Anomaly in RTT relation for DIM algebra and network matrix models

Abstract

We discuss the recent proposal of arXiv:1608.05351 about generalization of the RTT relation to network matrix models. We show that the RTT relation in these models is modified by a nontrivial, but essentially abelian anomaly cocycle, which we explicitly evaluate for the free field representations of the quantum toroidal algebra. This cocycle is responsible for the braiding, which permutes the external legs in the q-deformed conformal block and its 5d/6d gauge theory counterpart, i.e. the non-perturbative Nekrasov functions. Thus, it defines their modular properties and symmetry. We show how to cancel the anomaly using a construction somewhat similar to the anomaly matching condition in gauge theory. We also describe the singular limit to the affine Yangian (4d Nekrasov functions), which breaks the spectral duality.

21 pages

References in corpus (16)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• On AGT relation in the case of U(3)
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• Supersymmetric gauge theory and the Yangian
• Ding-Iohara-Miki symmetry of network matrix models
• Explicit examples of DIM constraints for network matrix models
• Notes on Ding-Iohara algebra and AGT conjecture
• Kernel function and quantum algebras
• Holomorphic blocks and the 5d AGT correspondence
• Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• Quantum toroidal gl(1) and Bethe ansatz
• On Factorization of Generalized Macdonald Polynomials
• Polynomials associated with fixed points on the instanton moduli space
• Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory
• A presentation of the deformed W_{1+\infty} algebra

Cited by in corpus (37)

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• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• A slow review of the AGT correspondence
• Towards topological quantum computer
• $\mathcal{W}$-algebra Modules, Free Fields, and Gukov-Witten Defects
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• Instanton R-matrix and W-symmetry
• (q,t)-KZ equation for Ding-Iohara-Miki algebra
• A Note on Quiver Quantum Toroidal Algebra
• Higgsed network calculus
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• 3d Mirror Symmetry from S-duality
• Quantum toroidal algebras and solvable structures in gauge/string theory
• Maulik-Okounkov's R-matrix from Ding-Iohara-Miki algebra
• An analogue of Schur functions for the plane partitions
• Fiber-base duality from the algebraic perspective
• Mixed network calculus
• Cauchy formula and the character ring
• Orthogonal Polynomials in Mathematical Physics
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Self-dual form of Ruijsenaars-Schneider models and ILW equation with discrete Laplacian
• Gauge origami and quiver W-algebras
• The $R$-matrix of the quantum toroidal algebra $U_{q,t}({\stackrel{..}{\mathrm{gl}}}_1)$ in the Fock module
• Argyres-Douglas Theories, S-duality and AGT Correspondence
• Refined toric branes, surface operators and factorization of generalized Macdonald polynomials
• Polynomial representations of classical Lie algebras and flag varieties
• Kac determinant and singular vector of the level N representation of Ding-Iohara-Miki algebra
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• Irreducible representations of simple Lie algebras by differential operators
• MacMahon KZ equation for Ding-Iohara-Miki algebra
• Quasi-Hopf twist and elliptic Nekrasov factor
• Invariants of knots and links at roots of unity
• The Grassmannian VOA
• Elliptic algebra, Frenkel-Kac construction and root of unity limit
• Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations
• Singular Vector of Ding-Iohara-Miki Algebra and Hall-Littlewood Limit of 5D AGT Conjecture