Conformal Anomaly Research Articles

On adiabatic renormalization with a physically motivated infrared cut-off

We introduce a new approach to renormalize physical quantities in curved space-time by adiabatic subtraction. We use a comoving infrared cut-off in defining the adiabatic counterpart of the physical quantity under consideration, building on the fact that the adiabatic approximation is ill-defined in the infrared tail of the spectrum.We show how this infrared cut-off should be used to obtain a completely well-defined renormalization scheme and how it is fundamental to avoid unphysical divergences that can be generated by a pathological behavior of the adiabatic subtraction extended to the infrared tail. The infrared cut-off appears as a new degree of freedom introduced in the theory and its actual value has to be consistently fixed by a physical prescription. As an example, we show how such degree of freedom can be set to obtain the correct value of the conformal anomaly in the symptomatic case of an inflationary model with gauge fields coupled to a pseudo-scalar inflaton.

AdS5 holography and higher-derivative supergravity

We study four-derivative corrections to 5d mathcal{N} = 2 minimal gauged supergravity using tools from conformal supergravity. There are two supersymmetric invariants at the four-derivative order and we show explicitly how to write down the action in the Poincaré frame in terms of their coefficients. We apply our results in the context of holography where the four-derivative terms in the supergravity action translate into corrections to the conformal anomalies of the dual 4d mathcal{N} = 1 SCFT. We use these results to study corrections to the entropy of the supersymmetric black string solutions of minimal supergravity. We also discuss explicit embeddings of the 5d supergravity in string and M-theory and use them to fix the higher-derivative coefficients in terms of the microscopic parameters.

A conformal anomaly detection based industrial fleet monitoring framework: A case study in district heating

The monitoring infrastructure of an industrial fleet can rely on the so-called unit-level and subfleet-level models to observe the behavior of a target unit. However, such infrastructure has to confront several challenges. First, from an anomaly detection perspective of monitoring a target unit, unit-level and subfleet-level models can give different information about the nature of an anomaly, and which approach or level model is appropriate is not always clear. Second, in the absence of well-understood prior models of unit and subfleet behavior, the choice of a base model at their respective levels, especially in an online/streaming setting, may not be clear. Third, managing false alarms is a major problem. To deal with these challenges, we proposed to rely on the conformal anomaly detection framework. In addition, an ensemble approach was deployed to mitigate the knowledge gap in understanding the underlying data-generating process at the unit and subfleet levels. Therefore, to monitor the behavior of a target unit, a unit-level ensemble model (ULEM) and a subfleet-level ensemble model (SLEM) were constructed, where each member of the respective ensemble is based on a conformal anomaly detector (CAD). However, since the information obtained by these two ensemble models through their p-values may not always agree, a combined ensemble model (CEM) was proposed. The results are based on real-world operational data obtained from district heating (DH) substations. Here, it was observed that CEM reduces the overall false alarms compared to ULEM or SLEM, albeit at the cost of some detection delay. The analysis demonstrated the advantages and limitations of ULEM, SLEM, and CEM. Furthermore, discords obtained from the state-of-the-art matrix-profile (MP) method and the combined calibration scores obtained from ULEM and SLEM were compared in an offline setting. Here, it was observed that SLEM achieved a better overall precision and detection delay. Finally, the different components related to ULEM, SLEM, and CEM were put together into what we refer to as TRANTOR: a conformal anomaly detection based industrial fleet monitoring framework. The proposed framework is expected to enable fleet operators in various domains to improve their monitoring infrastructure by efficiently detecting anomalous behavior and controlling false alarms at the target units.

LogCAD: An Efficient and Robust Model for Log-Based Conformal Anomaly Detection

Log files are usually semistructured files that record the historical operation information of systems or devices. Researchers often find anomalies by analyzing logs, so as to identify system operation faults and cyberattacks. Traditional classification-based methods, especially deep learning methods, can effectively solve the problem of static log anomaly detection. However, when addressing dynamic unstable logs caused by concept drift and noise, the performance of those methods decreased significantly, and false positives are prone to occur. Retraining model is a choice to solve the log instability problem, but this will greatly increase the computational complexity for deep learning models. The log-based conformal anomaly detection (LogCAD) builds a confidence evaluation mechanism for multiple labels, which can achieve good detection results by making collaborative decisions based on multiple weak classifiers without deep learning. Moreover, LogCAD can be easily extended to dynamic unstable logs. It incrementally updates the trained model with conformal detection results of new samples. Experimental results show that LogCAD can achieve excellent detection results for both dynamic unstable logs and static stable logs. Compared with LogRobust and other deep learning models, it has higher efficiency and wider application scope.

Precision holography for 5D Super Yang-Mills

We study 1/2-BPS Wilson loop (WL) operators in maximally supersymmetric Yang-Mills theory (MSYM) on S5. Their vacuum expectation value (vev) can be computed exactly at large N thanks to supersymmetric localization. The holographic dual to MSYM on S5 is geometrically realized by a stack of N D4-branes with spherical worldvolume in ten dimensions. We compute the vev of the circular WL using holography by evaluating the partition function of a fundamental string in this background. Our focus is on the next-to-leading order correction to the string partition function which is composed of two parts; the dilaton coupling to the worldsheet and the one-loop fluctuations of the worldsheet itself. We face a variety of issues, some related to the presence of a non-constant dilaton, and others that are common to its AdS analogue. However, the universality of UV divergences as well as the importance of a proper choice of an IR regulator have been recently stressed in the literature. Inspired by this, we resolve our issues by first carefully treating the Weyl anomaly which receives contributions from the non-constant dilaton, and then by computing the ratio of our partition function and the one of a string in AdS4×CP3, which is dual to a 1/2-BPS WL in ABJM. Crucially, this approach yields a finite result which matches the corresponding ratio of WL vevs on the gauge theory side.

Logarithmic correction to black hole entropy from the nonlocality of quantum gravity

It has been known for many years that the leading correction to the black hole entropy is a logarithmic term, which is universal and closely related to conformal anomaly. A fully consistent analysis of this issue has to take quantum backreactions to the black hole geometry into account. However, it was always unclear how to naturally derive the modified black hole metric especially from an effective action, because the problem refers to the elusive non-locality of quantum gravity. In this paper, we show that this problem can be resolved within an effective field theory (EFT) framework of quantum gravity. Our work suggests that the EFT approach provides a powerful and self-consistent tool for studying the quantum gravitational corrections to black hole geometries and thermodynamics.

Weyl anomalies of four dimensional conformal boundaries and defects

Motivated by questions about quantum information and classification of quantum field theories, we consider Conformal Field Theories (CFTs) in spacetime dimension d ≥ 5 with a conformally-invariant spatial boundary (BCFTs) or 4-dimensional conformal defect (DCFTs). We determine the boundary or defect contribution to the Weyl anomaly using the standard algorithm, which includes imposing Wess-Zumino consistency and fixing finite counterterms. These boundary/defect contributions are built from the intrinsic and extrinsic curvatures, as well as the pullback of the ambient CFT’s Weyl tensor. For a co-dimension one boundary or defect (i.e. d = 5), we reproduce the 9 parity-even terms found by Astaneh and Solodukhin, and we discover 3 parity-odd terms. For larger co-dimension, we find 23 parity-even terms and 6 parity-odd terms. The coefficient of each term defines a “central charge” that characterizes the BCFT or DCFT. We show how several of the parity-even central charges enter physical observables, namely the displacement operator two-point function, the stress-tensor one-point function, and the universal part of the entanglement entropy. We compute several parity-even central charges in tractable examples: monodromy and conical defects of free, massless scalars and Dirac fermions in d = 6; probe branes in Anti-de Sitter (AdS) space dual to defects in CFTs with d ≥ 6; and Takayanagi’s AdS/BCFT with d = 5. We demonstrate that several of our examples obey the boundary/defect a-theorem, as expected.

