Cluster integrable systems, q-Painleve equations and their quantization

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.

arXiv comment28 pages; v2 30 pages references added, misprints corrected; v3 small changes, references added, to appear in JHEP

References in corpus (38)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Five Dimensional SUSY Field Theories, Non-trivial Fixed Points and String Dynamics
• Integrability and Seiberg-Witten Exact Solution
• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• Five Dimensional Gauge Theories and Relativistic Integrable Systems
• N=2 Quantum Field Theories and Their BPS Quivers
• Non-Lagrangian Theories from Brane Junctions
• Topological Strings and Nekrasov's formulas
• How instanton combinatorics solves Painlevé VI, V and III's
• Isomonodromic tau-functions from Liouville conformal blocks
• Geometric Aspects of Painlevé Equations
• Brane Tilings and Reflexive Polygons
• Bilinear equations on Painleve tau functions from CFT
• Q-deformed Painleve tau function and q-deformed conformal blocks
• Kernel function and quantum algebras
• Isomonodromic $\tau$-functions and $W_N$ conformal blocks
• Five-Dimensional Gauge Theories and Local Mirror Symmetry
• Analyticity of Nekrasov Partition Functions
• A Comprehensive Survey of Brane Tilings
• Lie Groups, Cluster Variables and Integrable Systems
• A compendium on the cluster algebra and quiver package in sage
• Cluster X-varieties, amalgamation and Poisson-Lie groups
• Bilinear equations and $q$-discrete Painlevé equations satisfied by variables and coefficients in cluster algebras
• Backlund transformation of Painleve III($D_8$) tau function
• Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations
• Loop groups, Clusters, Dimers and Integrable systems
• CFT approach to the $q$-Painlevé VI equation
• Lattice equations arising from discrete Painlevé systems (I): $(A_2+A_1{)}^{(1)}$ and $(A_1+A_1^{\prime }{)}^{(1)}$ cases
• q-Painleve equation from Virasoro constraints
• W(E_10) Symmetry, M-Theory and Painleve Equations
• Norm of the Whittaker vector of the deformed Virasoro algebra
• Universal character and q-difference Painlevé equations with affine Weyl groups
• Discrete Painlevé equations from Y-systems
• Symmetries of quantum Lax equations for the Painlevé equations
• Quantum groups and quantization of Weyl group symmetries of Painlevé systems
• On Lie Groups and Toda Lattices
• Co-primeness preserving higher dimensional extension of $q$-discrete Painlevé I, II equations

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• Birational Transformations on Dimer Integrable Systems
• Classification and Birational Equivalence of Dimer Integrable Systems for Reflexive Polygons
• On a 5D UV completion of Argyres-Douglas theories
• Probing Quantum Curves and Transitions in 5d SQFTs via Defects and Blowup Equations
• Blowup relations on ${\mathbb{C}}^2/{\mathbb{Z}}_2$ from Nakajima-Yoshioka blowup relations
• Periodicity, linearizability and integrability in seed mutations of type $A_N^{(1)}$
• Birational maps conjugate to the rank 2 cluster mutations of affine types and their geometry
• Okamoto's symmetry on the representation space of the sixth Painlevé equation
• Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux