Universal two-parameter even spin <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="script">W</mml:mi></mml:mrow><mml:mrow><mml:mo>∞</mml:mo></mml:mrow></mml:msub></mml:math>-algebra

Kanade, Shashank; Linshaw, Andrew R.

doi:10.1016/j.aim.2019.106774

Cited by 22 publications

(31 citation statements)

References 49 publications

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“…For special values of parameters its quotients are W N algebras associated to A k series of simple Lie algebras. It also contains two-parametric family of even spin W ev∞ whose quotients are orthosymplectic W-algebras associated B k , C k and D k series [15,49]. The W 1+∞ family admits a completely different description as Yangian of gl (1), an associative algebra given by generators and relations [50,51,52].…”

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confidence: 99%

Instanton R-matrix and $$ \mathcal{W} $$-symmetry

Procházka¹

2019

J. High Energ. Phys.

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We study the relation between W 1+∞ algebra and Arbesfeld-Schiffmann-Tsymbaliuk Yangian using the Maulik-Okounkov R-matrix. The central object linking these two pictures is the Miura transformation. Using the results of Nazarov and Sklyanin we find an explicit formula for the mixed R-matrix acting on two Fock spaces associated to two different asymptotic directions of the affine Yangian. Using the free field representation we propose an explicit identification of Arbesfeld-Schiffmann-Tsymbaliuk generators with the generators of Maulik-Okounkov Yangian. In the last part we use the Miura transformation to give a conformal field theoretic construction of conserved quantities and ladder operators in the quantum mechanical rational and trigonometric Calogero-Sutherland models on which a vector representation of the Yangian acts.

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confidence: 99%

Instanton R-matrix and $$ \mathcal{W} $$-symmetry

Procházka¹

2019

J. High Energ. Phys.

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“…This truncation curve was computed in [MPS] and is easily seen to coincide with the specialization of the curve for W k (so 2n ) Z 2 to the case n = 2. Therefore [KL,Thm. 9.4] holds for all n ≥ 1 and m ≥ 2, and other results of this kind can be similarly improved.…”

Section: Vertex Algebrasmentioning

confidence: 95%

“…, let W 0 and W 1 be the simple quotients of W ev,I 0 (c, λ) and W ev,I 1 (c, λ). A criterion for W 0 and W 1 to be isomorphic is given by [KL,Thm. 8.1]; aside from a few degenerate cases, we must have c 0 = c 1 and λ 0 = λ 1 .…”

Section: Vertex Algebrasmentioning

confidence: 99%

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N= 4 Superconformal Algebras and Diagonal Cosets

Creutzig

Feigin

Linshaw

2020

International Mathematics Research Notices

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Coset constructions of ${{\mathcal{W}}}$-algebras have many applications and were recently given for principal ${{\mathcal{W}}}$-algebras of $A$, $D$, and $E$ types by Arakawa together with the 1st and 3rd authors. In this paper, we give coset constructions of the large and small $N=4$ superconformal algebras, which are the minimal ${{\mathcal{W}}}$-algebras of ${{\mathfrak{d}}}(2,1;a)$ and ${{\mathfrak{p}}}{{\mathfrak{s}}}{{\mathfrak{l}}}(2|2)$, respectively. From these realizations, one finds a remarkable connection between the large $N=4$ algebra and the diagonal coset $C^{k_1, k_2} = \textrm{Com}(V^{k_1+k_2}({{\mathfrak{s}}}{{\mathfrak{l}}}_2), V^{k_1}({{\mathfrak{s}}}{{\mathfrak{l}}}_2) \otimes V^{k_2}({{\mathfrak{s}}}{{\mathfrak{l}}}_2))$, namely, as two-parameter vertex algebras, $C^{k_1, k_2}$ coincides with the coset of the large $N=4$ algebra by its affine subalgebra. We also show that at special points in the parameter space, the simple quotients of these cosets are isomorphic to various ${{\mathcal{W}}}$-algebras. As a corollary, we give new examples of strongly rational principal ${{\mathcal{W}}}$-algebras of type $C$ at degenerate admissible levels.

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“…Although we collected some evidence for the existence of the four-parametric family of algebras unifying the Grassmannian cosets with the orthosymplectic cosets, the four-parametric algebra itself (if it exists) has not been constructed. One possibility would be to use the bootstrap approach as in [47,80], but due to large number of primary fields in Grassmannian this approach seems to be much more complicated. The W 1+∞ can be also constructed from its free field representations and coproduct, but the coproduct structure is not manifest in the coset description and for this reason it is not clear at this moment if such a structure exists in the Grassmannian or in the hypothetical four-parametric family.…”

Section: Jhep09(2020)150mentioning

confidence: 99%

The Grassmannian VOA

Eberhardt

Procházka²

2020

J. High Energ. Phys.

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We study the 3-parametric family of vertex operator algebras based on the Grassmannian coset CFT $$ \mathfrak{u} $$ u (M + N )k /($$ \mathfrak{u} $$ u (M )k×$$ \mathfrak{u} $$ u (N )k ). This VOA serves as a basic building block for a large class of cosets and generalizes the $$ {\mathcal{W}}_{\infty } $$ W ∞ algebra. We analyze representations and their characters in detail and find surprisingly simple character formulas for the representations in the generic parameter regime that admit an elegant combinatorial formulation. We also discuss truncations of the algebra and give a conjectural formula for the complete set of truncation curves. We develop a theory of gluing for these algebras in order to build more complicated coset and non-coset algebras. We demonstrate the power of this technology with some examples and show in particular that the $$ \mathcal{N} $$ N = 2 supersymmetric Grassmannian can be obtained by gluing three bosonic Grassmannian algebras in a loop. We finally speculate about the tantalizing possibility that this algebra is a specialization of an even larger 4-parametric family of algebras exhibiting pentality symmetry. Specializations of this conjectural family should include both the unitary Grassmannian family as well as the Lagrangian Grassmannian family of VOAs which interpolates between the unitary and the orthosymplectic cosets.

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Universal two-parameter even spin W∞-algebra

Cited by 22 publications

References 49 publications

Instanton R-matrix and $$ \mathcal{W} $$-symmetry

Instanton R-matrix and $$ \mathcal{W} $$-symmetry

N= 4 Superconformal Algebras and Diagonal Cosets

The Grassmannian VOA

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