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

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.

31 pages

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Cited by in corpus (14)

• Quiver Yangian and Supersymmetric Quantum Mechanics
• Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• Quantum toroidal algebras and solvable structures in gauge/string theory
• $q$-Deformation of Corner Vertex Operator Algebras by Miura Transformation
• Mixed network calculus
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Gauge origami and quiver W-algebras
• Argyres-Douglas Theories, S-duality and AGT Correspondence
• Polynomial representations of classical Lie algebras and flag varieties
• Irreducible representations of simple Lie algebras by differential operators
• Elliptic algebra, Frenkel-Kac construction and root of unity limit
• A Representation-Theoretic Approach to $qq$-Characters
• Intertwining operator and integrable hierarchies from topological strings