Pontryagin Density Research Articles

Trace anomaly of weyl fermions via the path integral

We compute the trace, diffeomorphism and Lorentz anomalies of a free Weyl fermion in a gravitational background field by path integral methods. This is achieved by regularising the variation of the determinant of the Weyl operator building on earlier work by Leutwyler. The trace anomaly is found to be one half of the one of a Dirac fermion. Most importantly we establish that the potential parity-odd curvature term RR~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ R\\overset{\\sim }{R} $$\\end{document}, corresponding to the Pontryagin density, vanishes. This is to the contrary of some recent findings in the literature which gave rise to a controversy. We verify, that the regularisation does not lead to (spurious) anomalies in the Lorentz and diffeomorphism symmetries. We argue that in d = 2 (mod 4) P- and CP-odd terms cannot appear and that for d = 4 (mod 4) they are absent at least at leading order.

Chiral current induced by torsional Weyl anomaly

Torsion can be realized as dislocation in the crystal lattice of material. It is particularly interesting if the material has fermions in the spectrum, such as graphene, topological insulators, and Dirac and Weyl semimetals, as it's transport properties can be affected by the torsion. In this paper, we find that, due to Weyl anomaly, torsion in Dirac and Weyl semimetals can induce novel chiral currents, either near a boundary or in a ``conformally flat space.'' We briefly discuss how to measure this interesting effect in experiment. It is remarkable that these experiments can help to clarify the theoretical controversy of whether an imaginary Pontryagin density could appear in the Weyl anomaly.

Slow-rotating charged black hole solution in dynamical Chern-Simons modified gravity

Chern-Simons (CS) gravity is a modified theory of Einstein's general relativity. The CS theory arises from the low energy limit of string theory which involves anomaly correction to the Einstein-Hilbert action. The CS term is given by the product of the Pontryagin density with a scalar field. In this study, we derive a charged slowly rotating black hole (BH) solution. The main incentives of this BH solution are axisymmetric and stationary and form distortion of the Kerr-Newman BH solution with a dipole scalar field. Additionally, we investigate the asymptotic correction of the metric with the inverse seventh power of the radial distance to the BH solution, This indicates that it will escape any meaningful constraints from weak field experiments. To find this kind of BH by observations, we investigate the propagation of the photon near the BH and we show that the difference between the left-rotated polarization and the right-handed one could be observed as stronger than the case of the Kerr-Newman BH. Finally, we derived the stability condition using the geodesic deviations.

Gravitational leptogenesis from metric perturbations

In this work, we make the observation that the gravitational leptogenesis mechanism can be implemented without invoking new axial couplings in the inflaton sector. We show that, in the perturbed Robertson-Walker background emerging after inflation, the spacetime metric itself breaks parity symmetry and generates a nonvanishing Pontryagin density which can produce a matter-antimatter asymmetry. We analyze the produced asymmetry in different inflationary and reheating scenarios. We show that the generated asymmetry can be locally comparable to observations in certain cases, although the size of the matter-antimatter regions is typically much smaller than the present Hubble radius.

Slow-rotating black holes with potential in dynamical Chern-Simons modified gravitational theory

The Chern-Simons amended gravity theory appears as a low-energy effective theory of string theory.The effective theory includes an anomaly-cancelation correction to the Einstein-Hilbert action.The Chern-Simons expression consists of the product φRR̃ of the Pontryagin density RR̃ with a scalar field φ, where the latter is considered a background field(dynamical construction or non-dynamical construction).Many different solutions to Einstein's general relativity continue to be valid in the amended theories.The Kerr metric is, however, considered an exceptional case that raised a search for rotating black hole solutions.We generalize the solution presented in Phys. Rev. D 77 (2008) 064007 by allowing the potential V to have a non-vanishing value,and we discuss three different cases of the potential, that is, V = const., V ∝ φ, and V ∝ φ 2 cases.This study presents, for the first time, novel solutions prescribing rotating black holes in the frame of the dynamical formulation of the Chern-Simons gravity,where we include a potential and generalize the previously derived solutions.We derive solutions in the slow-rotation limit, where we write the parameter of the slow-rotation expansion by ε.These solutions are axisymmetric and stationary, and they make a distortion of the Kerr solution by a dipole scalar field.Furthermore, we investigate that the correction to the metric behaves in the inverse of the fourth order of radial distance from the centerof the black hole as V ∝ φ. This suggests that any meaningful limits from the weak-field experiments could be passed.We show that the energy conditions associated with the scalar field of the case V ∝ φ are non-trivial and have non-trivial values to the leading order.These non-trivial values come mainly from the contribution of the potential. Finally, we derived the stability condition using the geodesic deviations.We conclude this study by showing that other choices of the potential, i.e., V ∝ φn , where n > 2 are not allowedbecause all the solutions to these cases will be of order \U0001d4aa(ε 2), which is not covered in this study.

