Quadratic relations of the deformed $W$-superalgebra ${\cal W}_{q, t}\bigl(A(M,N)\bigr)$

Kojima, Takeo

doi:10.48550/arxiv.2101.01110

Cited by 4 publications

(7 citation statements)

References 13 publications

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“…Free field realizations were derived in [40,52]. We can consider a q-deformation of the CVOA [51,[53][54][55][56], and it is obtained by taking tensor products of Fock representations. These Fock representations are associated with the divisors of C 3 and have a nontrivial central charge of C. We expect this will be the same situation for general toric Calabi-Yau manifolds.…”

Section: Summary and Discussionmentioning

confidence: 99%

A Note on Quiver Quantum Toroidal Algebra

Noshita,

Watanabe

2021

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Recently, Li and Yamazaki proposed a new class of infinite-dimensional algebras, quiver Yangian, which generalizes the affine Yangian. The characteristic feature of the algebra is the action on BPS states for non-compact toric Calabi-Yau threefolds, which are in one-to-one correspondence with the crystal melting models. These algebras can be bootstrapped from the action on the crystals and have various truncations.In this paper, we propose a q-deformed version of the quiver Yangian, referred to as the quiver quantum toroidal algebra (QQTA). We examine some of the consistency conditions of the algebra. In particular, we show that QQTA is a Hopf superalgebra with a formal super coproduct, like known quantum toroidal algebras. QQTA contains an extra central charge. When it is trivial, QQTA has the three-dimensional representation acting on the three-dimensional crystals, like Li-Yamazaki's quiver Yangian. When Calabi-Yau threefolds have compact 4-cycles, the action seems to have some inconsistency, whose nature is not clear at this moment.

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Section: Summary and Discussionmentioning

confidence: 99%

A Note on Quiver Quantum Toroidal Algebra

Noshita,

Watanabe

2021

Preprint

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“…• Studying horizontal representations (C = 1) [10,[16][17][18][19][20][21][22][23][24]69] of shifted QQTA is also one of the studies that must be done. Studying generalized intertwiners [29,30,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] with shift parameters as Φ r,r : (vertical) r ⊗ (horizontal) r → (horizontal) r+r is interesting.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…On the other hand, representations with C = 1 are called horizontal representations. We will not discuss about these representations, so see [10,[16][17][18][19][20][21][22][23][24] for details. See also [69] for a good review.…”

Section: Degenerate Limitmentioning

confidence: 99%

“…Infinite-dimensional alebra has been one of the most powerful tools to study supersymmetric gauge theories [1][2][3][4][5]. In particular, quantum toroidal algebras [6][7][8][9][10][11][12][13][14][15] and their truncations [10,[16][17][18][19][20][21][22][23][24] have played significant roles in the context of 5d AGT correspondence [25][26][27][28][29][30][31][32] and topological vertex [29,30,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]…”

Section: Introductionmentioning

confidence: 99%

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Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

Noshita,

Watanabe

2021

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Recently, new classes of infinite-dimensional algebras, quiver Yangian (QY) and shifted QY, were introduced, and they act on BPS states for non-compact toric Calabi-Yau threefolds. In particular, shifted QY acts on general subcrystals of the original BPS crystal. A trigonometric deformation called quiver quantum toroidal algebra (QQTA) was also proposed and shown to act on the same BPS crystal. Unlike QY, QQTA has a formal Hopf superalgebra structure which is useful in deriving representations.In this paper, we define the shifted QQTA and study a class of their representations. We define 1d and 2d subcrystals of the original 3d crystal by removing a few arrows from the original quiver diagram and show how the shifted QQTA acts on them. We construct the 2d crystal representations from the 1d crystal representations by utilizing a generalized coproduct acting on different shifted QQTAs. We provide a detailed derivation of subcrystal representations of C 3 , C 3 /Z n (n ≥ 2), conifold, suspended pinch point, and C 3 /(Z 2 × Z 2 ).

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“…While preparing this draft, we noticed a paper[35], which seems to be very relevant to this question.…”

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confidence: 99%

$q$-Deformation of Corner Vertex Operator Algebras by Miura Transformation

Harada,

Matsuo,

Noshita

et al. 2021

Preprint

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Recently, Gaiotto and Rapcak proposed a generalization of W N algebra by considering the symmetry at the corner of the brane intersection (corner vertex operator algebra). The algebra, denoted as Y L,M,N , is characterized by three non-negative integers L, M, N . It has a manifest triality automorphism which interchanges L, M, N , and can be obtained as a reduction of W 1+∞ algebra with a "pit" in the plane partition representation. Later, Prochazka and Rapcak proposed a representation of Y L,M,N in terms of L + N + M free bosons by a generalization of Miura transformation, where they use the fractional power differential operators.In this paper, we derive a q-deformation of the Miura transformation. It gives a free field representation for q-deformed Y L,M,N , which is obtained as a reduction of the quantum toroidal algebra. We find that the q-deformed version has a "simpler" structure than the original one because of the Miki duality in the quantum toroidal algebra. For instance, one can find a direct correspondence between the operators obtained by the Miura transformation and those of the quantum toroidal algebra. Furthermore, we can show that the both algebras share the same screening operators.

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Quadratic relations of the deformed $W$-superalgebra ${\cal W}_{q, t}\bigl(A(M,N)\bigr)$

Cited by 4 publications

References 13 publications

A Note on Quiver Quantum Toroidal Algebra

A Note on Quiver Quantum Toroidal Algebra

Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

$q$-Deformation of Corner Vertex Operator Algebras by Miura Transformation

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