Riemann Surface Research Articles

Nonsymmorphic spin-space cubic groups and SU(2)$_1$ conformal invariance in one-dimensional spin-1/2 models

Recently, extended gapless phases with emergent SU(2)_11 conformal invariance occupying finite regions in the phase diagrams have been found in one-dimensional spin-1/2 models with nonsymmorphic O_hOh symmetry groups. In this work, we investigate the question of whether the conditions for emergent SU(2)_11 invariance can be loosened. We find that besides the nonsymmorphic O_hOh group, the other four smaller nonsymmorphic cubic groups including OO, T_hTh, T_dTd and TT can also give rise to emergent SU(2)_11 invariance. Minimal spin-1/2 models having these nonsymmorphic cubic groups as symmetry groups are constructed, and numerical evidences for the emergent SU(2)_11 invariance are provided. Our work is useful for understanding gapless phases in one-dimensional spin systems with nonsymmorphic symmetries.

Hyperbolic Lattices and Two-Dimensional Yang-Mills Theory.

Hyperbolic lattices are a new type of synthetic quantum matter emulated in circuit quantum electrodynamics and electric-circuit networks, where particles coherently hop on a discrete tessellation of two-dimensional negatively curved space. While real-space methods and a reciprocal-space hyperbolic band theory have been recently proposed to analyze the energy spectra of those systems, discrepancies between the two sets of approaches remain. In this work, we reconcile those approaches by first establishing an equivalence between hyperbolic band theory and U(N) topological Yang-Mills theory on higher-genus Riemann surfaces. We then show that moments of the density of states of hyperbolic tight-binding models correspond to expectation values of Wilson loops in the quantum gauge theory and become exact in the large-N limit.

Moving holographic boundaries

In this paper, we show that for one sign of the deformation coupling single-trace TT‾ deformation moves the holographic screen in Gödel universe radially inward. For the other sign of the coupling it moves the holographic screen radially outward. We (thus) argue, on general grounds, that in holography (single-trace) TT‾ deformation can be generally thought of as either moving the holographic boundary into the bulk or washing it away to infinity. We explain in what sense. In Anti-de Sitter this breaks the spacetime conformal symmetry. We further note that moving timelike holographic boundary into bulk creates (at onset) a curvature singularity. In the boundary the singularity is understood by states with imaginary energies. To define or make the bulk theory sensible we introduce an ultraviolet cutoff and thereby move the boundary into the bulk. However, it is not clear whether suitable boundary conditions that lead to a consistent string theory exist. In this paper we first obtain the Penrose limit of the single-trace TT‾ deformed string background and then perform T-duality along a space-like isometry to obtain a class of deformed Gödel universes. The string background we consider is AdS3×S3×M4. The single-trace TT‾ deformation is a particular example of the more general O(d,d) transformations.

Structure of the Kuranishi spaces of pairs of Kähler manifolds and polystable Higgs bundles

AbstractLet be a compact Kähler manifold and be a Higgs bundle over it. We study the structure of the Kuranishi space for the pair when the Higgs bundle admits a harmonic metric or equivalently when the Higgs bundle is polystable and the Chern classes are 0. Under such assumptions, we show that the Kuranishi space of the pair is isomorphic to the direct product of the Kuranishi space of and the Kuranishi space of . Moreover, when is a Riemann surface and is stable and the degree is 0, we show that the deformation of the pair is unobstructed and calculate the dimension of the Kuranishi space.

Estimates of Kähler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

Estimates of Kähler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

Explicit entropic proofs of irreversibility theorems for holographic RG flows

We revisit the existence of monotonic quantities along renormalization group flows using only the Null Energy Condition and the Ryu-Takayanagi formula for the entanglement entropy of field theories with anti-de Sitter gravity duals. In particular, we consider flows within the same dimension and holographically reprove the c-, F -, and a-theorems in dimensions two, three, and four. We focus on the family of maximally spherical entangling surfaces, define a quasi-constant of motion corresponding to the breaking of conformal invariance, and use a properly defined distance between minimal surfaces to construct a holographic c-function that is monotonic along the flow. We then apply our method to the case of flows across dimensions: there, we reprove the monotonicity of flows from AdSD+1 to AdS3 and prove the novel case of flows from AdS5 to AdS4.

