Fuchsian Equations Research Articles

Reduction of singularities in Fuchsian equations, and Heun equations for perturbations of Kerr–Newman–de Sitter and anti-de sitter spacetimes

For Fuchsian differential equations with N singularities at finite points we display two simply expressed conditions on coefficients which guarantee equivalence, under a certain inversion transformation, to a differential equation with N−1 singularities at finite points. For the radial and angular differential equations that govern perturbations of Kerr–Newman–de Sitter and anti-de Sitter spacetimes we find that these two conditions are satisfied identically for arbitrary values of all the parameters in the differential equations, including pole locations, with regular singularities and non-zero cosmological constant. From this we generalize the results of Suzuki, Tarasugi, and Umetsu, Prog. Theor. Phys. 100, 491 (1998) and find 128 Heun equations which are equivalent to the equations which govern the perturbations of the KN(A)dS spacetimes.

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.

Bootstrapping closed string field theory

The determination of the string vertices of closed string field theory is shown to be a conformal field theory problem solvable by combining insights from Liouville theory, hyperbolic geometry, and conformal bootstrap. We first demonstrate how Strebel differentials arise from hyperbolic string vertices by performing a WKB approximation to the associated Fuchsian equation, which we subsequently use it to derive a Polyakov-like conjecture for Strebel differentials. This result implies that the string vertices are generated by the interactions of n zero momentum tachyons, or equivalently, a certain limit of suitably regularized on-shell Liouville action. We argue that the latter can be related to the interaction of three zero momentum tachyons on a generalized cubic vertex through classical conformal blocks. We test this claim for the quartic vertex and discuss its generalization to higher-string interactions.

Study of the Hypergeometric Equation via Data Driven Koopman-EDMD Theory

We consider a data-driven method, which combines Koopman operator theory with Extended Dynamic Mode Decomposition. We apply this method to the hypergeometric equation which is the Fuchsian equation with three regular singular points. The space of solutions at any of its singular points is a two-dimensional linear vector space on the field of reals when the independent variable is restricted to take values in the real axis and the unknown function is restricted to be a real-valued function of a real variable. A basis of the linear vector space of solutions is spanned by the hypergeometric function and its products with appropriate powers of the independent variable or the logarithmic function depending on the roots of the indicial equation of the hypergeometric equation. With our work, we obtain a new representation of the fundamental solutions of the hypergeometric equation and relate them to the spectral analysis of the finite approximation of the Koopman operator associated with the hypergeometric equation. We expect that the usefulness of our results will come more to the fore when we extend our study into the complex domain.

Elementary integral series for Heun functions: Application to black-hole perturbation theory

Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide such representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly confluent, and triconfluent. All the series are illustrated with concrete examples of use, and Python implementations are available for download. We demonstrate the utility of the integral series by providing the first representation of the solution to the Teukolsky radial equation governing the metric perturbations of rotating black holes that is convergent everywhere from the black hole horizon up to spatial infinity.

Love numbers and magnetic susceptibility of charged black holes

The response of black holes to companions is of fundamental importance in the context of their dynamics and of gravitational-wave emission. Here, we explore the effect of charge on the static response of black holes. With a view to constraining broader setups, we consider charged geometries in an arbitrary number of spacetime dimensions $D\geq4$. Tensor tidal Love numbers are shown to follow a power law in the black hole temperature $\sim T_{H}^{2l+1}$, and thus vanish at extremality. In contrast, the black hole charge $Q$ excites new modes of polarisation in the vector sector that are otherwise not responsive in the neutral limit. In four dimensions, Love numbers and magnetic susceptibilities vanish for all values of the charge that respect the extremality bound. Using the theory of Fuchsian equations we are able to obtain analytical results in most cases, even beyond the hypergeometric instances.

