Conformal Structure Research Articles

Degenerating sequences of conformal classes and the conformal Steklov spectrum

AbstractLet $\Sigma $ be a compact surface with boundary. For a given conformal class c on $\Sigma $ the functional $\sigma _k^*(\Sigma ,c)$ is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of conformal classes on $\Sigma $ . A precise formula for the limit of $\sigma _k^*(\Sigma ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander–Nadirashvili invariants of closed manifolds defined as $\inf _{c}\sigma _k^*(\Sigma ,c)$ , where the infimum is taken over all conformal classes c on $\Sigma $ . We show that these quantities are equal to $2\pi k$ for any surface with boundary. As an application of our techniques we obtain new estimates on the kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components.

An ambient approach to conformal geodesics

Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third-order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article, we give a third description using ideas from holography. A conformal [Formula: see text]-manifold [Formula: see text] can be seen (formally at least) as the asymptotic boundary of a Poincaré–Einstein [Formula: see text]-manifold [Formula: see text]. We show that any curve [Formula: see text] in [Formula: see text] has a uniquely determined extension to a surface [Formula: see text] in [Formula: see text], which we call the ambient surface of[Formula: see text]. This surface meets the boundary [Formula: see text] in right angles along [Formula: see text] and is singled out by the requirement that it be a critical point of renormalized area. The conformal geometry of [Formula: see text] is encoded in the Riemannian geometry of [Formula: see text]. In particular, [Formula: see text] is a conformal geodesic precisely when [Formula: see text] is asymptotically totally geodesic, i.e. its second fundamental form vanishes to one order higher than expected. We also relate this construction to tractors and the ambient metric construction of Fefferman and Graham. In the [Formula: see text]-dimensional ambient manifold, the ambient surface is a graph over the bundle of scales. The tractor calculus then identifies with the usual tensor calculus along this surface. This gives an alternative compact proof of our holographic characterization of conformal geodesics.

Quasi-continuous linear phase-gradient metamaterial based on conformal spoof surface plasmon polaritons

In this work, we propose a method of achieving quasi-continuous linear phase gradient for transmitted waves based on conformal spoof surface plasmon polariton (SSPP). To this end, a SSPP structure with high transmission is firstly designed as the unit cell of the metamaterial. To obtain the phase gradient, SSPP structures are arranged delicately in a way that they are conformal to the brachistochrone curve. In this way, quasi-continuous linear Pancharatnam-Berry (PB) phase profile can be realized strictly along one of the two transverse directions. To verify this idea, a dual-band transmissive metamaterial operating in X and Ku band was designed, fabricated and measured. Due to the phase gradient imparted by the conformal SSPP structures, high-efficiency anomalous refraction can be realized within the two bands. Different from the general PGM, the phase gradient of the conformal SSPP structure allows us to achieve the desired anomalous refraction angle without reconstructing the PB phase. Both the simulation and measurement results are well consistent with theoretical predictions. This work provides another strategy of achieving anomalous refraction and may find applications in beam steering, digital beam forming, etc.

An invariant related to the existence of conformally compact Einstein fillings

We define an invariant for compact spin manifolds X X of dimension 4 k 4k equipped with a metric h h of positive Yamabe invariant on its boundary. The vanishing of this invariant is a necessary condition for the conformal class of h h to be the conformal infinity of a conformally compact Einstein metric on X X .

New aspects of Argyres-Douglas theories and their dimensional reduction

Argyres-Douglas (AD) theories constitute an infinite class of superconformal field theories in four dimensions with a number of interesting properties. We study several new aspects of AD theories engineered in A-type class mathcal{S} with one irregular puncture of Type I or Type II and also a regular puncture. These include conformal manifolds, structures of the Higgs branch, as well as the three dimensional gauge theories coming from the reduction on a circle. We find that the latter admits a description in terms of a linear quiver with unitary and special unitary gauge groups, along with a number of twisted hypermultiplets. The origin of these twisted hypermultiplets is explained from the four dimensional perspective. We also propose the three dimensional mirror theories for such linear quivers. These provide explicit descriptions of the magnetic quivers of all the AD theories in question in terms of quiver diagrams with unitary gauge groups, together with a collection of free hypermultiplets. A number of quiver gauge theories presented in this paper are new and have not been studied elsewhere in the literature.

An Efficient Conformal Stacked Antenna Array Design and 3D-Beamforming for UAV and Space Vehicle Communications.

