Riemann Surface Research Articles

Gravitational waves of nonextremal Kerr black holes from conformal symmetry

In the low-energy limit, the near region of a generic Kerr black hole has been conjectured to be holographically dual to a two-dimensional conformal field theory. In this paper, we consider a test object orbiting in the near region of a nonextremal Kerr black hole. We first couple it to a massless scalar field. The resulting scalar radiation at the horizon is computed from the perspectives of gravity and dual conformal field theory. Considering the influence of nonzero temperature, the agreement of both computations is found. Then, we generalize the analysis to the gravitational radiation and find agreement again. The agreement supports the conjectured holography and provides a potential theoretical tool for gravitational wave computations. Published by the American Physical Society 2024

Topological recursion for hyperbolic string field theory

We derive an analog of Mirzakhani’s recursion relation for hyperbolic string vertices and investigate its implications for closed string field theory. Central to our construction are systolic volumes: the Weil-Petersson volumes of regions in moduli spaces of Riemann surfaces whose elements have systoles L ≥ 0. These volumes can be shown to satisfy a recursion relation through a modification of Mirzakhani’s recursion as long as L ≤ 2 sinh−1 1. Applying the pants decomposition of Riemann surfaces to off-shell string amplitudes, we promote this recursion to hyperbolic string field theory and demonstrate the higher order vertices are determined by the cubic vertex iteratively for any background. Such structure implies the solutions of closed string field theory obey a quadratic integral equation. We illustrate the utility of our approach in an example of a stubbed scalar theory.

Riemann surfaces and winding numbers of Rényi phase structure of charged-flat black holes

It’s widely recognized that the free energy landscape captures the essentials of thermodynamic phase transitions. In this work, we extend the findings of [1] by incorporating the nonextensive nature of black hole entropy. Specifically, the connection between black hole phase transitions and the winding number of Riemann surfaces derived through complex analysis is extended to the Rényi entropy framework. This new geometrical and nonextensive formalism is employed to predict the phase portraits of charged-flat black holes within both the canonical and grand canonical ensembles. Furthermore, we elucidate novel relations between the number of sheets comprising the Riemann surface of the Hawking–Page and Van der Waals transitions and the dimensionality of black hole spacetimes. Notably, these new numbers are consistent with those found for charged-AdS black holes in Gibbs–Boltzmann statistics, providing another significant example of the potential connection between the cosmological constant and the nonextensive Rényi parameter.

Crossed modules, non-abelian extensions of associative conformal algebras and wells exact sequences

In this paper, we introduce the notions of crossed modules of associative conformal algebras, two-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the third Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer–Cartan elements of a suitable differential graded Lie algebra, and prove that there is a one-one correspondence between the Deligne groupoid of this differential graded Lie algebra and the non-abelian cohomology. For any two associative conformal algebras [Formula: see text] and [Formula: see text], we provide the condition that a pair of automorphisms in Aut(A) × Aut(B) can be extended to an automorphisms of the non-abelian extension algebra of [Formula: see text] by [Formula: see text], and give the fundamental sequence of Wells in the context of associative conformal algebras.

Spin(8,C)-Higgs Bundles and the Hitchin Integrable System

Let M(Spin(8,C)) be the moduli space of Spin(8,C)-Higgs bundles over a compact Riemann surface X of genus g≥2. This admits a system called the Hitchin integrable system, induced by the Hitchin map, the fibers of which are Prym varieties. Moreover, the triality automorphism of Spin(8,C) acts on M(Spin(8,C)), and those Higgs bundles that admit a reduction in the structure group to G2 are fixed points of this action. This defines a map of moduli spaces of Higgs bundles M(G2)→M(Spin(8,C)). In this work, the action of triality automorphism is extended to an action on the Hitchin integrable system associated with M(Spin(8,C)). In particular, it is checked that the map M(G2)→M(Spin(8,C)) is restricted to a map at the level of the Prym varieties induced by the Hitchin map. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of G2 and Spin(8,C)-Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved.

On Holomorphic Tubular Neighborhoods of Compact Riemann Surfaces

On Holomorphic Tubular Neighborhoods of Compact Riemann Surfaces

Lie–Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications

Abstract We propose a generalization of two classes of Lie–Hamilton systems on the Euclidean plane to two-dimensional curved spaces, leading to novel Lie–Hamilton systems on Riemannian spaces (flat 2-torus, product of hyperbolic lines, sphere and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes), as well as to semi-Riemannian spaces (Newtonian or non-relativistic spacetimes). The vector fields, Hamiltonian functions, symplectic form and constants of the motion of the Euclidean classes are recovered by a contraction process. The construction is based on the structure of certain subalgebras of the so-called conformal algebras of the two-dimensional Cayley–Klein spaces. These curved Lie–Hamilton classes allow us to generalize naturally the Riccati, Kummer–Schwarz and Ermakov equations on the Euclidean plane to curved spaces, covering both the Riemannian and Lorentzian possibilities, and where the curvature can be considered as an integrable deformation parameter of the initial Euclidean system.

