Covariant and Consistent Anomalies in Gauged Supergravity

This article reviews some properties of the low-energy effective actions for gauged supergravity models. We summarize the current state of knowledge regarding gravity theories with minimal supersymmetry. The difference between the covariant and consistent anomalies is carefully explained in terms of their different origins. The gauged supergravity provides an interesting theoretical framework to the physics beyond the standard model. Gauged supergravities, where the global isometries of the matter lagrangian are promoted to local symmetries, have been widely explored and by now almost all allowed models for diverse spacetime dimensions. The geometry of curved superspace is shown to allow the existence of a large family of supermultiplets that can be used to describe supersymmetric matter, including vector, tensor and hypermultiplets. The main theories of interest in this post are four and six-dimensional supergravities with a minimal amount of supersymmetry. In the first part of our work we use the symplectic structure of four-dimensional minimal supergravities to study the possibility of gauged axionic shift symmetries. This leads to the introduction of generalized Chern-Simons terms, and a Green-Schwarz cancellation mechanism for gauge anomalies. Our models constitute the supersymmetric framework for string compactifications with axionic shift symmetries, generalized Chern-Simons terms and quantum anomalies. We construct the complete coupling of (1, 0) supergravity in six dimensions to tensor multiplets, extending previous results to all orders in the fermi fields. The resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops, and is determined by corresponding Wess-Zumino consistency conditions, aside from a quartic coupling for the gaugini. In addition, we show how to revert to a supersymmetric formulation in terms of covariant field equations that embody corresponding covariant anomalies. The subsequent work of some authors has developed the consistent formulation, but one can actually revert to a covariant formulation, at the price of having non-integrable field equations. The relation between the two sets of equations is one more instance of the link between covariant and consistent anomalies in field theory. This is a remarkable laboratory for current algebra, where one can play explicitly with anomalous symmetries and their consequences.

The Anomalies in D=4 Supergravity

The gauge anomalies manifest themselves as a non-invariance of the effective action under gauge transformations. We will concentrate on anomalous gauge symmetries, which are more problematic. Since gauge symmetries are needed to decouple the unphysical states of the theory, a violation of these symmetries renders the theory inconsistent. Our goal is to make generalization for theories with quantum anomalies. These anomalies depend only on the gauge vectors. We use the symplectic structure of four-dimensional minimal supergravities to study the possibility of gauged axionic shift symmetries. The total Lagrangian is

 
The kinetic Lagrangian is 

 
The gauge invariance is made by adding topological terms linear and quadratic in the tensor field to the gauge kinetic term, namely   
 

We introduce a generalized Chern-Simons term of the form

Variation of the total action is

Variation of the Lagrangian is

Expanding this result using and a partial integration, the variation of the Lagrangian is

which describes the quantum gauge anomalies due to anomalous chiral fermions

where the covariant anomaly is

The corresponding expression formally looks very similar to a symplectically covariant generalization of the electric consistent quantum anomaly. Even if the classical action is gauge invariant, a non-invariance of the path integral measure may occur and violate the gauge invariance, leading to a quantum anomaly  

More explicitly, for an arbitrary non-Abelian gauge group, the consistent form of the anomaly is given by

Similarly there are supersymmetry anomalies, such that the final non-invariance of the one-loop effective action is

  
This anomaly should satisfy the Wess-Zumino consistency conditions, which are the statement that these variations should satisfy the symmetry algebra. For the gauge anomalies these are

  
An anomalous spectrum of chiral fermions induces a gauge non-invariance of the quantum effective action, where the consistent anomaly is

Consequently, the low-energy theory is determined by the Wess-Zumino consistency conditions, rather than by the requirement of supersymmetry, and this procedure does not fix a quartic coupling for the gauginos. We have shown how gauge invariance can be restored in the presence gauged axionic shift symmetries in general N = 1 supersymmetric gauge theories. The reason is that the above classical action is not gauge invariant. As we are working in the Wess-Zumino gauge, this will also imply a non-invariance under supersymmetry. Thus, the entire classical plus quantum theory is indeed supersymmetric. 


The Anomalies in D=6 Supergravity

We review some properties of the field equations of six-dimensional (1,0) supergravity coupled to tensor and vector multiplets, and in particular their relation to covariant, consistent and gravitational anomalies. The complete coupling of (1,0) six-dimensional supergravity to non-abelian vector and tensor multiplets requiring the closure of the Wess-Zumino conditions, has revealed another related aspect of these six-dimensional models: a quartic coupling for the gauginos is undetermined, and the construction is consistent for any choice of this coupling. Correspondingly, the commutator of two supersymmetry transformations on the gauginos contains an extension, that plays a crucial role in ensuring that the Wess-Zumino consistency conditions close on-shell. In formulating the low-energy couplings between tensor and vector multiplets, one has two natural options. The first is related to covariant field equations and to the corresponding covariant anomalies. It has the virtue of respecting gauge covariance and supersymmetry, but the resulting field equations are not integrable. The second is related to consistent, and thus integrable field equations. These may be derived from an action principle that satisfies Wess-Zumino consistency conditions, and as a result embody a supersymmetry anomaly.

