Generalized Knizhnik-Zamolodchikov equation for Ding-Iohara-Miki algebra

Hidetoshi Awata, Hiroaki Kanno, Andrei Mironov, Alexei Morozov, Andrey Morozov, Yusuke Ohkubo, and Yegor Zenkevich

Phys. Rev. D 96, 026021 – Published 28 July 2017

Abstract

We derive the generalization of the Knizhnik-Zamolodchikov equation (KZE) associated with the Ding-Iohara-Miki algebra $U_{q, t} ({\hat{\hat{gl}}}_{1})$ . We demonstrate that certain refined topological string amplitudes satisfy these equations and find that the braiding transformations are performed by the $R$ matrix of $U_{q, t} ({\hat{\hat{gl}}}_{1})$ . The resulting system is the uplifting of the ${\hat{u}}_{1}$ Wess-Zumino-Witten model. The solutions to the $(q, t)$ KZE are identified with the (spectral dual of) building blocks of the Nekrasov partition function for five-dimensional linear quiver gauge theories. We also construct an elliptic version of the KZE and discuss its modular and monodromy properties, the latter being related to a dual version of the KZE.

Received 25 May 2017

DOI:https://doi.org/10.1103/PhysRevD.96.026021

Physics Subject Headings (PhySH)

Compactification Conformal field theory Strings & branes

Particles & Fields

Authors & Affiliations

Hidetoshi Awata^1,*, Hiroaki Kanno^1,2,†, Andrei Mironov^3,4,5,10,‡, Alexei Morozov^4,10,§, Andrey Morozov^4,5,6,10,∥, Yusuke Ohkubo^1,¶, and Yegor Zenkevich^{4,7,8,9,10,**}

¹Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
²KMI, Nagoya University, Nagoya 464-8602, Japan
³Lebedev Physics Institute, Moscow 119991, Russia
⁴ITEP, Moscow 117218, Russia
⁵Institute for Information Transmission Problems, Moscow 127994, Russia
⁶Laboratory of Quantum Topology, Chelyabinsk State University, Chelyabinsk 454001, Russia
⁷Institute of Nuclear Research, Moscow 117312, Russia
⁸Dipartimento di Fisica, Università di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
⁹INFN, sezione di Milano-Bicocca, I-20126 Milano, Italy
¹⁰National Research Nuclear University MEPhI, Moscow 115409, Russia

^*awata@math.nagoya-u.ac.jp
^†kanno@math.nagoya-u.ac.jp
^‡mironov@lpi.ru; mironov@itep.ru
^§morozov@itep.ru
^∥andrey.morozov@itep.ru
^¶m12010t@math.nagoya-u.ac.jp
^**yegor.zenkevich@gmail.com

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Issue

Vol. 96, Iss. 2 — 15 July 2017

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