Holographic stress tensor correlators on higher genus Riemann surfaces

In this work, we present a comprehensive study of holographic stress tensor correlators on general Riemann surfaces, extending beyond the previously well-studied torus cases to explore higher genus conformal field theories (CFTs) within the framework of the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We develop a methodological approach to compute holographic stress tensor correlators, employing the Schottky uniformization technique to address the handlebody solutions for higher genus Riemann surfaces. Through rigorous calculations, we derive four-point stress tensor correlators, alongside recurrence relations for higher-point correlators, within the AdS3/CFT2 context. Additionally, our research delves into the holography of cutoff AdS3 spaces, offering novel insights into the lower-point correlators of the TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document}-deformed theories on higher genus Riemann surfaces up to the first deformation order.

Types of connected components of the singular locus of the moduli space of compact Riemann surfaces

AbstractIn 2012 G. Bartolini and M. Izquierdo showed that all equisymmetric strata of Broughton’s stratification, corresponding to the actions of groups of order 2 and 3 on Riemann surfaces of genus g are contained in the same connected component $$\mathcal {C}_g^{2,3}$$ C g 2 , 3 of the singular locus $$\mathcal {S}_g$$ S g . The other components that were known to the literature were those composed of points of certain single strata. This motivated the question of whether the singular locus can contain connected components different from $$\mathcal {C}_g^{2,3}$$ C g 2 , 3 and being the union of at least two strata. The existence of such components was proved for infinitely many genera by G. Gromadzki and the author. In this paper, we will describe and classify all types of connected components of the singular locus. In particular, we prove that every connected component different than $$\mathcal {C}_g^{2,3}$$ C g 2 , 3 coincides with some Cornalba component, and we give necessary and sufficient conditions for a cyclic action to define such a component.

Galilean fluids from non-relativistic gravity

The 1/c-expansion of general relativity appropriately sourced by matter can be used to derive an action principle for Newtonian gravity. The gravitational part of this action is known as non-relativistic gravity (NRG). It is possible to source NRG differently and in such a way that one can construct solutions that are not described by Newtonian gravity (as they do not admit a notion of absolute time). It is possible to include a negative cosmological constant such that NRG admits a non-relativistic AdS solution. This non-relativistic AdS vacuum has Killing vectors that form the Galilean conformal algebra and a boundary that admits a conformal class of Newton-Cartan geometries. This begs the question of whether there exists an analogue of the fluid/gravity correspondence for NRG. In this paper we derive a non-relativistic AdS brane solution of NRG and confirm that it corresponds to the 1/c2-expansion of the AdS black brane geometry. We perform a Galilean boost of the non-relativistic AdS brane and derive the associated boundary energy-momentum tensor. We then show that this is the energy-momentum tensor of a massless Galilean fluid and explain how this is linked to the conformal isometries of the boundary. Along the way, we also present several new results for the theory of non-relativistic gravity itself. In particular we present a rewriting that greatly shortens and simplifies the equations of motion of the NRG action.

Inflationary non-Gaussianities in alpha vacua and consistency with conformal symmetries

We study the conformal invariance of inflationary non-Gaussianities associated with scalar fluctuations in a non-Bunch-Davies initial state, known as the α-vacuum, in single-field slow-roll inflation. The α-vacuum is a one-parameter family of states, including the Bunch-Davies one, that preserves the conformal symmetry of inflationary dynamics in a nearly de-Sitter space-time. Working within the leading slow-roll approximation, we compute the four-point scalar correlator (the trispectrum) in α-vacuum using the in-in formalism. We check that the conformal Ward identities are met between the three and four-point scalar α-vacua correlators. Surprisingly, this contrasts the previously reported negative result of the Ward identities being violated between the two and the three-point correlators. We have also extended the wave-functional method, previously used for correlators with Bunch-Davies initial condition, to compute the three and four-point scalar correlators in α-vacua. The results obtained from the wave-function method match the corresponding in-in results, adding further justification to our check of Ward identities with α-vacua correlators.

