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 $2\times 2$ 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 $\mathrm{III}_3$ 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 ($ε_1+ε_2=0$). Through blowup relations, we also find Nekrasov-Shatashvili type of quantizations ($ε_2=0$).
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 $ε_2\to 0$ limit of the blowup relations from the regularized action functional and CFT arguments.
Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus
International School of Advanced Studies (SISSA)（高等研究学院）；Institute for Geometry and Physics (IGAP)（几何与物理研究所）；INFN, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Center for Advanced Studies, Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院高级研究中心）；NRU HSE, International Laboratory of Representation Theory and Mathematical Physics（国立高等经济大学表示论与数学物理国际实验室）；Bogolyubov Institute for Theoretical Physics（博戈留博夫理论物理研究所）
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 $W_N$-algebra with central charge $c=N-1$. 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 $W_N$-algebra for generic central charge and prove several propertie…
The Brezin-Gross-Witten tau function is a tau function of the KdV hierarchy which arises in the weak coupling phase of the Brezin-Gross-Witten model. It falls within the family of generalized Kontsevich matrix integrals, and its algebro--geometric interpretation has been unveiled in recent works of Norbury. We prove that a suitably generalized Brezin-Gross-Witten tau function is the isomonodromic tau function of a $2\times 2$ isomonodromic system and consequently present a study of this tau function purely by means of this isomonodromic interpretation. Within this approach we derive effective …
Classical conformal blocks naturally appear in the large central charge limit of 2D Virasoro conformal blocks. In the $AdS_{3}/CFT_{2}$ correspondence, they are related to classical bulk actions and are used to calculate entanglement entropy and geodesic lengths. In this work, we discuss the identification of classical conformal blocks and the Painlevé VI action showing how isomonodromic deformations naturally appear in this context. We recover the accessory parameter expansion of Heun's equation from the isomonodromic $τ$-function. We also discuss how the $c = 1$ expansion of the $τ$-function…
In this paper we study the isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the Takiff algebra (i.e. truncated current algebra). Our motivation is to produce confluent versions of the celebrated Knizhnik--Zamolodchikov equations and explain how their quasiclassical solution can be expressed via the isomonodromic $τ$-function. In order to achieve this, we study the confluence cascade of $r+ 1$ simple poles to give rise to a singularity of arbitrary Poincaré rank $r$ as a Pois…
We construct a fundamental system of a $q$-difference Lax pair of rank $N$ in terms of 5d Nekrasov functions with $q=t$. Our fundamental system degenerates by the limit $q\to 1$ to a fundamental system of a differential Lax pair, which yields the Fuji-Suzuki-Tsuda system. We introduce tau functions of our system as Fourier transforms of 5d Nekrasov functions. Using asymptotic expansions of the fundamental system at $0$ and $\infty$, we obtain several determinantal identities of the tau functions.
In this paper, we show that it is always possible to deform a differential equation $\partial_x Ψ(x) = L(x) Ψ(x)$ with $L(x) \in \mathfrak{sl}_2(\mathbb{C})(x)$ by introducing a small formal parameter $\hbar$ in such a way that it satisfies the Topological Type properties of Bergère, Borot and Eynard. This is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce $\hbar$. The topological recursion is then proved to provide a formal series expansion of the corresponding tau-function whose coefficients can thus be expr…
We consider deformations of a differential system with Poincare' rank 1 at infinity and Fuchsian singularity at zero along a stratum of a coalescence locus. We give necessary and sufficient conditions for the deformation to be strongly isomonodromic, both as an explicit Pfaffian system (integrable deformation) and as a non linear system of PDEs on the residue matrix A at the Fuchsian singularity. This construction is complementary to that of [13]. For the specific system here considered, the results generalize those of [26], by giving up the generic conditions, and those of [3], by giving up t…
Hamiltonian representation of isomonodromic deformations of general rational connections on $\mathfrak{gl}_2(\mathbb{C})$
Université Jean Monnet Saint-Étienne, CNRS, Institut Camille Jordan UMR 5208（圣艾蒂安让·莫内大学，法国国家科学研究中心，卡米尔·乔丹研究所）；Institut Universitaire de France（法国高等研究院）；Université de Genève, Section de mathématiques（日内瓦大学数学系）
Irregular SLE(4) martingales and isomonodromic deformations
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Institute of Mathematics, EPFL（洛桑联邦理工学院数学研究所）；Department of Mathematics and Systems Analysis, Aalto University（阿尔托大学数学与系统分析系）；Division of Mathematics, University of Cologne（科隆大学数学系）
In this short note, we give a self-contained derivation of the formula for the $2$-point full-plane Ising spin correlation function under massive scaling limit in terms of a third Painlevé transcendant. This formula, first derived in a celebrated work of Wu, McCoy, Tracy, and Barouch, was subsequently reformulated in terms of the theory of isomonodromic deformation by Sato, Miwa, and Jimbo. In view of recent developments in the discrete analysis which have enabled, in particular, a convergence proof of spin correlation functions on isoradial lattice, we give a concise and rigorous account of t…
Dual Isomonodromic Deformations and Moment Maps to Loop Algebras
