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Gabriel Lopes Cardoso

Publications and source records attributed to Gabriel Lopes Cardoso.

At least 19 recordsLinked to original sources
arXiv2026

Holography from number theory: Emergent holographic AdS$_2$ space-time from exact BPS black hole microstate counting

In this note, starting from the exact counting formulae for 4D BPS black hole degeneracies in the cases of 1/2 BPS $\mathcal{N}=4$ and 1/8 BPS $\mathcal{N}=8$ solutions, expressed as Rademacher expansions for coefficients of modular and Jacobi forms respectively, we explicitly show how the number theoretic data in each Rademacher summand is encoded by a Segal-Bargmann heat kernel calculation in the de Alfaro-Fubini-Furlan (DFF) model of conformal quantum mechanics, thus demonstrating the holographic origin of the BPS Bekenstein-Hawking entropy, and all logarithmic and power law suppressed corrections to it. We then derive the 2D holographically dual bulk space-time for this CQM and show that it is an AdS$_2$ space-time with radial length scale correlated to the energy scale of the DFF model. We further show how the Bekenstein-Hawking entropy emerges from an entanglement entropy computation in this bulk space-time. Hence, for the two classes of 4D BPS black holes under consideration, we have inferred the holographic CQM relevant for microscopic state counting from the exact number-theoretic degeneracy formulae and demonstrated the emergence of the near-horizon AdS$_2$ attractor geometry that is holographically dual to it.

hep-th↗
arXiv2026

Solving gravitational field equations by Wiener-Hopf matrix factorisation, and beyond

By viewing Einstein's field equations -- reduced to two dimensions -- as an integrable system, one can simultaneously obtain exact solutions to both the equations themselves and their associated Lax pair via a canonical Wiener-Hopf factorisation of a so-called monodromy matrix. In this article, we review this remarkable interplay between gravitational field equations, integrable systems, Riemann-Hilbert problems, and Wiener-Hopf factorisation theory, with particular emphasis on developments from the past decade enabled by advances in Wiener-Hopf factorisation techniques arising from the study of singular integral equations and Toeplitz operators. Through a variety of concrete examples, we illustrate how Wiener-Hopf factorisation yields explicit, exact solutions to the field equations of gravitational theories, and how its generalisation through a so-called $τ$-invariance property provides a new solution-generating method. Along the way, we aim to demonstrate the importance of an interdisciplinary approach -- grounded in General Relativity, Complex Analysis, and Operator Theory -- for the study of gravitational field equations.

math-ph↗
arXiv2025

Exact BPS black hole microstate counting from holographic conformal quantum mechanics

In this note, we develop a prescription for describing BPS black hole microstates in terms of a holographic conformal quantum mechanics (CQM) model dual to the near-horizon $AdS_2$ geometry of the black hole. We use 1/2 BPS small black holes in a 4D ${\cal N}=4$ toroidal heterotic compactification as well as 1/8 BPS large black holes in a 4D ${\cal N}=8$ Type II toroidal compactification as test cases for our approach. In each case, the $SL(2,\mathbb{Z})$ modular symmetries of the known generating function of the exact microstate degeneracies enables the latter to be expressed as a Rademacher series expansion, with each summand consisting of phases and a modified Bessel function of the first kind. We make a motivated ansatz that the de Alfaro-Fubini-Furlan model (DFF) coupled to a simple harmonic oscillator is a universal sector of the holographic CQM dual to the BPS black hole's near-horizon $AdS_2$ geometry, and demonstrate how in both cases, the two parameters of this putative CQM, the DFF coupling as well as the oscillator frequency, exactly encode both the index and the argument of the Bessel function. Consequently, we extract the leading, logarithmic and all sub-leading power law black hole entropy contributions from calculations in the CQM. In the ${\cal N}=4$ case, the DFF ansatz is sufficient to successfully reproduce the exact microscopic generating function from the CQM.

hep-th↗
arXiv2025

Bulk-boundary entanglement correspondence and the Ryu-Takayanagi conjecture in an $AdS_2/CFT_1$ setup

Using recent developments in expressing one-loop partition functions in Euclidean $AdS_2$ space-times in terms of character integrals, we relate the one-loop effective action for a free field theory in $AdS_2$ (comprised of a massless scalar field and a massless Majorana fermion field) to the partition function of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics (CQM) models on the two global $AdS_2$ boundaries. The equal number of bosonic and fermionic degrees in the field theory guarantee that the one-loop calculation is free of all UV divergences except a logarithmic one consistent with the expected entanglement entropy behaviour in a CQM. Via a thermofield double representation, we compute the entanglement entropy between two copies of the $CFT_1$ (CQM), each living near one of the two boundaries of global $AdS_2$, in a state at global time $τ\rightarrow - \infty$. This entanglement entropy is expressed in terms of the logarithm of the regularised length of a closed particle trajectory infinitesimally near the rim of the Euclidean $AdS_2$ disc. We view this relation between boundary quantum entanglement and a bulk geometrical quantity as the $AdS_2/CFT_1$ version of the Ryu-Takayanagi conjecture in our setup. The boundary entanglement entropy is equal to 4 times the thermodynamic entropy read off from the regularised one-loop effective action in $AdS_2$. Further, we compute the bulk entanglement entropy associated with black hole horizons in Lorentzian $AdS_2$ and show that it precisely matches the boundary entanglement entropy.

