Spectral Duality Between Heisenberg Chain and Gaudin Model

We consider Dotsenko-Fateev matrix models associated with compactified Calabi-Yau threefolds. They can be constructed with the help of explicit expressions for refined topological vertex, i.e. are directly related to the corresponding topological string amplitudes. We describe a correspondence between these amplitudes, elliptic and affine type Selberg integrals and gauge theories in five and six dimensions with various matter content. We show that the theories of this type are connected by spectral dualities, which can be also seen at the level of elliptic Seiberg-Witten integrable systems. The most interesting are the spectral duality between the XYZ spin chain and the Ruijsenaars system, which is further lifted to self-duality of the double elliptic system.

Section: Contentsmentioning

confidence: 99%

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Миронов¹,

Морозов²,

Zenkevich³

2016

“…Moreover, now we can also consider deformations of the section which do not preserve verticality, like the dotted one in figure 3, c), and everything can still be calculated. This should provide a qualitatively new insight into spectral dualities [89][90][91][92][93][94] associated with global rotations of the network graph.…”

Section: Jhep07(2016)103mentioning

confidence: 99%

Explicit examples of DIM constraints for network matrix models

Awata

Kanno

Matsumoto

et al. 2016

Self Cite

118

Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.

“…1 S corresponds to rotation of the integer lattice by π 2 clockwise. The action of this element on the algebra realizes the spectral duality [72][73][74][75][76][77][78][79][80] of different representations: in particular, the central charge vector is rotated; the horizontal representations become the vertical ones and vice versa. The action of S is illustrated in figure 1.…”

Section: Jhep10(2016)047mentioning

confidence: 99%

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

Awata

Kanno

Миронов

et al. 2016

Self Cite

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.