Toric Calabi-Yau threefolds as quantum integrable systems. ℛ-matrix and ℛTT relations

• Published in: The journal of high energy physics 2016-10, Vol.2016 (10)
• Main Authors: Awata, Hidetoshi, Kanno, Hiroaki, Mironov, Andrei, Morozov, Alexei, Morozov, Andrey, Ohkubo, Yusuke, Zenkevich, Yegor
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Elementary Particles; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory
• ISSN: 1029-8479
• DOI: 10.1007/JHEP10(2016)047

A bstract ℛ -matrix is explicitly constructed for simplest representations of the Ding-Iohara-Miki algebra. Calculation is straightforward and significantly simpler than the one through the universal ℛ -matrix used for a similar calculation in the Yangian case by A. Smirnov but less general. We investigate the interplay between the ℛ -matrix structure and the structure of DIM algebra intertwiners, i.e. of refined topological vertices and show that the ℛ -matrix is diagonalized by the action of the spectral duality belonging to the SL(2, ℤ) group of DIM algebra automorphisms. We also construct the T -operators satisfying the ℛ T T relations with the ℛ -matrix from refined amplitudes on resolved conifold. We thus show that topological string theories on the toric Calabi-Yau threefolds can be naturally interpreted as lattice integrable models. Integrals of motion for these systems are related to q -deformation of the reflection matrices of the Liouville/Toda theories.