Shifted quantum groups and matter multiplets in supersymmetric gauge theories

Abstract

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 $\mathfrak{gl}(1)$ and quantum affine $\mathfrak{sl}(2)$ algebras (resp. denoted $\ddot{U}_{q_1,q_2}^\boldsymbolμ(\mathfrak{gl}(1))$ and $\dot{U}_q^\boldsymbolμ(\mathfrak{sl}(2))$). It defines several new representations, including finite dimensional highest $\ell$-weight representations for the toroidal algebra, and a vertex representation of $\dot{U}_q^\boldsymbolμ(\mathfrak{sl}(2))$ acting on Hall-Littlewood polynomials. It also explores the relations between representations of $\dot{U}_q^\boldsymbolμ(\mathfrak{sl}(2))$ and $\ddot{U}_{q_1,q_2}^\boldsymbolμ(\mathfrak{gl}(1))$ in the limit $q_1\to \infty$ ($q_2$ fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d $\mathcal{N}=1$ and 3d $\mathcal{N}=2$ 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.

arXiv commentv2: 75 pages, presentation improved, version to appear in CMP

References in corpus (40)

• Mirror Symmetry in Three Dimensional Gauge Theories
• Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics
• Webs of (p,q) 5-branes, Five Dimensional Field Theories and Grid Diagrams
• Instantons on noncommutative R^4, and (2,0) superconformal six dimensional theory
• Vortices, Instantons and Branes
• Branes, Superpotentials and Superconformal Fixed Points
• Integrable Structure of Conformal Field Theory III. The Yang-Baxter Relation
• Mirror Symmetry in Three-Dimensional Gauge Theories, SL(2,Z) and D-Brane Moduli Spaces
• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters
• A commutative algebra on degenerate CP^1 and Macdonald polynomials
• Walls, Lines, and Spectral Dualities in 3d Gauge Theories
• Quantum Algebraic Approach to Refined Topological Vertex
• On Three Dimensional Quiver Gauge Theories and Integrability
• Quantum toroidal ${\mathfrak{gl}}_1$ algebra : plane partitions
• Ding-Iohara-Miki symmetry of network matrix models
• Explicit examples of DIM constraints for network matrix models
• Asymptotic representations and Drinfeld rational fractions
• Quantum continuous ${\mathfrak{gl}}_{\infty }$: Semi-infinite construction of representations
• BPS/CFT correspondence II: Instantons at crossroads, Moduli and Compactness Theorem
• String theory of the Omega deformation
• Quiver W-algebras
• Higgs branch localization of 3d N=2 theories
• (p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters
• Plane partitions with a "pit": generating functions and representation theory
• Flipping the head of T[SU(N)]: mirror symmetry, spectral duality and monopoles
• Finite type modules and Bethe Ansatz for quantum toroidal gl(1)
• Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver
• Off-shell Bethe vectors and Drinfeld currents
• A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces
• Coulomb branches of quiver gauge theories with symmetrizers
• Liouville reflection operator, affine Yangian and Bethe ansatz
• Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence
• Holomorphic field realization of SH${}^c$ and quantum geometry of quiver gauge theories
• A note on the algebraic engineering of 4D $\mathcal{N}=2$ super Yang-Mills theories
• 3d Mirror Symmetry from S-duality
• Fiber-base duality from the algebraic perspective
• An Elliptic Vertex of Awata-Feigin-Shiraishi type for M-strings
• Connecting Mirror Symmetry in 3d and 2d via Localization
• Factorization of the Universal R-matrix for $U_q({\widehat{sl}}_2)$
• Quantum Q systems: From cluster algebras to quantum current algebras

Cited by in corpus (4)

• Quantum toroidal algebras and solvable structures in gauge/string theory
• ABCD of qq-characters
• Gauge origami and quiver W-algebras
• Supersymmetric polynomials and algebro-combinatorial duality