Painlevé Functions and Conformal Blocks

Iorgov, N. Z.; Lisovyy, O.; Shchechkin, A.; Tykhyy, Yuriy

doi:10.1007/s00365-013-9226-y

Cited by 11 publications

(16 citation statements)

References 18 publications

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“…[KK, LT] for explicit elliptic and algebraic examples. A number of such checks has already been reported in [ILT13,ILST].…”

Section: Connection Problem For Painlevé VI Tau Function Proof Of Thmentioning

confidence: 99%

Monodromy dependence and connection formulae for isomonodromic tau functions

Its¹,

Lisovyy²,

Prokhorov³

2018

Duke Math. J.

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We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for generic Painlevé VI tau function. The result proves the conjectural formula for this constant proposed in [ILT13]. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painlevé II tau function.

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“…[KK, LT] for explicit elliptic and algebraic examples. A number of such checks has already been reported in [ILT13,ILST].…”

Section: Connection Problem For Painlevé VI Tau Function Proof Of Thmentioning

confidence: 99%

Monodromy dependence and connection formulae for isomonodromic tau functions

Its¹,

Lisovyy²,

Prokhorov³

2018

Duke Math. J.

Self Cite

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“…Painlevé equations are second-order differential equations with no movable brunching points except poles. We recall several facts about them following [17], [20]. The Painlevé VI equation has the form 2 ,…”

Section: Bilinear Relations On Conformal Blocks 41 Painlevé Equationmentioning

confidence: 99%

“…We will see that in this case the isomonodromic τ function coincides with Painlevé τ function. We follow [20] in the presentation.…”

Section: Bilinear Relations On Conformal Blocks 41 Painlevé Equationmentioning

confidence: 99%

Bilinear Equations on Painlevé τ Functions from CFT

Bershtein

Shchechkin

2015

Commun. Math. Phys.

116

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In 2012 Gamayun, Iorgov, 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 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 C 2 /Z 2 (similarly to Nakajima-Yoshioka blow-up equations).

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“…The first one is its short-distance (r → 0) expansion. This expansion has been known already [18,19] and is given by the convergent series…”

Section: Introductionmentioning

confidence: 99%

Connection Problem for the Sine-Gordon/Painlevé III Tau Function and Irregular Conformal Blocks: Fig. 1.

Its

Lisovyy

Tykhyy

2014

Int Math Res Notices

106

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Abstract. The short-distance expansion of the tau function of the radial sine-Gordon/Painlevé III equation is given by a convergent series which involves irregular c = 1 conformal blocks and possesses certain periodicity properties with respect to monodromy data. The long-distance irregular expansion exhibits a similar periodicity with respect to a different pair of coordinates on the monodromy manifold. This observation is used to conjecture an exact expression for the connection constant providing relative normalization of the two series. Up to an elementary prefactor, it is given by the generating function of the canonical transformation between the two sets of coordinates.

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Painlevé Functions and Conformal Blocks

Cited by 11 publications

References 18 publications

Monodromy dependence and connection formulae for isomonodromic tau functions

Monodromy dependence and connection formulae for isomonodromic tau functions

Bilinear Equations on Painlevé τ Functions from CFT

Connection Problem for the Sine-Gordon/Painlevé III Tau Function and Irregular Conformal Blocks: Fig. 1.

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