quantum hamiltonian reduction question has been done from a 4D superconformal field theoretic perspective [34].…”

Section: S W- W--mentioning

confidence: 99%

“…This connection is expressed beautifully in the associated variety of the subregular W-algebra [14]. In light of these and many more motivations, much recent work has been done to improve our understanding of the structure and representation theory of subregular W-algebras [15][16][17][18][19][20][21].In general, an understanding of W-algebras for nonregular nilpotents is highly desirable. One means of achieving this is to leverage the well-understood W-algebras to learn about others: It is strongly suspected that in addition to the usual quantum hamiltonian reduction, one can also perform a 'partial quantum hamiltonian reduction' between two W-algebras as long as their corresponding nilpotent orbits are related by a certain partial ordering [22].…”

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confidence: 99%

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Subregular W-algebras of type A

Fehily

2021

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Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the +1 subregular W-algebra can be realised in terms of the +1 regular W-algebra and the half lattice vertex algebra Π. This generalises the realisations found for = 1 and 2 in [1,2] and can be interpreted as an inverse quantum hamiltonian reduction in the sense of Adamović. We use this realisation to explore the representation theory of +1 subregular W-algebras. Much of the structure encountered for 2 and 3 is also present for +1 . Particularly, the simple +1 subregular W-algebra at nondegenerate admissible levels can be realised purely in terms of the W +1 minimal model vertex algebra and Π.1. I 1.1. Background. Given a simple Lie superalgebra , a complex number k and a nilpotent element ∈ , there exists a cochain complex of vertex operator algebras whose zeroth cohomology has the structure of a vertex operator algebra W k ( , ) called a W-algebra [3,4]. This process is called quantum hamiltonian reduction of the universal affine vertex algebra V k ( ). By choosing the nilpotent element carefully, one obtains vertex operator algebras that are particularly easy to work with. Historically, a popular choice of nilpotent element has been the regular (or principal) nilpotent element reg ∈ . The associated W-algebra W k ( , reg ) has been studied extensively and has been at the forefront of many developments in mathematics and physics. See [5-9] for example.Recent work in physics and mathematics has indicated the importance of the W-algebra W k ( , sub ) associated to subregular nilpotent elements sub ∈ . For example, subregular W-algebras appear in the Schur-index of 4 superconformal field theories known as Argyres-Douglas theories [10][11][12]. Subregular nilpotent elements and their nilpotent orbits also play a crucial role in singularity theory: the ADE classification of simple surface singularities connects to the ADE classification of simply-laced Lie algebras through the geometry of the Slodowy slice corresponding to the subregular nilpotent orbit [13]. This connection is expressed beautifully in the associated variety of the subregular W-algebra [14]. In light of these and many more motivations, much recent work has been done to improve our understanding of the structure and representation theory of subregular W-algebras [15][16][17][18][19][20][21].In general, an understanding of W-algebras for nonregular nilpotents is highly desirable. One means of achieving this is to leverage the well-understood W-algebras to learn about others: It is strongly suspected that in addition to the usual quantum hamiltonian reduction, one can also perform a 'partial quantum hamiltonian reduction' between two W-algebras as long as their corresponding nilpotent orbits are related by a certain partial ordering [22]. In fact, a partial reduction for finite W-algebras has been defined for certain cases [23]. This is a good sign as finite W-algebras arise as the Zhu algebras of W-a...

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Rationality of the exceptional W-algebras $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$ associated with subregular nilpotent elements of $\mathfrak{sp}_4$

Abstract: We prove the rationality of the exceptional W-algebras W k (g, f ) associated with the simple Lie algebra g = sp 4 and a subregular nilpotent element f = f subreg of sp 4 , proving a new particular case of a conjecture of Kac-Wakimoto.

Cited by 3 publications

References 28 publications

On rationality for $C_2$-cofinite vertex operator algebras

On rationality for $C_2$-cofinite vertex operator algebras

Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras

Subregular W-algebras of type A

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