Riemann Surface Research Articles

Actions of some mapping classes on the closed Riemann surfaces of genus two

Actions of some mapping classes on the closed Riemann surfaces of genus two

Cosmological implications of the Weyl geometric gravity theory

We consider the cosmological implications of the Weyl geometric gravity theory. The basic action of the model is obtained from the simplest conformally invariant gravitational action, constructed, in Weyl geometry, from the square of the Weyl scalar, the strength of the Weyl vector, and a matter term, respectively. The total action is linearized in the Weyl scalar by introducing an auxiliary scalar field. To maintain the conformal invariance of the action the trace condition is imposed on the matter energy–momentum tensor, thus making the matter sector of the action conformally invariant. The field equations are derived by varying the action with respect to the metric tensor, the Weyl vector field, and the scalar field, respectively. We investigate the cosmological implications of the theory, and we obtain first the cosmological evolution equations for a flat, homogeneous and isotropic geometry, described by Friedmann–Lemaitre–Robertson–Walker metric, which generalize the Friedmann equations of standard general relativity. In this context we consider two cosmological models, corresponding to the vacuum state, and to the presence of matter described by a linear barotropic equation of state. In both cases we perform a detailed comparison of the predictions of the theory with the cosmological observational data, and with the standard Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} CDM model. By assuming that the presence of the Weyl geometric effects induce small perturbations in the homogeneous and isotropic cosmological background, and that the anisotropy parameter is small, the equations of the cosmological perturbations due to the presence of the Weyl geometric effects are derived. The time evolution of the metric and matter perturbations are explicitly obtained. Therefore, if Weyl geometric effects are present, the Universe would acquire some anisotropic characteristics, and its geometry will deviate from the standard FLRW one.

Conformal invariants of curves via those for inscribed polygons with circular edges

Abstract The conformal nature of smooth curves in R^ 3 is characterised by conformal length,
curvature and torsion. We present a derivation of these conformal parameters via
a limiting process using inscribed polygons with circular edges . The procedure is
based on elementary geometry in R^3 only and similar to the rectification of curves in
the metrical case. It seems to be not available in the literature so far.

Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3

Abstract An osculating curve is a type of curve in space that holds significance in the study of differential geometry. In this article, we investigate certain geometric invariants of osculating curves on smooth and regularly immersed surfaces under conformal transformations in Euclidean space R 3 {{\mathbb{R}}}^{3} . The primary objective of this article is to explore conditions sufficient for the conformal invariance of the osculating curve under both conformal transformations and isometries. We also compute the tangential and normal components of the osculating curves, demonstrating that they remain invariant under the isometry of the surfaces in R 3 {{\mathbb{R}}}^{3} .

Class S theories on S2

We study two-dimensional N=(0,2) and N=(0,4) theories derived from compactifying class S theories on S2 with a topological twist. We present concise expressions for the elliptic genera of both classes of theories, revealing the topological quantum field theory structure on Riemann surfaces Cg,n. Furthermore, our study highlights the relationship between the left-moving sector of the (0,2) theory and the chiral algebra of the four-dimensional N=2 theory. Notably, we propose that the (0,2) elliptic genus of a theory of this class can be expressed as a linear combination of characters of the corresponding chiral algebra. Published by the American Physical Society 2024

Contact instantons with Legendrian boundary condition: A priori estimates, asymptotic convergence and index formula

In this paper, we establish nonlinear ellipticity of the equation of contact instantons with Legendrian boundary condition on punctured Riemann surfaces by proving the a priori elliptic coercive estimates for the contact instantons with Legendrian boundary condition, and prove an asymptotic exponential [Formula: see text]-convergence result at a puncture under the uniform [Formula: see text] bound. We prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes. This eliminates the phenomenon of the appearance of spiraling cusp instanton along a Reeb core, which removes the only remaining obstacle towards the compactification and the Fredholm theory of the moduli space of contact instantons in the open string case, which plagues the closed string case. Leaving the study of [Formula: see text]-estimates and details of Gromov-Floer-Hofer style compactification of contact instantons to [27], we also derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the sequels [27]–[29] and [36] containing applications to contact topology and contact Hamiltonian dynamics.