In this paper, a new conformal array structure and beamforming technique are proposed to provide efficient communication performance for unmanned aerial vehicles (UAVs) and space vehicles. The proposed array is formed by conformally stacking cylindrical, conical, and concentric circular (CSC4) arrays which are all coaxially aligned with the same axis of the conformed body and with uniform interelement spacing. The array elements are then fed by a weighting vector that has an adaptive cosine tapered profile where the maximum amplitude coefficient is oriented with the mainlobe direction to improve the scanning capabilities of the array and increase the array effective area. In addition, for very large, conformed body structures such as space vehicles, a frontal mainlobe-oriented partial CSC4 array beamforming technique is proposed to efficiently utilize the large CSC4 structure, reduce the processing requirements for mainlobe electronic steering, and to provide very low sidelobe levels with reduced backlobe levels. Simulation results show that the proposed CSC4 design can provide wide scanning angles of up to angular range in the -direction with only change in the beamwidth, without increasing array size and with achievable sidelobe level of −45 dB and backlobe levels less than −10 dB.

ECM Enabled Whale optimization assisted facile design of dual-band conformal FSS for WLAN shielding applications

A new modified Equivalent Circuit Model (M-ECM) enabled Whale Optimization (WO) assisted electromagnetic (EM) approach is presented for an efficient design of a double square loop frequency selective surface (DSL-FSS) based conformal shielding structure. A simple single-layer DSL-FSS is designed, fabricated, and measured to suppress both wireless local area network (WLAN) bands without affecting the rest of the considered spectrum using ECM enabled WO. The analytical and measured results have been found in quite good congruence with each other. The measured transmission coefficient at 2.4 and 5.0 GHz shows suppression of 35 and 38 dB, respectively. Further, the angular stability, polarization insensitivity, and physical interpretation of the DSL-FSS shielding structure are investigated. Finally, the conformal behavior of the structure has been investigated for different bending radius and good agreement between simulated and measured results shows the effectiveness of the aforementioned technique for planar and conformal WLAN shielding applications.

A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

Motivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in {mathbb {R}}^{n} and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than {mathbb {Z}}^{*}={mathbb {Z}}{setminus }{0}) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.

On the Trace Anomaly of the Chaudhuri–Choi–Rabinovici Model

Recently a non-supersymmetric conformal field theory with an exactly marginal deformation in the large N limit was constructed by Chaudhuri–Choi–Rabinovici. On a non-supersymmetric conformal manifold, the c coefficient of the trace anomaly in four dimensions would generically change. In this model, we, however, find that it does not change in the first non-trivial order given by three-loop diagrams.

An Obata-type theorem on compact Einstein manifolds with boundary

We show a kind of Obata-type theorem on a compact Einstein n-manifold (W, bar{g}) with smooth boundary partial W. Assume that the boundary partial W is minimal in (W, bar{g}). If (partial W, bar{g}|_{partial W}) is not conformally diffeomorphic to (S^{n-1}, g_S), then for any Einstein metric check{g} in [bar{g}] with the minimal boundary condition, we have that, up to rescaling, check{g} = bar{g}. Here, g_S and [bar{g}] denote respectively the standard round metric on the (n-1)-sphere S^{n-1} and the conformal class of bar{g}. Moreover, if we assume that partial W subset (W, bar{g}) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of (W, partial W, [bar{g}]).

Discreteness and integrality in Conformal Field Theory

Various observables in compact CFTs are required to obey positivity, discreteness, and integrality. Positivity forms the crux of the conformal bootstrap, but understanding of the abstract implications of discreteness and integrality for the space of CFTs is lacking. We systematically study these constraints in two-dimensional, non-holomorphic CFTs, making use of two main mathematical results. First, we prove a theorem constraining the behavior near the cusp of integral, vector-valued modular functions. Second, we explicitly construct non-factorizable, non-holomorphic cuspidal functions satisfying discreteness and integrality, and prove the non-existence of such functions once positivity is added. Application of these results yields several bootstrap-type bounds on OPE data of both rational and irrational CFTs, including some powerful bounds for theories with conformal manifolds, as well as insights into questions of spectral determinacy. We prove that in rational CFT, the spectrum of operator twists tge frac{c}{12} is uniquely determined by its complement. Likewise, we argue that in generic CFTs, the spectrum of operator dimensions Delta >frac{c-1}{12} is uniquely determined by its complement, absent fine-tuning in a sense we articulate. Finally, we discuss implications for black hole physics and the (non-)uniqueness of a possible ensemble interpretation of AdS3 gravity.