Moduli spaces in CFT: large charge operators

Using the large-charge expansion, we prove a necessary condition for a CFT to exhibit conformal symmetry breaking, under the assumption that a continuous global symmetry is also broken on the moduli space: there must be a tower of charged local operators whose scaling dimensions are asymptotically linear in the charge. In supersymmetric theories with a continuous R-symmetry and a holomorphic moduli space, the existence of such a tower of operators follows trivially from a BPS condition: their scaling dimensions are then exactly linear in the R-charge. We illustrate the more general statement in several examples of three-dimensional 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 CFTs, where the leading linear behavior receives nontrivial corrections. By considering a suitable scaling limit, we also relate the spectrum of states with large charge on the cylinder (isomorphic to local operators) to the spectrum of massive particles on the moduli space.

Approximate CFTs and random tensor models

A key issue in both the field of quantum chaos and quantum gravity is an effective description of chaotic conformal field theories (CFTs), that is CFTs that have a quantum ergodic limit. We develop a framework incorporating the constraints of conformal symmetry and locality, allowing the definition of ensembles of ‘CFT data’. These ensembles take on the same role as the ensembles of random Hamiltonians in more conventional quantum ergodic phases of many-body quantum systems. To describe individual members of the ensembles, we introduce the notion of approximate CFT, defined as a collection of ‘CFT data’ satisfying the usual CFT constraints approximately, i.e. up to small deviations. We show that they generically exist by providing concrete examples. Ensembles of approximate CFTs are very natural in holography, as every member of the ensemble is indistinguishable from a true CFT for low-energy probes that only have access to information from semi-classical gravity. To specify these ensembles, we impose successively higher moments of the CFT constraints. Lastly, we propose a theory of pure gravity in AdS3 as a random matrix/tensor model implementing approximate CFT constraints. This tensor model is the maximum ignorance ensemble compatible with conformal symmetry, crossing invariance, and a primary gap to the black-hole threshold. The resulting theory is a random matrix/tensor model governed by the Virasoro 6j-symbol.

Anatomy of null contractions

We introduce null contractions of the Poincare and relativistic conformal algebras. The longitudinal null contraction involves writing the algebra in lightcone coordinates and contracting one of the null directions. For the Poincare algebra, this yields two non-overlapping co-dimension one Carroll algebras. The transverse contraction is a limit on the spatial dimensions and yields two non-overlapping co-dimension one Galilean algebras. We find, similar to Susskind’s original observation of the non-relativistic case, that the Poincare algebra, written in the lightcone coordinates, naturally contains Carrollian sub-algebras in one lower dimension. The effect of the longitudinal contraction, which essentially focusses on the null direction, is to disentangle the two Carroll algebras that now correspond to the symmetries of the two null boundaries. The transverse contraction similarly separates the overlapping Galilean sub-algebras of the original Poincare algebra. We discuss aspects of the conformal case, where we get lower dimensional Carroll Conformal algebras and Schrödinger algebras.

Differential equations for Carrollian amplitudes

Differential equations are powerful tools in the study of correlation functions in conformal field theories (CFTs). Carrollian amplitudes behave as correlation functions of Carrollian CFT that holographically describes asymptotically flat spacetime. We derive linear differential equations satisfied by Carrollian MHV gluon and graviton amplitudes. We obtain non-distributional solutions for both the gluon and graviton cases. We perform various consistency checks for these differential equations, including compatibility with conformal Carrollian symmetries.

