Conical Singularities Research Articles

Effects of conical singularity and Wu-Yang magnetic monopole on thermodynamic properties of harmonic oscillator interacting with potential

Abstract This paper investigates the thermodynamic properties of the harmonic oscillator system in a conical singularities background. Moreover, we incorporate a Wu-Yang magnetic monopole (WYMM) and inverse square potential into the quantum system and analyze the resulting effects. Using analytical methods, we derive the energy eigenvalues and study the thermodynamic properties, such as the partition function, Helmholtz free energy, Gibbs's free energy and specific heat capacity at finite temperature $T \neq 0$. Our findings demonstrate that these physical parameters are influenced by the topological parameter, strength of WYMM and potential parameters. These results provide valuable insights into the interplay between thermodynamic properties of a quantum mechanical problem and the gravitational effects.

Mean curvature flow from conical singularities

AbstractWe prove Ilmanen’s resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface $M^{n}\subset \mathbb{R}^{n+1}$ M n ⊂ R n + 1 , $2\leq n\leq 6$ 2 ≤ n ≤ 6 , with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens.

Perelman’s functionals on manifolds with non-isolated conical singularities

Perelman’s functionals on manifolds with non-isolated conical singularities

Superconformal monodromy defects in ABJM and mABJM theory

Abstract We study D = 11 supergravity solutions which are dual to one-dimensional superconformal defects in d = 3 SCFTs. We consider defects in ABJM theory with monodromy for U(1)4 ⊂ SO(8) global symmetry, as well as in 𝒩 = 2 mABJM SCFT, which arises from the RG flow of a mass deformation of ABJM theory, with monodromy for U(1)3 ⊂ SU(3) × U(1) global symmetry. We show that the defects of the two SCFTs are connected by a line of bulk marginal mass deformations and argue that they are also related by bulk RG flow. In all cases we allow for the possibility of conical singularities at the location of the defect. Various physical observables of the defects are computed including the defects conformal weight and the partition function, as well as associated supersymmetric Renyi entropies.

Accelerating Kerr-Taub-NUT spacetime in the low energy limit of heterotic string theory

We construct a novel solution in the low energy limit of heterotic string theory describing accelerating charged and rotating black holes with NUT parameter. Some aspects of the new spacetime are discussed, such as locations of horizons, ergoregions, conic singularity, closed timelike curves, and area temperature product. We find some of the properties resemble the ones belong to accelerating Kerr-Newman-Taub-NUT spacetime.

Dirac points and inverse problems of quantum graphs associated with Archimedean tilings

Abstract One interesting phenomenon of graphene is the presence of the conical singularity
or Dirac points. Using the quantum graph model, we show that there exist three
classes of possible Dirac points for all of the periodic quantum graphs associated with
Archimedean tilings, when the potentials are identical and even. They occur at the
periodic eigenvalues, anti-periodic eigenvalues and other double eigenvalues of the dispersion
relations respectively. We also characterize their associated potentials. Moreover,
we show that there are no other possible Dirac points. Finally, we solve an
inverse spectral problem for the potential, given the knowledge of the pure point and
absolutely continuous spectra.

Kerr/CFT correspondence for the extremal accelerating Kerr–Taub–NUT black hole

In this paper, we explore the application of Kerr/CFT correspondence to the accelerating Kerr–Taub–NUT spacetime. We demonstrate that the correspondence can be maintained if the near-horizon angular coordinate is appropriately rescaled. This rescaling allows the Cardy formula to accurately reproduce the extremal Bekenstein–Hawking entropy, a departure from the typical direct recovery of extremal Bekenstein–Hawking temperature in Kerr/CFT literature. This discrepancy is attributed to the presence of multiple sources of conic singularity in the spacetime, specifically the acceleration and NUT parameters. To support the Kerr/CFT holography, we investigate the hidden conformal symmetry in the near-horizon region of the extremal black hole geometry. Our findings confirm that the angular coordinate rescaling does not introduce inconsistencies within the Kerr/CFT framework, as evidenced by the preserved hidden conformal symmetry and consistent entropy matching calculations in the near-horizon geometry of the extremal accelerating Kerr–Taub–NUT black hole.

