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

Abstract

R-matrix is explicitly constructed for simplest representations of the DingIohara-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 RT 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.

Fiber-base duality from the algebraic perspective

“…The expression of the universal R-matrix with generic c andc follows from the Corollary 1.2 in [44] (see also [17,19,45,46]). 6 Miki's automorphism preserves both the linear and multiplicative structure of the algebra, therefore it can be used to twist the co-algebraic structure [47].…”

“…However, strictly speaking, we do not need the existence of the universal object F S , but 6 I would like to thank A. Negut for this clarification. 7 This coproduct ∆S differs from ∆ as it is possible to compute, for instance, 17) only its realization in specific representations. The latter will be constructed explicitly below, up to a normalization factor, using a product of intertwiners.…”

“…through the legs bearing the representation ( i ,¯ i ) = ( * i ,¯ * i ) with either i = 1 or i = 2. We call this type of objects T -operators in reference to those constructed from linear quiver gauge theories and identified with the Baxter T -operator of an underlying integrable system [17,20]. 16 These operators will be denoted T (i) = Φ · i Φ * where the index i in the product · i refers to the coupling channel.…”

“…This paper explores the realization of S-duality, also called fiber-base duality (see below), among 5D N = 1 quiver gauge theories using an algebraic formalism developed recently in the series of papers [15][16][17][18][19][20][21], and based on the Ding-Iohara-Miki (DIM) algebra [22,23]. This formalism, referred here as algebraic engineering, highlights the integrable properties of the gauge theories' BPS sector, and, at the same time, expresses in a very elegant manner the covariance properties under the (q-deformed) W-algebra at the root of AGT correspondence.…”

“…As a result, a representation ρ with levels (ρ(c), ρ(c)) = ( ,¯ ) composed with this automorphism becomes a representation ρ of different levels (ρ (c), ρ (c)) = (−¯ , ). Since the two levels encode the branes charges, the composition by Miki's automorphism renders the S-duality transformation in the algebraic formalism [17]. 3 Building on this fact, we investigate here the transformation of intertwiners, and their composite objects (T -operators, Lax matrices).…”