318 research outputs found
Quantum spectral problems and isomonodromic deformations
Full text linkWe develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of linear systems (Riemann–Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé equation. Then we extend the analysis to the 2-particle elliptic Calogero–Moser operator, which corresponds to a linear system on the torus. By using the Kyiv formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (). Through blowup relations, we also find Nekrasov–Shatashvili type of quantizations (). In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the limit of the blowup relations from the regularized action functional and CFT arguments.We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of linear systems (Riemann-Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé equation. Then we extend the analysis to the 2-particle elliptic Calogero-Moser operator, which corresponds to a linear system on the torus. By using the Kiev formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (). Through blowup relations, we also find Nekrasov-Shatashvili type of quantizations (). In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the limit of the blowup relations from the regularized action functional and CFT arguments
Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus
Full text linkWe compute the monodromy dependence of the isomonodromic tau function on a
torus with Fuchsian singularities and residue matrices by using its
explicit Fredholm determinant representation. We show that the exterior
logarithmic derivative of the tau function defines a closed one-form on the
space of monodromies and times, and identify it with the generating function of
the monodromy symplectomorphism. As an illustrative example, we discuss the
simplest case of the one-punctured torus in detail. Finally, we show that
previous results obtained in the genus zero case can be recovered in a
straightforward manner using the techniques presented here.Comment: 24 pages, 3 figure
Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions
Full text linkWe prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in GL(N, C) can be written in terms of a Fred holm determinant of Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of SL(2, C) this combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results we also propose a definition of the tau function of the Riemann-Hilbert problem on a torus with generic jump on the A-cycle
N=2*Gauge Theory, Free Fermions on the Torus and Painleve VI
No full textIn this paper we study the extension of Painlevé/gauge theory correspondence to circular quivers by focusing on the special case of SU(2) N= 2 ∗ theory. We show that the Nekrasov–Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of SL2 flat connections on the one-punctured torus. This is achieved by reformulating the Riemann–Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the SU(2) N= 2 ∗ theory on self-dual Ω -background and, in the Seiberg–Witten limit, an elegant relation between the IR and UV gauge couplings. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature
Crossing invariant correlation functions at c = 1 from isomonodromic τ functions
Full text linkWe 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 SLspace A404(s,C) 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 RunkelWatts theory. Among the possible crossing-invariant theories, we obtain also the analytic Liouville theory, whose consistence was assumed only on the basis of numerical tests
Irregular conformal blocks, Painleve III and the blow-up equations
Full text linkWe study the relation of irregular conformal blocks with the Painleve III3 equation. The functional representation for the quasiclassical irregular block is shown to be consistent with the BPZ equations of conformal field theory and the Hamilton-Jacobi approach to Painleve III3. It leads immediately to a limiting case of the blow-up equations for dual Nekrasov partition function of 4d pure supersymmetric gauge theory, which can be even treated as a defining system of equations for both c = 1 and c -> infinity conformal blocks. We extend this analysis to the domain of strong-coupling regime where original definition of conformal blocks and Nekrasov functions is not known and apply the results to spectral problem of the Mathieu equations. Finally, we propose a construction of irregular conformal blocks in the strong coupling region by quantization of Painleve III3 equation, and obtain in this way a general expression, reproducing c = 1 and quasiclassical c -> infinity results as its particular cases. We have also found explicit integral representations for c = 1 and c = -2 irregular blocks at infinity for some special points
Het'man Pavlo Skoropads'kyi and Foundation of Ukrainian Academy of Sciences
No full textАвтор намагається з’ясувати, яку роль відіграв гетьман Павло Скоропадський у справі заснування Української академії наук та причини, що призвели до затьмарення ролі глави Української Держави в «академічному процесі». Аналізуються самооцінка Павла Скоропадського, інвективи Миколи Василенка та Володимира Вернадського щодо його ролі в заснуванні УАН, а також офіційна документація та публічні акти 1918 р. Автор статті доходить висновку, що П.Скоропадський має бути залучений до кола «батьків-засновників» УАН.The author is trying to find out which role hetman Pavlo Skoropads’kyi performed in the case of foundation of Ukrainian Academy of Sciences and the causes that led to diminishing of the part of Ukrainian state’s head in «academic process». The article analyzes Pavlo Skoropads’kyi’s self-assessment, Mykola Vasylenko’s and Volodymyr Vernads’kyi’s invectives as to his role in foundation of UAS, and official documents and public acts of 1918. The author of the article comes to the conclusion that P.Skoropads’kyi has to be enlisted to the circle of founders of UAS
Analytical Review of the Biobibliography of Pavlo Zahrebelnyi
No full textThe purpose of the article is to conduct a critical study of the bibliography of Pavlo Zahrebelnyi in the indexes contained in the collection of V. I. Vernadsky National Library of Ukraine (herein after referred to as VNLU) and available in the Internet environment, to analyze the selected bibliographic indexesusing modern methodological developments; to evaluate and provide evidence of their scientific, informational, and cultural and historical value. Methodology. The study used survey-bibliographic, analytical-critical, historical, and comparative scientific methods. Scientific novelty. For the first time, a study of the bibliography, divided into many editions in the form of indexes, was carried out; the artistic and literary orientation of Pavlo Zahrebelnyi’s works was taken into account using new theoretical and methodological principles. The article meets the current tasks of modern bibliographic studies. Conclusions. The works by Pavlo Zahrebelnyi are known outside Ukraine, have been translated into many languages of the world, and have been repeatedly filmed. Not all of them have become outstanding artistic achievements of Ukrainian prose, but all of them have been actively perceived by readers, discussed by critics, and in one way or another influenced the development of the modern literary process. The critical analysis of the indexes logically led the author of the article to the opinion that the compilation of acomplete consolidated bibliographic index is still waiting for its author. The publication of the writer’s private diaries is still ahead, and there is information on the Internet about holding scientific conferences dedicated to the work of Pavlo Zahrebelnyi and, as a result, the publication of articles in periodicals and collections
Higher rank isomonodromic deformations and -algebras
Full text linkInternational audienceWe 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 -algebra with central charge . The simplest example is given by the tau function of the Fuji–Suzuki–Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respect to intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the minimal set of matrix elements of vertex operators of the -algebra for generic central charge and prove several properties of semi-degenerate vertex operators and conformal blocks for
On solutions of the Fuji-Suzuki-Tsuda system
Full text linkInternational audienceWe 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 Painleve 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 initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for
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