Fuchsian Singularities Research Articles

Non-spinning tops are stable

We consider coupled gravitational and electromagnetic perturbations of a family of five-dimensional Einstein-Maxwell solutions that describes both magnetized black strings and horizonless topological stars. We find that the odd perturbations of this background lead to a master equation with five Fuchsian singularities and compute its quasinormal mode spectrum using three independent methods: Leaver, WKB and numerical integration. Our analysis confirms that odd perturbations always decay in time, while spherically symmetric even perturbations may exhibit for certain ranges of the magnetic fluxes instabilities of Gregory-Laflamme type for black strings and of Gross-Perry-Yaffe type for topological stars. This constitutes evidence that topological stars and black strings are classically stable in a finite domain of their parameter space.

Nakayama Algebras and Fuchsian Singularities

Nakayama Algebras and Fuchsian Singularities

Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus

We compute the monodromy dependence of the isomonodromic tau function on a torus with n Fuchsian singularities and SL(N) 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.

Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions

We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in GL(N,{mathbb {C}}) can be written in terms of a Fredholm 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,{mathbb {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.

CFT description of BH’s and ECO’s: QNMs, superradiance, echoes and tidal responses

Using conformal field theory and localization tecniques we study the propagation of scalar waves in gravity backgrounds described by Schrödinger like equations with Fuchsian singularities. Exact formulae for the connection matrices relating the asymptotic behaviour of the wave functions near the singularities are obtained in terms of braiding and fusion rules of the CFT. The results are applied to the study of quasi normal modes, absorption cross sections, amplification factors, echoes and tidal responses of black holes (BH) and exotic compact objects (ECO) in four and five dimensions. In particular, we propose a definition of dynamical Love numbers in gravity.

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.

On the tau function of the hypergeometric equation

The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formulæ for Gauss’ hypergeometric functions. We define the tau-function of such a system as the generating function of the extended monodromy symplectomorphism, using an idea recently developed. This formulation allows us to determine the dependence of the tau-function on the monodromy data. Using the explicit solution of the monodromy problem, the tau-function is then explicitly written in terms of Barnes G-function. In particular, if the Fuchsian singularities are placed to 0, 1 and ∞, this gives the structure constants of the asymptotical formula of Iorgov–Gamayun–Lisovyy for solutions of Painlevé VI equation.

Calculation of special functions arising in the problem of diffraction by a dielectric ball

To apply the incomplete Galerkin method to the problem of the scattering of electromagnetic waves by lenses, it is necessary to study the differential equations for the field amplitudes. These equations belong to the class of linear ordinary differential equations with Fuchsian singularities and, in the case of the Lneburg lens, are integrated in special functions of mathematical physics, namely, the Whittaker and Heun functions. The Maple computer algebra system has tools for working with Whittaker and Heun functions, but in some cases this system gives very large values for these functions, and their plots contain various kinds of artifacts. Therefore, the results of calculations in the Maple11 and Maple2019 systems of special functions related to the problem of scattering by a Lneburg lens need additional verification. For this purpose, an algorithm for finding solutions to linear ordinary differential equations with Fuchsian singular points by the method of Frobenius series was implemented, designed as a software package Fucsh for Sage. The problem of scattering by a Lneburg lens is used as a test case. The calculation results are compared with similar results obtained in different versions of CAS Maple. Fuchs for Sage allows computing solutions to other linear differential equations that cannot be expressed in terms of known special functions.

Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters u=(u_1,ldots ,u_n), which are eigenvalues of the leading matrix at the irregular singularity. 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,ldots ,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 singularities. We construct isomonodromic selected and singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connection coefficients, so extending a result of Balser et al. (I SIAM J Math Anal 12(5): 691–721, 1981) and Guzzetti (Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case, including confluence of singularities. Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowing to obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed in terms of connection coefficients. These facts, in addition to extending (Balser et al. in I SIAM J Math Anal 12(5): 691–721, 1981; Guzzetti in Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case (with coalescences/confluences), allow to prove by means of Laplace transform the main result of Cotti et al. (Duke Math J arXiv:1706.04808, 2017), namely the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.

On Inner Expansions for a Singularly Perturbed Cauchy Problem with Confluent Fuchsian Singularities

A nonlinear singularly perturbed Cauchy problem with confluent Fuchsian singularities is examined. This problem involves coefficients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has recently been investigated by the author, for which sets of holomorphic inner and outer solutions were built up and expressed as a Laplace transform with logarithmic kernel. Here, a family of holomorphic inner solutions are constructed by means of exponential transseries expansions containing infinitely many Laplace transforms with special kernel. Furthermore, asymptotic expansions of Gevrey type for these solutions relatively to the perturbation parameter are established.

On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities

We consider a family of nonlinear singularly perturbed PDEs whose coefficients involve a logarithmic dependence in time with confluent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished finite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the first set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second.

On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities

A family of linear singularly perturbed difference differential equations is examined. These equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs q-Gevrey and Gevrey bounds.

