Fermionic coset, critical level $ \mathcal{W}_4^{{(2)}} $ -algebra and higher spins

Creutzig, Thomas; Gao, Peng; Linshaw, Andrew R.

doi:10.1007/jhep04(2012)031

Cited by 3 publications

(2 citation statements)

References 102 publications

(147 reference statements)

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“…One can use these results to also study the representation theory of W 3−p+(p−1) −1 (sl(p − 1|1)). Beyond this there have appeared interesting realizations of the subregular W-algebra at the critical level [CGL1,CGL2,GK] together with some connection to geometry and physics. At the critical level the subregular W-algebra has a large center.…”

Section: 5mentioning

confidence: 99%

Duality of subregular W-algebras and principal W-superalgebras

Creutzig,

Genra,

Nakatsuka

2020

Preprint

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We prove Feigin-Frenkel type dualities between subregular Walgebras of type A, B and principal W-superalgebras of type sl(1|n), osp(2|2n). The type A case proves a conjecture of Feigin and Semikhatov.Let (g 1 , g 2 ) = (sl n+1 , sl(1|n + 1)) or (so 2n+1 , osp(2|2n)) and let r be the lacity of g 1 . Let k be a complex number and ℓ defined by r(ki the dual Coxeter numbers of the g i . Our first main result is that the Heisenberg cosets C k (g 1 ) and C ℓ (g 2 ) of these W-algebras at these dual levels are isomorphic, i.e. C k (g 1 ) ≃ C ℓ (g 2 ) for generic k. We determine the generic levels and furthermore establish analogous results for the cosets of the simple quotients of the W-algebras.Our second result is a novel Kazama-Suzuki type coset construction: We show that a diagonal Heisenberg coset of the subregular W-algebra at level k times the lattice vertex superalgebra V Z is the principal W-superalgebra at the dual level ℓ. Conversely a diagonal Heisenberg coset of the principal W-superalgebra at level ℓ times the lattice vertex superalgebra V √ −1Z is the subregular W-algebra at the dual level k. Again this is proven for the universal W-algebras as well as for the simple quotients.We show that a consequence of the Kazama-Suzuki type construction is that the simple principal W-superalgebra and its Heisenberg coset at level ℓ are rational and/or C 2 -cofinite if the same is true for the simple subregular W-algebra at dual level ℓ. This gives many new C 2 -cofiniteness and rationality results.

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Section: 5mentioning

confidence: 99%

Duality of subregular W-algebras and principal W-superalgebras

Creutzig,

Genra,

Nakatsuka

2020

Preprint

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“…The precise form of the conjecture has recently been formulated in [20]. Support for this conjecture has been given in [21,22], where the W (2) n -algebra at critical level has been constructed. At critical level the quantum Hamiltonian reduction is guaranteed to have a large center, and indeed also the W (2) n -algebra at critical level has such a large center.…”

Section: Introductionmentioning

confidence: 96%

Unitary W-algebras and three-dimensional higher spin gravities with spin one symmetry

Afshar

Creutzig

Grumiller

et al. 2014

J. High Energ. Phys.

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We investigate whether there are unitary families of W-algebras with spin one fields in the natural example of the Feigin-Semikhatov W (2) n -algebra. This algebra is conjecturally a quantum Hamiltonian reduction corresponding to a non-principal nilpotent element. We conjecture that this algebra admits a unitary real form for even n. Our main result is that this conjecture is consistent with the known part of the operator product algebra, and especially it is true for n = 2 and n = 4. Moreover, we find certain ranges of allowed levels where a positive definite inner product is possible. We also find a unitary conformal field theory for every even n at the special level k + n = (n + 1)/(n − 1). At these points, the W (2) n -algebra is nothing but a compactified free boson. This family of W-algebras admits an 't Hooft limit. Further, in the case of n = 4, we reproduce the algebra from the higher spin gravity point of view. In general, gravity computations allow us to reproduce some leading coefficients of the operator product.

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A Commutant Realization of 𝒲n(2) at Critical Level

Creutzig

Gao

Linshaw

2012

International Mathematics Research Notices

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For n ≥ 2, there is a free field realization of the affine vertex superalgebra A associated to psl(n|n) at critical level inside the bcβγ system W of rank n 2 . We show that the commutant C = Com(A, W) is purely bosonic and is freely generated by n + 1 fields. We identify the Zhu algebra of C with the ring of invariant differential operators on the space of n × n matrices under SL n × SL n , and we classify the irreducible, admissible C-modules with finite dimensional graded pieces. For n ≤ 4, C is isomorphic to the W (2) n -algebra at critical level, and we conjecture that this holds for all n.

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Fermionic coset, critical level $ \mathcal{W}_4^{{(2)}} $ -algebra and higher spins

Cited by 3 publications

References 102 publications

Duality of subregular W-algebras and principal W-superalgebras

Duality of subregular W-algebras and principal W-superalgebras

Unitary W-algebras and three-dimensional higher spin gravities with spin one symmetry

A Commutant Realization of 𝒲n(2) at Critical Level

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