Probability of a Single Current

The Riemann surface associated with counting the current between two states of an underlying Markov process is hyperelliptic. We explore the consequences of this property for the time-dependent probability of that current for Markov processes with generic transition rates. When the system is prepared in its stationary state, the relevant meromorphic differential is in particular fully characterized by the precise identification of all its poles and zeroes.

AdS3×S3$AdS_3 \times S^3$ Background From Poisson–Lie T‐Duality

AbstractThe author proceed to construct a dual pair for the background by applying non‐Abelian T‐duality (here as Poisson–Lie [PL] T‐duality on a semi‐Abelian double). By using a certain parametrization of the 4‐dimensional Lie group and by a suitable choice of spectator‐dependent matrices the original ‐model including the metric and a non‐trivial ‐field are constructed. The dual background constructed by means of the PL T‐duality with the spectators is an asymptotically flat one with a potential black hole interpretation supported by a non‐trivial ‐flux whose metric contains the true singularity with a single horizon. The question of classical integrability of the non‐Abelian T‐dual ‐models under consideration is addressed, and their corresponding Lax pairs are found, depending on some spectral parameters. Finally, the conformal invariance conditions of the models are checked up to two‐loop order, and it has been concluded that the resulting model is indeed a solution of supergravity.

Light-ray wave functions and integrability

Using integrability, we construct (to leading order in perturbation theory) the explicit form of twist-three light-ray operators in planar 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. This construction allows us to directly compute analytically continued CFT data at complex spin. We derive analytically the “magic” decoupling zeroes previously observed numerically. Using the Baxter equation, we also show that certain Regge trajectories merge together into a single unifying Riemann surface. Perhaps more surprisingly, we find that this unification of Regge trajectories is not unique. If we organize twist-three operators differently into what we call “cousin trajectories” we find infinitely more possible continuations. We speculate about which of these remarkable features of twist-three operators might generalize to other operators, other regimes and other theories.

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.

Bootstrapping multi-wound twist effects in symmetric orbifold CFTs

We investigate the effects of the twist-2 operator in 2D symmetric orbifold CFTs. The twist operator can join together a twist-M state and a twist-N state, creating a twist-(M + N) state. This process involves three effects: pair creation, propagation, and contraction. We study these effects by using a Bogoliubov ansatz and conformal symmetry. In this multi-wound scenario, pair creation no longer decouples from propagation, in contrast to the previous study where M = N = 1. We derive equations for these effects, which organize themselves into recursion relations and constraints. Using the recursion relations, we can determine the infinite number of coefficients in the effects through a finite number of inputs. Moreover, the number of required inputs can be further reduced by applying constraints.

On mappings with the inverse Poletsky inequality on Riemannian surfaces

On mappings with the inverse Poletsky inequality on Riemannian surfaces

Symmetry analysis in multi scalar-torsion cosmological model with quantum description

A detailed symmetry analysis has been done for the present multi scalar-torsion gravity theory in the background of flat Friedmann Lemai^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{\ ext{ i }}$$\\end{document}tre Robertson Walker (FLRW) space-time. Not only the Noether symmetry from the invariance of the Lagrangian has been determined and associated conserved quantities are evaluated, but also the conformal symmetry of the physical metric (i.e., metric of the kinetic part) has been investigated to evaluate homothetic and killing vector fields. Finally, a through study of quantum cosmology both by canonical approach and by considering quantum trajectories has been presented.

Riemannian Manifolds, Closed Geodesic Lines, Topology and Ramsey Theory

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely (M1,g1) and (M2,g2), represented by the Riemann surfaces which intersect along the curve (M1,g1)∩(M2,g2)=ℒ were addressed. Curve ℒ does not contain geodesic lines in either of the manifolds (M1,g1) and (M2,g2). Consider six points located on the ℒ: {1,…6}⊂ℒ. The points {1,…6}⊂ℒ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold (M1,g1)/red links, and, alternatively, with the geodesic lines belonging to the manifold (M2,g2)/green links. Points {1,…6}⊂ℒ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented.

Comments on Global Symmetries and Anomalies of 5d SCFTs

We study various aspects of global symmetries in five-dimensional superconformal field theories. Whenever a supersymmetry-preserving relevant deformation is available, the infrared gauge theory description might exhibit a finite order mixed ’t Hooft anomaly between a 1-form symmetry and the instantonic symmetry. This anomaly constrains the flavor symmetry group acting faithfully on the SCFT and the consistency of certain RG flows. As an additional example, we consider the instructive case of three-dimensional N=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}=4$$\\end{document} SQED. Finally, we discuss the compatibility between conformal invariance and the presence of 1-form and 2-group global symmetries.