Gravity, axion Chern-Simons Einstein, and black holes

The low-energy limit of string theory involves an anomaly-removing correction of the action of Einstein-Hilbert, which defines a practical idea: Chern-Simons (CS) modified gravity. The CS correction composes of the product of a scalar field with the Pontryagin density, in which the scalar field is considered as a background field (non-dynamical formulation) or as an evolving field (dynamical formulation). Different solutions of Einstein’s general relativity (GR) continue in the amended theory; a keyable exception is the Kerr metric, which has affected a search for rotating black hole solutions. In this study, we derive a black solution characterizing a rotating black hole within the non-dynamical and dynamical frame of CS axion Einstein gravity (CSAEG). The case of non-dynamical has no new physics different from GR, which is consistent with what is presented in the literature. The main merit of this study is that in the dynamical framework of CSAEG, we show that the scalar field has a dynamical value up to the first-order approximation and under some constraints between the unknown functions that characterize this theory. Moreover, we show that the function U that characterizes CS can help to weaken the singularity for the invariant ^{*} ! R R.

A comment on metric vs metric-affine gravity

We consider the sum of the Einstein-Hilbert action and a Pontryagin density (PD) in arbitrary even dimension D≥4. All curvatures are functions of independent affine (torsionless) connections only. In arbitrary even dimension, not only in D=4n, these first order PD terms are shown to be covariant divergences of “Chern-Simons” currents. The field equation for the connection leads to it being Levi-Civita, and to the metric and affine field equations being equivalent to the second order metric theory. This result is a counterexample to the theorem stating that purely metric and metric-affine models can only be equivalent for Lovelock theories.

Noncommutative D=5 Chern–Simons gravity: Kaluza–Klein reduction and chiral gravitational anomaly

Actions for noncommutative (NC) gauge field theories can be expanded perturbatively in powers of the noncommutativity parameter theta using the Seiberg–Witten map between ordinary classical fields and their NC counterparts. The leading order term represents classical (theta =0) action while higher-order terms give us theta -dependent NC corrections that ought to capture some aspects of quantum gravity. Building on previous work of Aschieri and Castellani on NC Chern–Simons (CS) gauge and gravity theories, showing that non-trivial theta -dependence exists only for spacetime dimensions Dge 5, we investigate a correlated effect of these extra spatial dimensions and noncommutativity on four-dimensional physics, up to first-order in theta . Assuming that one spatial dimension is compactified into a circle, we apply the Kaluza–Klein reduction procedure on the NC D=5 CS theory for the conformal gauge group SO(4, 2), to obtain an effective, theta -dependent four-dimensional theory of gravity that has Einstein–Hilbert gravity with negative cosmological constant as its commutative limit. We derive field equations for this modified theory of gravity and study the effect of NC interactions on some classical geometries, such as the AdS-Schwarzschild black hole. We find that this NC background spacetime gives rise to chiral gravitational anomaly due to the nonvanishing theta -dependent Pontryagin density.

Torsional anomaly in Kerr black hole and quantization in Einstein–Cartan gravity

Recently, presence of gravitational and axial anomalies in Riemann–Cartan (RC) spacetime [Garcia de Andrade, Class. Quantum Gravity 38 (2021)] indicates that topological densities expressed in terms of torsion may be very useful in understanding the physics involved. Pontryagin and Euler density may be presented in terms of torsion in teleparallelism. In this paper, computation of these topological torsional densities in Kerr black holes is given. The geometric quantisation of torsion is also discussed in terms of the teleparallel metric. Actually, the recent work of Del Grosso and Poplawski [arXiv: 2107.06112] showed that the torsion quantization appears when torsion is the generator of momentum quantum mechanical commutator. Moreover, Poplawski et al. showed that use of quantum torsion in quantum electrodynamic (QED) may avoid the divergences in Feynmann integrals, process called, a torsion regularization. Here, we compute the momenta components in terms of teleparallel Kerr black hole torsion generator, like a Casimir tensor. The quantum generator, while not yet related by torsion, is seen to be naturally associated to axial torsion skew-symmetric in the teleparallel geometry. An important fact is that from torsion quantization, torsion chirality appears naturally in the Einstein–Cartan spin-torsion analogy, where now spin-angular momentum of the black hole is connected to torsion. Torsional chiral anomalies are shown to diverge at black hole singularity, and then they cannot be canceled at singularity.

