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.     

Point-splitting analysis of commutator anomalies in non-abelian chiral gauge theories

A gauge covariant point-splitting regularisation is employed to calculate different anomalous commutators in four dimensional chiral gauge theories. For an external gauge field the fixed time anomalous commutator of the gauge group generators is seen to violate the Jacobi identity. The cohomological prediction can be confirmed provided the electric fields do not commute. Other commutators like the current-current and current-electric field are consistent with the Bjorken-Johnson-Low (BJL) derivation.     

Regularization of chiral gauge theories

The regularization of chiral gauge theories is reviewed from the “overlap” point of riew. This is a brief and biased review containing no references. Contribution dedicated to Lawrence Horwitz on the occasion of his 65th birthday     

Quark confinement and topology of gauge theories

The phenomenon of quark confinement is known to be connected with the restoration of apparently broken gauge symmetry. In this paper we focus on a special mechanism which is responsible for such restoration. The major suggestion is that in the treatment of infrared problems certain classical field trajectories are of paramount importance, which trajectories connect seemingly degenerate vacua thereby eliminating the degeneracy. Initially, we demonstrate this mechanism in certain simple non-gauge models. In order of increasing difficulty, we next examine compact quantum electro-dynamics in 2 + 1 space-time dimensions- essentially a variant of unified models of the Georgi-Glashow type. In this case we prove that charge is confined and show that the force between two charges is independent of the distance between them. For small values of the fine structure constant this force is explicitly evaluated. Finally, we turn to the more realistic case of a 3 + 1 dimensional non-Abelian gauge theory, and analyse the contribution of a single pseudoparticle to the correlation functions. It is proposed that the quantum fluctuations of the pseudoparticle are inessential, and that the one-loop approximation is effectively correct even for the large scale pseudoparticles. The emergent conclusion then is that the renormalized Yang-Mills theory is reduced to the problem of evaluating the effects of those configurations which involve many pseudoparticles, due account being taken of the interactions between them. Some aspects of this last problem are also discussed.     

Berry's phase in chiral gauge theories

Berry's phase in chiral gauge theories is computed. The result implies that the anomalous commutator of gauge transformations is the one previously obtained by Faddeev.     

Quantization of chiral gauge theories in two-dimensions

It is shown that both abelian and non abelian chiral gauge theories in two dimensions can be made gauge invariant at the quantum level by adding a scalar field. In the bosonized form of the theory, the scalar field appears in a gauged Wess-Zumino action. The current algebra of the extended abelian theory is shown to be free of anomalous terms.     

Anomalous commutators in two-dimensional chiral gauge theories

It is shown that in two-dimensional models of chiral fermions coupled with background gauge fields, anomalous commutators of currents as well as of generators of time-independent gauge transformations are completely determined by the anomaly if the gauge group is semisimple. The four-dimensional case is commented on.     

Gauge anomaly cancellation in chiral gauge theories

We consider chiral fermions interacting minimally with abelian and non-abelian gauge fields. Using a path integral approach and exploring the consequences of a mechanism of symmetry restoration, we show that the gauge anomaly has null expectation value in the vacuum for both cases (abelian and non-abelian). We argue that the same mechanism has no possibility to cancel the chiral anomaly, what eliminates competition between chiral and gauge symmetry at full quantum level. We also show that the insertion of the gauge anomaly in arbitrary gauge invariant correlators gives a null result, which points towards anomaly cancellation in the subspace of physical state vectors.     

Anomalous commutators in vector and chiral gauge theories

A fixed-time point-splitting regularisation employing suitable phase factors is used to derive various anomalous commutators - both in usual vector gauge theories and non-abelian chiral gauge theories in four dimensions. In the latter case, new structures for the commutators involving currents have been found. Consistency of these expressions is established by reproducing some familiar results.     

Quantization of gauge theories with global anomalies

Global anomalies, which obstruct the quantization of certain gauge theories in the temporal gauge, get bypassed in canonical quantization.     

Consistent and covariant anomalies in gauge and gravitational theories

The gauge structure of anomalies and the related currents is analyzed in detail. We construct the covariant forms for both the currents and the anomalies for general gauge theories in even-dimensional space-time. The results are then extended to determine the structure of gravitational anomalies. These can always be interpreted as anomalies for local Lorentz transformations.     

General chiral gauge theories on the lattice

We still extend the large class of Dirac operators decribing massless fermions on the lattice found recently, only requiring that such operators decompose into Weyl operators. After deriving general relations and constructions of operators, we study the basis representations of the chiral projections. We then investigate correlation functions of Weyl fermions for any value of the index, stressing the related conditions for basis transformations and their consequences, and getting the precise behaviors under gauge transformations and CP transformations. Various further developments include considerations of the explicit form of the effective action and of a representation of the general correlation functions in terms of alternating multilinear forms. For comparison we also consider gauge-field variations and their respective applications. Finally we compare with continuum perturbation theory.     

On Schwinger terms in nonabelian chiral gauge theories

Chiral fermionic currents, coupled with nonabelian background gauge fields, are known to have Schwinger terms in their commutators. It is shown that if the gauge group is semisimple, the anomaly is completely determined by these Schwinger terms. Violation of Jacobi identities can also be demonstrated using them.     

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     

On the absence of supersymmetry anomalies in gauge theories

It is shown that N = 2 non-abelian gauge theory can be renormalized consistent with gauge invariance, supersymmetry and unitarity. There is no perturbative supersymmetry anomaly: the proof is independent of any specific regulator.