Gravitational Chern-Simons Term Research Articles

Holographic description of gravitational anomalies

The holographic duality can be extended to include quantum theories with broken coordinate invariance leading to the appearance of the gravitational anomalies. On the gravity side one adds the gravitational Chern-Simons term to the bulk action which gauge invariance is only up to the boundary terms. We analyze in detail how the gravitational anomalies originate from the modified Einstein equations in the bulk. As a side observation we find that the gravitational Chern-Simons functional has interesting conformal properties. It is invariant under conformal transformations. Moreover, its metric variation produces conformal tensor which is a generalization of the Cotton tensor to dimension $d+1=4k-1, k\in Z$. We calculate the modification of the holographic stress-energy tensor that is due to the Chern-Simons term and use the bulk Einstein equations to find its divergence and thus reproduce the gravitational anomaly. Explicit calculation of the anomaly is carried out in dimensions $d=2$ and $d=6$. The result of the holographic calculation is compared with that of the descent method and agreement is found. The gravitational Chern-Simons term originates by Kaluza-Klein mechanism from a one-loop modification of M-theory action. This modification is discussed in the context of the gravitational anomaly in six-dimensional $(2,0)$ theory. The agreement with earlier conjectured anomaly is found.

A kink in a funny place

When the 3-dimensional gravitational Chern-Simons term is reduced to two dimensions a dilaton-like gravity theory emerges. Its solutions involve kinks, which therefore describe 3-dimensional, conformally flat spaces.

Holographic gravitational anomalies

In the AdS/CFT correspondence one encounters theories that are not invariant under diffeomorphisms. In the boundary theory this is a gravitational anomaly, and can arise in 4k+2 dimensions. In the bulk, there can be gravitational Chern-Simons terms which vary by a total derivative. We work out the holographic stress tensor for such theories, and demonstrate agreement between the bulk and boundary. Anomalies lead to novel effects, such as a nonzero angular momentum for global AdS3. In string theory such Chern-Simons terms are known with exact coefficients. The resulting anomalies, combined with symmetries, imply corrections to the Bekenstein-Hawking entropy of black holes that agree exactly with the microscopic counting.

Birefringent gravitational waves and the consistency check of inflation

In this work we show that the gravitational Chern-Simons term, aside from being a key ingredient in inflationary baryogenesis, modifies superhorizon gravitational waves produced during inflation. We compute the super-Hubble gravitational power spectrum in the slow-roll approximation and show that its overall amplitude is modified while its spectral index remains unchanged (at leading order in the slow-roll parameters). Then, we calculate the correction to the tensor to scalar ratio, $T/S$. We find a correction of $T/S$ which is dependent on $\mathcal{N}$ (more precisely quadratic in $\mathcal{N}$), the parameter characterizing the amplitude of the Chern-Simons terms. In a stringy embedding of the leptogenesis mechanism, $\mathcal{N}$ is the ratio between the Planck scale and the fundamental string scale. Thus, in principle, we provide a direct probe of leptogenesis due to stringy dynamics in the cosmic microwave background. However, we demonstrate that the corresponding correction of $T/S$ is in fact very small and not observable in the regime where our calculations are valid. To obtain a sizable effect, we argue that a nonlinear calculation is necessary.

Leptogenesis and tensor polarisation from a gravitational Chern-Simons term

Centro de Investigaciones, Universidad Antonio Narin˜o,Cll 58A # 37-94, Bogot´a D.C., ColombiaJanuary 2005AbstractWithin an effective field theory derived from string theory, the universal axionhas to be coupled to the the gravitational Chern-Simons (gCS) term. Duringany era when the axion field is varying, the vacuum fluctuation of the gravi-tational wave amplitude will then be circularly polarised, generating an expec-tation value for the gCS term. The polarisation may be observable throughthe Cosmic Microwave Background, and the vacuum expectation value of thegCS term may generate the baryon asymmetry of the Universe. We argue herethat such effects cannot be computed without further input from string theory,since the ‘vacuum’ in question is unlikely to be the field-theoretic one.

Chemistry of Chern-Simons Supergravity: reduction to a BPS kink, oxidation to M-theory and thermodynamical aspects

We construct a supersymmetric extension of the two dimensional Kaluza-Klein-reduced gravitational Chern-Simons term, and globally study its solutions, labelled by mass and U(1) charge c. The kink solution is BPS, and in an appropriate conformal frame all solutions asymptotically approach AdS. The thermodynamics of the Hawking effect yields interesting behavior for the specific heat and hints at a Hawking-Page-like transition at Tcritical ~ c3/2. We address implications for higher dimensions (``oxidation''), in particular D = 3,4 and 11, and comment briefly on AdS/CFT aspects of the kink.