New construction of a vacuum doubly rotating black ring by the Ehlers transformation

Using the Ehlers transformation, we derive an exact solution for a doubly rotating black ring in five-dimensional vacuum Einstein theory. It is well known that the vacuum Einstein theory with three commuting Killing vector fields can be reduced to a nonlinear sigma model with SL(3,R) target space symmetry. As shown previously by Giusto and Saxena, the SO(2,1) subgroup in the SL(3,R) can generate a rotating solution from a static solution while preserving asymptotic flatness. This so-called Ehlers transformation actually transforms the five-dimensional Schwarzschild black hole into the five-dimensional Myers-Perry black hole. However, unlike the case with the black hole, applying this method directly to the static black ring or the Emparan-Reall black ring, does not yield a regular rotating black ring due to the emergence of a Dirac-Misner string singularity. To solve this undesirable issue, we use a singular vacuum solution of a rotating black ring/lens that already possesses a Dirac-Misner string singularity as the seed solution for the Ehlers transformation. The resulting solution is regular, indicating the absence of curvature singularities, conical singularities, orbifold singularities, Dirac-Misner string singularities, and closed timelike curves both on and outside the horizon. We show that this solution obtained by the Ehlers transformation coincides precisely with the Pomeransky-Sen’kov solution. We expect that applying this method to other theories may lead to the finding of new exact solutions, such as solutions for black lenses and capped black holes, as well as black ring configurations. Published by the American Physical Society 2024

On the class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} origin of spindle solutions

We analyse the backreacted geometry corresponding to a stack of M5-branes wrapped on a spindle, with a view towards precision tests of the dual N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 superconformal field theory. We carefully study the singular loci of the uplifted geometry and show that these correspond to C3/Zn conical singularities. Therefore, these solutions present one of the first explicit realisations of honest locally N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 preserving punctures in class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document}. Additionally we study the symmetries and anomalies of the dual field theory through anomaly inflow and compute a variety of holographic observables including dimensions of BPS operators. This work paves the way for advancements in the study and identification of the precise dual field theories.

Casimir energy of hyperbolic orbifolds with conical singularities

In this article, we obtain the explicit expression of the Casimir energy for compact hyperbolic orbifold surfaces in terms of the geometrical data of the surfaces with the help of zeta-regularization techniques. The orbifolds may have finitely many conical singularities. In computing the contribution to the energy from a conical singularity, we derive an expression of an elliptic orbital integral as an infinite sum of special functions. We prove that this sum converges exponentially fast. Additionally, we show that under a natural assumption known to hold asymptotically on the growth of the lengths of primitive closed geodesics of the (2, 3, 7)-triangle group orbifold, its Casimir energy is positive (repulsive).

Superconformal monodromy defects in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document}=4 SYM and LS theory

We study type IIB supergravity solutions that are dual to two-dimensional superconformal defects in d = 4 SCFTs which preserve N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = (0, 2) supersymmetry. We consider solutions dual to defects in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM theory that have non-trivial monodromy for U(1)3 ⊂ SO(6) global symmetry and we also allow for the possibility of conical singularities. In addition, we consider the addition of fermionic and bosonic mass terms that have non trivial dependence on the spatial directions transverse to the defect, while preserving the superconformal symmetry of the defect. We compute various physical quantities including the central charges of the defect expressed as a function of the monodromy, the on-shell action as well as associated supersymmetric Rényi entropies. Analogous computations are carried out for superconformal defects in the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1, d = 4 Leigh-Strassler SCFT. We also show that the defects of the two SCFTs are connected by a line of bulk marginal mass deformations and argue that they are also related by bulk RG flow.