Two-point function of the energy-momentum tensor and generalised conformal structure

Theories with generalised conformal structure contain a dimensionful parameter, which appears as an overall multiplicative factor in the action. Examples of such theories are gauge theories coupled to massless scalars and fermions with Yukawa interactions and quartic couplings for the scalars in spacetime dimensions other than 4. Many properties of such theories are similar to that of conformal field theories (CFT), and in particular their 2-point functions take the same form as in CFT but with the normalisation constant now replaced by a function of the effective dimensionless coupling g constructed from the dimensionful parameter and the distance separating the two operators. Such theories appear in holographic dualities involving non-conformal branes and this behaviour of the correlators has already been observed at strong coupling. Here we present a perturbative computation of the two-point function of the energy-momentum tensor to two loops in dimensions d= 3, 5, confirming the expected structure and determining the corresponding functions of g to this order, including the effects of renormalisation. We also discuss the hbox {d}=4 case for comparison. The results for d=3 are relevant for holographic cosmology, and in this case we also study the effect of a Phi ^6 coupling, which while marginal in the usual sense it is irrelevant from the perspective of the generalised conformal structure. Indeed, the effect of such coupling in the 2-point function is washed out in the IR but it modifies the UV.

TCSA and the finite volume boundary state

We develop a new way to calculate the overlap of a boundary state with a finite volume bulk state in the truncated conformal space approach. We check this method in the thermally perturbed Ising model analytically, while in the scaling Lee-Yang model numerically by comparing our results to excited state g-functions, which we obtained by the analytical continuation method. We also give a simple argument for the structure of the asymptotic overlap between the finite volume boundary state and a periodic multiparticle state, which includes the ratio of Gaudin type determinants.

Wide-Angle Broadband Rasorber for Switchable and Conformal Application

A novel wide-angle and broadband rasorber is presented incorporating several advances. To broaden the bandwidth, we introduce a new type of frequency selective surface (FSS) in the bottom layer of the rasorber. This strategy increases the upper and lower absorption bandwidths while reducing the insertion loss at transmission window. Furthermore, a new top resistive layer is proposed via the loading of electric field coupled resonator elements on a cross-dipole structure. The proposed design offers both, an angularly stable performance and a thin structure that can be fabricated on a single dielectric substrate. To enable switchability, p-i-n diodes are employed and a reconfigurable rasorber/absorber is designed where a new concept of “lossy/lossless” top layer is investigated and established as the best choice among other switchable designs. Equivalent circuit models are constructed for both active and passive designs to provide physical insight into their operation. Finally, to enable deployment of the rasorber over curved geometries, these advances are incorporated into a conformal structure. Three separate prototypes are fabricated and good agreement is obtained between design predictions and experimental results.

Harmonic cubic homogeneous polynomials such that the norm-squared of the Hessian is a multiple of the Euclidean quadratic form

There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean metric. Solutions are constructed in all dimensions and solutions are classified in dimension at most $4$. Techniques are given for determining when two solutions are linearly conformally inequivalent.

The contact structure on the space of null geodesics of causally simple spacetimes

It is shown that the space of null geodesics of a star-shaped causally simple subset of Minkowski space is contactomorphic to the canonical contact structure in the spherical cotangent bundle of Rn. In the 3-dimensional case we prove a similar result for a large class of causally simple contractible subsets of an arbitrary globally hyperbolic spacetime applying methods from the theory of contact-convex surfaces. Moreover we prove that under certain assumptions the space of null geodesics of a causally simple spacetime embeds with smooth boundary into the space of null geodesics of a globally hyperbolic spacetime. The characteristic foliation of this boundary provides an invariant of the conformal class of the causally simple spacetime.

A general method to construct invariant PDEs on homogeneous manifolds

Let [Formula: see text] be an [Formula: see text]-dimensional homogeneous manifold and [Formula: see text] be the manifold of [Formula: see text]-jets of hypersurfaces of [Formula: see text]. The Lie group [Formula: see text] acts naturally on each [Formula: see text]. A [Formula: see text]-invariant partial differential equation of order [Formula: see text] for hypersurfaces of [Formula: see text] (i.e., with [Formula: see text] independent variables and [Formula: see text] dependent one) is defined as a [Formula: see text]-invariant hypersurface [Formula: see text]. We describe a general method for constructing such invariant partial differential equations for [Formula: see text]. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup [Formula: see text] of the [Formula: see text]-prolonged action of [Formula: see text]. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space [Formula: see text] and in the conformal space [Formula: see text]. Our method works under some mild assumptions on the action of [Formula: see text], namely: A1) the group [Formula: see text] must have an open orbit in [Formula: see text], and A2) the stabilizer [Formula: see text] of the fiber [Formula: see text] must factorize via the group of translations of the fiber itself.

