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## Painleve functions and conformal blocks

Constructive Approximation. 2014. Vol. 39. No. 1. P. 255-272.

Iorgov N., Lisovyy O., Tykhyy Y., A. Shchechkin

We outline recent developments relating Painlev ́e equations and 2D conformal field theory. Generic tau functions of Painlev ́e VI and Painlev ́e III_3 are written as linear combinations of c= 1 conformal blocks and their irregular limits. This provides explicit combinatorial series representations of the tau functions, and helps to establish connection formula for the tau function in the Painlev ́e VI case.

M.A. Bershtein, A.I.Shchechkin, Communications in Mathematical Physics 2015 Vol. 339 No. 3 P. 1021-1061

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 ...

Added: August 14, 2015

Gavrylenko P., Iorgov N., Lisovyy O., Letters in Mathematical Physics 2020 Vol. 110 No. 2 P. 327-364

We construct the general solution of a class of Fuchsian systems of rank N as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of WN-algebra with central charge c = N − 1. The simplest example is given by the tau function of the FujiSuzuki-Tsuda system, expressed as a Fourier ...

Added: August 20, 2020

Bershtein M., Gavrylenko P., Marshakov A., Journal of High Energy Physics 2018 Vol. 08 No. 108 P. 1-54

We study the twist-field representations of W-algebras and generalize construction of the corresponding vertex operators to D- and B-series. It is shown, how the computation of characters of these representations leads to nontrivial identities involving lattice theta-functions. We also propose a way to calculate their exact conformal blocks, expressing them for D-series in terms of ...

Added: September 11, 2018

Gavrylenko P., Lisovyy O., / arXiv.org. Series arXiv.org "math-ph". 2017. No. 1705.01869.

We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painlev\'e III equation of type $D_8$ (radial sine-Gordon equation). In particular, ...

Added: May 5, 2017

Gavrylenko P., Lisovyy O., Communications in Mathematical Physics 2018 Vol. 363 No. 1 P. 1-58

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL (N,ℂ). The corresponding operator acts in the direct sum of N (n − 3) copies of L2 (S1). Its kernel has a block integrable form and is expressed in ...

Added: September 12, 2018

Belavin A. A., Bershtein M., Feigin B. L. et al., Communications in Mathematical Physics 2012 P. 1-33

The recently proposed relation between conformal field theories in two dimensions and supersymmetric gauge theories in four dimensions predicts the existence of the distinguished basis in the space of local fields in CFT. This basis has a number of remarkable properties: one of them is the complete factorization of the coefficients of the operator product ...

Added: February 7, 2013

Bezrukavnikov R., Kazhdan D., Representation Theory 2015

Geometry of second adjointness for p-adic groups. ...

Added: July 27, 2015

Bershtein M., Shchechkin A., Letters in Mathematical Physics 2019 Vol. 109 No. 11 P. 2359-2402

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 ...

Added: August 31, 2020

Two-dimensional abelian BF theory in Lorenz gauge as a twisted N = (2,2) superconformal field theory

Losev A. S., Mnev P., Youmans D., Journal of Geometry and Physics 2018 Vol. 131 P. 122-137

We study the two-dimensional topological abelian BF theory in the Lorenz gauge and, surprisingly, we find that the gauged-fixed theory is a free type B twisted N = (2, 2) superconformal theory with odd linear target space, with the ghost field c being the pullback of the linear holomorphic coordinate on the target. The Q(BRST) ...

Added: October 5, 2018

Glutsyuk A., Bibilo Y., / Cornell University. Series arXiv "math". 2021. No. 2011.07839.

We study family of dynamical systems on 2-torus modeling over-damped Josephson junction in superconductivity. It depends on three parameters (B,A;ω): B (abscissa), A(ordinate), ω (a fixed frequency).We study the rotation numberρ(B,A;ω) as a function of (B,A) withfixedω. Aphase-lock areais the level set Lr:={ρ=r}, if it has an on-empty interior. This holds for r∈Z (a result ...

Added: November 26, 2020

Alkalaev K. B., Geiko R., Rappoport V. B., Journal of High Energy Physics 2017 P. 1-21

We study four types of one-point torus blocks arising in the large central charge regime. There are the global block, the light block, the heavy-light block, and the linearized classical block, according to different regimes of conformal dimensions. It is shown that the blocks are not independent being connected to each other by various links. ...

Added: April 17, 2017

Belavin V., Geiko R., Journal of High Energy Physics 2018 No. 112 P. 1-23

We develop a recursive approach to computing Neveu-Schwarz conformal blocks associated with n-punctured Riemann surfaces. This work generalizes the results of [1] obtained recently for the Virasoro algebra. The method is based on the analysis of the analytic properties of the superconformal blocks considered as functions of the central charge c. It consists of two main ...

