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

  • Hidetoshi Awata
  • Hiroaki Kanno
  • Andrei Mironov
  • Alexei Morozov
  • Kazuma Suetake
  • Yegor Zenkevich
Open Access
Regular Article - Theoretical Physics
  • 19 Downloads

Abstract

We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody \( {U}_q{\left(\widehat{\mathfrak{g}}\right)}_k \) to generic quantum toroidal algebras. A nearly exhaustive presentation is given for both \( {U}_{q,t}\left({\widehat{\widehat{\mathfrak{gl}}}}_1\right) \) and \( {U}_{q,t}\left({\widehat{\widehat{\mathfrak{gl}}}}_n\right) \) when the 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 ℛ-matrix. The 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.

Keywords

Conformal and W Symmetry Conformal Field Theory Supersymmetric Gauge Theory Topological Strings 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Hidetoshi Awata
    • 1
  • Hiroaki Kanno
    • 1
    • 2
  • Andrei Mironov
    • 3
    • 4
    • 5
    • 6
  • Alexei Morozov
    • 4
    • 5
    • 6
  • Kazuma Suetake
    • 1
  • Yegor Zenkevich
    • 4
    • 7
    • 8
  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  2. 2.KMI, Nagoya UniversityNagoyaJapan
  3. 3.Lebedev Physics InstituteMoscowRussia
  4. 4.ITEPMoscowRussia
  5. 5.Institute for Information Transmission ProblemsMoscowRussia
  6. 6.National Research Nuclear University MEPhIMoscowRussia
  7. 7.Dipartimento di FisicaUniversità di Milano-BicoccaMilanoItaly
  8. 8.INFN, sezione di Milano-BicoccaMilanoItaly

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