Defect a-theorem and a-maximization

Conformal defects describe the universal behaviors of a conformal field theory (CFT) in the presence of a boundary or more general impurities. The coupled critical system is characterized by new conformal anomalies which are analogous to, and generalize those of standalone CFTs. Here we study the conformal a- and c-anomalies of four dimensional defects in CFTs of general spacetime dimensions greater than four. We prove that under unitary defect renormalization group (RG) flows, the defect a-anomaly must decrease, thus establishing the defect a-theorem. For conformal defects preserving minimal supersymmetry, the full defect symmetry contains a distinguished U(1)R subgroup. We derive the anomaly multiplet relations that express the defect a- and c-anomalies in terms of the defect (mixed) ’t Hooft anomalies for this U(1)R symmetry. Once the U(1)R symmetry is identified using the defect a-maximization principle which we prove, this enables a non-perturbative pathway to the conformal anomalies of strongly coupled defects. We illustrate our methods by discussing a number of examples including boundaries in five dimensions and codimension-two defects in six dimensions. We also comment on chiral algebra sectors of defect operator algebras and potential conformal collider bounds on defect anomalies.

Gravitational decoupling, hairy black holes and conformal anomalies

Hairy black holes in the gravitational decoupling setup are studied from the perspective of conformal anomalies. Fluctuations of decoupled sources can be computed by measuring the way the trace anomaly-to-holographic Weyl anomaly ratio differs from the unit. Therefore the gravitational decoupling parameter governing three hairy black hole metrics is then bounded to a range wherein one can reliably emulate AdS/CFT with gravitational decoupled solutions, in the tensor vacuum regime.

Weyl anomalies on conformal manifolds and moduli spaces

A Weyl (conformal) anomaly signals a subtle quantum breaking of classical conformal invariance in conformal field theory. Over the years, Weyl anomalies have been used to characterize nonperturbative properties of conformal field theory. Anomalies associated with the energy–momentum tensor, like the coefficients [Formula: see text] and [Formula: see text] in four space–time dimensions, are generic and have been studied extensively. More generally, in even dimensions, there are also conformal anomalies associated with any primary operator that has integer scaling dimension. Some of the most interesting features of Weyl anomalies have to do with their behavior under continuous deformations or in vacua with spontaneously broken conformal symmetry. In this review, we summarize the defining properties of conformal anomalies, their classification into A- and B-type, and their implications on the structure of correlation functions. We point out that type-B anomalies can exhibit complicated dynamics and review recent progress in the study of this dynamics with special focus on four-dimensional [Formula: see text] superconformal field theories. We emphasize two applications of type-B anomalies in this context: potential constraints on the holonomy of superconformal manifolds and the deconstruction of anomalies in higher dimensions from anomalies in broken phases of lower-dimensional conformal field theories.

Quantum Field Theory and its Anomalies for Topological Matter

Topology enters in quantum field theory (qft) in multiple forms: one of the most important, in non-abelian gauge theories, being in the identification of the θ vacuum in QCD. A very relevant aspect of this connection is through the phenomenon of chiral and conformal qft anomalies. It has been realized that a class of materials, comprising topological insulators and Weyl semimetals, also exhibit the phenomenon of anomalies, which are responsible for several exotic phenomena, such as the presence of edge currents, resilient under perturbations and scattering by impurities. Another example comes from the response functions of these materials under thermal and mechanical stresses, that may be performed using correlation functions of stress energy tensors in General Relativity. In this case the conformal anomaly plays an important role. We briefly illustrate some salient features of this correspondence, and the effective action describing the long-range interactions that may account for such topological effects.

Obstruction tensors in Weyl geometry and holographic Weyl anomaly

Recently a generalization of the Fefferman-Graham gauge for asymptotically locally AdS spacetimes, called the Weyl-Fefferman-Graham (WFG) gauge, has been proposed. It was shown that the WFG gauge induces a Weyl geometry on the conformal boundary. The Weyl geometry consists of a metric and a Weyl connection. Thus, this is a useful setting for studying dual field theories with background Weyl symmetry. Working in the WFG formalism, we find the generalization of obstruction tensors, which are Weyl-covariant tensors that appear as poles in the Fefferman-Graham expansion of the bulk metric for even boundary dimensions. We see that these Weyl-obstruction tensors can be used as building blocks for the Weyl anomaly of the dual field theory. We then compute the Weyl anomaly for $6d$ and $8d$ field theories in the Weyl-Fefferman-Graham formalism, and find that the contribution from the Weyl structure in the bulk appears as cohomologically trivial modifications. Expressed in terms of the Weyl-Schouten tensor and extended Weyl-obstruction tensors, the results of the holographic Weyl anomaly up to $8d$ also reveal hints on its expression in any dimension.

