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## $\mathcal{N}=2^*$ gauge theory, free fermions on the torus and Painlevé VI

Gavrylenko P., Bonelli G., Del Monte F., Tanzini A.

Bonelli G., Gavrylenko P., Tanzini A. et al., Working papers by Cornell University. Series math "arxiv.org" 2019

Added: November 13, 2019

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

Bershtein M., Gavrylenko P., Marshakov A., Theoretical and Mathematical Physics 2019 Vol. 198 No. 2 P. 157-188

We extend the relation between cluster integrable systems and q-difference equations beyond the Painlevé case. We consider the class of hyperelliptic curves where the Newton polygons contain only four boundary points. We present the corresponding cluster integrable Toda systems and identify their discrete automorphisms with certain reductions of the Hirota difference equation. We also construct ...

Added: November 13, 2019

Gavrylenko P., Marshakov A., / Cornell University. Series "Working papers by Cornell University". 2015. No. 1507.08794.

We consider the conformal blocks in the theories with extended conformal W-symmetry for the integer Virasoro central charges. We show that these blocks for the generalized twist fields on sphere can be computed exactly in terms of the free field theory on the covering Riemann surface, even for a non-abelian monodromy group. The generalized twist ...

Added: October 14, 2015

Iorgov N., Lisovyy O., Tykhyy Y. et al., Constructive Approximation 2014 Vol. 39 No. 1 P. 255-272

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 ...

Added: August 14, 2015

Gavrylenko P., Marshakov A., Theoretical and Mathematical Physics 2016 Vol. 87 No. 2 P. 649-677

We consider the theory of multicomponent free massless fermions in two dimensions and use it to construct representations of W-algebras at integer Virasoro central charges. We define the vertex operators in this theory in terms of solutions of the corresponding isomonodromy problem. We use this construction to obtain some new insights into tau functions of the ...

Added: September 16, 2016

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

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

Cafasso M., Gavrylenko P., Lisovyy O., Communications in Mathematical Physics 2019 Vol. 365 No. 2 P. 741-772

We define a tau function for a generic Riemann–Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed as the Fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary ...

Added: September 12, 2018

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

Gavrylenko P., Iorgov N., Lisovyy O., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2018 Vol. 14 P. 1-27

We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlevé VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also ...

Added: November 22, 2018

Гонцов Р.Р., Побережный В.А., В кн. : Дифференциальные уравнения и смежные вопросы математики. Труды XIII Приокской научной конференции. : Государственный социально-гуманитарный университет, 2021. Гл. 8. С. 67-84.

Дан вводный обзор явления изомонодромности, играющего важнейшую роль в современной аналитической теории дифференциальных уравнений и возникающего во множестве смежных с ней областей математики и физики. Выделено различие между определениями и свойствами локальной и глобальной изомонодромности, расммотрен ряд примеров с использовангием шлезингеровской деформации. ...

Added: December 6, 2022

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

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

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