$(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces

Abstract

We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody $U_q(\hat{\mathfrak{g}}{)}_k$ to generic quantum toroidal algebras. A nearly exhaustive presentation is given for the two series $U_{q,t}({\widehat{\hat{\mathfrak{gl}}}}_1)$ and $U_{q,t}({\widehat{\hat{\mathfrak{gl}}}}_n)$, when screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations. Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of the Fock representations of the quantum toroidal algebra of type $A_n$. The correlation functions of these operators satisfy the $(q,t)$-Knizhnik-Zamolodchikov (KZ) equation, which features the $\mathcal{R}$-matrix. Matching with the Nekrasov function for the instanton counting on the ALE space is worked out explicitly. We also present an important application of the DIM formalism to the study of $6d$ gauge theories described by the double elliptic integrable systems. We show that the modular and periodicity properties of the gauge theories are neatly explained by the network matrix models providing solutions to the elliptic $(q,t)$-KZ equations.

arXiv comment56 pages

References in corpus (97)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Supersymmetric Yang-Mills Systems And Integrable Systems
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Integrability and Seiberg-Witten Exact Solution
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• On AGT relation in the case of U(3)
• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems
• An Algorithm for the Microscopic Evaluation of the Coefficients of the Seiberg-Witten Prepotential
• Refined BPS state counting from Nekrasov's formula and Macdonald functions
• Collective fields, Calogero-Sutherland model and generalized matrix models
• Conformal Matrix Models as an Alternative to Conventional Multi-Matrix Models
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Refined Topological Vertex and Instanton Counting
• Para-Liouville/Toda central charges from M5-branes
• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• The Quiver Matrix Model and 2d-4d Conformal Connection
• A commutative algebra on degenerate CP^1 and Macdonald polynomials
• Quantum Algebraic Approach to Refined Topological Vertex
• Correlation functions in conformal Toda field theory II
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• Integrable Structures in Supersymmetric Gauge and String Theory
• Excited States of Calogero-Sutherland Model and Singular Vectors of the $W_N$ Algebra
• Proving AGT conjecture as HS duality: extension to five dimensions
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• Five-dimensional AGT Relation and the Deformed beta-ensemble
• Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model
• Instanton counting, Macdonald function and the moduli space of D-branes
• Instanton counting with a surface operator and the chain-saw quiver
• Quantum toroidal ${\mathfrak{gl}}_1$ algebra : plane partitions
• Multi instanton calculus on ALE spaces
• Ding-Iohara-Miki symmetry of network matrix models
• Penner Type Matrix Model and Seiberg-Witten Theory
• Matrix model version of AGT conjecture and generalized Selberg integrals
• Explicit examples of DIM constraints for network matrix models
• Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra
• M5-branes, toric diagrams and gauge theory duality
• Super Liouville conformal blocks from N=2 SU(2) quiver gauge theories
• Quantum continuous ${\mathfrak{gl}}_{\infty }$: Semi-infinite construction of representations
• Instanton moduli spaces and bases in coset conformal field theory
• Seiberg-Witten theory, matrix model and AGT relation
• 3d & 5d gauge theory partition functions as q-deformed CFT correlators
• 5D partition functions, q-Virasoro systems and integrable spin-chains
• Matrix models for $\beta$-ensembles from Nekrasov partition functions
• On Double-Elliptic Integrable Systems. 1. A Duality Argument for the case of SU(2)
• Spectral Duality Between Heisenberg Chain and Gaudin Model
• Spectral Duality in Integrable Systems from AGT Conjecture
• (p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters
• An A_r threesome: Matrix models, 2d CFTs and 4d N=2 gauge theories
• Notes on Ding-Iohara algebra and AGT conjecture
• Finite type modules and Bethe Ansatz for quantum toroidal gl(1)
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Yangian Gelfand-Zetlin Bases, gl(N)-Jack Polynomials and computation of Dynamical Correlation Functions in the Spin Calogero-Sutherland Model
• Gauge/Liouville Triality
• Commuting Hamiltonians from Seiberg-Witten Theta-Functions
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Anomaly in RTT relation for DIM algebra and network matrix models
• Coherent states in quantum ${\mathcal{W}}_{1+\infty }$ algebra and qq-character for 5d Super Yang-Mills
• Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity
• Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver
• Kernel function and quantum algebras
• Spectral dualities in XXZ spin chains and five dimensional gauge theories
• Decomposing Nekrasov Decomposition
• Holomorphic blocks and the 5d AGT correspondence
• Integrals of motion from quantum toroidal algebras
• 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space
• An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems
• Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings
• On elementary proof of AGT relations from six dimensions
• A Combinatorial Study on Quiver Varieties
• The orthogonal eigenbasis and norms of eigenvectors in the Spin Calogero-Sutherland Model
• ADE Little String Theory on a Riemann Surface (and Triality)
• (q,t)-KZ equation for Ding-Iohara-Miki algebra
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence
• Five-dimensional SU(2) AGT conjecture and recursive formula of deformed Gaiotto state
• Quantum toroidal gl(1) and Bethe ansatz
• Three-particle Integrable Systems with Elliptic Dependence on Momenta and Theta Function Identities
• Branching rules for quantum toroidal gl(n)
• Seiberg-Witten curves and double-elliptic integrable systems
• Maulik-Okounkov's R-matrix from Ding-Iohara-Miki algebra
• Generalized Whittaker states for instanton counting with fundamental hypermultiplets
• Representations of quantum toroidal $g{l}_n$
• The q-deformed Knizhnik-Zamolodchikov-Bernard heat equation
• Integrable Many-Body Systems and Gauge Theories
• Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory
• q-Vertex Operator from 5D Nekrasov Function
• Quantum Algebras and Cyclic Quiver Varieties
• The integral representation of solutions of KZ equation and a modification by $\mathcal{K}$ operator insertion
• Modular properties of 6d (DELL) systems
• New non-linear equations and modular form expansion for double-elliptic Seiberg-Witten prepotential
• Integrability in N=2 Gauge Theory: A Proof
• Screening Currents Ward Identity and Integral Formulas for the WZNW Correlation Functions
• Traces of intertwiners for quantum groups and difference equations, II
• Traces of intertwiners for quantum affine algebras and difference equations (after Etingof-Schiffmann-Varchenko)

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• MacMahon KZ equation for Ding-Iohara-Miki algebra
• Quasi-Hopf twist and elliptic Nekrasov factor
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• Towards the theory of Yangians
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• Quantum $W_{1+\infty }$ subalgebras of BCD type and symmetric polynomials
• Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations
• Coloured refined topological vertices and parafermion conformal field theories
• Intertwining operator and integrable hierarchies from topological strings
• Large N limit of spectral duality in classical integrable systems
• Bethe/Gauge correspodence: a short review on an aspect of the integrability nature of supersymmetric gauge theories
• Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal ${\mathfrak{gl}}_2$