Detecting scale anomaly in chiral phase transition of QCD: new critical endpoint pinned down

Violation of scale symmetry, scale anomaly, being a radical concept in quantum field theory, is of importance to comprehend the vacuum structure of QCD, and should potentially contribute to the chiral phase transition in thermal QCD, as well as the chiral and U(1) axial symmetry. Though it should be essential, direct evidence of scale anomalies has never been observed in the chiral phase transition. We propose a methodology to detect a scale anomaly in the chiral phase transition, which is an electromagnetically induced scale anomaly: apply a weak magnetic field background onto two-flavor massless QCD with an extremely heavy strange quark, first observe the chiral crossover; second, adjusting the strange quark mass to be smaller and smaller, observe the second-order chiral phase transition, and then the first-order one in the massless-three flavor limit. Thus, the second-order chiral phase transition, observed as the evidence of the quantum scale anomaly, is a new critical endpoint. It turns out that this electromagnetic scale anomaly gets most operative in the weak magnetic field regime, rather than a strong field region. We also briefly address accessibility of lattice QCD, a prospected application to dense matter system, and implications to astrophysical observations, such as gravitational wave productions provided from thermomagnetic QCD-like theories.

Di-gluonium sum rules, I = 0 scalar mesons and conformal anomaly

We revisit, scrutinize, improve, confirm and complete our previous results [1–4] from the scalar di-gluonium sum rules within the standard SVZ-expansion at N2LO without instantons and beyond the minimal duality ansatz: “one resonance ⊕ QCD continuum” parametrization of the spectral function which is necessary for a better understanding of the complex spectra of the I=0 scalar mesons. We select different (un)subtracted sum rules (USR) moments of degree ≤ 4 for extracting the two lowest gluonia masses and couplings. We obtain: [MσB,fσB]=[1.07(13),0.46(16)],[MG1,fG1]=[1.55(12),0.37(11)] GeV and the corresponding masses of the radial excitations: MσB′=1.11(12) and MG1′=1.56(14) GeV which are (unexpectedly) almost degenerated with the ground states. The 2nd radial excitation is found to have a much heavier mass: MG2≃ 2.99(22) GeV. Combining these results with some Low-Energy Vertex Sum Rules (LEV-SR), we predict some hadronic widths and classify these states into two groups: – The σ-like (σB,σB′) which decay copiously to ππ from OZI-violating process and the σB′ to 2(ππ)S through σσ. – The G-like (G1,G1′ and eventually G2) which decay into η′η,ηη through the U(1)A gluonic vertex. Besides some eventual mixings with quarkonia states, we may expect that the observed σ/f0(500) and f0(1.37) are σ-like while the f0(1.5) and f0(1.7) are G-like gluonia. The high mass G2(2.99) can also mix with the G1,G1′ to bring the gluon component of the gluonia candidates above 2 GeV. We also estimate the conformal charge ψG(0)=2.09(29) GeV4 and its slope 102×ψG′(0)=−22(29) GeV2. Our results are summarized in Table 1.

Cosmological constant, axial anomalies and photon mass dynamos from chiral fermionic torsion