New solutions in 3D gravity

We study gravitational theory in 1+2 spacetime dimensions which is determined by the Lagrangian constructed as a sum of the Einstein-Hilbert term plus the two (translational and rotational) gravitational Chern-Simons terms. When the corresponding coupling constants vanish, we are left with the purely Einstein theory of gravity. We obtain new exact solutions for the gravitational field equations with the nontrivial material sources. Special attention is paid to plane-fronted gravitational waves (in case of the Maxwell field source) and to the circularly symmetric as well as the anisotropic cosmological solutions which arise for the ideal fluid matter source.

Chern-Simons modification of general relativity

General relativity is extended by promoting the three-dimensional gravitational Chern-Simons term to four dimensions. This entails choosing an embedding coordinate ${v}_{\ensuremath{\mu}}$---an external quantity, which we fix to be a nonvanishing constant in its time component. The theory is identical to one in which the embedding coordinate is itself a dynamical variable, rather than a fixed, external quantity. Consequently diffeomorphism symmetry breaking is hidden in the modified theory: the Schwarzschild metric is a solution; gravitational waves possess two polarizations, each traveling at the velocity of light; a conserved energy-momentum (pseudo)tensor can be constructed. The modification is visible in the intensity of gravitational radiation: the two polarizations of a gravity wave carry intensities that are suppressed or enhanced by the extension.

Dimensionally reduced gravitational Chern–Simons term and its kink

When the gravitational Chern–Simons term is reduced from 3 to 2 dimensions, the lower dimensional theory supports a symmetry breaking solution and an associated kink. Kinks in general relativity bear a close relation to flat-space kinks, governed by identical potentials.

The classical solutions of the dimensionally reduced gravitational Chern–Simons theory

The Kaluza–Klein reduction of the 3d gravitational Chern–Simons term to a 2d theory is equivalent to a Poisson-sigma model with 4d target space and degenerate Poisson tensor of rank 2. Thus two constants of motion (Casimir functions) exist, namely charge and energy. The application of well-known methods developed in the framework of first order gravity allows to construct all classical solutions straightforwardly and to discuss their global structure. For a certain fine tuning of the values of the constants of motion the solutions of hep-th/0305117 are reproduced. Possible generalizations are pointed out.

Brane resolution and gravitational Chern–Simons terms

We show that gravitational Chern–Simons corrections, associated with the σ-model anomaly on the M5-brane worldvolume, can resolve the M2-brane solution with Ricci-flat, special holonomy transverse space. We explicitly find smooth solutions in the cases when the transverse space is a manifold of Spin(7) holonomy and SU(4) holonomy. We comment on the consequences of these results for the holographically related three-dimensional theories living on the worldvolume of a stack of such resolved M2 branes.

Propagator, tree-level unitarity and effective nonrelativistic potential for higher-derivative gravity theories in D dimensions

A prescription for computing the propagator for D-dimensional higher-derivative gravity theories, based on the Barnes–Rivers operators, is presented. A systematic study of the tree-level unitarity of these theories is developed and the agreement of their linearized versions with Newton’s law is investigated by computing the corresponding effective nonrelativistic potential. Three-dimensional quadratic gravity with a gravitational Chern–Simons term is also analyzed. A discussion on the issue of light bending within the framework of both D-dimensional quadratic gravity and three-dimensional quadratic gravity with a Chern–Simons term is provided as well.

Propagator, tree-level unitarity and effective nonrelativistic potential for three-dimensional higher-derivative gravity augmented by a gravitational Chern–Simons term

An algorithm for computing the propagator for three-dimensional quadratic gravity with a gravitational Chern–Simons term, based on an extension of the three-dimensional Barnes–Rivers operators, is proposed. A systematic study of the tree-level unitarity of this theory is developed and its agreement with Newton's law is investigated by computing the effective nonrelativistic potential.

M5-branes, special Lagrangian submanifolds and $\sigma$-models

We study M-theory fivebranes wrapped on Special Lagrangian submanifolds ($§_n$) in Calabi-Yau three- and fourfolds. When the M5 wraps a four-cycle, the resulting theory is a two-dimensional domain wall embedded in three-dimensional bulk with four supercharges. The theory on the wall is specified in terms of the geometry of the CY manifold and the cycle $§_4$. It is chiral and anomalous, however the presence of a three-dimensional gravitational Chern-Simons terms with a coefficient that jumps when crossing the wall allows to cancel the anomaly by inflow. Kahler manifolds of special type, where the potential depends only on the real part of the complex coordinate, are shown to emerge as the target spaces of two-dimensional sigma-models when the M5 is wrapped on $§_3 \times S^1$, thus providing a physical realization of some recent symplectic construction by Hitchin.

