Zumino Consistency Condition Research Articles

Weyl invariance, non-compact duality and conformal higher-derivative sigma models

We study a system of n Abelian vector fields coupled to frac{1}{2} n(n+1) complex scalars parametrising the Hermitian symmetric space {textsf{Sp}}(2n, {mathbb {R}})/ {textsf{U}}(n). This model is Weyl invariant and possesses the maximal non-compact duality group {textsf{Sp}}(2n, {mathbb {R}}). Although both symmetries are anomalous in the quantum theory, they should be respected by the logarithmic divergent term (the “induced action”) of the effective action obtained by integrating out the vector fields. We compute this induced action and demonstrate its Weyl and {textsf{Sp}}(2n, {mathbb {R}}) invariance. The resulting conformal higher-derivative sigma -model on {textsf{Sp}}(2n, {mathbb {R}})/ {textsf{U}}(n) is generalised to the cases where the fields take their values in (i) an arbitrary Kähler space; and (ii) an arbitrary Riemannian manifold. In both cases, the sigma -model Lagrangian generates a Weyl anomaly satisfying the Wess–Zumino consistency condition.

Crawling technicolor

We analyze the Callan-Symanzik equations when scale invariance at a nontrivial infrared (IR) fixed point $\alpha^{}_{\mathrm{IR}}$ is realized in the Nambu-Goldstone (NG) mode. As a result, Green's functions at $\alpha^{}_{\mathrm{IR}}$ do not scale in the same way as for the conventional Wigner-Weyl (WW) mode. This allows us to propose a new mechanism for dynamical electroweak symmetry breaking where the running coupling $\alpha$ "crawls" towards (but does not pass) $\alpha^{}_{\mathrm{IR}}$ in the exact IR limit. The NG mechanism at $\alpha^{}_{\mathrm{IR}}$ implies the existence of a massless dilaton $\sigma$, which becomes massive for IR expansions in $\epsilon \equiv \alpha^{}_{\mathrm{IR}} - \alpha$ and is identified with the Higgs boson. Unlike "dilatons" that are close to a WW-mode fixed point or associated with a Coleman-Weinberg potential, our NG-mode dilaton is genuine and hence naturally light. Its (mass)$^2$ is proportional to $\epsilon \beta'(4+\beta')F_\sigma^{-2} \langle\hat{G}^2\rangle_{\text{vac}}$, where $\beta'$ is the (positive) slope of the beta function at $\alpha^{}_{\mathrm{IR}}$, $F_\sigma$ is the dilaton decay constant and $\langle\hat{G}^2\rangle_{\text{vac}}$ is the technigluon condensate. Our effective field theory for this works because it respects Zumino's consistency condition for dilaton Lagrangians. We find a closed form of the Higgs potential with $\beta'$-dependent deviations from that of the Standard Model. Flavor-changing neutral currents are suppressed if the crawling region $\alpha \lesssim \alpha^{}_{\mathrm{IR}}$ includes a sufficiently large range of energies above the TeV scale. In Appendix A, we observe that, contrary to folklore, condensates protect fields from decoupling in the IR limit.

General covariance from the quantum renormalization group

The Quantum renormalization group (QRG) is a realisation of holography through a coarse graining prescription that maps the beta functions of a quantum field theory thought to live on the `boundary' of some space to holographic actions in the `bulk' of this space. A consistency condition will be proposed that translates into general covariance of the gravitational theory in the $D + 1$ dimensional bulk. This emerges from the application of the QRG on a planar matrix field theory living on the $D$ dimensional boundary. This will be a particular form of the Wess--Zumino consistency condition that the generating functional of the boundary theory needs to satisfy. In the bulk, this condition forces the Poisson bracket algebra of the scalar and vector constraints of the dual gravitational theory to close in a very specific manner, namely, the manner in which the corresponding constraints of general relativity do. A number of features of the gravitational theory will be fixed as a consequence of this form of the Poisson bracket algebra. In particular, it will require the metric beta function to be of gradient form.

Massive gravity from descent equations

Both massless and massive gravity are derived from descent equations (Wess–Zumino consistency conditions). The massive theory is a continuous deformation of the massless one.

The CFT dual of AdS gravity with torsion

We consider the Mielke–Baekler model of three-dimensional AdS gravity with torsion, which has gravitational and translational Chern–Simons terms in addition to the usual Einstein–Hilbert action with cosmological constant. It is shown that the topological nature of the model leads to a finite Fefferman–Graham expansion. We derive the holographic stress tensor and the associated Ward identities and show that, due to the asymmetry of the left- and right-moving central charges, a Lorentz anomaly appears in the dual conformal field theory. Both the consistent and the covariant Weyl and Lorentz anomaly are determined, and the Wess–Zumino consistency conditions for the former are verified. Moreover we consider the most general solution with flat boundary geometry, which describes left- and right-moving gravitational waves on AdS3 with torsion, and show that in this case the holographic energy–momentum tensor is given by the wave profiles. The anomalous transformation laws of the wave profiles under diffeomorphisms preserving the asymptotic form of the bulk solution yield the central charges of the dual CFT and confirm the results that appeared earlier on in the literature. Finally we comment on some points concerning the microstate counting for the Riemann–Cartan black hole.