In order to obtain the bosonic equations, it is convenient to associate the fermionic equations to the Lagrangian

In order to derive the bosonic equations, one can add

In a similar fashion, the scalar equation is

Requiring that the commutator of two supersymmetry transformations on the Fermi fields close on-shell then determines the complete Fermi field equations. The equations obtained in this way are

for the gravitino

for the tensorinos

for the hyperinos.
 
To lowest order in the fermi fields, we produce the construction adding the hypermultiplet couplings. The equations for all fields can be obtained from     
                                                           

after imposing the (anti)self-duality conditions. With this prescription, its variation under the supersymmetry transformations gives the supersymmetry anomaly

related by the Wess-Zumino consistency conditions 

     
to the consistent anomaly

All the  field equations may then be derived from the lagrangian

The variation of this lagrangian with respect to gauge transformations gives the abelian gauge anomaly

The variation of the Lagrangian with respect to the supersymmetry transformations

gives the supersymmetry anomaly

It is well known that consistent and covariant gauge anomalies are related by the divergence of a local functional. The covariant vector field equation completes the results to all orders in the Fermi fields. In six dimensions the covariant gauge anomaly is related to the consistent anomaly by a local counterterm

Analogous to the consistent anomaly the covariant anomaly is defined by the covariant divergence of the covariant current

where the covariant anomaly contains higher-order Fermi terms

For models without gravitational anomalies one would expect that the divergence of the energy-momentum tensor vanish. Actually, this is no longer true if other anomalies are present, since all fields, not only the metric, have derivative variations under coordinate transformations. In a theory with gauge and supersymmetry anomalies, the gravitational anomaly is

Starting again from the consistent equations, one finds

In particular, in our case we are not accounting for gravitational anomalies, that would result in higher-derivative couplings, and indeed one can verify that the divergence of the energy-momentum tensor does not vanish, but satisfies the relation 

Reverting to the covariant form eliminates the divergence of the Rarita-Schwinger equation and alters the vector equation, so that the third term has to be retained. The final result is

and is nicely verified by our equations. In particular, this implies that, to lowest order in the fermi couplings, the divergence of energy-momentum tensor vanishes. In particular, it vanishes to lowest order in the fermi couplings, while it gives a covariant non-vanishing result if all fermion couplings are taken into account.



The main purpose of this article was to investigate the covariant, consistent and gravitational anomalies in gauged supergravity. We have shown how general gauge theories with axionic shift symmetries, generalized Chern-Simons terms and quantum anomalies can be formulated in a way that is covariant with respect to electric-magnetic duality transformations. We performed our analysis first in rigid supersymmetry. Using superconformal techniques, we could then show that only one cancellation had to be checked to extend the results to supergravity. It turns out that the Chern-Simons term does not need any gravitino corrections and can thus be added as such to the matter-coupled supergravity actions. Our results provides thus an extension to the general framework of coupled chiral and vector multiplets in N = 1 supergravity. We have completed the coupling of (1, 0) six-dimensional supergravity to tensor and vector multiplets. The coupling to tensor multiplets is of a more conventional nature, and parallels similar constructions in other supergravity models. The Yang-Mills currents are not conserved, and the consistent residual gauge anomaly is accompanied by a corresponding anomaly in the supersymmetry current. In completing these results to all orders in the fermi fields, we have come to terms with another peculiar feature of anomalies, neatly displayed by these field equations: anomalous divergences of gauge currents are typically accompanied by corresponding anomalies in current commutators. All N = (1; 0) supersymmetric theories in 6D with one gravity and one tensor multiplet which are free of anomalies or other quantum inconsistencies admit a string construction. Explicit knowledge of this set of theories gives us a powerful tool for exploring the connection between string theory and low-energy physics. It would also be interesting to formulate the matter coupled anomaly-free supergravity theories in six dimensions such that the classically gauge invariant and supersymmetric part of the action is identified and the anomaly corrections are determined by means of the anomaly equations. The anomalies are the key to a deeper research and understanding of gauged supergravity. We hope to have conveyed the idea that anomalies play an important role in supergravity and their cancellation has been and still is a valuable guide for constructing consistent quantum supergravity theories. The treatment of anomalies makes fascinating contacts with several branches of modern theoretical physics.