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

Bourgine, Jean-Emile

doi:10.48550/arxiv.2107.10063

Cited by 6 publications

(24 citation statements)

References 64 publications

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“…In particular, we will show that 5d N = 1 hypermultiplets can be introduced in this framework using the shifted versions of the quantum toroidal gl(1) algebra. We also derive several technical results that justify the claims made in [11]. Finally, we will explore the relation between representations of shifted quantum toroidal gl(1) and quantum affine sl(2) algebras, and revisit the Higgsing procedure relating 3d and 5d theories from an algebraic perspective.…”

Section: Introductionmentioning

confidence: 82%

“…In [11], the author proposed an algebraic construction of certain BPS observables for a class of 3d N = 2 gauge theories based on the representation theory of the shifted quantum affine sl (2) algebras. This construction is a specific instance of the algebraic engineering which aims to extend the "CFT techniques" of vertex operators to supersymmetric gauge theories using the representation theory of quantum groups.…”

Section: Introductionmentioning

confidence: 99%

“…The algebraic construction of 3d N = 2 gauge theories presented in [11] is briefly recalled in the fifth section, and then extended to (A-type) quiver theories by analyzing the case of the gauge group U(1) × U (1). We also come back to the description of the Higgsing procedure using the results on the limit of representations obtained in the sections II and III.…”

Section: Introductionmentioning

confidence: 99%

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Shifted quantum groups and matter multiplets in supersymmetric gauge theories

Bourgine¹

2022

Preprint

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The notion of shifted quantum groups has recently played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory of quantum groups. The first part of this paper presents new mathematical results for the shifted quantum toroidal gl(1) and quantum affine sl(2) algebras (resp. denoted Ü µ q 1 ,q 2 (gl(1)) and U µ q (sl(2))). It defines several new representations, including finite dimensional highest ℓ-weight representations for the toroidal algebra, and a vertex representation of U µ q (sl(2)) acting on Hall-Littlewood polynomials. It also explores the relations between representations of U µ q (sl(2)) and Ü µ q 1 ,q 2 (gl(1)) in the limit q 1 → ∞ (q 2 fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d N = 1 and 3d N = 2 gauge theories. In particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be introduced in the algebraic engineering framework using shifted representations, and the Higgsing procedure is revisited from this perspective.

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Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Shifted quantum groups and matter multiplets in supersymmetric gauge theories

Bourgine¹

2022

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. Actually the original motivation of this work was to study two-dimensional crystal representations that might enter in the vertical representation part of the intertwiner.…”

Section: Conclusion and Discussionmentioning

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%

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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Gauge origami and quiver W-algebras

Kimura,

Noshita

2024

J. High Energ. Phys.

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We explore the quantum algebraic formalism of the gauge origami system in ℂ4, where D2/D4/D6/D8-branes are present. We demonstrate that the contour integral formulas have free field interpretations, leading to the operator formalism of qq-characters associated with each D-brane. The qq-characters of D2 and D4-branes correspond to screening charges and generators of the affine quiver W-algebra, respectively. On the other hand, the qq-characters of D6 and D8-branes represent novel types of qq-characters, where monomial terms are characterized by plane partitions and solid partitions. The composition of these qq-characters yields the instanton partition functions of the gauge origami system, eventually establishing the BPS/CFT correspondence.Additionally, we demonstrate that the fusion of qq-characters of D-branes in lower dimensions results in higher-dimensional D-brane qq-characters. We also investigate quadratic relations among these qq-characters. Furthermore, we explore the relationship with the representations, q-characters, and the Bethe ansatz equations of the quantum toroidal $$ {\mathfrak{gl}}_1 $$ gl 1 . This connection provides insights into the Bethe/Gauge correspondence of the gauge origami system from both gauge-theoretic and quantum-algebraic perspectives.We finally present conjectures regarding generalizations to general toric Calabi-Yau four-folds. These generalizations imply the existence of an extensive class of qq-characters, which we call BPS qq-characters. These BPS qq-characters offer a new systematic approach to derive a broader range of BPS/CFT correspondence and Bethe/Gauge correspondence.

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

Cited by 6 publications

References 64 publications

Shifted quantum groups and matter multiplets in supersymmetric gauge theories

Shifted quantum groups and matter multiplets in supersymmetric gauge theories

Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

Gauge origami and quiver W-algebras

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