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

TL;DR

This paper constructs an explicit R-matrix for the Ding-Iohara-Miki algebra, demonstrating its diagonalization via spectral duality and linking topological string theories on toric Calabi-Yau threefolds to integrable lattice models.

Contribution

It provides a simplified explicit R-matrix construction for the DIM algebra and connects topological string amplitudes to integrable systems through RTT relations.

Findings

R-matrix explicitly constructed for DIM algebra representations

R-matrix diagonalized by spectral duality in SL(2,Z)

Topological string theories interpreted as lattice integrable models

Abstract

R-matrix is explicitly constructed for simplest representations of the Ding-Iohara-Miki algebra. The 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 RTT 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.