Quasinormal modes of Schwarzschild black holes in projective invariant Chern-Simons modified gravity

We generalize the Chern-Simons modified gravity to the metric-affine case and impose projective invariance by supplementing the Pontryagin density with homothetic curvature terms which do not spoil topologicity. The latter is then broken by promoting the coupling of the Chern-Simons term to a (pseudo)-scalar field. The solutions for torsion and nonmetricity are derived perturbatively, showing that they can be iteratively obtained from the background fields. This allows us to describe the dynamics for the metric and the scalar field perturbations in a self-consistent way, and we apply the formalism to the study of quasinormal modes in a Schwarzschild black hole background. Unlike in the metric formulation of this theory, we show that the scalar field is endowed with dynamics even in the absence of its kinetic term in the action. Finally, using numerical methods we compute the quasinormal frequencies and characterize the late-time power law tails for scalar and metric perturbations, comparing the results with the outcomes of the purely metric approach.

N = 3 conformal supergravity in four dimensions

In this paper we derive the action for N = 3 conformal supergravity in four space-time dimensions. We construct a density formula for N = 3 conformal supergravity based on the superform action principle. Finally, we embed the N = 3 Weyl multiplet in the density formula to obtain the invariant action for N = 3 conformal supergravity. There are two inequivalent embeddings by changing a particular coefficient from real to imaginary. They lead to invariant actions, which will either be the supersymmetrization of the Weyl square term or the Pontryagin density in the eventuality of gauge fixing to Poincaré supergravity. As a consistency check of our formalism, we will show that the supersymmetrization of the Pontryagin density is a total derivative. We will demonstrate this for purely bosonic terms. We will also present the complete action for the supersymmetrization of Weyl square term. We also discuss consistent truncation of N = 4 Weyl multiplet to N = 3 Weyl multiplet and use it for a robust check of our results using the earlier known results in N = 4 conformal supergravity.

Effective electromagnetic actions for Lorentz violating theories exhibiting the axial anomaly

The CPT odd contribution to the effective electromagnetic action deriving from the vacuum polarization tensor in a large class of fermionic systems exhibiting Lorentz invariance violation (LIV) is calculated using thermal field theory methods, focusing upon corrections depending on the chemical potential. The systems considered exhibit the axial anomaly and their effective actions are described by axion electrodynamics whereby all the LIV parameters enter in the coupling Θ(x) to the unmodified Pontryagin density. A preliminary application to type-I tilted Weyl semimetals is briefly presented.

String gravity in D=4

We revisit the four-dimensional theory of gravity that arises from string theory with higher-derivative corrections. By compactifying and truncating the ten-dimensional effective action of heterotic string theory at first order in $\alpha'$, and carefully dealing with field redefinitions, we show that the four-dimensional theory takes the form of an axidilaton model where the scalars couple to the Gauss-Bonnet and Pontryagin densities. Thus, the actual string gravity is a generalization of the well-studied Einstein-dilaton-Gauss-Bonnet and dynamical Chern-Simons models. Using this action we compute the stringy corrections to the Kerr geometry and we obtain, for the first time, the corrections to the entropy of the Kerr black hole at order $\alpha'^2$. We check that the first law of black hole mechanics is satisfied and discuss several properties of the solution. Our results suggest that there exist black hole solutions with $J>M^2$ and therefore the extremal ratio $J/M^2$ must be modified positively.

First-order perturbations of Gödel-type metrics in non-dynamical Chern–Simons modified gravity

Gödel-type metrics that are homogeneous in both space and time remain, like the Schwarzschild metric, consistent within Chern–Simons modified gravity; this is true in both the non-dynamical and dynamical frameworks, each of which involves an additional pseudoscalar field coupled to the Pontryagin density. In this paper, we consider stationary first-order perturbations to these metrics in the non-dynamical framework. Under certain assumptions we find analytical solutions to the perturbed field equations. The solutions of the first-order field equations break the translational and cylindrical symmetries of the unperturbed metrics. The effective potential controlling planar geodesic orbits is also affected by the perturbation parameter, which changes the equilibrium radii for the orbits of both massive particles and massless photons.

Canonical structure of minimal varying Λ theories

Minimal varying Λ theories are defined by an action built from the Einstein–Cartan–Holst first order action for gravity with the cosmological constant Λ as an independent scalar field, and supplemented by the Euler and Pontryagin densities multiplied by 1/Λ. We identify the canonical structure of these theories which turn out to represent an example of irregular systems. We find five degrees of freedom on generic backgrounds and for generic values of parameters, whereas if the parameters satisfy a certain condition (which includes the most commonly considered Euler case) only three degrees of freedom remain. On de Sitter-like backgrounds the canonical structure changes, and due to an emergent conformal symmetry one degree of freedom drops from the spectrum. We also analyze the self-dual case with an holomorphic action depending only on the self-dual part of the connection. In this case we find two (complex) degrees of freedom, and further discuss the Kodama state, the restriction to de Sitter background and the effect of reality conditions.

