TOPOLOGY OF CURRENTS,EFFECTIVE LAGRANGIANS AND ANOMALIES—SOME RECENT PROGRESS IN TOPOLOGICAL INVESTIGATIONS ON GAUGE THEORIES

We briefly review some recent progress i.n topological investigations on quantum field theories with gauge symmetries.The subjects include fractional charges, topological currents,chiral effective Lagrangians and tt:cir gauge invariances,gauge anomalies in different dimensions etc, We explain tree generalization of the Chern-Simons secondary classes, the cohomologies of gauge groups realized upon these generalized classes and their cornbinatory forms which satisfy the generalized transgression formulas, and make use of them in the cohomological analyses on the structure of anomalies in different dimensionswhich represent Topological obstructions to the existence of consistent quantized gauge field theories.     

A chiral SU(4) × SU(4) gauge theory of weak and electromagnetic interactions

A lepton hadron analogy is considered based on the gauge group SU(4)_L × SU(4)_R × U(1), which is broken entirely spontaneously. The model satisfies the physical requirements of the V-A theory, muon-electron universality, no neutral strangeness changing currents, the Cabibbo structure for the SU(3) currents, and triangle anomalies can be avoided. The contribution of the existing neutral currents to various neutrino processes are calculated.     

Anomalies and the particle content of grand unified theories

For the largest class of grand unified models based on simple gauge groups, the presence of anomalies not only results in unrenormalizability, but in addition means that there are charged, massless fermions in the theory. Conversely, the absence of charged, massless particles in such models guarantees the absence of anomalies. For models outside this class, if there are anomalies but no charged massless fermions, the anomalies occur only for rather exotic currents.     

Ward identities and the physical interpretations of anomalies in stochastic quantization

The Ward identities are obtained for vector and axial currents through a functional integral representation of the Langvin equations for the stochastic quantization of fermions in an external gauge field. In this approach anomalies appear as due to fluctuations in the evolution of the system to its equilibrium regime.     

Covariant and consistent anomalies in even-dimensional chiral gauge theories

This paper is devoted to various aspects of anomalies in even-dimensional chiral gauge theories. The difference between the covariant and consistent anomalies is carefully explained in terms of their different origins. The consistent current is defined in terms of a gauge-variant effectve action constructed from the covariant current. An alternative scheme is set up where the covariant anomaly is unaltered but the consistent anomaly vanishes because the effective actions is gauge-invariant. A discussion of theories with vector and axial currents separately gauged is included: here, apart from the covariant anomalies, two different ways of constructing gauge-variant effective actions are possible, giving rise to different structures of the consistent anomalies.     

Local BRST Cohomology and Covariance

The paper provides a framework for a systematic analysis of the local BRST cohomology in a large class of gauge theories. The approach is based on the cohomology of s+d in the jet space of fields and antifields, s and d being the BRST operator and exterior derivative respectively. It relates the BRST cohomology to an underlying gauge covariant algebra and reduces its computation to a compactly formulated problem involving only suitably defined generalized connections and tensor fields. The latter are shown to provide the building blocks of physically relevant quantities such as gauge invariant actions, Noether currents and gauge anomalies, as well as of the equations of motion. Received: 25 July 1996 / Accepted: 23 April 1997     

Higher-spin gauge and trace anomalies in two-dimensional backgrounds

Two-dimensional quantum fields in electric and gravitational backgrounds can be described by conformal field theories, and hence all the physical (covariant) quantities can be written in terms of the corresponding holomorphic quantities. In this paper, we first derive relations between covariant and holomorphic forms of higher-spin currents in these backgrounds, and then, by using these relations, obtain higher-spin generalizations of the trace and gauge (or gravitational) anomalies up to spin 4. These results are applied to derive higher-moments of Hawking fluxes in black holes in a separate paper [S. Iso, T. Morita, H. Umetsu, Hawking radiation via higher-spin gauge anomalies, arXiv: 0710.0456 [hep-th]].     

The action of the group of bundle-automorphisms on the space of connections and the geometry of gauge theories

Some aspects of the geometry of gauge theories are sketched in this review. We deal essentially with Yang-Mills theory, discussing the structure of the space of gauge orbits and the geometrical interpretation of ghosts and anomalies. Occasionally we deal also with classical gauge theories of gravitation and in particular we study the action of the group of diffeomorphisms on the space of linear connections. Finally we comment on the mathematical interpretation of anomalies in field theories.     

Determination of covariant Schwinger terms in anomalous gauge theories

A functional integral method is used to determine equal time commutators between the covariant currents and the covariant Gauss-law operators in theories which are affected by an anomaly. By using a differential geometrical setup we show how the derivation of consistent- and covariant Schwinger terms can be understood on an equal footing. We find a modified consistency condition for the covariant anomaly. As a by-product the Bardeen-Zumino functional, which relates consistent and covariant anomalies, can be interpreted as connection on a certain line bundle over the space of all gauge potentials. Finally the convariant commutator anomalies are calculated for the two- and four dimensional case.     

