Quantum toroidal algebras and solvable structures in gauge/string theory

Abstract

This is a review article on the quantum toroidal algebras, focusing on their roles in various solvable structures of 2d conformal field theory, supersymmetric gauge theory, and string theory. Using $\mathcal{W}$-algebras as our starting point, we elucidate the interconnection of affine Yangians, quantum toroidal algebras, and double affine Hecke algebras. Our exploration delves into the representation theory of the quantum toroidal algebra of ${\mathfrak{gl}}_1$ in full detail, highlighting its connections to partitions, $\mathcal{W}$-algebras, Macdonald functions, and the notion of intertwiners. Further, we also discuss integrable models constructed on Fock spaces and associated $\mathcal{R}$-matrices, both for the affine Yangian and the quantum toroidal algebra of ${\mathfrak{gl}}_1$. The article then demonstrates how quantum toroidal algebras serve as a unifying algebraic framework that bridges different areas in physics. Notably, we cover topological string theory and supersymmetric gauge theories with eight supercharges, incorporating the AGT duality. Drawing upon the representation theory of the quantum toroidal algebra of ${\mathfrak{gl}}_1$, we provide a rather detailed review of its role in the algebraic formulations of topological vertex and $qq$-characters. Additionally, we briefly touch upon the corner vertex operator algebras and quiver quantum toroidal algebras.

151+42 pages, Comments are welcome, v2: fixed typos and added references

References in corpus (73)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• The Refined Topological Vertex
• Instantons in the Higgs Phase
• Asymptotically free N=2 theories and irregular conformal blocks
• Brane Tilings and Their Applications
• Crystal Melting and Toric Calabi-Yau Manifolds
• BPS/CFT correspondence II: Instantons at crossroads, Moduli and Compactness Theorem
• Ghost D-branes
• Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial
• Spiked Instantons from Intersecting D-branes
• Nekrasov prepotential with fundamental matter from the quantum spin chain
• Quiver Yangian and Supersymmetric Quantum Mechanics
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Shifted Quiver Yangians and Representations from BPS Crystals
• On W algebras commuting with a set of screenings
• Intersecting Solitons, Amoeba and Tropical Geometry
• Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver
• Kernel function and quantum algebras
• Rank-3 antisymmetric matter on 5-brane webs
• Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• Conformal blocks of W_N Minimal Models and AGT correspondence
• Gauge/Bethe correspondence from quiver BPS algebras
• A Note on Quiver Quantum Toroidal Algebra
• Construction of Gaiotto states with fundamental multiplets through Degenerate DAHA
• On AGT-W Conjecture and q-Deformed W-Algebra
• Instantons from Blow-up
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• On spectrum of ILW hierarchy in conformal field theory II: coset CFT's
• Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants
• More on topological vertex formalism for 5-brane webs with O5-plane
• Affine Yangian of $\mathfrak{gl}(2)$ and integrable structures of superconformal field theory
• Instanton counting and O-vertex
• $q$-Deformation of Corner Vertex Operator Algebras by Miura Transformation
• Intersecting Defects and Supergroup Gauge Theory
• R-matrix formulation of affine Yangian of $\widehat{\mathfrak{gl}}(1|1)$
• Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds
• Spherical Hecke algebra in the Nekrasov-Shatashvili limit
• Topological vertex for 6d SCFTs with ${\mathbb{Z}}_2$-twist
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Super instanton counting and localization
• Refined topological vertex with ON-planes
• Recursive Method for Nekrasov partition function for classical Lie groups
• BPS Lie algebras and the less perverse filtration on the preprojective CoHA
• A Note on Quiver Yangians and $\mathcal{R}$-Matrices
• Thermodynamic limit of Nekrasov partition function for 5-brane web with O5-plane
• Quadratic relations of the deformed $W$-superalgebra ${\mathcal{W}}_{q,t}(A(M,N))$
• ABCD of qq-characters
• 6d/5d exceptional gauge theories from web diagrams
• Crystal Melting, BPS Quivers and Plethystics
• Integrable structure of $\text{BCD}$ conformal field theory and boundary Bethe ansatz for affine Yangian
• Shifted quantum groups and matter multiplets in supersymmetric gauge theories
• Double Quiver Gauge Theory and BPS/CFT Correspondence
• Shuffle approach towards quantum affine and toroidal algebras
• More on Affine Dynkin Quiver Yangians
• Parallel surface defects, Hecke operators, and quantum Hitchin system
• $DE$-type little strings from glued brane webs
• Twisted reduction of quiver W-algebras
• Branes, Quivers and BPS Algebras
• BPS Lie algebras for totally negative 2-Calabi-Yau categories and nonabelian Hodge theory for stacks
• MacMahon KZ equation for Ding-Iohara-Miki algebra
• Orthosymplectic Superinstanton Counting and Brane Dynamics
• Engineering 3D $\mathcal{N}=2$ theories using the quantum affine $\mathfrak{sl}(2)$ algebra
• 3D Bosons, 3-Jack polynomials and affine Yangian of $\mathfrak{gl}(1)$
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• Quiver algebras and their representations for arbitrary quivers
• BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories
• Computing the R-matrix of the quantum toroidal algebra
• Branes and DAHA Representations
• A Survey of Toric Quivers and BPS Algebras
• Hanany-Witten brane crossing and Ding-Iohara-Miki algebra
• Quantum toroidal algebras and motivic Hall algebras I. Hall algebras for singular elliptic curves
• Orbifolds of Gaiotto-Rapčák $Y$-algebras

Cited by in corpus (9)

• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Gauge origami and quiver W-algebras
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues
• An Overview of Crystals and Double Quiver Yangians
• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• Bethe/Gauge Correspondence for $A_N$ Spin Chains with Integrable Boundaries
• Free field realization of the Ding-Iohara algebra at general levels
• Proof of 5D $A_n$ AGT conjecture at $\beta =1$