Quantum Field Theory on Noncommutative Curved Space-times and Noncommutative Gravity

In the noncommutative gauge-theoretical formulation of Langman and Szabo, apparently it appears that the torsion generated there is a generalized one i.e. it may containvector, axial vector and tensor components. However, when we transcribe the noncommutative gauge theory in terms of the Maxwell gauge theory using the Seiberg-Wittencorrespondence, we have noted that upto the first order in the noncommutative parameter, this effectively can be taken to induce a change in chiral anomaly and hence theassociated torsion should be an axial vector one. The noncommutative gauge symmetriesgive a very natural and explicit realizations of the mixing of space-time and internal symmetries which is a characteristic feature of the conventional gauge theory of gravity. Thegauge fields of the dimensionally reduced noncommutative Yang-Mills theory map ontoaWeitzenbӧck space time and a teleparallel theory of gravity arises as the zero curvature reduction of a Poincare gauge theory which induces an Einstein-Cartan space-timecharacterized by connections with both nonvanishing torsion and curvature. However, theteleparallelism equivalent of general relativity involves all the components of torsion. Thechiral anomaly in the Einstein-Cartan space U₄ is characterized by the topological invariants like Pontryagin density as well as the Nieh-Yan density when the latter term involvesthe length scale governed by the measure of noncommutativity of space points. It is shownthat we have discussed the equivalence of this formalism with noncommutative U(1) Yang Mills theory.

Resonant gravitational waves in dynamical Chern–Simons–axion gravity

In this paper, we consider dynamical Chern–Simons gravity with the identification of the scalar field coupled though the Pontryagin density with the axion dark matter, and we discuss the effects of the parametric resonance on gravitational waves (GWs). When we consider GWs in a coherently oscillating axion cloud, we confirm that significant resonant amplification of GWs occurs in a narrow frequency band, and the amplification is restricted to the late epoch after the passage of the incident waves. We also identify the condition that an axion cloud spontaneously emits GWs. Once we take into account the randomness of the spatial phase distribution of the axion oscillations, we find that the amplification is suppressed compared with the coherent case, but significant amplification of GWs can still occur. We also examine whether or not the amplification of GWs is possible in the present Universe, taking into account the history of the Universe. We find that resonant amplification is difficult to be tested from GW observations in the standard scenario of the axion DM model, in which the axion is the dominant component of DM. However, there is some parameter window in which the resonant amplification of GWs might be observed, if the axion is subdominant component of DM, and the axion cloud formation is delayed until the Hubble rate becomes much smaller than the axion mass.

Atiyah–Patodi–Singer index theorem for domain walls

We consider the index of a Dirac operator on a compact even dimensional manifold with a domain wall. The latter is defined as a co-dimension one submanifold where the connection jumps. We formulate and prove an analog of the Atiyah–Patodi–Singer theorem that relates the index to the bulk integral of Pontryagin density and η-invariants of auxiliary Dirac operators on the domain wall. Thus the index is expressed through the global chiral anomaly in the volume and the parity anomaly on the wall.

CP -violating super-Weyl anomaly

In CP-violating conformal field theories in four dimensions, the Pontryagin density can appear in the Weyl anomaly. The Pontryagin density in the Weyl anomaly is consistent, but it has a peculiar feature that the parent three-point function of the energy-momentum tensor can violate CP only (semi-)locally. In this paper, we study the supersymmetric completion of the Pontryagin density in the Weyl anomaly, where the central charge $c$ effectively becomes a complex number. The supersymmetry suggests that it accompanies the graviphoton $\theta$ term associated with the R-symmetry gauging in the Weyl anomaly. It also accompanies new CP-violating terms in the R-current anomaly. While there are no conclusive perturbative examples of CP-violating super Weyl anomaly, we construct explicit supersymmetric dilaton effective action which generates these anomalies.

Trace anomaly for chiral fermions via Hadamard subtraction

We calculate the trace (conformal) anomaly for chiral fermions in a general curved background using Hadamard subtraction. While in intermediate steps of the calculation imaginary terms proportional to the Pontryagin density appear, imposing a vanishing divergence of the stress tensor these terms completely cancel, and we recover the well- known result equal to half the trace anomaly of a Dirac fermion. We elaborate in detail on the advantages of the Hadamard method for the general definition of composite operators in general curved spacetimes, and speculate on possible causes for the appearance of the Pontryagin density in other calculations.