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## Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations

Cornell University
,
2015.
No. 1507.08794.

Gavrylenko P., Marshakov A.

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 fields are identified with particular primary fields of the W-algebra, and we propose a straightforward way to compute their W-charges. We demonstrate how these exact conformal blocks can be effectively computed using the technique arisen from the gauge theory/CFT correspondence. We discuss also their direct relation with the isomonodromic tau-function for the quasipermutation monodromy data, which can be an encouraging step on the way of definition of generic conformal blocks for W-algebra using the isomonodromy/CFT correspondence.

Priority areas:
mathematics

Language:
English

Keywords: интегрируемые иерархииintegrable hierarchiesизомонодромные деформацииconformal and W symmetryisomonodromic deformationsконформная теория поля

Publication based on the results of:

Gavrylenko P., Journal of High Energy Physics 2015 No. 09 P. 167

We study the solution of the Schlesinger system for the 4-point $\mathfrak{sl}_N$ isomonodromy problem and conjecture an expression for the isomonodromic τ-function in terms of 2d conformal field theory beyond the known N = 2 Painlevé VI case. We show that this relation can be used as an alternative definition of conformal blocks for the ...

Added: October 9, 2015

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

Bershtein M., Feigin B. L., Litvinov A., Letters in Mathematical Physics 2016 Vol. 106 No. 1 P. 29-56

We study the conformal vertex algebras which naturally arise in relation to the Nakajima–Yoshioka blow-up equations. ...

Added: November 8, 2017

Litvinov A., Spodyneiko L., Journal of High Energy Physics 2016 Vol. 1611 P. 1-17

We consider the problem of classification of all W algebras which commute with a set of exponential screening operators. Assuming that the W algebra has a nontrivial current of spin 3, we find equations satisfied by the screening operators and classify their solutions. ...

Added: November 7, 2017

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

Gavrylenko P., Santachiara R., Journal of High Energy Physics 2019 Vol. 2019 No. 11 P. 1-36

We present an approach that gives rigorous construction of a class of crossing invariant functions in c = 1 CFTs from the weakly invariant distributions on the moduli space \( {\mathcal{M}}_{0,4}^{\mathrm{SL}\left(s,\mathbb{C}\right)} \) of SL(2, ℂ) flat connections on the sphere with four punctures. By using this approach we show how to obtain correlation functions in the Ashkin-Teller and the Runkel- ...

Added: May 14, 2020

Geiko R., Belavin V., / Cornell. Series 1705.10950 "hep-th". 2017.

We continue to investigate the dual description of the Virasoro conformal blocks
arising in the framework of the classical limit of the AdS 3 /CFT 2 correspondence. To give such
an interpretation in previous studies, certain restrictions were necessary. Our goal here is to
consider a more general situation available through the worldline approximation to the dual
AdS gravity. ...

Added: June 3, 2017

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

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

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

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

Belavin V., Geiko R., Journal of High Energy Physics 2017 Vol. 125 P. 1-13

We continue to investigate the dual description of the Virasoro conformal blocks arising in the framework of the classical limit of the AdS3/CFT2 correspondence. To give such an interpretation in previous studies, certain restrictions were necessary. Our goal here is to consider a more general situation available through the worldline approximation to the dual AdS gravity. ...

Added: August 31, 2017

Zabrodin A., Zotov A., Journal of Mathematical Physics 2012 Vol. 53 No. 7 P. 073508-1-073508-19

This paper is a continuation of our previous paper where the Painlevé-Calogero correspondence has been extended to auxiliary linear problems associated with Painlevé equations. We have proved, for the first five equations from the Painlevé list, that one of the linear problems can be recast in the form of the non-stationary Schrödinger equation whose Hamiltonian ...

Added: September 19, 2012

Alexandrov A., Mironov A., Morozov A. et al., Journal of Physics A: Mathematical and Theoretical 2012 No. 45 P. 1-10

We construct partition functions that are tau-functions of integrable hierarchies. ...

Added: September 19, 2012

Alexandrov A., Mironov A., Morozov A. et al., Journal of High Energy Physics 2014 Vol. 11 No. 80 P. 1-31

There is now a renewed interest to a Hurwitz tau-function, counting the
isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and
Grothiendicks’s dessins d’enfant. It is distinguished by belonging to a particular family
of Hurwitz tau-functions, possessing conventional Toda/KP integrability properties. We
explain how the variety of recent observations about this function fits into the ...

Added: December 2, 2014

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., / 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

M.N. Alfimov, A.V. Litvinov, Journal of High Energy Physics 2015 No. 02 P. 150-162

We study integrable structure of the coset conformal field theory and define the system of Integrals of Motion which depends on external parameters. This system can be viewed as a quantization of the ILW type hierarchy. We propose a set of Bethe anzatz equations for its spectrum. ...

Added: November 7, 2017

Zabrodin A., Zotov A., Journal of Mathematical Physics 2012 Vol. 53 No. 7 P. 073507-1-073507-19

The Painlevé-Calogero correspondence is extended to auxiliary linear problems associated with Painlevé equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the Painlevé-Calogero correspondence. Namely, the linear problem responsible for the time evolution is brought into the form of non-stationary Schrödinger equation in imaginary ...

Added: September 19, 2012

M.N. Alfimov, Belavin A., Tarnopolsky G., Journal of High Energy Physics 2013 Vol. 08 No. 2013 P. 134-160

We study conformal field theory with the symmetry algebra ...

Added: October 20, 2016

Zabrodin A., Alexandrov A., Journal of Geometry and Physics 2013 Vol. 67 P. 37-80

We review the formalism of free fermions used for construction of tau-functions of classical integrable hierarchies and give a detailed derivation of group-like properties of the normally ordered exponents, transformations between different normal orderings, the bilinear relations, the generalized Wick theorem and the bosonization rules. We also consider various examples of tau-functions and give their ...

Added: February 16, 2013

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

BOSSY M., Jabir J. M., Electronic Communications in Probability 2018 Vol. 23 P. 1-14

In this paper, we prove a particle approximation, in the sense of the propagation of chaos, of a Lagrangian stochastic model submitted to specular boundary condition and satisfying the mean no-permeability condition. ...

Added: June 7, 2018

Буров А.А., Герман А. Д., Косенко И. И., Космические исследования 2014 Т. 52 № 4 С. 307-312

The problem of planar oscillations of a pendulum with variable length suspended on the Moon’s surface is considered. It is assumed that the Earth and Moon (or, in the general case, a planet and its satellite, or an asteroid and a spacecraft) revolve around the common center of mass in unperturbed elliptical Keplerian orbits. We ...

Added: November 8, 2014