Abstract
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.
Highlights
On the gauge theory side, the 5d Nekrasov partition functions are more closely related to the topological strings partition functions, see [14, 34]
Recall the meaning of other relations in this case: the spectral curve of relativistic Toda is the Seiberg-Witten curve of pure SU(2) 5d theory, the corresponding Nekrasov partition function equals to the Whittaker limit of conformal block for q-deformed Virasoro algebra
We present a formal solution for the tau-functions of these quantum equations — as a linear combination of Nekrasov partition functions with generic -background parameters (q1, q2)
Summary
A lattice polygon ∆ is a polygon in the plane R2 with all vertices in Z2 ⊂ R2. There is an action of the group SA(2, Z) = SL(2, Z) Z2 on the set of such polygons, which preserves the area, the number of interior points, and the discrete lengths of sides (number of points on side including vertices minus 1). Integrability means that the coefficients, corresponding to the boundary points of I ∈ ∆ ̄ , are Casimir functions for the Poisson bracket (2.2) (their total number is B − 3, since the equation (2.1) is defined modulo multiplicative renormalization of spectral parameters λ, 1This property means that for each vertex the numbers of outgoing and incoming edges coincide. This is always true for quivers, corresponding to the cluster integrable systems, constructed from dimer models on a torus, or to the integrable systems on the Poisson submanifolds in affine Lie groups. Formula (2.6) provides the most technically effective way for writing the spectral curve equation (2.1), and provides expression for the integrals of motion in terms of cluster variables [40, 43], which can be even sometimes generalized to other series beyond P GL(N )
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