hep-th↗
arXiv2024

Classical integrability in the presence of a cosmological constant: analytic and machine learning results

We study the integrability of two-dimensional theories that are obtained by a dimensional reduction of certain four-dimensional gravitational theories describing the coupling of Maxwell fields and neutral scalar fields to gravity in the presence of a potential for the neutral scalar fields. For a certain solution subspace, we demonstrate partial integrability by showing that a subset of the equations of motion in two dimensions are the compatibility conditions for a linear system. Subsequently, we study the integrability of these two-dimensional models from a complementary one-dimensional point of view, framed in terms of Liouville integrability. In this endeavour, we employ various machine learning techniques to systematise our search for numerical Lax pair matrices for these models, as well as conserved currents expressed as functions of phase space variables.

hep-th↗
arXiv2024

Factorisation of symmetric matrices and applications in gravitational theories

We consider the canonical Wiener-Hopf factorisation of $2 \times 2$ symmetric matrices $\mathcal M$ with respect to a contour $Γ$. For the case that the quotient $q$ of the two diagonal elements of $\mathcal M$ is a rational function, we show that due to the symmetric nature of the matrix $\mathcal M$, the second column in each of the two matrix factors that arise in the factorisation is determined in terms of the first column in each of these matrix factors, by multiplication by a rational matrix, and we give a method for determining the second columns of these factors. We illustrate our method with two examples in the context of a Riemann-Hilbert approach to obtaining solutions to the Einstein field equations.

math.FA↗
arXiv2024

Weyl-Lewis-Papapetrou coordinates, self-dual Yang-Mills equations and the single copy

We consider the dimensional reduction to two dimensions of certain gravitational theories in $D \geq 4$ dimensions at the two-derivative level. It is known that the resulting field equations describe an integrable system in two dimensions which can also be obtained by a dimensional reduction of the self-dual Yang-Mills equations in four dimensions. We use this relation to construct a single copy prescription for classes of gravitational solutions in Weyl-Lewis-Papapetrou coordinates. In contrast with previous proposals, we find that the gauge group of the Yang-Mills single copy carries non-trivial information about the gravitational solution. We illustrate our single copy prescription with various examples that include the extremal Reissner-Nordstrom solution, the Kaluza-Klein rotating attractor solution, the Einstein-Rosen wave solution and the self-dual Kleinian Taub-NUT solution.

hep-th↗
arXiv2024

Generating new gravitational solutions by matrix multiplication

Explicit solutions to the non-linear field equations of some gravitational theories can be obtained, by means of a Riemann-Hilbert approach, from a canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices. In this paper we describe other types of factorisation from which solutions can be constructed in a similar way. Our approach is based on an invariance problem, which does not constitute a Riemann-Hilbert problem and allows to construct solutions that could not have been obtained by Wiener-Hopf factorisation of a monodromy matrix. It gives rise to a novel solution generating method based on matrix multiplications. We show, in particular, that new solutions can be obtained by multiplicative deformation of the canonical Wiener-Hopf factorisation, provided the latter exists, and that one can superpose such solutions. Examples of applications include Kasner, Einstein-Rosen wave and gravitational pulse wave solutions.

math-ph↗
arXiv2024

Riemann-Hilbert problems, Toeplitz operators and ergosurfaces

The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix $M$ depending on the Weyl coordinates $ρ$ and $v$, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the $(ρ,v)$ plane on which some elements of $M(ρ,v)$ tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces.

math-ph↗
arXiv2024

Rademacher expansion of a Siegel modular form for ${\cal N}= 4$ counting

The degeneracies of $1/4$ BPS states with unit torsion in heterotic string theory compactified on a six-torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form $Φ_{10}$ of weight $10$. We use the symplectic symmetries of the latter to construct a fine-grained Rademacher type expansion which expresses these BPS degeneracies as a regularized sum over residues of the poles of $1/Φ_{10}$. The construction uses two distinct ${\rm SL}(2, \mathbb{Z})$ subgroups of ${\rm Sp}(2, \mathbb{Z})$ which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein. Additionally, it shows how the polar data are explicitly built from the Fourier coefficients of $1/η^{24}$ by means of a continued fraction structure.

