Isomonodromic Deformations Research Articles

Description of Generalized Isomonodromic Deformations of Rank-Two Linear Differential Equations Using Apparent Singularities

In this paper, we consider the generalized isomonodromic deformations of rank-two irregular connections on the Riemann sphere. We introduce Darboux coordinates on the parameter space of a family of rank-two irregular connections by apparent singularities. Using the Darboux coordinates, we describe the generalized isomonodromic deformations as Hamiltonian systems.

Isomonodromic and isospectral deformations of meromorphic connections: the sl2(C) case

Abstract We consider non-twisted meromorphic connections in sl 2 ( C ) and the associated symplectic Hamiltonian structure. In particular, we provide explicit expressions of the Lax pair in the geometric gauge supplementing results of Marchal et al (2022 arXiv:2212.04833) where explicit formulas have been obtained in the oper gauge. Expressing the geometric Lax matrices requires the introduction of specific Darboux coordinates for which we provide the explicit Hamiltonian evolutions. These expressions allow to build bridges between the isomonodromic deformations and the isospectral ones. More specifically, we propose an explicit change of Darboux coordinates to obtain isospectral coordinates for which Hamiltonians match the spectral invariants. This result solves the issue left opened in Bertola et al (2023 J. Math. Phys. 64 083502) in the case of sl 2 ( C ) .

Four-Dimensional Painlevé-Type Difference Equations

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 isomonodromic deformation equations with four-dimensional phase space. We compute an example of discrete isomonodromic deformation equations of a certain Fuchsian equation.

Painlevé Kernels and Surface Defects at Strong Coupling

AbstractIt is well established that the spectral analysis of canonically quantized four-dimensional Seiberg–Witten curves can be systematically studied via the Nekrasov–Shatashvili functions. In this paper, we explore another aspect of the relation between $${\mathcal {N}}=2$$ N = 2 supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlevé equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg–Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an $${{\,\mathrm{O(2)}\,}}$$ O ( 2 ) matrix model. We then show that these eigenfunctions are computed by surface defects in $${{\,\mathrm{SU(2)}\,}}$$ SU ( 2 ) super Yang–Mills in the self-dual phase of the $$\Omega $$ Ω -background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.

Deflection of light by a Reissner–Nordström black hole and Painlevé VI equation

We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner–Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard–Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

Scalar quasi-normal modes of accelerating Kerr-Newman-AdS black holes

We study linear scalar perturbations of slowly accelerating Kerr-Newman-anti-de Sitter black holes using the method of isomonodromic deformations. The conformally coupled Klein-Gordon equation separates into two second-order ordinary differential equations with five singularities. Nevertheless, the angular equation can be transformed into a Heun equation, for which we provide an asymptotic expansion for the angular eigenvalues in the small acceleration and rotation limit. In the radial case, we recast the boundary value problem in terms of a set of initial conditions for the isomonodromic tau function of Fuchsian systems with five regular singular points. For the sake of illustration, we compute the quasi-normal modes frequencies.

Geometric local systems on very general curves and isomonodromy

We show that the minimum rank of a non-isotrivial local system of geometric origin on a suitably general n n -pointed curve of genus g g is at least 2 g + 1 2\sqrt {g+1} . We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformations, which additionally answers questions of Biswas, Heu, and Hurtubise.

Hamiltonian structure of rational isomonodromic deformation systems

The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincaré rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space L*gl(r) of the loop algebra Lgl(r). Nonautonomous isomonodromic counterparts of isospectral systems generated by spectral invariants are obtained by identifying deformation parameters as Casimir elements on the phase space. These are shown to coincide with higher Birkhoff invariants determining local asymptotics near to irregular singular points, together with the pole loci. Pairs consisting of Birkhoff invariants, together with the corresponding dual spectral invariant Hamiltonians, appear as “mirror images” matching, at each pole, the negative power coefficients in the principal part of the Laurent expansion of the fundamental meromorphic differential on the associated spectral curve with the corresponding positive power terms in the analytic part. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves may be identified as the orbits of a free Abelian local group action. The projection of infinitesimal isomonodromic deformation vector fields to the quotient manifold under this action gives commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci.

