Painleve equations from Nakajima-Yoshioka blowup relations

Abstract

Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation is equal to the series of $c=1$ Virasoro conformal blocks. We study similar series of $c=-2$ conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima-Yoshioka blowup relations on Nekrasov partition functions. We also study series of $q$-deformed $c=-2$ conformal blocks and relate it to $q$-Painlevé equation. As an application, we prove formula for the tau function of $q$-Painlevé $A_7^{(1{)}^{\prime }}$ equation.

arXiv commentv1 36 pages; v2 38 pages, many small corrections (including the title), references added

References in corpus (19)

• Instanton counting on blowup. I. 4-dimensional pure gauge theory
• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• Symplectic Fermions
• Logarithmic extensions of minimal models: characters and modular transformations
• How instanton combinatorics solves Painlevé VI, V and III's
• Isomonodromic tau-functions from Liouville conformal blocks
• Conformal field theory of Painlevé VI
• Fiber-Base Duality and Global Symmetry Enhancement
• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• Bilinear equations on Painleve tau functions from CFT
• Cluster integrable systems, q-Painleve equations and their quantization
• Five-dimensional Chern-Simons terms and Nekrasov's instanton counting
• Q-deformed Painleve tau function and q-deformed conformal blocks
• Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations
• Perverse coherent sheaves on blow-up. III. Blow-up formula from wall-crossing
• On determinant representation and integrability of Nekrasov functions
• Backlund transformation of Painleve III($D_8$) tau function
• Coupling of two conformal field theories and Nakajima-Yoshioka blow-up equations
• Free fermions, W-algebras and isomonodromic deformations

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• Quantum spectral problems and isomonodromic deformations
• SU(N) q-Toda equations from mass deformed ABJM theory
• Irregular conformal blocks, Painlevé III and the blow-up equations
• M2-branes and $\mathfrak{q}$-Painlevé equations
• S-duality and supersymmetry on curved manifolds