Hidden conformal symmetry in AdS2×S2 beyond tree level

Correlators of N=2 hypermultiplets with two-derivative interactions in AdS2×S2 exhibit a hidden four-dimensional conformal symmetry which allows one to repackage all tree-level four-point correlators into a single four-dimensional object corresponding to a contact diagram arising from a massless ϕ4 theory in AdS2×S2. This theory serves as a toy model for IIB string theory in AdS5×S5 and is interesting in its own right because AdS2×S2 describes the near-horizon limit of extremal black holes in four dimensions. We argue that, after acting with an SU(1,1)×SU(2) Casimir, all one-loop correlators can similarly be encoded by a four-dimensional function which arises from a one-loop scalar bubble diagram in AdS2×S2, explaining how the hidden conformal symmetry extends beyond tree level. Finally, we conjecture a scalar effective field theory with a two-derivative interaction in AdS2×S2 whose Witten diagrams should directly reproduce four-point correlators to all loops without acting with Casimirs. Published by the American Physical Society 2024

Pion gravitational form factors in the QCD instanton vacuum. I

The pion form factors of the QCD energy-momentum tensor (EMT) are studied in the instanton liquid model (ILM) of the QCD vacuum. In this approach, the breaking of conformal symmetry is encoded in the form of stronger-than-Poisson fluctuations in the number of instantons. For the trace of the EMT, it is shown that the gluonic trace anomaly term contributes half the pion mass, with the other half coming from the quark-mass-dependent σ term. The Q2 dependence of the form factors is governed by glueball and scalar meson exchanges. For the EMT, the spin-0 (trace) and spin-2 (traceless rank-2 tensor) form factors are computed at next-to-leading order in the instanton density using effective quark operators. Relations between the gluon and quark contributions to the EMT form factors are derived. The form factors are also expressed in terms of the pion light-front wave functions in the ILM. The results at the low resolution scale of the inverse instanton size are evolved to higher scales using the renormalization group equation. The ILM results compare well with those of recent lattice QCD calculations. Published by the American Physical Society 2024

Finite simple permutation groups acting with fixity 4

Abstract Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we classify all finite simple groups that have a faithful transitive action with fixity 4. We also give details about all possible such actions.

Homogenization of quasi-periodic conformal architectured materials and applications to chiral lattices

In this study, we propose to extend asymptotic periodic homogenization for non-periodic continuous microstructured media, assuming that the non-periodic geometry (called quasi-periodic) can be designed by a conformal planar transformation of a periodic parent domain architectured media with periodically disposed unit cells. Conformal transformations are shown to play a privileged role in the design of circular macroscopic heterogeneous domains tessellated with non-periodic unit cells, obtained from a periodic parent domain architectured with these unit cells. The conditions for conformal invariance are established, leading to the general form of conformal transformation in their dependencies upon the periodic coordinates. It is shown that any conformal map can be decomposed into the product of an isotropic dilatation function of the first periodic spatial position of decreasing exponential type and a rotation characterized by an angular function linear in the second periodic position. A general theory of quasi-periodic homogenization in the framework of conformal transformations is established for the first time, leading to an expression of the tensor of quasi-periodic moduli which is fully evaluated from the solution of the elasticity boundary value problem posed over the periodic unit cell. The influence of microcurvature distortion of individual unit cells on their effective properties is evaluated. Closed-form solutions are confronted to numerical examples issued from the implementation of circular periodicity in a finite element solver, showing overall a good agreement with the identified homogenized moduli.

Perturbed conformal invariance and Mei adiabatic invariants of the generalized perturbed Hamiltonian systems with additional terms

AbstractThis study investigates the relationship between perturbed conformal invariance and Mei symmetry in the generalized perturbed Hamiltonian systems with additional terms. A necessary and sufficient condition is derived to determine whether perturbed conformal invariance can be considered an approximate Mei symmetry. Furthermore, the Mei adiabatic invariants are also obtained. Lastly, an example is presented to demonstrate the key findings.

Jacobian varieties with group algebra decomposition not affordable by Prym varieties

The action of a finite group G on a compact Riemann surface X naturally induces another action of G on its Jacobian variety J(X). In many cases, each component of the group algebra decomposition of J(X) is isogenous to a Prym varieties of an intermediate covering of the Galois covering πG:X→X/G; in such a case, we say that the group algebra decomposition is affordable by Prym varieties. In this article, we present an infinite family of groups that act on Riemann surfaces in a manner that the group algebra decomposition of J(X) is not affordable by Prym varieties; namely, affine groups Aff(Fq) with some exceptions: q=2, q=9, q a Fermat prime, q=2n with 2n−1 a Mersenne prime and some particular cases when X/G has genus 0 or 1. In each one of this exceptional cases, we give the group algebra decomposition of J(X) by Prym varieties.