Transition of large R-charge operators on a conformal manifold

We study the transition between phases at large R-charge on a conformal manifold. These phases are characterized by the behaviour of the lowest operator dimension ∆(QR) for fixed and large R-charge QR. We focus, as an example, on the D = 3, mathcal{N} = 2 Wess-Zumino model with cubic superpotential W= XYZ+frac{tau }{6}left({X}^3+{Y}^3+{Z}^3right) , and compute ∆(QR, τ) using the ϵ-expansion in three interesting limits. In two of these limits the (leading order) result turns out to beΔQR,τ=BPSbound1+Oϵτ2QR,QR≪1ϵ1ϵτ298ϵτ22+τ21D−1QRDD−11+Oϵτ2QR−2D−1,QR≫1ϵ,1ϵτ2\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Delta \\left({Q}_{R,\\tau}\\right)=\\left\\{\\begin{array}{ll}\\left(\\mathrm{BPS}\\;\\mathrm{bound}\\right)\\left[1+O\\left(\\epsilon {\\left|\\tau \\right|}^2{Q}_R\\right)\\right],& {Q}_R\\ll \\left\\{\\frac{1}{\\epsilon },\\kern0.5em \\frac{1}{\\epsilon {\\left|\\tau \\right|}^2}\\right\\}\\\\ {}\\frac{9}{8}{\\left(\\frac{\\epsilon {\\left|\\tau \\right|}^2}{2+{\\left|\\tau \\right|}^2}\\right)}^{\\frac{1}{D-1}}{Q}_R^{\\frac{D}{D-1}}\\left[1+O\\left({\\left(\\epsilon {\\left|\\tau \\right|}^2{Q}_R\\right)}^{-\\frac{2}{D-1}}\\right)\\right],& {Q}_R\\gg \\left\\{\\begin{array}{ll}\\frac{1}{\\epsilon },& \\frac{1}{\\epsilon {\\left|\\tau \\right|}^2}\\end{array}\\right\\}\\end{array}\\right. $$\\end{document}which leads us to the double-scaling parameter, ϵ|τ|2QR, which interpolates between the “near-BPS phase” (∆(Q) ∼ Q) and the “superfluid phase” (∆(Q) ∼ QD/(D−1)) at large R-charge. This smooth transition, happening near τ = 0, is a large-R-charge manifestation of the existence of a moduli space and an infinite chiral ring at τ = 0. We also argue that this behavior can be extended to three dimensions with minimal modifications, and so we conclude that ∆(QR, τ) experiences a smooth transition around QR ∼ 1/|τ|2. Additionally, we find a first-order phase transition for ∆(QR, τ) as a function of τ, as a consequence of the duality of the model. We also comment on the applicability of our result down to small R-charge.

(Mis-)matching type-B anomalies on the Higgs branch

Building on [1], we uncover new properties of type-B conformal anomalies for Coulomb-branch operators in continuous families of 4D mathcal{N} = 2 SCFTs. We study a large class of such anomalies on the Higgs branch, where conformal symmetry is spontaneously broken, and compare them with their counterpart in the CFT phase. In Lagrangian the- ories, the non-perturbative matching of the anomalies can be determined with a weak coupling Feynman diagram computation involving massive multi-loop banana integrals. We extract the part corresponding to the anomalies of interest. Our calculations support the general conjecture that the Coulomb-branch type-B conformal anomalies always match on the Higgs branch when the IR Coulomb-branch chiral ring is empty. In the opposite case, there are anomalies that do not match. An intriguing implication of the mismatch is the existence of a second covariantly constant metric on the conformal manifold (other than the Zamolodchikov metric), which imposes previously unknown restrictions on its holonomy group.

Initial data in general relativity described by expansion, conformal deformation and drift

The conformal method is a technique for finding Cauchy data in general relativity solving the Einstein constraint equations, and its parameters include a conformal class, a conformal momentum (as measured by a densitized lapse), and a mean curvature. Although the conformal method is successful in generating constant mean curvature (CMC) solutions of the constraint equations, it is unknown how well it applies in the non-CMC setting, and there have been indications that it encounters difficulties there. We are therefore motivated to investigate alternative generalizations of the CMC conformal method. Introducing a densitized lapse into the ADM Lagrangian, we find that solutions of the momentum constraint can be described in terms of three parameters. The first is conformal momentum as it appears in the standard conformal method. The second is volumetric momentum, which appears as an explicit parameter in the CMC conformal method, but not in the non-CMC formulation. We have called the third parameter drift momentum, and it is the conjugate momentum to infinitesimal motions in superspace that preserve conformal class and volume form up to independent diffeomorphisms. This decomposition of solutions of the momentum constraint leads to extensions of the CMC conformal method where conformal and volumetric momenta both appear as parameters. There is more than one way to treat drift momentum, in part because of an interesting duality that emerges, and we identify three candidates for incorporating drift into a variation of the conformal method.