The Wess Zumino consistency condition: a paradigm in renormalized perturbation theory

Talk given at Symposium in Honor of Julius Wess on the Occasion of his 70th Birthday, 10-11 January 2005 at Max Planck Institute for Physics (Werner Heisenberg Institut) Fohringer Ring 6 - D-80805 MUENCHEN

Results on the Wess–Zumino consistency condition for arbitrary Lie algebras

The so-called covariant Poincaré lemma on the induced cohomology of the space–time exterior derivative in the cohomology of the gauge part of the BRST differential is extended to cover the case of arbitrary, nonreductive Lie algebras. As a consequence, the general solution of the Wess–Zumino consistency condition with a nontrivial descent can, for arbitrary (super) Lie algebras, be computed in the small algebra of the one-form potentials, the ghosts and their exterior derivatives. For particular Lie algebras that are the semidirect sum of a semisimple Lie subalgebra with an ideal, a theorem by Hochschild and Serre is used to characterize more precisely the cohomology of the gauge part of the BRST differential in the small algebra. In the case of an Abelian ideal, this leads to a complete solution of the Wess–Zumino consistency condition in this space. As an application, the consistent deformations of 2+1 dimensional Chern–Simons theory based on iso(2,1) are rediscussed.

All couplings of minimal six-dimensional supergravity

We describe the complete coupling of (1,0) six-dimensional supergravity to tensor, vector and hypermultiplets. The generalized Green–Schwarz mechanism implies that the resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops. Consequently, the low-energy theory is determined by the Wess–Zumino consistency conditions, rather than by the requirement of supersymmetry. As already shown for the case without hypermultiplets, this procedure does not fix a quartic coupling for the gauginos. With respect to these previous results, the inclusion of charged hypermultiplets gives additional terms in the supersymmetry anomaly. We also consider the case in which abelian vectors are present. As in the absence of hypermultiplets, abelian vectors allow additional couplings. Finally, we apply the Pasti–Sorokin–Tonin prescription to this model.

Higher order anomaly consistency conditions: renormalization and non-locality

We study the Slavnov–Taylor Identities (STI) breaking terms, up to the second order in perturbation theory. We investigate which requirements are needed for the first order Wess–Zumino consistency condition to hold true at the next order in perturbation theory. We find that: (a) If the cohomologically trivial contributions to the first order ST breaking terms are not removed by a suitable choice of the first order normalization conditions, the Wess–Zumino consistency condition is modified at the second order. (b) Moreover, if one fails to remove the cohomologically trivial part of the first order STI breaking local functional, the second order anomaly actually turns out to be a non-local functional of the fields and the external sources. By using the Wess–Zumino consistency condition and the Quantum Action Principle, we show that the cohomological analysis of the first order STI breaking terms can actually be performed also in a model (the massive Abelian Higgs–Kibble one) where the BRST transformations are not nilpotent.

Anomaly cancellation condition in lattice gauge theory

We study the gauge anomaly A defined on a 4-dimensional infinite lattice while keeping the lattice spacing finite. We assume that (I) A depends smoothly and locally on the gauge potential, (II) A reproduces the gauge anomaly in the continuum theory in the classical continuum limit, and (III) U (1) gauge anomalies have a topological property. It is then shown that the gauge anomaly A can always be removed by local counterterms to all orders in powers of the gauge potential, leaving possible breakings proportional to the anomaly in the continuum theory. This follows from an analysis of nontrivial local solutions to the Wess–Zumino consistency condition in lattice gauge theory. Our result is applicable to the lattice chiral gauge theory based on the Ginsparg–Wilson Dirac operator, when the gauge field is sufficiently weak ∥ U ( n , μ )−1∥< ϵ ′ , where U ( n , μ ) is the link variable and ϵ ′ a certain small positive constant.

Remark on the consistent gauge anomaly in supersymmetric theories

We present a direct field theoretical calculation of the consistent gauge anomaly in the superfield formalism, on the basis of a definition of the effective action through the covariant gauge current. The scheme is conceptually and technically simple and the gauge covariance in intermediate steps reduces calculational labors considerably. The resultant superfield anomaly, being proportional to the anomaly d abc =tr T a { T b , T c }, is minimal without supplementing any counterterms. Our anomaly coincides with the anomaly obtained by Marinković as the solution of the Wess–Zumino consistency condition.

Becchi–Rouet–Stora cohomology of zero curvature systems. I. The complete ladder case

We present here the zero curvature formulation for a wide class of field theory models. This formalism, which relies on the existence of an operator δ which decomposes the exterior space–time derivative as a BRS commutator, turns out to be particularly useful in order to solve the Wess–Zumino consistency condition. The examples of the topological theories and of the B−C string ghost system are considered in detail.

Algebraic structure of gravity with torsion

The BRST transformations for gravity with torsion are discussed by using the Maurer--Cartan horizontality conditions. With the help of an operator which allows the decomposition of the exterior spacetime derivative as a BRST commutator we solve the Wess--Zumino consistency condition corresponding to invariant Lagrangians and anomalies.

P-forms and supergravity: Gauge symmetries in curved space

We display a general algebraic construction of gauge symmetries in curved space. We explicitly give the gauge symmetries of totally antisymmetric massless tensors, described by differential forms, in the presence of gravity. The way in which we include the invariance under diffeomorphisms allows for a simplification of the construction of the closed supersymmetry algebras in supergravities. We illustrate this procedure on the case of N = 1, d = 4 supergravity. This construction appears useful in discussing the anomalies defined as the solutions of the Wess and Zumino consistency condition.

A simple algebraic construction of the symmetries of supergravity

We display a direct construction of the complete BRS symmetry of a simple N = 1 supergravity. Unlike the Noether construction, this method separates the determination of the symmetry and the construction of an invariant lagrangian. The Wess and Zumino consistency condition is made explicit, providing a firm ground for the study of supergravitational anomalies.