Erosion of surfaces by trapped vortices

Two two-dimensional free boundary problems describing the erosion of solid surfaces by the flow of inviscid fluid in the presence of trapped vortices are considered. The first problem tackles an initially flat, infinite fluid-solid interface with uniform flow at infinity and a vortex in equilibrium above the surface. The second involves flow around a finite body with a trailing Föppl-type vortex pair. The conformal invariance of the complex potential permits both problems to be formulated as a Polubarinova–Galin (PG) type equation in which the time-dependent eroding surface in the physical z-plane is mapped to the fixed boundary of the ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta $$\\end{document}-disk. The Hamiltonian governing the equilibrium position of the vortex (or vortex pair in the second problem) is also found from the same map. In each problem, the PG equation giving the conformal map is found numerically and the time-dependent evolution of the interface and vortex location is determined. Different models governing the erosion of the interface are investigated in which the normal velocity of the boundary depends on some given function of the fluid flow velocity at the boundary. Typically, in the infinite surface case, erosion leads to the formation of a symmetric valley beneath the vortex which, in turn, moves downward toward the interface. A finite body undergoes erosion which is asymmetric in the flow direction leading to a flattening of the lee surface of the body so displaying some similarity to the experiments and associated viscous theory of Ristroph et al, Moore et al (Proc Natl Acad Sci 109(48):19606–19609, 2012, Phys Fluids 25(11):116602, 2013).

Morse index bound of simple closed geodesics on 2-spheres and strong Morse inequalities

Abstract We give a Morse-theoretic characterization of simple closed geodesics on Riemannian 2-spheres. On any Riemannian 2-sphere endowed with a generic metric, we show there exists a simple closed geodesic with Morse index 1, 2 and 3. In particular, for an orientable Riemannian surface, we prove strong Morse inequalities for the length functional applied to the space of simple closed curves.

Nonlinearly charging the conformally dressed black holes preserving duality and conformal invariance

We start this paper by concisely rederiving ModMax, which is nothing but the unique nonlinear extension of Maxwell’s equations preserving conformal and duality invariance. The merit of this new derivation is its transparency and simplicity since it is based on an approach where the elusive duality invariance is manifest. In the second part, we couple the ModMax electrodynamics to Einstein gravity with a cosmological constant together with a standard conformal scalar field, and new stationary spacetimes with dyonic charges are found. These solutions are later used as seed configurations to generate nonlinearly charged (super-)renormalizably dressed spacetimes by means of a known generating method that we extend to include any nonlinear conformal electrodynamics. We end by addressing the issue of how to generalize some of these results to include the recently studied non-Noetherian conformal scalar fields, whose equation of motion still enjoys conformal symmetry even though its action does not. It turns out that the static non-Noetherian conformally dressed black holes also become amenable to being charged by ModMax. Published by the American Physical Society 2024

Analyticity and the Unruh effect: a study of local modular flow

The Unruh effect can be formulated as the statement that the Minkowski vacuum in a Rindler wedge has a boost as its modular flow. In recent years, other examples of states with geometrically local modular flow have played important roles in understanding energy and entropy in quantum field theory and quantum gravity. Here I initiate a general study of the settings in which geometric modular flow can arise, showing (i) that any geometric modular flow must be a conformal symmetry of the background spacetime, and (ii) that in a well behaved class of “weakly analytic” states, geometric modular flow must be future-directed. I further argue that if a geometric transformation is conformal but not isometric, then it can only be realized as modular flow in a conformal field theory. Finally, I discuss a few settings in which converse results can be shown — i.e., settings in which a state can be constructed whose modular flow reproduces a given vector field.