Holography of linear dilaton spacetimes from the bottom up

The linear dilaton (LD) background is the keystone of a string-derived holographic correspondence beyond AdSd+1/CFTd. This motivates an exploration of the (d+1)-dimensional linear dilaton spacetime (LDd+1) and its holographic properties from the low-energy viewpoint. We first notice that the LDd+1 space has simple conformal symmetries, that we use to shape an effective field theory (EFT) on the LD background. We then place a brane in the background to study holography at the level of quantum fields and gravity. We find that the holographic correlators from the EFT feature a pattern of singularities at certain kinematic thresholds. We argue that such singularities can be used to bootstrap the putative d-dimensional dual theory using techniques analogous to those of the cosmological bootstrap program. Turning on finite temperature, we study the holographic fluid emerging on the brane in the presence of a bulk black hole. We find that the holographic fluid is pressureless for any d due to a cancellation between Weyl curvature and dilaton stress tensor, and verify consistency with the time evolution of the theory. From the fluid thermodynamics, we find a universal temperature and Hagedorn behavior for any d. This matches the properties of a CFT2 with large TT¯ deformation, and of little string theory for d=6. We also find that the holographic fluid entropy exactly matches the bulk black hole Bekenstein-Hawking entropy. Both the fluid equation of state and the spectrum of quantum fluctuations suggest that the d-dimensional dual theory arising from LDd+1 is generically gapped. Published by the American Physical Society 2024

Boundary Points, Minimal L2 Integrals and Concavity Property III—Linearity on Riemann Surfaces and Fibrations over Open Riemann Surfaces

Boundary Points, Minimal L2 Integrals and Concavity Property III—Linearity on Riemann Surfaces and Fibrations over Open Riemann Surfaces

Geometry of the spectral parameter and renormalisation of integrable sigma-models

In the past few years, the unifying frameworks of 4-dimensional Chern-Simons theory and affine Gaudin models have allowed for the systematic construction of a large family of integrable σ-models. These models depend on the data of a Riemann surface C (here of genus 0 or 1) and of a meromorphic 1-form ω on C, which encodes the geometry of their spectral parameter and the analytic structure of their Lax connection. The main subject of this paper is the renormalisation of these theories and in particular two conjectures describing their 1-loop RG-flow in terms of the 1-form ω. These conjectures were put forward in [1] and [2] and were proven in a variety of cases. After extending the proposal of [1] to the elliptic setup (with C of genus 1), we establish the equivalence of these two conjectures and discuss some of their applications. Moreover, we check their veracity on an explicit example, namely an integrable elliptic deformation of the Principal Chiral Model on \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ ext{SL}}}_{\\mathbb{R}}\\left(N\\right)$$\\end{document}.

Trace anomaly form factors from lattice QCD

The hadron mass can be obtained through the calculation of the trace of the energy-momentum tensor in the hadron which includes the trace anomaly and sigma terms. The anomaly due to conformal symmetry breaking is believed to be an important ingredient for hadron mass generation and confinement. In this work, we will present the calculation of the glue part of the trace anomaly form factors of the pion up to Q2∼4.3 GeV2 and the nucleon up to Q2∼1 GeV2. The calculations are performed on a domain wall fermion ensemble with overlap valence quarks at seven valence pion masses varying from ∼250 to ∼540 MeV, including the unitary point ∼340 MeV. We calculate the radius of the glue trace anomaly for the pion and the nucleon from the z expansion. By performing a two-dimensional Fourier transform on the glue trace anomaly form factors in the infinite momentum frame with no energy transfer, we also obtain their spatial distributions for several valence quark masses. The results are qualitatively extrapolated to the physical valence pion mass with systematic errors from the unphysical sea quark mass, discretization effects in the renormalization sum rule, and finite-volume effects to be addressed in the future. We find the pion’s form factor changes sign, as does its spatial distribution, for light quark masses. This explains how the trace anomaly contribution to the pion mass approaches zero toward the chiral limit. Published by the American Physical Society 2024

Regge Spectroscopy of Higher-Twist States in N=4 Supersymmetric Yang-Mills Theory

We study a family of higher-twist Regge trajectories in N=4 supersymmetric Yang-Mills theory using the quantum spectral curve. We explore the many-sheeted Riemann surface connecting the different trajectories and show the interplay between the degenerate nonlocal operators known as (near-)horizontal trajectories. We resolve their degeneracy analytically by computing the first nontrivial order of the Regge intercept at weak coupling, which exhibits new behavior: It depends linearly on the coupling. This is consistent with our numerics, which interpolate all the way to strong coupling. Published by the American Physical Society 2024

Conformal theory of central surface density for galactic dark halos

Numerous dark matter studies of galactic halo gravitation depend on models with core radius r0 and central density ρ0. Central surface density product ρ0r0 is found to be nearly a universal constant for a large range of galaxies. Standard variational field theory with Weyl conformal symmetry postulated for gravitation and the Higgs scalar field, without dark matter, implies nonclassical centripetal acceleration ∆a, for a = aN +∆a, where Newtonian acceleration aN is due to observable baryonic matter. Neglecting a halo cutoff at very large galactic radius, conformal ∆a is constant over the entire halo and a = aN + ∆a is a universal function, consistent with a recent study of galaxies, with independently measured mass, that constrains acceleration due to dark matter or to alternative theory. An equivalent dark matter source would be a pure cusp distribution with cutoff parameter determined by a halo boundary radius. This is shown to imply universal central surface density for any dark matter core model.