Gauging of chiral bosonized actions

We present bosonic actions which are equivalent to various chiral fermion theories. For the case of one chiral fermion coupled to an abelian gauge field, we present two bosonized actions, one corresponding to regularizing in the vector conserving scheme and the other in the left-right scheme. We then propose an action for the non-abelian bosonization of Weyl fermions which is a WZW action coupled to a fixed curved background. The chiral WZW action is then coupled to non-abelian gauge fields. We derive the anomalies of the axial current (in the vector conserving scheme) and the left-right currents (in the left-right regularization scheme), both for the abelian and non-abelian bosonized actions. The expressions for the anomalies are identical to those derived in the corresponding fermionic theories.     

Anomalles,cohomology, and finite-mode regularization in higher dimensions

The structure of anomalies in a V-A gauge theory is studied, using a cohomological method. The results are checked with the finite mode regularization method.     

Anomalies and odd dimensions

The parity-violating effective action for theories of fermions coupled to external gauge and gravitational fields in odd dimensions is computed exactly. This action is then used to compute gauge and gravitational anomalies in even dimensions. This derivation of the anomalies elucidates the relation of covariant to consistent anomalies as well as the relation between the Abelian anomaly and the non-Abelian anomaly in two lower dimensions.     

Dynamical supersymmetry breaking and gauge anomalies

Some aspects of supersymmetric gauge theories and discussed. It is shown that dynamical supersymmetry breaking does not occur in supersymmetric QED in higher dimensions. The cancellation of both local (perturbative) and global (non-perturbative) gauge anomalies are also discussed in supersymmetric gauge theories. We argue that there is no dynamical supersymmetry breaking in higher dimensions in any supersymmetric gauge theories free of gauge anomalies. It is also shown that for supersymmetric gauge theories in higher dimensions with a compact connected simple gauge group, when the local anomaly-free condition is satisfied, there can be at most a possibleZ ₂ global gauge anomaly in extended supersymmetricSO(10) (or spin (10)) gauge theories inD=10 dimensions containing additional Weyl fermions in a spinor representation ofSO(10) (or spin (10)). In four dimensions with local anomaly-free condition satisfied, the only possible global gauge anomalies in supersymmetric gauge theories areZ ₂ global gauge anomalies for extended supersymmetricSP(2N) (N=rank) gauge theories containing additional Weyl fermions in a representation ofSP(2N) with an odd 2nd-order Dynkin index.     

Decoupling structure of the principal sigma model-Maxwell interactions

The principal sigma model and Abelian gauge fields coupling is studied. By expressing the first-order formulation of the gauge field equations an implicit on-shell scalar-gauge field decoupling structure is revealed. It is also shown that due to this decoupling structure the scalars of the theory belong to the pure sigma model and the gauge fields sector consists of a number of coupled Maxwell theories with currents partially induced by the scalars.     

Origins of global anomalies in quantum mechanics

Simple and tractable examples of abelian and non-abelian gauge systems with global anomalies are presented in quantum mechanics. Explicit calculations are done both in the path-integral and hamiltonian formalism. Algebraic criteria are given for the existence of global gauge anomalies. These criteria are applied for every gauge group and many representations. The inconsistency of theories with a global gauge anomaly is discussed.     

A geometrical description of local and global anomalies

The general topological framework for testing the possible occurrence of anomalies in gauge theories can be constructed in terms of the theory of group actions on line bundles through the introduction of a suitable group cohomology. In this Letter, we generalize this construction in such a way that it can be applied to a larger class of theories, allowing for a noncontractible configuration space and a nonconnected ‘gauge’ group. This construction find applications to the problem of the lifts of principal group actions. As a physical application, we compare the mechanisms of the anomalies cancelation in gauge and string theories, through a geometrical splitting of local and global anomalies.     

Removal of Chiral Anomalies in Abelian Gauge Theories

It is shown that chiral anomalies can be removed in abelian gauge theories. After a discussion of the two dimensional case where exact solutions are available we study the four dimensional theory. We use perturbation theory, i. e. analyse the triangle Feynman integrals, and determine the general subtraction structure of the gauge current. Then we show that gauges exist for which current conservation holds and the theory is gauge invariant. As far as the generating functional is concerned the anomaly is employed first as gauge fixing condition. After rewriting the interaction in a gauge invariant form the gauge fixing condition can be imposed as usual. In our approach the integration over the gauge group remains trivial.     

Cancellation of global anomalies in spontaneously broken gauge theories

I discuss the generalization to global gauge anomalies of the familiar procedure for the cancellation of local gauge anomalies in effective theories of spontaneously broken symmetries