Quasi-Hopf twist and elliptic Nekrasov factor

Abstract We investigate the quasi-Hopf twist of the quantum toroidal algebra of $$ {\mathfrak{gl}}_1 $$ 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 ℝ4× T2. 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.

[1]  H. Umezawa,et al.  THERMO FIELD DYNAMICS , 1996 .

[2]  E. Rains,et al.  A Nekrasov–Okounkov formula for Macdonald polynomials , 2016, 1606.04613.

[3]  O. Foda,et al.  An elliptic topological vertex , 2018, Journal of Physics A: Mathematical and Theoretical.

[4]  Jintai Ding,et al.  Generalization of Drinfeld Quantum Affine Algebras , 1997 .

[5]  O. Schiffmann.  Drinfeld realization of the elliptic Hall algebra , 2010, 1004.2575.

[6]  Elliptic Algebra : Drinfeld Currents and Vertex Operators , 1998, math/9802002.

[7]  Kota Yoshioka,et al.  Instanton counting on blowup. I. 4-dimensional pure gauge theory , 2003, math/0306198.

[8]  R. Poghossian,et al.  Instantons and the 5D U(1) gauge theory with extra adjoint , 2008, 0804.3564.

[9]  Ke Wu,et al.  Representation of elliptic Ding-Iohara algebra , 2020 .

[10]  P. Alam.  ‘S’ , 2021, Composites Engineering: An A–Z Guide.

[11]  Hiraku Nakajima.  Lectures on Hilbert Schemes of Points on Surfaces , 1999 .

[12]  Deformations of W-algebras associated to simple Lie algebras , 1997, q-alg/9708006.

[13]  Rui-Dong Zhu.  An elliptic vertex of Awata-Feigin-Shiraishi type for M-strings , 2017, Journal of High Energy Physics.

[14]  Nikita A. Nekrasov.  Seiberg-Witten prepotential from instanton counting , 2002 .

[15]  Vertex operators in solvable lattice models , 1993, hep-th/9305100.

[16]  The refined topological vertex , 2007, hep-th/0701156.

[17]  Erik Carlsson,et al.  Five dimensional gauge theories and vertex operators , 2013, 1308.2465.

[18]  N. Nekrasov.  BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters , 2015, 1512.05388.

[19]  Fabrizio Nieri.  An elliptic Virasoro symmetry in 6d , 2015, Letters in Mathematical Physics.

[20]  A. Mironov,et al.  Generalized Knizhnik-Zamolodchikov equation for Ding-Iohara-Miki algebra , 2017 .

[21]  H. Awata,et al.  Five-dimensional AGT conjecture and the deformed Virasoro algebra , 2009, 0910.4431.

[22]  Michio Jimbo,et al.  Algebraic Analysis of Solvable Lattice Models. , 1994 .

[23]  Werner Israel,et al.  Thermo-field dynamics of black holes☆ , 1976 .

[24]  L. Alday,et al.  Liouville Correlation Functions from Four-Dimensional Gauge Theories , 2009, 0906.3219.

[25]  C. Vafa,et al.  Superconformal partition functions and non-perturbative topological strings , 2012, Journal of High Energy Physics.

[26]  M. Nishizawa.  An elliptic analogue of the multiple gamma function , 2001 .

[27]  V. Pestun,et al.  Quantum Geometry and Quiver Gauge Theories , 2013, 1312.6689.

[28]  Quasi-Hopf twistors for elliptic quantum groups , 1997, q-alg/9712029.

[29]  Kernel function and quantum algebras , 2010, 1002.2485.

[30]  Kota Yoshioka,et al.  Instanton counting on blowup, I , 2003 .

[31]  A. Mironov,et al.  The MacMahon R-matrix , 2018, Journal of High Energy Physics.

[32]  C. Vafa,et al.  Matrix models, geometric engineering and elliptic genera , 2003, hep-th/0310272.

[33]  A. Sciarappa,et al.  Quantum hydrodynamics from large-n supersymmetric gauge theories , 2015, 1510.00972.