hep-th↗
arXiv2023

The gravitational path integral for $ N=4$ BPS black holes from black hole microstate counting

We use the exact degeneracy formula of single-centred $\frac14$ BPS dyonic black holes with unit torsion in 4D $N=4$ toroidally compactified heterotic string theory to improve on the existing formulation of the corresponding quantum entropy function obtained using supersymmetric localization. The result takes the form of a sum over Euclidean backgrounds including orbifolds of the Euclidean $AdS_2 \times S^2$ attractor geometry. Using an $N=2$ formalism, we determine the explicit form of the Abelian gauge potentials supporting these backgrounds. We further show how a rewriting of the degeneracy formula is amenable, at a semi-classical level, to a gravitational interpretation involving 2D Euclidean wormholes. This alternative picture is useful to elucidate different aspects of the gravitational path integral capturing the microstate degeneracies. We also comment on the relation between the associated 1D holographic models.

hep-th↗
arXiv2022

Observations on holographic aspects of four-dimensional asymptotically flat ${\cal N}=2$ black holes

In this note, we explore holographic attributes of four-dimensional near-extremal Reissner-Nordstrom black hole solutions in ungauged ${\cal N}=2$ supergravity theories at the two-derivative level by recasting them as a specific first-order deformation in solution space, associated with an infinitesimal Harrison transformation, of black holes in an $AdS_2$ space-time. Specifically, we use this link to exhibit how bulk properties, such as mass and entropy, of four-dimensional near-extremal black holes are holographically encoded in the one-dimensional boundary theory dual to gravity in an infinitesimally deformed $AdS_2$ space-time. We do so for the case of four-dimensional near-extremal black holes that arise as deformations in solution space of BPS black holes by changing the non-extremality parameter. For these near-extremal black holes, we further show that the nAdS$_2$ attractor mechanism can be recast as a specific deformation of the BPS flow equations in four dimensions. Additionally, we also discuss time-dependent perturbations of the four-dimensional near-extremal Reissner-Nordstrom solutions from a two-dimensional point of view.

hep-th↗
arXiv2021

$R^2$ corrected AdS$_2$ holography

We approach the problem of constructing an explicit holographic dictionary for the AdS$_2$/CFT$_1$ correspondence in the context of higher derivative gravitational actions in AdS$_2$ space-times. These actions are obtained by an $S^2$ reduction of four-dimensional ${\cal N}=2$ Wilsonian effective actions with Weyl squared interactions restricted to constant scalar backgrounds. BPS black hole near-horizon space-times fall into this class of backgrounds, and by identifying the boundary operators dual to the bulk fields, we explicitly show how the Wald entropy of the BPS black hole is holographically encoded in the anomalous transformation of the operator dual to a composite bulk field. Additionally, using a 2d/3d lift, we show that the CFT holographically dual to AdS$_2$ is naturally embedded in the chiral half of the CFT$_2$ dual to the AdS$_3$ space-time, and we identify the specific operator in CFT$_1$ that encodes the chiral central charge of the CFT$_2$.

hep-th↗
arXiv2020

Comments on the double copy construction for gravitational theories

We revisit the double copy description for linearized gravity and point out various technical issues and subtleties, associated with setting up the double copy description, including the problem of matching degrees of freedom on both sides of the double copy dictionary and the related issue of the constraint between graviton and dilaton sources. We introduce and discuss possible resolutions of these issues.

hep-th↗
arXiv2020

Special Geometry, Hessian Structures and Applications

The target space geometry of abelian vector multiplets in ${\cal N}= 2$ theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometries of four- and five-dimensional vector multiplets. We formulate ${\cal N}= 2$ theories in terms of Hessian structures and give various concrete applications of Hessian geometry, ranging from static BPS black holes in four and five space-time dimensions to topological string theory, emphasizing the role of the Hesse potential. We also discuss the r-map and c-map which relate the special geometries of vector multiplets to each other and to hypermultiplet geometries. By including time-like dimensional reductions, we obtain theories in Euclidean signature, where the scalar target spaces carry para-complex versions of special geometry.

hep-th↗
arXiv2017

Multi-centered ${\mathcal N}=2$ BPS black holes: a double copy description

We present the on-shell double copy dictionary for linearised ${\mathcal N}=2$ supergravity coupled to an arbitrary number of vector multiplets in four dimensions. Subsequently, we use it to construct a double copy description of multi-centered BPS black hole solutions in these theories in the weak-field approximation.

hep-th↗
arXiv2016

Hessian geometry and the holomorphic anomaly

We present a geometrical framework which incorporates higher derivative corrections to the action of N = 2 vector multiplets in terms of an enlarged scalar manifold which includes a complex deformation parameter. This enlarged space carries a deformed version of special Kahler geometry which we characterise. The holomorphic anomaly equation arises in this framework from the integrability condition for the existence of a Hesse potential.

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