Zeros of the isomonodromic tau functions in constructive conformal mapping of polycircular arc domains: the n-vertex case

The prevertices of the conformal map between a generic, n-vertex, simply connected, polycircular arc domain and the upper half plane are determined by finding the zeros of an isomonodromic tau function. The accessory parameters of the associated Fuchsian equation are then found in terms of logarithmic derivatives of this tau function. Using these theoretical results a constructive approach to the determination of the conformal map is given and the particular case of five vertices is considered in detail. A computer implementation of a construction of the isomonodromic tau function described by Gavrylenko and Lisovyy (2018 Commun. Math. Phys. 363 1–58) is used to calculate some illustrative examples. A procedural guide to constructing the conformal map to a given polycircular arc domain using the method presented here is also set out.

On the splitting type of holomorphic vector bundles induced from regular systems of differential equation

Abstract We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.

A Fuchsian viewpoint on the weak null condition

We analyze systems of semilinear wave equations in 3 + 1 dimensions whose associated asymptotic equation admit bounded solutions for suitably small choices of initial data. Under this special case of the weak null condition, which we refer to as the bounded weak null condition , we prove the existence of solutions to these systems of wave equations on neighborhoods of spatial infinity under a small initial data assumption. Existence is established using the Fuchsian method. This method involves transforming the wave equations into a Fuchsian equation defined on a bounded spacetime region. The existence of solutions to the Fuchsian equation then follows from an application of the existence theory developed in [11] . This, in turn, yields, by construction, solutions to the original system of wave equations on a neighborhood of spatial infinity.

Hyperbolic three-string vertex

We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

A numerical algorithm for Fuchsian equations and fluid flows on cosmological spacetimes

We consider a class of Fuchsian equations that, for instance, describes the evolution of compressible fluid flows on a cosmological spacetime. Using the method of lines, we introduce a numerical algorithm for the singular initial value problem when data are imposed on the cosmological singularity and the evolution is performed from the singularity hypersurface. We approximate the singular Cauchy problem of Fuchsian type by a sequence of regular Cauchy problems, which we next discretize by pseudo-spectral and Runge-Kutta techniques. Our main contribution is a detailed analysis of the numerical error which has two distinct sources, and our main proposal here is to keep in balance the errors arising at the continuum and at the discrete levels of approximation. We present numerical experiments which strongly support our theoretical conclusions. This strategy is finally applied to compressible fluid flows evolving on a Kasner spacetime, and we numerically demonstrate the nonlinear stability of such flows, at least in the so-called sub-critical regime identified earlier by the authors.

Computing period integrals of rigid double octic Calabi-Yau threefolds with Picard-Fuchs operator

We present a method for numerical computation of period integrals of a rigid Calabi-Yau threefold using Picard-Fuchs operator of a one-parameter smoothing. Our method gives a possibility of computing the lattice of period integrals of a rigid double octic without any explicit knowledge of its geometric properties , employing only simple facts from the theory of Fuchsian equations and computations in MAPLE with a library for differential equations. As a surprising consequence we also get approximations of additional integrals related to a singular (nodal) model of the considered Calabi-Yau threefold.

The locus of the representation of logarithmic connections by Fuchsian equations

The generic element of the moduli space of logarithmic connections with parabolic points on holomorphic vector bundle over the Riemann sphere can be represented by a Fuchsian equation with some singularities and some apparent singularities. We analyze the case of rank 3 vector bundle which leads to third order Fuchsian equation. We find coordinates on an open subset of the moduli space and we construct a non-trivial part of the moduli space by blowing up along a variety in a special case.

Relations Between Second-Order Fuchsian Equations and First-Order Fuchsian Systems

Each component of any solution of a Fuchsian differential system satisfies a Fuchsian differential equation. The set of Fuchsian systems is fibered into equivalence classes. Each class consists of systems with similar sets of matrix residues, the conjugation matrix being the same for all elements of the set. We investigate the corresponding classes of scalar equations.