[34]  Finite Type Modules and Bethe Ansatz for Quantum Toroidal $${\mathfrak{gl}_1}$$gl1 , 2016, 1603.02765.

[35]  N. Reshetikhin,et al.  Quantum affine algebras and holonomic difference equations , 1992 .

[36]  Alexandr Garbali,et al.  The R-Matrix of the Quantum Toroidal Algebra $$U_{q,t}(\overset{..}{gl}_1)$$ in the Fock Module , 2020, Communications in Mathematical Physics.

[37]  O. Schiffmann,et al.  On the Hall algebra of an elliptic curve, I , 2005, math/0505148.

[38]  A. Mironov,et al.  On AGT relation in the case of U(3) , 2009, 0908.2569.

[39]  L. Clavelli,et al.  Pomeron factorization in general dual models , 1973 .

[40]  H. Awata,et al.  REFINED BPS STATE COUNTING FROM NEKRASOV'S FORMULA AND MACDONALD FUNCTIONS , 2008, 0805.0191.

[41]  C. Vafa,et al.  Orbifolds of M-strings , 2013, 1310.1185.

[42]  Jan de Gier,et al.  The R-Matrix of the Quantum Toroidal Algebra Uq,t(gl..1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{q,t}(\overs , 2020, Communications in Mathematical Physics.

[43]  K. Miki.  A (q,γ) analog of the W1+∞ algebra , 2007 .

[44]  Andrei Okounkov,et al.  Seiberg-Witten theory and random partitions , 2003, hep-th/0306238.

[45]  B. Feigin,et al.  A commutative algebra on degenerate CP^1 and Macdonald polynomials , 2009, 0904.2291.

[46]  Panupong Cheewaphutthisakun,et al.  MacMahon KZ equation for Ding-Iohara-Miki algebra , 2021, Journal of High Energy Physics.

[47]  H. Awata,et al.  Changing the preferred direction of the refined topological vertex , 2009, 0903.5383.

[48]  I. Ojima.  Gauge fields at finite temperatures—“Thermo field dynamics” and the KMS condition and their extension to gauge theories , 1981 .

[49]  A. Mironov,et al.  Anomaly in RTT relation for DIM algebra and network matrix models , 2016, 1611.07304.

[50]  B. Feigin,et al.  Quantum algebraic approach to refined topological vertex , 2012, Journal of High Energy Physics.

[51]  A. Iqbal,et al.  Refined topological vertex, cylindric partitions and U(1) adjoint theory , 2008, 0803.2260.

[52]  M. Zabzine,et al.  Elliptic modular double and 4d partition functions , 2017, 1703.04614.

[53]  A. Mironov,et al.  (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces , 2017, 1712.08016.

[54]  Elliptic Ding–Iohara Algebra and the Free Field Realization of the Elliptic Macdonald Operator , 2013, 1301.4912.

[55]  A. Sciarappa,et al.  On elliptic algebras and large-n supersymmetric gauge theories , 2016, 1601.08238.

[56]  Elliptic quantum groups , 1994, hep-th/9412207.

[57]  이현주.  Q. , 2005 .

[58]  Taro Kimura,et al.  Quiver elliptic W-algebras , 2016, 1608.04651.

[59]  S. Narison,et al.  Gluonium nature of the σ/f0(600) from its coupling to KK¯ , 2009, 0904.2555.

[60]  C. Vafa,et al.  M-Strings , 2013, 1305.6322.

[61]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[62]  Taro Kimura,et al.  Quiver W-algebras , 2015, 1512.08533.

[63]  M. van der Ploeg,et al.  Matrix models , 2004, Histochemistry.

[64]  M. Jimbo,et al.  K-6 DPSU-982 Elliptic algebra U q , p ( ŝl 2 ) : Drinfeld currents and vertex operators , 1998 .

[65]  N. Wyllard.  A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories , 2009, 0907.2189.

[66]  A. Mironov,et al.  Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings , 2016, 1603.00304.