The Grassmannian VOA

Eberhardt, Lorenz; Procházka, Tomáš

doi:10.1007/jhep09(2020)150

Cited by 6 publications

(14 citation statements)

References 104 publications

(195 reference statements)

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“…Finally, the OPE between the charged spin-3 current and itself provides the following singular terms. 18 Therefore, we observe that the presence of a neutral spin-1 current with Kronecker delta symbols appears in the OPEs between the nonsinglet currents where the sum of spins of the left hand side is given by odd integer numbers. Although there are some higher order terms which do not appear in the coset realization, we observe that by simply ignoring the above uncharged spin-1 current, all the linear terms in the free field realization arise in the coset realization.…”

Section: Jhep03(2021)037mentioning

confidence: 78%

“…The Thielemans package [16] is used together with the mathematica package [17]. The similar coset in the work of [18] where the possibility of four parameters in the specific coset is described is studied. 2 1 The integer M = 4 is the lowest value in order to have an independent SU(M ) invariant tensors [2].…”

Section: Jhep03(2021)037mentioning

confidence: 99%

“…If we do not expect a new quasi primary operator, then we should manage to rewrite each singular term in terms of the multiple product of known currents. 18 The sixth order pole is proportional to the Kronecker delta symbols and is given by 2 3 δ ab . The fifth order pole contains the spin-1 current and is 2 3 i f abc W −1,c (w).…”

Section: Jhep03(2021)037mentioning

confidence: 99%

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The Grassmannian-like coset model and the higher spin currents

Ahn

2021

J. High Energ. Phys.

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In the Grassmannian-like coset model, $$ \frac{\mathrm{SU}{\left(N+M\right)}_k}{\mathrm{SU}{(N)}_k\times \mathrm{U}{(1)}_{kNM\left(N+M\right)}} $$ SU N + M k SU N k × U 1 kNM N + M , Creutzig and Hikida have found the charged spin-2, 3 currents and the neutral spin-2, 3 currents previously. In this paper, as an extension of Gaberdiel-Gopakumar conjecture found ten years ago, we calculate the operator product expansion (OPE) between the charged spin-2 current and itself, the OPE between the charged spin-2 current and the charged spin-3 current and the OPE between the neutral spin-3 current and itself for generic N, M and k. From the second OPE, we obtain the new charged quasi primary spin-4 current while from the last one, the new neutral primary spin-4 current is found implicitly. The infinity limit of k in the structure constants of the OPEs is described in the context of asymptotic symmetry of MM matrix generalization of AdS3 higher spin theory. Moreover, the OPE between the charged spin-3 current and itself is determined for fixed (N, M) = (5, 4) with arbitrary k up to the third order pole. We also obtain the OPEs between charged spin-1, 2, 3 currents and neutral spin-3 current. From the last OPE, we realize that there exists the presence of the above charged quasi primary spin-4 current in the second order pole for fixed (N, M) = (5, 4). We comment on the complex free fermion realization.

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Section: Jhep03(2021)037mentioning

confidence: 78%

Section: Jhep03(2021)037mentioning

confidence: 99%

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The Grassmannian-like coset model and the higher spin currents

Ahn

2021

J. High Energ. Phys.

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“…For M = 2, the above coset arises in the context of the large N = 4 holography [46]. See also previous works on this holography [2,44,.…”

Section: Introductionmentioning

confidence: 79%

“…For generic k, N and M, this OPE is further studied in [12] and see also [13,14]. 2 Why do we add the complex fermions into the above coset model (1.1)? This is one of the ways to make the bosonic theory to be the supersymmetric theory.…”

