Lattice equations arising from discrete Painlevé systems. I. (<i>A</i>2 + <i>A</i>1)(1) and (A1+A1′)(1) cases

Joshi, Nalini; Nakazono, Nobutaka; Shi, Yang

doi:10.1063/1.4931481

Cited by 18 publications

(41 citation statements)

References 31 publications

(90 reference statements)

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“…One can also say, that {ỹ i = 1} in tropical semifield, or this is the integrable q = 1 case, moreover restricted to a special case, when all Casimir functions are put to unities. In this case action of the subgroup of G Q presented above is equivalent to the action on tau-functions, see [37] for the case of A (1) 6 surface, [54] for the case of A 3 surface was proposed in [36], see also [44] and [9] (see Discussion).…”

Section: Remark 33mentioning

confidence: 99%

Cluster integrable systems, q-Painlevé equations and their quantization

Bershtein

Gavrylenko

Marshakov

2018

J. High Energ. Phys.

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We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.We also define quantum q-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using q-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.

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Section: Remark 33mentioning

confidence: 99%

Cluster integrable systems, q-Painlevé equations and their quantization

Bershtein

Gavrylenko

Marshakov

2018

J. High Energ. Phys.

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“…(1) symmetry, which has been obtained from a similar type of reduction of a system of quad-equations [16]. In Section 4, we provide an algebro-geometric description of Equation (4.43).…”

Section: System (315) Is a Sub-case Of The Q-discrete Painlevé Equatmentioning

confidence: 99%

Reflection groups and discrete integrable systems

2015

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Abstract. We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups. Discrete integrable systems are associated with space-filling polytopes arise from the geometric representation of the Weyl groups in the n-dimensional real Euclidean space R n . The "multi-dimensional consistency" property of the discrete integrable system is shown to be inherited from the combinatorial properties of the polytope; while the dynamics of the system is described by the affine translations of the polytopes on the weight lattices of the Weyl groups. The connections between some well-known discrete systems such as the multi-dimensional consistent systems of quad-equations [1] and discrete Painlevé equations [28] are obtained via the geometric constraints that relate the polytope of one symmetry group to that of another symmetry group, a procedure which we call geometric reduction.

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“…Details will be given in a subsequent paper. This work is motivated by our previous findings [14], where quad-equations were observed on what is called the ω-lattice, constructed from the τfunction theory of the A (1) 5 -surface q-Painlevé system. The present paper begins at the other end of the story with quad-equations on an n-cube.…”

Section: Introductionmentioning

confidence: 99%

Geometric reductions of ABS equations on ann-cube to discrete Painlevé systems

Joshi¹,

Nakazono²,

Shi³

2014

J. Phys. A: Math. Theor.

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In this paper, we show how to relate n-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlevé equations. We here focus on the reduction from the 4-dimensional cube to the q-discrete third Painlevé equation, which is a dynamical system on a rational surface of type A(1) 5 with the extended affine Weyl group W (A2 + A1) (1) . We provide general theorems to show that this reduction also extends to other discrete Painlevé equations at least of type A.

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Lattice equations arising from discrete Painlevé systems. I. (A2 + A1)(1) and (A1+A1′)(1) cases

Cited by 18 publications

References 31 publications

Cluster integrable systems, q-Painlevé equations and their quantization

Cluster integrable systems, q-Painlevé equations and their quantization

Reflection groups and discrete integrable systems

Geometric reductions of ABS equations on ann-cube to discrete Painlevé systems

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