# Toric Calabi-Yau threefolds as quantum integrable systems. $$\mathrm{\mathcal{R}}$$ -matrix and $$\mathrm{\mathcal{R}}\mathcal{T}\mathcal{T}$$ relations

Journal of High Energy Physics, Oct 2016

Abstract $$\mathrm{\mathcal{R}}$$-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 $$\mathrm{\mathcal{R}}$$-matrix used for a similar calculation in the Yangian case by A. Smirnov but less general. We investigate the interplay between the $$\mathrm{\mathcal{R}}$$-matrix structure and the structure of DIM algebra intertwiners, i.e. of refined topological vertices and show that the $$\mathrm{\mathcal{R}}$$-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 $$\mathcal{T}$$-operators satisfying the $$\mathrm{\mathcal{R}}\mathcal{T}\mathcal{T}$$ relations with the $$\mathrm{\mathcal{R}}$$-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.

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Hidetoshi Awata, Hiroaki Kanno, Andrei Mironov, Alexei Morozov, Andrey Morozov, Yusuke Ohkubo, Yegor Zenkevich. Toric Calabi-Yau threefolds as quantum integrable systems. $$\mathrm{\mathcal{R}}$$ -matrix and $$\mathrm{\mathcal{R}}\mathcal{T}\mathcal{T}$$ relations, Journal of High Energy Physics, 2016, 47, DOI: 10.1007/JHEP10(2016)047