The sixth Painlevé equation as isomonodromy deformation of an irregular system: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds

The sixth Painlevé equation PVI is both the isomonodromy deformation condition of a 2-dimensional isomonodromic Fuchsian system and of a 3-dimensional irregular system. Only the former has been used in the literature to solve the nonlinear connection problem for PVI, through the computation of invariant quantities . We prove a new simple formula expressing the invariants p jk in terms of the Stokes matrices of the irregular system, making the irregular system a concrete alternative for the nonlinear connection problem. We classify the transcendents such that the Stokes matrices and the p jk can be computed in terms of special functions, providing a full non-trivial class of 3-dim. examples such that the theory of non-generic isomonodromy deformations of Cotti et al (2019 Duke Math. J. 168 967–1108) applies. A sub-class of these transcendents realises the local structure of all the 3-dim Dubrovin–Frobenius manifolds with semisimple coalescence points of the type studied in Cotti et al (2020 SIGMA 16 105). We compute all the monodromy data for these manifolds (Stokes matrix, Levelt exponents and central connection matrix).

Moduli Space of Factorized Ramified Connections and Generalized Isomonodromic Deformation

We introduce the notion of factorized ramified structure on a generic ramified irregular singular connection on a smooth projective curve. By using the deformation theory of connections with factorized ramified structure, we construct a canonical 2-form on the moduli space of ramified connections. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-form becomes nondegenerate. We prove that the 2-form on the moduli space of ramified connections is d -closed via constructing an unfolding of the moduli space. Based on the Stokes data, we introduce the notion of local generalized isomonodromic deformation for generic unramified irregular singular connections on a unit disk. Applying the Jimbo-Miwa-Ueno theory to generic unramified connections, the local generalized isomonodromic deformationis equivalent to the extendability of the family of connections to an integrable connection. We give the same statement for ramified connections. Based on this principle of Jimbo-Miwa-Ueno theory, we construct a global generalized isomonodromic deformation on the moduli space of generic ramified connections by constructing a horizontal lift of a universal family of connections. As a consequence of the global generalized isomonodromic deformation, we can lift the relative symplectic form on the moduli space to a total closed form, which is called a generalized isomonodromic 2-form.

On a Connection Formula of a Higher Rank Analog of Painlevé VI

Abstract In this paper, we derive a connection formula for a higher rank analog of Painlevé VI arising from the isomonodromic deformation theory and the study of Frobenius manifolds.

Isomonodromic Deformations: Confluence, Reduction and Quantisation

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 truncated current algebra, also called generalised Takiff 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 tau -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 Poisson morphism and explicitly compute the isomonodromic Hamiltonians. In loving memory of Igor KricheverA great man and outstanding mathematician

Stokes Phenomenon and Reflection Equations

In this paper, we study the Stokes phenomenon of the cyclotomic Knizhnik–Zamolodchikov equation associated to the symmetric pair \((\mathfrak {gl}_n,\mathfrak {so}_n)\), and prove that its two types of Stokes matrices satisfy the Yang–Baxter and reflection equations respectively. We briefly discuss its isomonodromy deformation, and its relations with cyclotomic associators, twists and quantum symmetric pairs.

Isomonodromic deformations along a stratum of the coalescence locus

We consider deformations of a differential system with Poincaré 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 (Cotti et al 2019 Duke Math. J. 168 967–1108). For the specific system here considered, the results generalize those of (Jimbo et al 1981 Physica D 2 306), by giving up the generic conditions, and those of (Bertola and Mo 2005 Int. Math. Res. Pap. 2005 565–635), by giving up the Lidskii generic assumption. The importance of the case here considered originates form its applications in the study of strata of Dubrovin-Frobenius manifolds and F-manifolds.

Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries

For more than a century, the Painlevé I equation has played an important role in both physics and mathematics. Its two-parameter family of solutions was studied in many different ways, yet still leads to new surprises and discoveries. Two popular tools in these studies are the theory of isomonodromic deformation that uses the exact WKB method, and the asymptotic description of transcendents in terms of two-parameter transseries. Combining methods from both schools of thought, we follow work by Takei and collaborators to formulate complete, two-parameter connection formulae for solutions when they cross arbitrary Stokes lines in the complex plane. These formulae allow us to study Stokes phenomenon for the full two-parameter family of transseries solutions. In particular, we recover the exact expressions for the Stokes data that were recently found by Baldino, Schwick, Schiappa and Vega and compare our connection formulae to theirs. We also explain several ambiguities in relating transseries parameter choices to actual Painlevé transcendents, study the monodromy of formal solutions, and provide high-precision numerical tests of our results.

Isomonodromic quantization of the second Painlevé equation by means of conservative Hamiltonian systems with two degrees of freedom

For the three nonstationary Schrödinger equations i ℏ Ψ τ = H ( x , y , − i ℏ ∂ ∂ x , − i ℏ ∂ ∂ y ) Ψ , \begin{equation*} i\hbar \Psi _{\tau }=H(x,y,-i\hbar \frac {\partial }{\partial x},-i\hbar \frac {\partial }{\partial y})\Psi , \end{equation*} solutions are constructed that correspond to conservative Hamiltonian systems with two degrees of freedom whose general solutions can be represented by those of the second Painlevé equation. These solutions of the Schrödinger equations are expressed via fundamental solutions of systems of linear equations arising in the isomonodromic deformations method, the compatibility condition of which is the second Painlevé equation. The constructed solutions of two nonstationary Schrödinger equations are globally smooth. Some of the smooth solutions in question of one of these two equations exponentially tend to zero as x 2 + y 2 → ∞ x^2+y^2\to \infty if the corresponding solutions of linear systems that are used in the method of isomonodromic deformations are compatible on the so-called 1-tronquée solutions of the second Painlevé equation.

On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation* *Supported by RSF Grant 18-41-05003.

The tunnelling effect predicted by Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modelled by a family of differential equations on two-torus depending on three parameters: B (abscissa), A (ordinate), ω (frequency). We study its rotation number ρ(B, A; ω) as a function of (B, A) with fixed ω. The phase-lock areas are the level sets L r ≔ {ρ = r} with non-empty interiors; they exist for (Buchstaber, Karpov, Tertychnyi). Each L r is an infinite chain of domains going vertically to infinity and separated by points. Those separating points for which A ≠ 0 are called constrictions. We show that: (1) all the constrictions in L r lie on the axis {B = ωr}; (2) each constriction is positive: this means that some its punctured neighbourhood on the axis {B = ωr} lies in Int(L r ). These results confirm experiments by physicists (1970ths) and two mathematical conjectures. We first prove deformability of each constriction to another one, with arbitrarily small ω and the same , using equivalent description of model by linear systems of differential equations on (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painlevé 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with , or of non-positive type) with a given ℓ for small ω is proved by slow-fast methods.

Fredholm Pfaffian $$\tau $$-Functions for Orthogonal Isospectral and Isomonodromic Systems

We extend the approach to \(\tau \)-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld–Sokolov reductions and isomonodromic deformations systems. The combinatorial expansion of the \(\tau \)-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 dependence on the time parameters. When reduced to the orthogonal case, these correlators become finite Pfaffians and the determinantal \(\tau \)-functions, both in the Drinfeld–Sokolov and isomonodromic case, become squares of \(\tau \)-functions of Pfaffian type. The results are illustrated by several examples, consisting of polynomial \(\tau \)-functions of orthogonal Drinfeld–Sokolov type and isomonodromic ones with four regular singular points.

Quantum Spectral Problems and Isomonodromic Deformations

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 2times 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 Kyiv formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (epsilon _1+epsilon _2=0). Through blowup relations, we also find Nekrasov–Shatashvili type of quantizations (epsilon _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 epsilon _2rightarrow 0 limit of the blowup relations from the regularized action functional and CFT arguments.

A short proof of a theorem of Cotti, Dubrovin and Guzzetti

We give a short proof of a theorem of G. Cotti, B. Dubrovin and D. Guzzetti (arXiv:1706.04808 and arXiv:2101.03397) asserting the vanishing of some entries of the Stokes matrices at coalescing points of an isomonodromic deformation.