The spectrum of an operator associated with G 2-instantons with 1-dimensional singularities and Hermitian Yang–Mills connections with isolated singularities

abstract: This is the first step in an attempt at a deformation theory for $G_{2}$-instantons with $1$-dimensional conic singularities. Under a set of model data, the linearization yields a Dirac operator $P$ on a certain bundle over $\mathbb{S}^{5}$, called the \textit{link operator}. As a dimension reduction, the link operator also arises from Hermitian Yang--Mills connections with isolated conic singularities on a Calabi--Yau $3$-fold. Using the quaternion structure in the Sasakian geometry of $\mathbb{S}^{5}$, we describe the set of all eigenvalues of $P$, denoted by $\Spec P$. We show that $\Spec P$ consists of finitely many integers induced by certain sheaf cohomologies on $\mathbb{P}^{2}$, and infinitely many real numbers induced by the spectrum of the rough Laplacian on the pullback endomorphism bundle over $\mathbb{S}^{5}$. The multiplicities and the form of an eigensection can be described fairly explicitly. In particular, there is a relation between the spectrum on $\mathbb{S}^{5}$ to certain sheaf cohomologies on~$\mathbb{P}^{2}$. Moreover, on a Calabi--Yau $3$-fold, the index of the linearized operator for admissible singular Hermitian Yang--Mills connections is also calculated, in terms of these sheaf cohomologies. Using the representation theory of $\SU(3)$ and the subgroup $S[U(1)\times U(2)]$, we show an example in which $\Spec P$ and the multiplicities can be completely determined.

Practical Integer-Constrained Cone Construction for Conformal Parameterizations.

We propose a practical method to construct sparse integer-constrained cone singularities with low distortion constraints for conformal parameterizations. Our solution for this combinatorial problem is a two-stage procedure that first enhances sparsity for generating an initialization and then optimizes to reduce the number of cones and the parameterization distortion. Central to the first stage is a progressive process to determine the combinatorial variables, i.e., numbers, locations, and angles of cones. The second stage iteratively conducts adaptive cone relocations and merges close cones for optimization. We extensively test our method on a data set containing 3885 models, demonstrating practical robustness and performance. Our method achieves fewer cone singularities and lower parameterization distortion than state-of-the-art methods.

The quasilocal energy and thermodynamic first law in accelerating AdS black holes

We scrutinize the conserved energy of an accelerating AdS black hole by employing the off-shell quasilocal formalism, which amalgamates the ADT formalism with the covariant phase space approach. In the presence of conical singularities in the accelerating black hole, the energy expression is articulated through the surface term derived from our formalism. The essence of our analysis of the quasilocal energy resides in the surface contributions coming from the conical singularities as well as the conventional radial boundary. Consequently, the resultant conserved quasilocal energy naturally conforms the thermodynamic first law for the black hole without necessitating any augmentation of thermodynamic variables.

Schwarzschild deformed supergravity background: Possible geometry origin of fermion generations and mass hierarchy

The problem of fermion masses hierarchy in the Standard Model is considered on a toy model of a 10-dimensional space-time with a IIA supergravity type background. The Dirac equation written on this background, after compactification of extra four- and one-dimensional subspaces, gives the spectrum of Fermi fields which profiles in five dimensions and corresponding Higgs generated masses in four dimensions depend on the eigenvalues of Dirac operator on the named compact subspaces. Schwarzschild Euclidean deformation of the supergravity throat with the “apple-shaped” conical singularity permits to leave only three nondivergent angular modes interpreted as three generations of the down-type quarks. The resulting expressions for the quark masses are a geometry version of the Froggatt-Nielsen mechanism. Published by the American Physical Society 2024

Microstates of Accelerating and Supersymmetric AdS_{4} Black Holes from the Spindle Index.

We provide a first principles derivation of the microscopic entropy of a very general class of supersymmetric, rotating, and accelerating black holes in AdS_{4}. This is achieved by analyzing the large-N limit of the spindle index and completes the construction of the first example of a holographic duality involving supersymmetric field theories defined on orbifolds with conical singularities.