On Singularly Perturbed Linear Initial Value Problems with Mixed Irregular and Fuchsian Time Singularities

We consider a family of linear singularly perturbed PDE depending on a complex perturbation parameter $$\epsilon $$ . As in the former study (Lastra and Malek in J Differ Equ 259(10):5220–5270, 2015) of the authors, our problem possesses an irregular singularity in time located at the origin but, in the present work, it also entangles differential operators of Fuchsian type acting on the time variable. As a new feature, a set of sectorial holomorphic solutions are built up through iterated Laplace transforms and Fourier inverse integrals following a classical multisummability procedure introduced by Balser. This construction has a direct consequence on the Gevrey bounds of their asymptotic expansions w.r.t $$\epsilon $$ which are shown to increase the order of the leading term which combines both irregular and Fuchsian types operators.

Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in \Delta$. We analyse the case when the leading matrix remains diagonalisable at $\Delta$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as $t$ varies in the polydisc, and their limits for $t$ tending to points of $\Delta$. When the deformation is isomonodromic away from $\Delta$, it is well known that a fundamental matrix solution has singularities at $\Delta$. When the system also has a Fuchsian singularity at $z=0$, we show under minimal vanishing conditions on the residue matrix at $z=0$ that isomonodromic deformations can be extended to the whole polydisc, including $\Delta$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed $t=0$. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting $\Delta$, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that $\Delta$ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

The Kontsevich–Penner Matrix Integral, Isomonodromic Tau Functions and Open Intersection Numbers

We identify the Kontsevich-Penner matrix integral, for finite size $n$, with the isomonodromic tau function of a $3\times 3$ rational connection on the Riemann sphere with $n$ Fuchsian singularities placed in correspondence with the eigenvalues of the external field of the matrix integral. By formulating the isomonodromic system in terms of an appropriate Riemann-Hilbert boundary value problem, we can pass to the limit $n\to\infty$ (at a formal level) and identify an isomonodromic system in terms of the Miwa variables, which play the role of times of the KP hierarchy. This allows to derive the String and Dilaton equations via a purely Riemann-Hilbert approach. The expression of the formal limit of the partition function as an isomonodromic tau function allows us to derive explicit closed formul\ae\ for the correlators of this matrix model in terms of the solution of the Riemann-Hilbert problem with all times set to zero. These correlators have been conjectured to describe the intersection numbers for Riemann surfaces with boundaries, or open intersection numbers.

Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .], discussed in our talk [G. Cotti, Monodromy of semisimple Frobenius coalescent structures, in Int. Workshop Asymptotic and Computational Aspects of Complex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an [Formula: see text] linear system of ODEs with an irregular singularity of Poincaré rank 1 at [Formula: see text] and Fuchsian singularity at [Formula: see text], holomorphically depending on parameter [Formula: see text] within a polydisk in [Formula: see text] centered at [Formula: see text]. The eigenvalues of the leading matrix at [Formula: see text], which is diagonal, coalesce along a coalescence locus [Formula: see text] contained in the polydisk. Under minimal vanishing conditions on the residue matrix at [Formula: see text], we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .] that isomonodromic deformations can be extended to the whole polydisk, including [Formula: see text], in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at point of [Formula: see text], where it simplifies. Conversely, if the [Formula: see text]-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting [Formula: see text], and if suitable entries of the Stokes matrices vanish, then [Formula: see text] is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlevé equations.

Zero Level Perturbation of a Certain Third-Order Linear Solvable ODE with an Irregular Singularity at the Origin of Poincaré Rank 1

We study an irregular singularity of Poincare rank 1 at the origin of a certain third-order linear solvable homogeneous ODE. We perturb the equation by introducing a small parameter $\varepsilon \in ({\mathbb R}_{+},0)$ (e < 1), which causes the splitting of the irregular singularity into two finite Fuchsian singularities. We show that when the solutions of the perturbed equation contain logarithmic terms, the Stokes matrices of the initial equation are limits of the part of the monodromy matrices around the finite resonant Fuchsian singularities of the perturbed equation.

Non dégénérescence et singularités des métriques d’Einstein asymptotiquement hyperboliques en dimension 4

We prove that desingularizations of non degenerate Poincare–Einstein metrics with $$A_1$$ singularities remain non degenerate. In principle this enables a recursive procedure to desingularize the other Fuchsian singularities. We illustrate this procedure by the $$A_2$$ case.

Fuchsian systems for Dotsenko–Fateev multipoint correlation functions and similar integrals of hypergeometric type

The Dotsenko-Fateev integral is an analytic function of one complex variable expressing the amplitude in the 4-point correlator of the 2D conformal field theory. Dotsenko-Fateev found ODE of third order with Fuchsian singularities satisfied by their integral. In the present paper, this work is extended to generalized Dotsenko-Fateev integrals, in particular those associated to arbitrary multipoint correlators, and Pfaff systems of PDE of Fuchsian type are constructed for them. The ubiquity of the Fuchsian systems is in that they permit to obtain local expansions of solutions in the neighborhoods of singularities of the system.

On a certain generalization of triangle singularities

Triangle singularities are Fuchsian singularities associated with von Dyck groups, which are index two subgroups of Schwarz triangle groups. Hypersurface triangle singularities are classified by Dolgachev, and give 14 exceptional unimodal singularities classified by Arnold. We introduce a generalization of triangle singularities to higher dimensions, show that there are only finitely many hypersurface singularities of this type in each dimension, and give a complete list in dimension 3.