Riemann Surface Research Articles

Quasi-abelian group as automorphism group of Riemann surfaces

Quasi-abelian group as automorphism group of Riemann surfaces

Stability conditions and Teichmüller space

AbstractWe consider a 3-Calabi–Yau triangulated category associated to an ideal triangulation of a marked bordered surface. Using the theory of harmonic maps between Riemann surfaces, we construct a natural map from a component of the space of Bridgeland stability conditions on this category to the enhanced Teichmüller space of the surface. We describe a relationship between the central charges of objects in the category and shear coordinates on the Teichmüller space.

Tropical curves and solitons in nonlinear integrable systems

We construct families of rationally degenerating Riemann surfaces corresponding to tropical curves with trivial weights, and give explicit formulas of the associated quasi-periodic solutions and soliton solutions as their regularized limits to the KP hierarchy and the 2D Toda lattice hierarchy. This result especially gives rise to soliton solutions in extensive classes to these nonlinear integrable systems which are expected to cover all solutions expressed by compositions of elementary functions. For these quasi-periodic solutions, we also discuss their regularity and limits to the mixtures with solitons obtained from tropical curves with nontrivial weights.

Graviton-photon oscillations in an expanding universe

Through the Gertsenshtein effect, the presence of a large external B-field may allow photons and gravitons to mix in a way that resembles neutrino oscillations and is even more similar to axion-photon mixing. Assuming a background B-field (or any field that behaves like one), we examine the Gertsenshtein mechanism in the context of FLRW expanding universe models, as well as de Sitter space. The conformal invariance of Maxwell’s equations and the conformal noninvariance of the Einstein equations preclude the operation of the Gertsenshtein effect at all scales. In general we find for the matter- and radiation-dominated cases, graviton-oscillations are possible only at late conformal times or when the wavelengths are much shorter than the horizon (κη≫1), but that the time-dependent oscillations eventually damp out in any case. The presence of charged particles additionally damps out the oscillations. For the de Sitter universe, we find that oscillations are possible only at early conformal times (η≪H−1) and for wavelengths short compared to the Hubble radius, but eventually freeze in when wavelengths become longer than the Hubble radius. In principle a Gertsenshtein-like mechanism might influence the balance of particle species in an inflationary phase before freezing in; however, we find that in all our models the mixing length is larger than the Hubble radius. We discuss several possible remedies to this situation.

Principal SO(2n,ℂ)-Bundle Fixed Points over a Compact Riemann Surface

Principal SO(2n,ℂ)-Bundle Fixed Points over a Compact Riemann Surface

Geometric Quantization Results for Semi-positive Line Bundles on a Riemann Surface

In earlier work (Marinescu and Savale in Math Ann. https://doi.org/10.1007/s00208-023-02750-3, 2023) the authors proved the Bergman kernel expansion for semi-positive line bundles over a Riemann surface whose curvature vanishes to at most finite order at each point. Here we explore the related results and consequences of the expansion in the semi-positive case including: Tian’s approximation theorem for induced Fubini-Study metrics, leading-order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections, and the asymptotics of holomorphic torsion.

Faber Series for $$L^2$$ Holomorphic One-Forms on Riemann Surfaces with Boundary

Faber Series for $$L^2$$ Holomorphic One-Forms on Riemann Surfaces with Boundary

Riemann–Hilbert approach for the inhomogeneous discrete nonlinear Schrödinger equation with nonzero boundary conditions

In this paper, we systematically investigate the Riemann–Hilbert (RH) approach and obtain the soliton solutions for the inhomogeneous discrete nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBCs). Starting from the spectral problem and introducing the uniformization variable κ to avoid the complexity of double-valued function and Riemann surface, we deduce the analyticity, asymptotics and symmetries of the eigenfunctions and scattering coefficients, then the RH problem and reconstruction formula for the potential are successfully constructed. Under reflectionless condition and combining the time evolution of the scattering coefficients and eigenfunctions, we obtain various first-order soliton solutions with different direction of propagation caused by the change of the coefficients. Based on the analytic solution and the choice of special parameter values, we obtain the collision mechanism of two soliton solutions. Furthermore, the important advantage of the RH problem is that it can be further used to study the soliton resolution and the long-time asymptotic behavior of the solutions.

Universal asymptotics for high energy CFT data

Equilibrium finite temperature observables of a CFT can be described by a local effective action for background fields — a “thermal effective action”. This effective action determines the asymptotic density of states of a CFT as a detailed function of dimension and spin. We discuss subleading perturbative and nonperturbative corrections to the density, comparing with free and holographic examples. We furthermore show how to use the thermal effective action on more complicated geometries at special locations called “hot spots”. The hot spot idea makes a prediction for a CFT partition function on a higher-dimensional version of a genus-2 Riemann surface, in a particular high temperature limit. By decomposing the partition function into a novel higher-dimensional version of genus-2 conformal blocks (which we compute at large scaling dimension), we extract the asymptotic density of heavy-heavy-heavy OPE coefficients in a higher-dimensional CFT. We also compute asymptotics of thermal 1-point functions using the same techniques.

