This note examines the geometry behind the Hamiltonian structure of isomonodromy deformations of connections on vector bundles over Riemann surfaces. The main point is that one should think of an open set of the moduli of pairs $(V,\nabla)$ of vector bundles and connections as being obtained by "twists" supported over points of a fixed vector bundle $V_0$ with a fixed connection $\nabla_0$; this gives two deformations, one, isomonodromic, of $(V,\nabla)$, and another induced from the isomonodromic deformation of $(V_0,\nabla_0)$. The difference between the two will be Hamiltonian.
Here we review some recent developments in the theory of isomonodromic deformations on Riemann sphere and elliptic curve. For both cases we show how to derive Schlesinger transformations together with their action on tau-function, and construct classes of solutions in terms of multi-dimensional theta-functions.
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 study symmetries of the Bethe equations for the Gaudin model appeared naturally in the framework of the geometric Langlands correspondence under the name of Hecke operators and under the name of Schlesinger transformations in the theory of isomonodromic deformations, and particularly in the theory of Painlevé transcendents.
In the present work we calculate the group structure of the Schlesinger transformations for isomonodromic deformations of order two Fuchsian differential equations. We perform these transformations as the isomorphisms between the moduli spaces of the logarithmic sl(2)- connections with fixed eigenvalues of residues at singular points. We give the geometrical interpretation of the Schlesinger transformations and perform the calculations using the techniques of the modifications of bundles with connections. In order to illustrate this result we present classical examples of symmetries of the hyp…
A class of Riemann-Hilbert problems corresponding to quasi-permutation monodromy matrices is solved in terms of Szegö kernel on auxiliary Riemann surfaces. The tau-function of Schlesinger system turns out to be closely related to determinant of Cauchy-Riemann operator. The link between theta-divisor and Malgrange's divisor in the theory of Schlesinger equations is established.
Anti-self-dual (ASD) solutions to the Yang-Mills equation (or instantons) over an anti-self-dual four manifold, which are invariant under an appropriate action of a three dimensional Lie group, give rise, via twistor construction, to isomonodromic deformations of connections on C P 1 having four simple singularities. As is well known this kind of deformations is governed by the sixth Painlevé equation P vi (α, \b{eta}, γ, δ) . We work out the particular case of the SU 2 -action on S 4 , obtained from the irreducible representation on R 5 . In particular, we express the pa- rameters (α, \b{eta}…
A comparison is made between bispectral systems and dual isomonodromic deformation equations. A number of examples are given, showing how bispectral systems may be embedded into isomonodromic ones. Sufficiency conditions are given for the construction of rational solutions of isomonodromic deformation equations through the Riemann-Hilbert problem dressing method, and these are shown, in certain cases, to reduce to bispectral systems.
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）
We consider the theory of multicomponent free massless fermions in two dimensions and use it for construction of 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 get some new insights on tau-functions of the multicomponent Toda type hierarchies for the class of solutions, given by the isomonodromy vertex operators and get useful representation for the tau-function of isomonodromic deformations.
In this paper we build a link between the Teichmuller theory of hyperbolic Riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the Fuchsian group associated to the given hyperbolic Riemann surface by the Poincare' uniformization. In the case of a one-sheeted hyperboloid with n orbifold points we show that the Poisson algebra D_n of geodesic length functions is the semiclassical limit of the twisted q-Yangian for the orthogonal Lie algebra defined by Molev, Ragoucy and Sorba. We give a representation of the braid group action on this algebra in terms of an…
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 …
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…
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…
The Hamiltonian approach to the theory of dual isomonodromic deformations is developed within the framework of rational classical R-matrix structures on loop algebras. Particular solutions to the isomonodromic deformation equations appearing in the computation of correlation functions in integrable quantum field theory models are constructed through the Riemann-Hilbert problem method. The corresponding $τ$-functions are shown to be given by the Fredholm determinant of a special class of integral operators.
This paper is devoted to two geometric constructions related to the isomonodromic method. We follow the Drinfeld ideas and develop them in the case of the curve $X=\mathbb{P}^1\setminus\{a_1,...,a_n\}$. Thus we generalize the results of Arinkin and Lysenko to the case of arbitrary number $n$ of points. First, we construct separated Darboux coordinated in terms of the Hecke correspondences between moduli spaces. In this way we present a geometric interpretation of the Sklyanin formulas. In the second part of the paper, we construct Drinfeld's compactification of the initial data space and descr…
In 1980 Jimbo and Miwa evaluated the diagonal two-point correlation function of the square lattice Ising model as a $τ$-function of the sixth Painlevé system by constructing an associated isomonodromic system within their theory of holonomic quantum fields. More recently an alternative isomonodromy theory was constructed based on bi-orthogonal polynomials on the unit circle with regular semi-classical weights, for which the diagonal Ising correlations arise as the leading coefficient of the polynomials specialised appropriately. Here we demonstrate that the next-to-diagonal correlations of the…
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.
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 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…
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…
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 gaug…
The Hamiltonian structure of the monodromy preserving deformation equations of Jimbo {\it et al } is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to ``dual'' pairs of matrix different…
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…
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…
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）
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…
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 cor…
We conisder time-dependent Schrödinger systems, which are quantizations of the Hamiltonian systems obtained from a similarity reduction of the Drinfeld-Sokolov hierarchy by K. Fuji and T. Suzuki, and a similarity reduction of the UC hierarchy by T.Tsuda, independently. These Hamiltonian systems describe isomonodromic deformations for certain Fuchsian systems. Thus, our Schrödinger systems can be regarded as quantum isomonodromic systems. Y. Yamada conjectured that our quantum isomonodromic systems determine instanton partition functions in N=2 SU(L) gauge theory. The main purpose of this paper…
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）
The Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators D_l. These have regular singular points at the 2m endpoints of the curve segments and a singular point of Poincare index 1 at infinity. The rank r of the vector bundle over the Riemann sphere on which they act equals the number of distinct terms in the exponential sums entering in the numerator of the integral kernels. The deformation equations may be viewed…
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…
[Note: important Corrigendum now available at arXiv:1601.04790] The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange's divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which is the same, a certain Riemann-Hilbert problem has no solution. In their original work, Jimbo, Miwa, Ueno did not derive the dependence on the (generalized) monodromy data (i.e. monodromy representation and Stokes' parameters). We fill the gap by providing a (simpler and more general) description in which all the parameters of the problem (monodromy-changing …
We study a generalization of the isomonodromic deformation to the case of connections with irregular singularities. We call this generalization Isostokes Deformation. A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points. We study this part of the deformation, giving an algebraic description. Then we show how to use loop groups and hypercohomology to write explicit hamiltonians. We work on an arbitrary complete algebraic curve, the structure group is an arbitrary semisimiple group.
We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax matrices are presented in a factored form. A system of discrete isomonodromic deformations is completely determined by commutation relations between the factors. We also reparameterize these systems in terms of the image and kernel vectors at singular points to obtain a separate birational form. A distinguishing feature of this study is the presence of a sym…
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。