Added: August 24, 2018

Poberezhny V. A., / ИТЭФ. Series "Препринты ИТЭФ". 2012. No. 57/12.

In this work we investigate the action of generalized Schlesinger transformation on the isomonodromic families of meromorphic connections on the linear bundles of rank two and degree zero over an elliptic curve. The main interest is the action of the gauge transformation on the moduli space of vector bundles. the central result is the explicit ...

Added: March 31, 2014

Gavrylenko P., Marshakov A., Journal of High Energy Physics 2014 No. 5 P. 97

We study the extended prepotentials for the S-duality class of quiver gauge theories, considering them as quasiclassical tau-functions, depending on gauge theory condensates and bare couplings. The residue formulas for the third derivatives of extended prepotentials are proven, which lead to effective way of their computation, as expansion in the weak-coupling regime. We discuss also ...

Added: October 20, 2014

Levin A., Olshanetsky M., Zotov A., Journal of Physics A: Mathematical and Theoretical 2016 Vol. 49 No. 39 P. 1-24

In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on the R-matrix description which provides Lax pairs in terms of quantum and classical R-matrices. First, we prove that for relativistic (and non-relativistic) tops, such Lax pairs with spectral parameters follow from the associative Yang–Baxter equation and its ...

Added: September 17, 2016

Parusnikova A., Vasilyev A., Journal of Mathematical Sciences 2019 Vol. 241 No. 3 P. 318-326

We examine asymptotic expansions of the third Painlevé transcendents for αδ ≠ 0 and γ = 0 in the neighborhood of infinity in a sector of aperture <2π by the method of dominant balance). We compare intermediate results with results obtained by methods of three-dimensional power geometry. We find possible asymptotics in terms of elliptic ...

Added: October 26, 2019

Bershtein M., Tarnopolsky G. M., Belavin A. A., JETP Letters 2011 Vol. 93 No. 2 P. 51-55

The one-matrix model is considered. The generating function of the correlation numbers is defined in such a way that this function coincide with the generating function of the Liouville gravity. Using the Kontsevich theorem we explain that this generating function is an analytic continuation of the generating function of the Topological gravity. We check the ...

Added: October 23, 2014

Bershtein M., Gavrylenko P., Marshakov A., / arXiv.org. Series arXiv.org "hep-th". 2017. No. 1705.00957.

We study twist-field representations of the W-algebras and generalize the construction of the corresponding vertex operators to D- and B-series. We demonstrate how the computation of characters of such representations leads to the nontrivial identities involving lattice theta-functions. We propose a construction of their exact conformal blocks, which for D-series express them in terms of ...

Added: May 4, 2017

Gavrylenko P., Lisovyy O., / Cornell University. Series math-ph "arXiv". 2016. No. 1608.00958.

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL(N,ℂ). The corresponding operator acts in the direct sum of N(n−3) copies of L2(S1). Its kernel has a block integrable form and is expressed in terms of fundamental solutions of ...

Added: September 20, 2016

Okounkov A., Rains E., Algebra and Number Theory 2015

Added: September 4, 2015

Losev A. S., Rosly A. A., Polubin I., Journal of High Energy Physics 2018 No. 41 P. 1-15

We compute the ultraviolet divergences in the self-dual Yang-Mills theory, both in the purely perturbative (zero instanton charge) and topologically non-trivial sectors. It is shown in particular that the instanton measure is precisely the same as the one-loop result in the standard Yang-Mills theory. ...

Added: March 28, 2018

Feigin B. L., Jimbo M., Mukhin E., Journal of Physics A: Mathematical and Theoretical 2017 Vol. 50 No. 46 Article 464001

We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors.
That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model.
We also discuss the (gl(m),gl(n)) duality of XXZ models in ...

Added: November 5, 2020

Poberezhny V. A., / ИТЭФ. Series "Препринты ИТЭФ". 2014. No. 50.14.

We prove that any non-resonant Fuchsian system with commutative monodromy is in fact a commutative system, that is a system with commuting residues. For logarithmic connection that Fuchsian system presents that implies the triviality of its isomonodromic deformations. ...

Added: March 26, 2015

Bershtein M., Алексеев О. В., Теоретическая и математическая физика 2010 Т. 164 № 1 С. 119-140

We consider the M(2,3) Minimal Liouville gravity, whose states in the gravity sector are represented by irreducible modules of the Virasoro algebra. We present a recursive construction for BRST cohomology classes. This construction is based on using an explicit form of singular vectors in irreducible modules of the Virasoro algebra. We construct an algebra of ...

Added: October 23, 2014