Entanglement asymmetry in CFT and its relation to non-topological defects

The entanglement asymmetry is an information based observable that quantifies the degree of symmetry breaking in a region of an extended quantum system. We investigate this measure in the ground state of one dimensional critical systems described by a CFT. Employing the correspondence between global symmetries and defects, the analysis of the entanglement asymmetry can be formulated in terms of partition functions on Riemann surfaces with multiple non-topological defect lines inserted at their branch cuts. For large subsystems, these partition functions are determined by the scaling dimension of the defects. This leads to our first main observation: at criticality, the entanglement asymmetry acquires a subleading contribution scaling as log ℓ/ℓ for large subsystem length ℓ. Then, as an illustrative example, we consider the XY spin chain, which has a critical line described by the massless Majorana fermion theory and explicitly breaks the U(1) symmetry associated with rotations about the z-axis. In this situation the corresponding defect is marginal. Leveraging conformal invariance, we relate the scaling dimension of these defects to the ground state energy of the massless Majorana fermion on a circle with equally-spaced point defects. We exploit this mapping to derive our second main result: the exact expression for the scaling dimension associated with n defects of arbitrary strengths. Our result generalizes a known formula for the n = 1 case derived in several previous works. We then use this exact scaling dimension to derive our third main result: the exact prefactor of the log ℓ/ℓ term in the asymmetry of the critical XY chain.

Monopoles in Dirac spin liquids and their symmetries from instanton calculus

The Dirac spin liquid (DSL) is a two-dimensional (2D) fractionalized Mott insulator featuring massless Dirac spinon excitations coupled to a compact U(1) gauge field, which allows for flux-tunneling instanton events described by magnetic monopoles in (2+1)D Euclidean spacetime. The state-operator correspondence of conformal field theory has been used recently to define associated monopole operators and determine their quantum numbers, which encode the microscopic symmetries of conventional ordered phases proximate to the DSL. In this work, we utilize semiclassical instanton methods not relying on conformal invariance to construct monopole operators directly in (2+1)D spacetime as instanton-induced ’t Hooft vertices, i.e., fermion-number-violating effective interactions originating from zero modes of the Euclidean Dirac operator in an instanton background. In the presence of a flavor-adjoint fermion mass, resummation of the instanton gas is shown to select the correct monopole to be proliferated, in accordance with predictions of the state-operator correspondence. We also show that our instanton-based approach is able to determine monopole quantum numbers on bipartite lattices.

Notes on emergent conformal symmetry for black holes

We examine the motion of the massless scalar field and nearly bound null geodesics in the near-ring region of a black hole, which may possess either acceleration or a gravitomagnetic mass. Around such black holes, the photon ring deviates from the equatorial plane. In the large angular momentum limit, we demonstrate that the massless scalar field exhibits an emergent conformal symmetry in this near-ring region. Additionally, in the nearly bound limit, we observe the emergence of a conformal symmetry for the null geodesics that constitute the photon ring in the black hole image. These findings suggest that the hidden conformal symmetry, associated with the Lie algebra 𝔰𝔩(2, ℝ), persists even for black holes lacking north-south reflection symmetry, thereby broadening the foundation of photon ring holography. Finally, we show that the conformal symmetry also emerges for nearly bound timelike geodesics and scalar fields in proximity to the particle ring, and with specific mass around a Schwarzschild black hole.

The Small-N Series in the Zero-Dimensional O(N) Model: Constructive Expansions and Transseries