Induced spin from the ISO(2, 1) gauge theory with the gravitational Chern-Simons term

In the context of ISO(2, 1) gauge theory, we consider (2 + 1)-dimensional gravity with the gravitational Chern-Simons term (CST). This formulation allows the ‘exact’ solution for the system coupled to a massive point particle (which is not the case in the conventional Chern-Simons gravity). The solution exhibits locally trivial structure even with the CST, although still shows globally nontrivialness such as the conical space and the helical time structure. Since the solution is exact, we can say that the CST induces spin even for the noncritical case of σ + αm ≠ 0.

Universality and ultraviolet regularizations of Chern-Simons theory

The universality of radiative corrections to the gauge coupling constant k of Chern-Simons theory is studied in a very general regularization scheme. We show that the effective coupling constant k induced by radiative corrections depends crucially on the balance between the ultraviolet behavior of scalar and pseudoscalar terms in the regularized action. There are three different regimes. When the ultraviolet leading term is scalar, the coupling k is shifted to k + h V. However, if the leading term is pseudoscalar the shift is k + sh V with s = 0 or s = 2 depending on the sign of such a term. In the borderline case when the scalar and pseudoscalar terms have the same ultraviolet behavior the shift of k becomes arbitrary (even non-integer) and depends on the parameters of the regularization. We also show that the coefficient of the induced gravitational Chern-Simons term is different for the three regimes and has the same universality properties as the effective coupling constant k. The results open the possibility of a connection with non-rational two-dimensional conformal theories in the borderline regime.

Fractional statistics from gravitation.

We show that the solitons of the SO(3) nonlinear sigma model in 2+1 dimensions, when coupled to gravitation with the gravitational Chern-Simons interaction, become anyons in the absence of the Hopf term. In particular we calculate the fractional statistical factor of the gravitating anyons, and prove that the gravitational Chern-Simons term itself can be interpreted as the Hopf term of the topological current of [Pi][sub 2]([ital S][sup 2]).

Gravitational anyonization.

The statistics of scalar matter changes when it is coupled to a U(1) gauge field with Chern-Simons dynamics, because all particles carry a magnetic flux and therefore give rise to Aharonov-Bohm phases when they move around each other. We argue that also the ``dual'' version of the Aharonov-Bohm effect, the Aharonov-Casher effect, can give rise to Berry phases which transmute ordinary particles into anyons. The Aharonov-Casher effect consists of an extra topological phase in the wave function of a magnetic moment moving around an electric charge. Considering (2+1)-dimensional Dirac fermions at low energies, both effects are present, and the fermions are turned into (interacting) anyons even though there is no Chern-Simons term included in the action. We study in detail the gravitational analogue of this mechanism. The post-Newtonian approximation is applied to the gravitational interaction of (2+1)-dimensional particles with spin, and to stringlike matter distributions with internal angular momentum in 3+1 dimensions. The action for gravity is taken to be the pure Einstein-Hilbert term. In the adiabatic limit one finds scrA\ensuremath{\cdot}v-type interactions where scrA is a long-range vortex field. These interactions give rise to various kinds of Berry phases, in particular to the gravitational analogues of the Aharonov-Bohm and the Aharonov-Casher phases. The former occurs when a mass moves around a particle with spin, and the latter arises when a particle with spin moves in the Newtonian scalar potential of a second (spinless) particle. These Berry phases lead to a ``self-anyonization'' of particles with nonzero spin. The topological term in their effective action has the same structure as the one which obtains when spinless particles are considered, but with a gravitational Chern-Simons term included in the action for the gravitational field.

Linearized fermion-graviton system in a (2+1)-dimensional spacetime with Chern-Simons term

The fermion-graviton system is studied at the linearized level in a (2+1)-dimensional spacetime with the gravitational Chern-Simons term. In this approximation it is shown that this system presents anomalous rotational properties and spin, in analogy with the gauge-field-matter system.

INDUCED GRAVITATIONAL CHERN-SIMONS TERMS IN PARITY-VIOLATING VECTOR MODELS

We compute the gravitational Chern-Simons terms induced by several vector models in D = 3. We also comment on the possible applications to the Bose-Fermi correspondence.