Section: Introductionmentioning

confidence: 99%

Adding complex fermions to the Grassmannian-like coset model

Ahn

2021

Eur. Phys. J. C

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In the $${{\mathcal {N}}}=2$$ N = 2 supersymmetric coset model, $$\frac{SU(N+M)_k \times SO(2 N M)_1}{ SU(N)_{k+M} \times U(1)_{ N M (N+M)(k+N+M)}}$$ S U ( N + M ) k × S O ( 2 N M ) 1 S U ( N ) k + M × U ( 1 ) N M ( N + M ) ( k + N + M ) , we construct the SU(M) nonsinglet $${{\mathcal {N}}}=2$$ N = 2 multiplet of spins $$(1, \frac{3}{2}, \frac{3}{2}, 2)$$ ( 1 , 3 2 , 3 2 , 2 ) in terms of coset fields. The next SU(M) singlet and nonsinglet $${{\mathcal {N}}}=2$$ N = 2 multiplets of spins $$(2, \frac{5}{2}, \frac{5}{2}, 3)$$ ( 2 , 5 2 , 5 2 , 3 ) are determined by applying the $${{\mathcal {N}}}=2$$ N = 2 supersymmetry currents of spin $$\frac{3}{2}$$ 3 2 to the bosonic singlet and nonsinglet currents of spin 3 in the bosonic coset model. We also obtain the operator product expansions (OPEs) between the currents of the $${{\mathcal {N}}}=2$$ N = 2 superconformal algebra and above three kinds of $${{\mathcal {N}}}=2$$ N = 2 multiplets. These currents in two dimensions play the role of the asymptotic symmetry, as the generators of $${{\mathcal {N}}}=2$$ N = 2 “rectangular W-algebra”, of the $$M \times M$$ M × M matrix generalization of $$\mathcal{N}=2$$ N = 2 $$AdS_3$$ A d S 3 higher spin theory in the bulk. The structure constants in the right hand sides of these OPEs are dependent on the three parameters k, N and M explicitly. Moreover, the OPEs between SU(M) nonsinglet $${{\mathcal {N}}}=2$$ N = 2 multiplet of spins $$(1, \frac{3}{2}, \frac{3}{2}, 2)$$ ( 1 , 3 2 , 3 2 , 2 ) and itself are analyzed in detail. The complete OPE between the lowest component of the SU(M) singlet $${{\mathcal {N}}}=2$$ N = 2 multiplet of spins $$(2, \frac{5}{2}, \frac{5}{2}, 3)$$ ( 2 , 5 2 , 5 2 , 3 ) and itself is described. In particular, when $$M=2$$ M = 2 , it is known that the above $${{\mathcal {N}}}=2$$ N = 2 supersymmetric coset model provides the realization of the extension of the large $${{\mathcal {N}}}=4$$ N = 4 nonlinear superconformal algebra. We determine the currents of the large $${{\mathcal {N}}}=4$$ N = 4 nonlinear superconformal algebra and the higher spin-$$\frac{3}{2}, 2$$ 3 2 , 2 currents of the lowest $${{\mathcal {N}}}=4$$ N = 4 multiplet for generic k and N in terms of the coset fields. For the remaining higher spin-$$\frac{5}{2},3$$ 5 2 , 3 currents of the lowest $$\mathcal{N}=4$$ N = 4 multiplet, some of the results are given.

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Webs of W-algebras

Procházka¹,

Rapčák

2018

J. High Energ. Phys.

187

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We associate vertex operator algebras to (p, q)-webs of interfaces in the topologically twisted N = 4 super Yang-Mills theory. Y-algebras associated to trivalent junctions are identified with truncations of W 1+∞ algebra. Starting with Y-algebras as atomic elements, we describe gluing of Y-algebras analogous to that of the topological vertex. At the level of characters, the construction matches the one of counting D0-D2-D4 bound states in toric Calabi-Yau threefolds. For some configurations of interfaces, we propose a BRST construction of the algebras and check in examples that both constructions agree. We define generalizations of W 1+∞ algebra and identify a large class of glued algebras with their truncations. The gluing construction sheds new light on the structure of vertex operator algebras conventionally constructed by BRST reductions or coset constructions and provides us with a way to construct new algebras. Many well-known vertex operator algebras, such as U(N) k affine Lie algebra, N = 2 superconformal algebra, N = 2 super-W ∞ , Bershadsky-Polyakov W (2) 3 , cosets and Drinfeld-Sokolov reductions of unitary groups can be obtained as special cases of this construction.

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The Grassmannian VOA

Cited by 6 publications

References 104 publications

The Grassmannian-like coset model and the higher spin currents

The Grassmannian-like coset model and the higher spin currents

Adding complex fermions to the Grassmannian-like coset model

Webs of W-algebras

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