Solution generation of a capped black hole

Utilizing the electric Harrison transformation developed in five-dimensional minimal supergravity, we construct an exact solution characterizing non-BPS (Bogomol’nyi-Prasad-Sommerfield) charged rotating black holes with a horizon cross section of a lens space L(n;1). Among these solutions, only the ones corresponding to n=0 and n=1 do not have any curvature singularities, conical singularities, Dirac-Misner string singularities, and orbifold singularities both on and outside the horizon; additionally, they are free from closed timelike curves. The solution for n=0 corresponds to the charged dipole black ring that we constructed in the previous paper. The specific solution for n=1, referred to as the “capped black hole,” was introduced in our previous article. This provides the first example of a non-BPS exact solution, representing an asymptotically flat, stationary spherical black hole with a domain of outer communication (DOC) having a nontrivial topology in five-dimensional minimal supergravity. We demonstrate that the DOC on a time slice has the topology of [R4#CP2]\B4. Differing from the well-known Myers-Perry and Cvetič-Youm black holes describing a spherical horizon topology and a DOC with a trivial topology of R4\B4 on a timeslice, the capped black hole’s horizon is capped by a disk-shaped bubble. We explicitly demonstrate that the capped black hole carries mass, two angular momenta, an electric charge, and a magnetic flux, with only three of these quantities being independent. Furthermore, we reveal that this black hole can possess identical conserved charges as the Cvetič-Youm black hole. The existence of this solution challenges black hole uniqueness beyond both the black ring and the BPS spherical black hole. Moreover, within specific parameter regions, the capped black hole can exhibit larger entropy than the Cvetič-Youm black hole. Published by the American Physical Society 2024

Gravitational Rényi entropy from corner terms

We provide a consistent first principles prescription to compute gravitational Rényi entropy using Hayward corner terms. For Euclidean solutions to Einstein gravity, we compute Rényi entropy of Hartle-Hawking and fixed-area states by cutting open a manifold containing a conical singularity into a wedge with a corner. The entropy functional for fixed–area states is equal to the corner term itself, having a flat-entanglement spectrum, while extremization of the functional follows from the variation of the corner term under diffeomorphisms. Notably, our method does not require regularization of the conical singularity, and naturally extends to higher-curvature theories of gravity. Published by the American Physical Society 2024

Cornering gravitational entropy

We present a new derivation of gravitational entropy functionals in higher-curvature theories of gravity using corner terms that are needed to ensure well-posedness of the variational principle in the presence of corners. This is accomplished by cutting open a manifold with a conical singularity into a wedge with boundaries intersecting at a corner. Notably, our observation provides a rigorous definition of the action of a conical singularity that does not require regularization. For Einstein gravity, we compute the Rényi entropy of gravitational states with either fixed-periodicity or fixed-area boundary conditions. The entropy functional for fixed-area states is equal to the corner term, whose extremization follows from the variation of the Einstein action of the wedge under transverse diffeomorphisms. For general Lovelock gravity the entropy functional of fixed-periodicity states is equal to the Jacobson-Myers (JM) functional, while fixed-area states generalize to fixed-JM-functional states, having a flat spectrum. Extremization of the JM functional is shown to coincide with the variation of the Lovelock action of the wedge. For arbitrary F(Riemann) gravity, under special periodic boundary conditions, we recover the Dong-Lewkowycz entropy for fixed-periodicity states. Since the variational problem in the presence of corners is not well-posed, we conjecture the generalization of fixed-area states does not exist for such theories without additional boundary conditions. Thus, our work suggests the existence of entropy functionals is tied to the existence of corner terms which make the Dirichlet variational problem well-posed.

A variational construction of Hamiltonian stationary surfaces with isolated Schoen–Wolfson conical singularities

AbstractWe construct using variational methods Hamiltonian stationary surfaces with isolated Schoen–Wolfson conical singularities. We obtain these surfaces through a convergence process reminiscent to the Ginzburg–Landau asymptotic analysis in the strongly repulsive regime introduced by Bethuel, Brezis and Hélein. We describe in particular how the prescription of Schoen–Wolfson conical singularities is related to optimal Wente constants.