利用从辛向量空间到两个不同loop代数的对偶空间的参数依赖矩对映射，解释了Jimbo { \it et al }的保单值变形方程的Hamilton结构.产生变形的非自治Hamilton系统通过拉回Poisson子空间上的谱不变量来获得，Poisson子空间由在回路参数中有理的元素组成，并通过确定矩映射来确定变形参数。这种结构导致“双”对矩阵微分算子的monodromy是保存在同一家庭的变形。作为说明性的例子，涉及离散和连续的减少，一个更高的排名推广的哈密顿方程的关联函数的不可穿透的玻色气体，以及对偶对isomonodromy表示的Painleve超越$P_{V}$和$P_{VI}$的方程。
We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters u=(u_1,...,u_n), which are eigenvalues of the leading matrix at the irregular singuilarity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters u_1,...,u_n. The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singulariti…
Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy
Université de Lyon（里昂大学）；Université Jean Monnet Saint-Étienne（圣艾蒂安让·莫内大学）；CNRS UMR 5208, Institut Camille Jordan（法国国家科学研究中心 5208 联合实验室，卡米尔·若当研究所）；Institut Universitaire de France（法兰西学院）
We extend the approach to $τ$-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld-Sokolov hierarchies and isomonodromic deformations systems. The combinatorial expansion of the $τ$-function as a sum of correlators, each expressed as products of finite determinants, follows from using multicomponent fermionic vacuum expectation values of certain dressing operators encoding the initial conditions and the dependence on the flow (or deformation) parameters. When reduced to the orthogonal case, these correlators become finite Pfaffians and the…
This is the second article in a suite of articles investigating relations between Stäckel-type systems and Painlevé-type systems. In this article we construct isomonodromic Lax representations for Painlevé-type systems found in the previous paper by Frobenius integrable deformations of Stäckel-type systems. We first construct isomonodromic Lax representations for Painlevé-type systems in the so called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painlevé-type systems in the non-magnetic represent…
The dependence on the monodromy data of the isomonodromic tau function
[Note：important Corrigendum now available at arXiv：1601.04790] Jimbo-Miwa-Ueno定义的等单值tau函数在广义单值数据的Malgrange因子上消失，对于广义单值数据，向量丛是非平凡的，或者，相同的，某个Riemann-Hilbert问题没有解。在他们的原始工作中，Jimbo，Miwa，Ueno没有推导出对（广义）单值数据的依赖性（即单值表示和斯托克斯参数）。我们填补了空白，提供了一个（更简单，更一般）的描述，其中所有的参数的问题（单值变化和单值保持）处理在同一水平。因此，我们提供变分公式的isomonodromic tau函数（广义）单值数据。这种构造适用于更普遍的情况：给定任何（足够好的）黎曼-希尔伯特问题族（RHP），其中跳跃矩阵任意依赖于变形参数，我们可以在变形空间（马尔格朗日微分）上构造一个单形式的Omega（不一定是封闭的），由马尔格朗日因子定义。然后，我们引入离散Schlesinger变换的概念：这意味着我们允许RHP的解在指定点处具有极点（或零点）。即使欧米茄是不封闭的，它的差异评估沿原来的解决方案和变换的一个，是对数微分（变形空间）的功能。作为Schlesinger变换的点的位置的函数，即使在没有tau函数的情况下，也产生了Baker-Akhiezer向量的Sato公式的自然推广，并且它实现…
We focus on Fuchsian equations with four accessory parameters and three singular points. We see that the Fuchsian equations admit a "degeneration scheme" in some sense, which is expected to give rise to a degeneration scheme of discrete isomodromic deformation equations with four-dimensional phase space. We compute an example of discrete isomonodromic deformation equations of a certain Fuchsian equation.
A symmetry reduction of the Painlevé IV hierarchy to the Flaschka-Newell Painlevé II hierarchy
Université de Genève（日内瓦大学）；Université Jean Monnet Saint-Étienne, CNRS, Institut Camille Jordan UMR 5208, Institut Universitaire de France（圣艾蒂安让·穆兰大学、法国国家科学研究中心、卡米尔·乔丹研究所UMR 5208、法国高等研究院）
研究偶数阶 Painlevé IV 层级对应的等单值变形问题，通过对称性约化得到 Flaschka-Newell Painlevé II 层级，并显式匹配 Lax 矩阵、Darboux 坐标和哈密顿结构。
We introduce and study the dynamics of Chebyshev polynomials on $d>2$ real intervals. We define isoharmonic deformations as a natural generalization of the Chebyshev dynamics. This dynamics is associated with a novel class of constrained isomonodromic deformations for which we derive the constrained Schlesinger equations. We provide explicit solutions to these equations in terms of differentials on an appropriate family of hyperelliptic curves of any genus $g=d-1\ge 2$. The verification of the obtained solutions relies on the combinatorial properties of the Bell polynomials and on the analysis…
We consider the moduli space of holomorphic principal bundles for reductive Lie groups over Riemann surfaces (possibly with boundaries) and equipped with meromorphic connections. We associate to this space a point-wise notion of quantum spectral curve whose generalized periods define a new set of moduli. We define homology cycles and differential forms of the quantum spectral curve, allowing to derive quantum analogs of the form-cycle duality and Riemann bilinear identities of classical geometry. A tau-function is introduced for this system in the form of a theta-series and in such a way that …
The extension of the Painlevé-Calogero coorespondence for n-particle Inozemtsev systems raises to the multi-particle generalisations of the Painlevé equations which may be obtained by the procedure of Hamiltonian reduction applied to the matrix or non-commutative Painlevé systems, which also gives isomonodromic formulation for these non-autonomous Hamiltonian systems. We provide here dual systems for the rational multi-particle Painlevé systems (PI,PII and PIV) by reduction from another intersection a coadjoint orbit of GL(n) action with the level set of moment map. We describe this duality in…
We consider a 3-dimensional Pfaffian system, whose z-component is a differential system with irregular singularity at infinity and Fuchsian at zero. In the first part of the paper, we prove that its Frobenius integrability is equivalent to the sixth Painlevé equation PVI. The coefficients of the system will be explicitly written in terms of the solutions of PVI. In this way, we remake a result of [44, 61]. We then express in terms of the Stokes matrices of the 3x3 irregular system the monodromy invariants p_{jk}=Tr(M_jM_k) of the 2-dimensional isomonodromic Fuchsian system with four singularit…
