Communications in Mathematical Physics

, Volume 339, Issue 3, pp 1021–1061 | Cite as

Bilinear Equations on Painlevé τ Functions from CFT



In 2012, Gamayun, Iorgov, and Lisovyy conjectured an explicit expression for the Painlevé VI τ function in terms of the Liouville conformal blocks with central charge c = 1. We prove that the proposed expression satisfies Painlevé VI τ function bilinear equations (and therefore prove the conjecture). The proof reduces to the proof of bilinear relations on conformal blocks. These relations were studied using the embedding of a direct sum of two Virasoro algebras into a sum of Majorana fermion and Super Virasoro algebra. In the framework of the AGT correspondence, the bilinear equations on the conformal blocks can be interpreted in terms of instanton counting on the minimal resolution of \({\mathbb{C}^2/\mathbb{Z}_2}\) (similarly to Nakajima–Yoshioka blow-up equations).


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Institute for Information Transmission ProblemsMoscowRussia
  3. 3.International Laboratory of Representation Theory and Mathematical PhysicsNational Research University Higher School of EconomicsMoscowRussia
  4. 4.Independent University of MoscowMoscowRussia
  5. 5.Department of PhysicsTaras Shevchenko National University of KievKievUkraine
  6. 6.Bogolyubov Institute for Theoretical PhysicsKievUkraine

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