Black holes, conformal symmetry, and fundamental fields

Cosmic censorship protects the outside world from black hole singularities and paves the way for assigning entropy to gravity at the event horizons. We point out a tension between cosmic censorship and the quantum backreacted geometry of Schwarzschild black holes, induced by vacuum polarization and driven by the conformal anomaly. A similar tension appears for the Weyl curvature hypothesis at the Big Bang singularity. We argue that the requirement of exact conformal symmetry resolves both conflicts and has major implications for constraining the set of fundamental constituents of the Standard Model.

On the length spectrums of Riemann surfaces given by generalized Cantor sets

On the length spectrums of Riemann surfaces given by generalized Cantor sets

Conformally Invariant Gravity and Gravitating Mirages

The action of an ideal fluid in Euler variables with a variable number of particles is used for the phenomenological description of the processes of particle creation in strong external fields. It has been demonstrated that the conformal invariance of the creation law imposes quite strict restrictions on the possible types of sources. It is shown that combinations with the particle number density in the creation law can be interpreted as dark matter within the framework of this model.

A note on the quality of dilatonic ultralight dark matter

Dilatons are pseudo-Nambu-Goldstone bosons arising from the breaking of conformal invariance. In this letter we point out that in general a dilaton mass has a power-law dependence on a small parameter related to the explicit breaking of conformal invariance whereas the ratio between the ultraviolet and infrared scales in the theory is exponentially related to the same parameter. We show that this scaling results in a separation between the dilaton mass and the infrared scale that can not be arbitrary large. Therefore a small dilaton mass necessarily is associated to a secluded conformal sector. We argue that the fact that the dilaton field must have a small displacement from the minimum of its effective potential generated near the infrared scale precludes a cosmologically interesting amount of dilatonic dark matter to be produced by a misalignment mechanism in the early Universe.

Wilson loops and wormholes

We analyse the properties of Wilson loop observables for holographic gauge theories, when the dual bulk geometries have a single and/or multiple boundaries (Euclidean spacetime wormholes). Such observables lead to a generalisation and refinement of the characterisation in [1] based on the compressibility of cycles and the pinching limit of higher genus Riemann surfaces, since they carry information about the dynamics and phase structure of the dual gauge theory of an arbitrary dimensionality. Finally, we describe how backreacting correlated observables such as Wilson loops can lead to wormhole saddles in the dual gravitational path integral, by taking advantage of a representation theoretic entanglement structure proposed in [13, 15].

Color-kinematics duality and dual conformal symmetry for a four-loop form factor in $$\mathcal{N}$$ = 4 SYM

Color-kinematics duality and dual conformal symmetry for a four-loop form factor in $$\mathcal{N}$$ = 4 SYM

Approximation by Meromorphic k-Differentials on Compact Riemann Surfaces

Approximation by Meromorphic k-Differentials on Compact Riemann Surfaces

Bootstrapping smooth conformal defects in Chern-Simons-matter theories

The expectation value of a smooth conformal line defect in a CFT is a conformal invariant functional of its path in space-time. For example, in large N holographic theories, these fundamental observables are dual to the open-string partition function in AdS. In this paper, we develop a bootstrap method for studying them and apply it to conformal line defects in Chern-Simons matter theories. In these cases, the line bootstrap is based on three minimal assumptions — conformal invariance of the line defect, large N factorization, and the spectrum of the two lowest-lying operators at the end of the line. On the basis of these assumptions, we solve the one-dimensional CFT on the line and systematically compute the defect expectation value in an expansion around the straight line. We find that the conformal symmetry of a straight defect is insufficient to fix the answer. Instead, imposing the conformal symmetry of the defect along an arbitrary curved line leads to a functional bootstrap constraint. The solution to this constraint is found to be unique.

Warped conformal symmetries of the accelerating Kerr black hole

Recent studies on the holographic descriptions of Kerr black holes indicate that the conformal or the warped conformal symmetries are responsible for the Kerr black hole physics at both background and perturbation levels. In the present paper, we extend the validity of these studies to the case of accelerating Kerr black hole. By invoking a set of non-trivial diffeomorphisms near the horizon bifurcation surface of the accelerating Kerr black hole, the Dirac brackets among charges of the diffeomorphisms form the symmetry algebra of a warped CFT which consists of one Virasoro and one Kac-Moody algebra with central extensions. This provides the evidence for warped CFTs being possible holographic dual to accelerating Kerr black holes. The thermal entropy formula of the warped CFT fixed by modular parameters and vacuum charges reproduces the entropy of the rotating black hole with acceleration.

A new braneworld with conformal symmetry breaking

We explore the conformal 5D braneworld, where warping emerges through conformal symmetry breaking. Our scenario seamlessly aligns with conventional brane approaches if conformal symmetry remains unbroken. It is shown that a model with a single conformal breaking parameter effectively localizes gravity on the brane, but it falls short in trapping gauge bosons. However, in scenarios with two parameters, gravity is localized, and the model also achieves the localization of zero modes for both gauge and Dirac fields.

Bootstrapping the effect of the twist operator in the D1D5 CFT

In the D1D5 CFT the twist operator of order 2 can twist together two copies in the untwisted sector into a single joined copy in the twisted sector. Traditionally, this effect is computed by using the covering map method. Recently, a new method was developed using the Bogoliubov ansatz and conformal symmetry to compute this effect in a toy model of one free boson. In this paper, we use this method with superconformal symmetry to compute the effect of the twist operator in the D1D5 CFT. This may provide more effective tools for computing correlation functions of twist operators in this system.