AbstractWe consider the zero-dimensional quartic O(N) vector model and present a complete study of the partition function Z(g, N) and its logarithm, the free energy W(g, N), seen as functions of the coupling g on a Riemann surface. We are, in particular, interested in the study of the transseries expansions of these quantities. The point of this paper is to recover such results using constructive field theory techniques with the aim to use them in the future for a rigorous analysis of resurgence in genuine quantum field theoretical models in higher dimensions. Using constructive field theory techniques, we prove that both Z(g, N) and W(g, N) are Borel summable functions along all the rays in the cut complex plane $$\mathbb {C}_{\pi } =\mathbb {C}{\setminus } \mathbb {R}_-$$ C π = C \ R - . We recover the transseries expansion of Z(g, N) using the intermediate field representation. We furthermore study the small-N expansions of Z(g, N) and W(g, N). For any $$g=|g| e^{\imath \varphi }$$ g = | g | e ı φ on the sector of the Riemann surface with $$|\varphi |<3\pi /2$$ | φ | < 3 π / 2 , the small-N expansion of Z(g, N) has infinite radius of convergence in N, while the expansion of W(g, N) has a finite radius of convergence in N for g in a subdomain of the same sector. The Taylor coefficients of these expansions, $$Z_n(g)$$ Z n ( g ) and $$W_n(g)$$ W n ( g ) , exhibit analytic properties similar to Z(g, N) and W(g, N) and have transseries expansions. The transseries expansion of $$Z_n(g)$$ Z n ( g ) is readily accessible: much like Z(g, N), for any n, $$Z_n(g)$$ Z n ( g ) has a zero- and a one-instanton contribution. The transseries of $$W_n(g)$$ W n ( g ) is obtained using Möbius inversion, and summing these transseries yields the transseries expansion of W(g, N). The transseries of $$W_n(g)$$ W n ( g ) and W(g, N) are markedly different: while W(g, N) displays contributions from arbitrarily many multi-instantons, $$W_n(g)$$ W n ( g ) exhibits contributions of only up to n-instanton sectors.

Correction to: Local index theorem for orbifold Riemann surfaces

Correction to: Local index theorem for orbifold Riemann surfaces

Complete Contraction of Tensor and Its Application

This article presents a mathematical model for tensor complete contraction utilizing permutation group theory. The model helps in identifying the index class that dictates the complete contraction of even-order tensors. Additionally, the concept of a complete contraction closed loop is introduced along with a method for calculating the number of completely contracted images of any even-order tensor under any λ type permutation. The model is applied to investigate the algebraic structure of global conformal invariants on three-dimensional hypersurfaces. Moreover, it is shown that the generalized Willmore functional is the sole global conformal invariant in three-dimensional hypersurfaces when the difference is a constant multiple.

Solving conformal defects in 3D conformal field theory using fuzzy sphere regularization

Defects in conformal field theory (CFT) are of significant theoretical and experimental importance. The presence of defects theoretically enriches the structure of the CFT, but at the same time, it makes it more challenging to study, especially in dimensions higher than two. Here, we demonstrate that the recently-developed theoretical scheme, fuzzy (non-commutative) sphere regularization, provides a powerful lens through which one can dissect the defect of 3D CFTs in a transparent way. As a notable example, we study the magnetic line defect of 3D Ising CFT and clearly demonstrate that it flows to a conformal defect fixed point. We have identified 6 low-lying defect primary operators, including the displacement operator, and accurately extract their scaling dimensions through the state-operator correspondence. Moreover, we also compute one-point bulk correlators and two-point bulk-defect correlators, which show great agreement with predictions of defect conformal symmetry, and from which we extract various bulk-defect operator product expansion coefficients. Our work demonstrates that the fuzzy sphere offers a powerful tool for exploring the rich physics in 3D defect CFTs.

Lie Symmetries of the Wave Equation on the Sphere Using Geometry

A semilinear quadratic equation of the form Aij(x)uij=Bi(x,u)ui+F(x,u) defines a metric Aij; therefore, it is possible to relate the Lie point symmetries of the equation with the symmetries of this metric. The Lie symmetry conditions break into two sets: one set containing the Lie derivative of the metric wrt the Lie symmetry generator, and the other set containing the quantities Bi(x,u),F(x,u). From the first set, it follows that the generators of Lie point symmetries are elements of the conformal algebra of the metric Aij, while the second set serves as constraint equations, which select elements from the conformal algebra of Aij. Therefore, it is possible to determine the Lie point symmetries using a geometric approach based on the computation of the conformal Killing vectors of the metric Aij. In the present article, the nonlinear Poisson equation Δgu−f(u)=0 is studied. The metric defined by this equation is 1 + 2 decomposable along the gradient Killing vector ∂t. It is a conformally flat metric, which admits 10 conformal Killing vectors. We determine the conformal Killing vectors of this metric using a general geometric method, which computes the conformal Killing vectors of a general 1+(n−1) decomposable metric in a systematic way. It is found that the nonlinear Poisson equation Δgu−f(u)=0 admits Lie point symmetries only when f(u)=ku, and in this case, only the Killing vectors are admitted. It is shown that the Noether point symmetries coincide with the Lie point symmetries. This approach/method can be used to study the Lie point symmetries of more complex equations and with more degrees of freedom.