Rigid Fuchsian Systems in 2-Dimensional Conformal Field Theories

We investigate Fuchsian equations arising in the context of 2-dimensional conformal field theory (CFT) and we apply the Katz theory of Fucshian rigid systems to solve some of these equations. We show that the Katz theory provides a precise mathematical framework to answer the question whether the fusion rules of degenerate primary fields are enough for determining the differential equations satisfied by their correlation functions. We focus on the case of W3 Toda CFT: we argue that the differential equations arising for four-point conformal blocks with one n-th level semi-degenerate field and a fully-degenerate one in the fundamental sl3 representation are associated to Fuchsian rigid systems. We show how to apply Katz theory to determine the explicit form of the differential equations, the integral expression of solutions and the monodromy group representation. The theory of twisted homology is also used in the analysis of the integral expression. This approach allows to construct the corresponding fusion matrices and to perform the whole bootstrap program: new explicit factorization of W3 correlation functions as well shift relations between structure constants are also provided.

Fuchsian Equations with Three Non-Apparent Singularities

We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients which maps the space of solutions of $H$ into the space of solutions of $E$. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations $E$ with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature $1$ on the punctured sphere with conic singularities, all but three of them having integer angles.

Symmetries and apparent singularities for the simplest Fuchsian equations

We consider the simplest Fuchsian second-order equations with particular attention to the role of apparent singularities. We show the relation to the Painleve equation and follow the matrix formulation of the problem.

Symmetries and apparent singularities for the simplest Fuchsian equations

Рассматриваются простейшие фуксовы уравнения второго порядка, особое внимание уделяется роли ложных сингулярностей. Показана связь с уравнением Пенлеве. Прослежена матричная формулировка проблемы.

The Klein–Gordon–Fock equation in the curved spacetime of the Kerr–Newman (anti) de Sitter black hole

Exact solutions of the Klein–Gordon–Fock (KGF) general relativistic equation that describe the dynamics of a massive, electrically charged scalar particle in the curved spacetime geometry of an electrically charged, rotating Kerr–Newman–(anti) de Sitter black hole are investigated. In the general case of a rotating, charged, cosmological black hole the solution of the KGF equation with the method of separation of variables results in Fuchsian differential equations for the radial and angular parts which for most of the parameter space contain more than three finite singularities and thereby generalise the Heun differential equations. For particular values of the physical parameters (i.e. mass of the scalar particle) these Fuchsian equations reduce to the case of the Heun equation and the closed form analytic solutions we derive are expressed in terms of Heun functions. For other values of the parameters some of the extra singular points are false singular points. We derive the conditions on the coefficients of the generalised Fuchsian equation such that a singular point is a false point. In such a case the exact solution of the Fuchsian equation can in principle be simplified and expressed in terms of Heun functions. This is the generalisation of the case of a Heun equation with a false singular point in which the exact solution of Heun’s differential equation is expressed in terms of Gauß hypergeometric function. We also derive the exact solutions of the radial and angular equations for a charged massive scalar particle in the Kerr–Newman spacetime. The analytic solutions are expressed in terms of confluent Heun functions. Moreover, we derived the constraints on the parameters of the theory such that the solution simplifies and expressed in terms of confluent Kummer hypergeometric functions. We also investigate the radial solutions in the KN case in the regions near the event horizon and far from the black hole. Finally, we construct several expansions of the solutions of the Heun equation in terms of generalised hypergeometric functions of Lauricella–Appell.

Spectral Types of Linear $q$-Difference Equations and $q$-Analog of Middle Convolution

We give a $q$-analog of middle convolution for linear $q$-difference equations with rational coefficients. In the differential case, middle convolution is defined by Katz, and he examined properties of middle convolution in detail. In this paper, we define a $q$-analog of middle convolution. Moreover, we show that it also can be expressed as a $q$-analog of Euler transformation. The $q$-middle convolution transforms Fuchsian type equation to Fuchsian type equation and preserves rigidity index of $q$-difference equations.