Antisymmetric tensor gauge theories and non-linear σ-models

Antisymmetric tensor fields B_μν subject to the gauge transformation δB_μν = ?_μξ_ν ? ?_νξ_μ can describe spinless particles. We investigate the properties of field theories with a “non-abelian generalization” of this invariance. One class of such theories is equivalent to non-linear principal chiral σ-models, another to massive Yang-Mills theories. A supersymmetric analogue in 2 + 2 superspace is constructed and leads to the supersymmetric σ-model defined on a general riemannian manifold.     

On the energy-momentum tensor in gauge field theories

The energy-momentum tensor in spontaneously broken non-Abelian gauge field theories is studied. The motivation is to show that recent results on the finiteness and gauge independence of S-matrix elements in gauge theories extends to observable amplitudes for transitions in a gravitational field. Path integral methods and dimensional regularization are used throughout. Green's functions Γ_μν^(j)(q; p₁,…,p_j) involving the energy-momentum tensor and j particle fields are proved finite to all orders in perturbation theory to zero and first order in q, and finite to one loop order for general q. Amputated Green's functions of the energy momentum tensor are proved to be gauge independent on mass shell.     

Gravitational anomalies

It is shown that in certain parity-violating theories in 4k+2 dimensions, general covariance is spoiled by anomalies at the one-loop level. This occurs when Weyl fermions of $\text{spin}\text{-}\text{1}\text{2}\text{or}\text{-3}\text{2}$ or self-dual antisymmetric tensor fields are coupled to gravity. (For Dirac fermions there is no trouble.) The conditions for anomaly cancellation between fields of different spin is investigated. In six dimensions this occurs in certain theories with a fairly elaborate field content. In ten dimensions there is a unique theory with anomaly cancellation between fields of different spin. It is the chiral n = 2 supergravity theory, which is the low-energy limit of one of the superstring theories. Beyond ten dimensions there is no way to cancel anomalies between fields of different spin.     

Gravitational anomalies

The effective action for fermions moving in external gravitational and gauge fields is analyzed in terms of the corresponding external field propagator. The central object in our approach is the covariant energy-momentum tensor which is extracted from the regular part of the propagator at short distances. It is shown that the Lorentz anomaly, the conformal anomaly and the gauge anomaly can be expressed in terms of the local polynomials which determine the singular part of the propagator. (There are no coordinate anomalies.) Except for the conformal anomaly, for which we give explicit representations only ind<=4, we consider an arbitrary number of dimensions.     

Gravitational gauge mediation

It is often the case that the naive introduction of the messenger sector to supersymmetry breaking models causes restoration of supersymmetry. We discuss a possibility of stabilizing the supersymmetry breaking vacuum by gravitational interaction.     

The energy-momentum tensor in scalar and gauge field theories

A previous study of the energy-momentum tensor in ?⁴ theory and spontaneously broken non-Abelian gauge field theories is extended here to show finiteness to all orders in perturbation theory. Divergences of Green's functions Γ_μν^(j) (q; p₁, …, p_j) involving the energy-momentum tensor θ_μν and j particle fields are removed by counterterms of the ordinary Lagrangian plus a renormalization of the coefficient of the Callan-Coleman-Jackiw improvement term in θ_μν. Physically the extra renormalization means that the mean square “mass radius” of elementary spin zero particles must be specified from experiment.     

Approaches to a non-abelian antisymmetric tensor gauge field theory

Two distinct attempts at constructing a theory of non-abelian antisymmetric tensor gauge fields (ATGF's) are considered. First, a recently proposed geometry of abelian ATGF's is reviewed and then generalized to the non-abelian case. The resulting geometric action is non-local and is invariant under non-local gauge transformations; in the local limit the action describes free fields. Lattice actions for both the abelian and non-abelian ATGF theories are also presented. In the second approach, a lattice action for non-abelian ATGF's is constructed using a plaquette variables that carry four internal indices. The continuum limit is also a non-interacting theory.     

On the new representation of the antisymmetric tensor gauge field

A formalism is presented which provides a framework for analysis of the abelian gauge theory of a second-rank anti-symmetric tensor in terms of the mutually dual auxiliary fields. This new formalism illustrates how the different gauge fixing conditions are related to the different field representations exhibiting the rich dynamical features of the antisymmetric tensor gauge fields. The derivation relies upon an analogy with the simpler case of the two-dimensional Maxwell field.     

Gravitational anomalies and entropy

A derivation of entropy from the expressions for two dimensional gravitation anomalies is given. Starting from the near horizon anomalous energy–momentum tensors corresponding to particular anomalies, the Virasoro algebra with central extension is obtained. The central charge is identified by comparing with the standard form of the algebra. Then the conserved charge in the ground state is computed. Finally, using the Cardy formula the entropy is obtained. Here both the vector and chiral theories are discussed.     