Motivated by Palle’s investigation on the handness of chirality of vorticity in Einstein–Cartan cosmology [Entropy 5 (2014)], several aspects of chiral torsional handness in magnetogenesis and cosmology are presented. In the first one, we obtain torsion bounds from massive photons and axial anomalies. In the second, we deal with magnetogenesis from photon mass and in the third, we discuss chiral torsion degrees of freedom to obtain a torsion cosmological constant dependent solution. The torsion solution decays fast and implies a strong suppression of torsion at present universe. Our result contains the Poplawski [Phys. Lett. B (2010)] results in the case axial torsion vector associated to Einstein–Cartan fermionic sector matter and conformal anomalies of quarks. In the third example, a magnetic field bound from chiral torsionic dynamos is obtained as [Formula: see text]. In the non-minimal cosmological models, chiral dynamos are sourced by massive photons, London currents and chiral magnetic effect (CME). Chiral chemical potential is found to be mimic by torsion. Cosmological constant bound [Formula: see text] is found. At the early universe, the cosmological constant [Formula: see text] is obtained. Torsion used in the present universe is [Formula: see text]. In the last and fourth example, chiral anisotropic currents are obtained and magnetic helicity is shown to depend upon torsion when the chiral chemical potential is non-constant.

Beta functions for the duality-invariant sigma model

The O(d, d) invariant worldsheet theory for bosonic string theory with d abelian isometries is employed to compute the beta functions and Weyl anomaly at one-loop. We show that vanishing of the Weyl anomaly coefficients implies the equations of motion of the Maharana-Schwarz action. We give a self-contained introduction into the required techniques, including beta functions, the Weyl anomaly for two-dimensional sigma models and the background field method. This sets the stage for a sequel to this paper on generalizations to higher loops and α′ corrections.

Mass radius of the proton

The mass radius is a fundamental property of the proton that so far has not been determined from experiment. Here we show that the mass radius of the proton can be rigorously defined through the formfactor of the trace of the energy-momentum tensor (EMT) of QCD in the weak gravitational field approximation, as appropriate for this problem. We then demonstrate that the scale anomaly of QCD enables the extraction of the formfactor of the trace of the EMT from the data on threshold photoproduction of $J/\psi$ and $\Upsilon$ quarkonia, and use the recent GlueX Collaboration data to extract the r.m.s. mass radius of the proton ${\rm R_m = 0.55 \pm 0.03 \ fm}$. The extracted mass radius is significantly smaller than the charge radius of the proton ${\rm R_C = 0.8409 \pm 0.0004 \ fm}$. We attribute this difference to the interplay of asymptotic freedom and spontaneous breaking of chiral symmetry in QCD, and outline future measurements needed to determine the mass radius more precisely.

Insensitivity of the complexity rate of change to the conformal anomaly and Lloyd’s bound as a possible renormalization condition

We determine the effect on the computational complexity of a conformal anomaly using the Complexity=Action prescription of the gauge/gravity correspondence. To allow the involvement of said anomaly, we extend previous studies to include arbitrary values for the anisotropic parameter and the magnetic field respectively on the Mateos-Trancanelli and the D'Hoker-Kraus holographic models. Our main result is that the rate of change of the computational complexity is independent of the conformal anomaly in both cases. In addition, this allows us to also show that, if so desired, the saturation of Lloyd's bound at infinite time can be used as a renormalization condition.

Spontaneously broken boosts in CFTs

Conformal Field Theories (CFTs) have rich dynamics in heavy states. We describe the constraints due to spontaneously broken boost and dilatation symmetries in such states. The spontaneously broken boost symmetries require the existence of new low-lying primaries whose scaling dimension gap, we argue, scales as O(1). We demonstrate these ideas in various states, including fluid, superfluid, mean field theory, and Fermi surface states. We end with some remarks about the large charge limit in 2d and discuss a theory of a single compact boson with an arbitrary conformal anomaly.

De Sitter entropy as holographic entanglement entropy

We review the results of refs. [1,2], in which the entanglement entropy in spaces with horizons, such as Rindler or de Sitter space, is computed using holography. This is achieved through an appropriate slicing of anti-de Sitter space and the implementation of a UV cutoff. When the entangling surface coincides with the horizon of the boundary metric, the entanglement entropy can be identified with the standard gravitational entropy of the space. For this to hold, the effective Newton's constant must be defined appropriately by absorbing the UV cutoff. Conversely, the UV cutoff can be expressed in terms of the effective Planck mass and the number of degrees of freedom of the dual theory. For de Sitter space, the entropy is equal to the Wald entropy for an effective action that includes the higher-curvature terms associated with the conformal anomaly. The entanglement entropy takes the expected form of the de Sitter entropy, including logarithmic corrections.