Quasi-Hopf twist and elliptic Nekrasov factor

Abstract

We investigate the quasi-Hopf twist of the quantum toroidal algebra of ${\mathfrak{gl}}_1$ as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter $p$ is identified as the elliptic modulus. Computing the quasi-Hopf twist of the $R$ matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on ${\mathbb{R}}^4\times T^2$. The same $R$ matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function.

arXiv comment45 pages, 3 figures

References in corpus (41)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• 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)
• Instanton counting on blowup. I. 4-dimensional pure gauge theory
• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• M-Strings
• Matrix Models, Geometric Engineering and Elliptic Genera
• Refined BPS state counting from Nekrasov's formula and Macdonald functions
• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters
• A commutative algebra on degenerate CP^1 and Macdonald polynomials
• Quantum Algebraic Approach to Refined Topological Vertex
• On orbifolds of M-Strings
• On the Hall algebra of an elliptic curve, I
• Schur duality in the toroidal setting
• Deformations of W-algebras associated to simple Lie algebras
• Quiver W-algebras
• Vertex operators in solvable lattice models
• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• Finite type modules and Bethe Ansatz for quantum toroidal gl(1)
• Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory
• Elliptic algebra U_{q,p}(^sl_2): Drinfeld currents and vertex operators
• Anomaly in RTT relation for DIM algebra and network matrix models
• Gluonium nature of the sigma/f_0(600) from its coupling to KK
• Quiver elliptic W-algebras
• Changing the preferred direction of the refined topological vertex
• Kernel function and quantum algebras
• Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings
• An elliptic Virasoro symmetry in 6d
• (q,t)-KZ equation for Ding-Iohara-Miki algebra
• The MacMahon R-matrix
• An Elliptic Vertex of Awata-Feigin-Shiraishi type for M-strings
• Quantum Hydrodynamics from Large-n Supersymmetric Gauge Theories
• On Elliptic Algebras and Large-n Supersymmetric Gauge Theories
• An elliptic topological vertex
• Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function
• Elliptic modular double and 4d partition functions
• Instantons and the 5D U(1) gauge theory with extra adjoint
• A Nekrasov-Okounkov formula for Macdonald polynomials
• The $R$-matrix of the quantum toroidal algebra $U_{q,t}({\stackrel{..}{\mathrm{gl}}}_1)$ in the Fock module
• MacMahon KZ equation for Ding-Iohara-Miki algebra
• Non-Stationary Ruijsenaars Functions for $\kappa =t^{-1/N}$ and Intertwining Operators of Ding-Iohara-Miki Algebra