Dimensional regularization of gauge theories in spherical spacetime: Free field trace anomalies

The O(n + 1) covariant formulation of massless quantum electrodynamics in spherical spacetime is further developed to allow a calculation of the energy-momentum tensor trace anomalies for the free Dirac, electromagnetic, and SU(2) gauge fields. The principal technical development is the construction of the Faddeev-Popov ghosts for electrodynamics and SU(2) Yang-Mills theory. This construction is unconventional first in that the gauge fixing term in the Lagrangian is not a perfect square, and second because it is necessary to remove radial as well as gauge degrees of freedom from the measure of the functional integral. The ghost fields are shown to satisfy a minimal scalar field equation. The free field effective action is found to be divergent in four dimensions, and is renormalized by the inclusion in the Lagrangian of a counterterm local in the gravitational fields. The energy-momentum tensor calculated from this renormalized effective action is shown to have a trace anomaly.     

Gravitational anomalies and Schwinger terms

A BRST transformation incorporating general coordinate, Lorentz and Weyl transformations is defined. The variation of the effective action for fermions moving in an external gravitational field is calculated using an extended BRST analysis. InD=2 dimensions the Schwinger terms appearing in the equal time commutators of energy momentum tensors are related to the gravitational and Weyl anomalies for a massless spin 0 and a massless spin 1/2 model, respectively.     

Discrete gauge symmetry anomalies

It has been recently argued that quantum gravity effects strongly violate all non-gauge symmetries. This would suggest that all low energy discrete symmetries should be gauge symmetries, either continuous or discrete. Acceptable continuous gauge symmetries are constrained by the condition they should be anomaly free. We show here that any discrete gauge symmetry should also obey certain “discrete anomaly cancellation” conditions. These conditions strongly constrains the massles fermion content of the theory and follow from the “parent” cancellation of the usual continuous gauge anomalies. They have interesting applications in model building. As an example we consider the constraints on the Z_N “generalized matter parities” of the supersymmetric standard model. We show that only a few (including the standard R-parity) are “discrete anomaly free” unless the fermion content of the minimal supersymmetric standard model is enlarged.     

On the cancellation of hard anomalies in gauge field models: a regularization independent proof

In this paper we prove Bardeen's conjecture that the anomaly of the Adler-Bardeen-Bell-Jackiw-Schwinger type in gauge models are definitely absent if they are cancelled at the first order of the perturbation expansion. Our analysis develops within the regularization independent B.P.H.Z. renormalization scheme. We discuss the possible appearance of anomalies in an enlarged class of gauge models admitting soft violations of the Slavnov-Taylor identities which prescribe the gauge transformation properties of the Green functions. By a repeated use of the Callan-Symanzik equation we conclude that the lowest non vanishing contributions to the anomalies must necessarily correspond to the first order in the perturbation expansion, hence if they are cancelled at this order the theory will be definitely anomaly free.     

Gravitational correction to running of gauge couplings

We calculate the contribution of graviton exchange to the running of gauge couplings at lowest non-trivial order in perturbation theory. Including this contribution in a theory that features coupling constant unification does not upset this unification, but rather shifts the unification scale. When extrapolated formally, the gravitational correction renders all gauge couplings asymptotically free.     

An energy-momentum tensor in gauge theories

We consider the renormalization of the twist two, dimension four gauge invariant operator O_μν⁽¹⁾ = − F_μσF_νσ − g_{μν 0}. By using the general theory of renormalization of gauge invariant operators, we find the gauge noninvariant operator O⁽²⁾ with which it mixes. We construct a finite combination of O⁽¹⁾ and O⁽²⁾ and show that it is an acceptable energy momentum tensor for gauge theories. We compare our energy momentum tensor with that constructed by Freedman, Muzinich, and Weinberg.     

Gravitational anomalies and the family's index theorem

We discuss the use of the family's index theorem in the study of gravitational anomalies. The geometrical framework required to apply the family's index theorem is presented and the relation to gravitational anomalies is discussed. We show how physics necessitates the introduction of the notion oflocal cohomology which is distinct from the ordinary topological cohomology. The recent results of Alvarez-Gaumé and Witten are derived by using the family's index theorem.This work was supported in part by the National Science Foundation under Contracts PHY81-18547 and MCS80-23356; and by the Director, Office of High Energy and Nuclear Physics of the US Department of Energy under Contracts DE-AC03-76SF00098 and AT0380-ER10617Alfred P. Sloan Foundation Fellow