Shifted Quiver Yangians and Representations from BPS Crystals

Galakhov, Dmitry; Li, Wei; Yamazaki, Masahito

doi:10.48550/arxiv.2106.01230

Cited by 7 publications

(30 citation statements)

References 52 publications

(139 reference statements)

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“…In particular, the generalized coproduct structure will be important in deriving the two-dimensional crystal representations from the one-dimensional crystal representations (see section 5 for various examples). In section 4.4, we discuss the relation with a recent paper [60]. The dark (light) gray region is mapped to the black (white) point.…”

Section: Subcrystal Representationsmentioning

confidence: 94%

“…Let us briefly discuss the relation with the shifted quiver Yangian defined in [60] (see section 2 and 3 of [60] for details). The Drinfeld currents of the shifted quiver Yangian are defined as…”

Section: General Subcrystalsmentioning

confidence: 99%

“…The defining relations in the current realization are the same as (2.0.18). It was shown in [60] that this algebra acts on general subcrystals of the original threedimensional BPS crystal (canonical crystal). General subcrystals can be decomposed into a superposition of multiple canonical crystals after translation and weighting the multiplicities properly 6 .…”

Section: General Subcrystalsmentioning

confidence: 99%

“…Recently, infinite-dimensional algebras associated with toric Calabi-Yau threefolds called quiver Yangian [56,57] was introduced (see also [58,59] for similar directions). As a generalization, a shifted version of quiver Yangian was defined [60]. Quiver Yangian acts on three-dimensional BPS crystals [61], while shifted quiver Yangian acts on the subcrystal of the mother BPS crystal.…”

Section: Introductionmentioning

confidence: 99%

“…While preparing this paper, new preprints [60,68] appeared in the arXiv. We discuss the relation to [60] in section 4. 4.…”

Section: Introductionmentioning

confidence: 99%

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Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

Noshita,

Watanabe

2021

Preprint

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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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Section: Subcrystal Representationsmentioning

confidence: 94%

Section: General Subcrystalsmentioning

confidence: 99%

Section: General Subcrystalsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…While preparing this paper, new preprints [60,68] appeared in the arXiv. We discuss the relation to [60] in section 4. 4.…”

Section: Introductionmentioning

confidence: 99%

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Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

Noshita,

Watanabe

2021

Preprint

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Engineering 3D $\mathcal{N}=2$ theories using the quantum affine $\mathfrak{sl}(2)$ algebra

Bourgine¹

2021

Preprint

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The algebraic engineering technique is applied to a class of 3D N = 2 gauge theories on the omega-deformed background R 2 ε × S 1 . The vortex partition function and the fundamental qqcharacter are obtained from a network of intertwiners between representations of the shifted (or asymptotic) quantum affine sl(2) algebra. This network involves two types of representations, the prefundamental representation of Hernandez-Jimbo, and a new vertex representation acting on a bosonic Fock space. The brane system associated to this network is identified: D3 branes carry the prefundamental module while NS5-branes (+D5) support the Fock module. In the process, we highlight the role of shifted quantum algebras in implementing the Higgsing procedure.

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Toroidal and Elliptic Quiver BPS Algebras and Beyond

Galakhov,

Li,

Yamazaki

2021

Preprint

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The quiver Yangian, an infinite-dimensional algebra introduced recently in [1], is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional N = 2 supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.

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Shifted Quiver Yangians and Representations from BPS Crystals

Cited by 7 publications

References 52 publications

Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

Engineering 3D $\mathcal{N}=2$ theories using the quantum affine $\mathfrak{sl}(2)$ algebra

Toroidal and Elliptic Quiver BPS Algebras and Beyond

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