The aetiology of sigma model anomalies

Certain nonlinear sigma models with fermions are ill-defined due to an anomaly which exhibits characteristics of both the nonabelian gauge theory anomaly and the SU(2) anomaly. The simplest way to diagnose the anomaly involves consideration of the global topology of the theory. We review the mathematical methods needed for this analysis and apply them to several supersymmetric sigma models. Some of these are found to be anomalous.     

Pion production by neutral currents in neutrino-nucleon collisions

It is argued that the apparent arbitrariness of the “anomaly” in Fujikawa's approach should be placed into the context of the regularization of the theory not just the regularization of the anomaly. When the theory is regularized at the level of the action the consistent anomaly arises. The converse also holds, namely that if the anomaly is not consistent then the regulator employed does not, by itself, regulate the theory at the level of the action. The criterion that the regularization presription be given at the outset is advocated, and a discussion of results of various two-dimensional models given.     

Conformally invariant Yang-Mills theory realizing the Virasoro-Kac-Moody algebra

We will discuss a conformally invariant Yang-Mills theory in two dimensions, which realizes the Virasoro-Kac-Moody algebra as the BRST symmetry. The theory will have the Virasoro anomaly with the central charge −26, but no anomaly for the Kac-Moody algebra.     

Axial anomaly in lattice abelian gauge theory in arbitrary dimensions

The axial anomaly of lattice abelian gauge theory on a hyper-cubic regular lattice in arbitrary even dimensions is investigated by applying the method of exterior differential calculus. The topological invariance, gauge invariance and locality of the axial anomaly determine the explicit form of the topological part. The anomaly is obtained up to a multiplicative constant for finite lattice spacing and can be interpreted as the Chern character of the abelian lattice gauge theory.     

Point-splitting derivation of anomalous divergence for the spinor current

The anomaly for the divergence of the spinor current in the supersymmetric Yang-Mills theory is obtained by the point-splitting regularization method. The results agree with those of the lowest order perturbation theory. The anomaly is removed by redifinition of the gauge invariant conserved current.     

Anomaly matching conditions and the moduli space of supersymmetric gauge theories

The structure of the moduli space of N = 1 supersymmetric gauge theories is analyzed from an algebraic geometric viewpoint. The connection between the fundamental fields of the ultraviolet theory, and the gauge-invariant composite fields of the infrared theory is explained in detail. The results are then used to prove an anomaly matching theorem. The theorem is used to study anomaly matching for supersymmetric QCD, and can explain all the known anomaly matching results for this case.     

Hamiltonian Anomalies from Extended Field Theories

We develop a proposal by Freed to see anomalous field theories as relative field theories, namely field theories taking value in a field theory in one dimension higher, the anomaly field theory. We show that when the anomaly field theory is extended down to codimension 2, familiar facts about Hamiltonian anomalies can be naturally recovered, such as the fact that the anomalous symmetry group admits only a projective representation on the Hilbert space, or that the latter is really an abelian bundle gerbe over the moduli space. We include in the discussion the case of non-invertible anomaly field theories, which is relevant to six-dimensional (2, 0) superconformal theories. In this case, we show that the Hamiltonian anomaly is characterized by a degree 2 non-abelian group cohomology class, associated to the non-abelian gerbe playing the role of the state space of the anomalous theory. We construct Dai-Freed theories, governing the anomalies of chiral fermionic theories, and Wess-Zumino theories, governing the anomalies of Wess-Zumino terms and self-dual field theories, as extended field theories down to codimension 2.     

Gravitational anomaly cancellation type I superstring theory

By explicit calculations we show that the gravitational anomaly of type I superstring theory vanishes at the string level. There are contributions from four topologically different diagrams to the anomaly: annulus, Möbius strip, torus, and Klein bottle. We explicitly show how the non-trivial cancellation occurs between the open (annulus and Möbius strip) and closed (Klein bottle) sectors. The anomaly of the torus diagram has the same form of type II superstring theory and vanishes because of the modular invariance.     

Discrete quantum gravity and anomalies

A new version of quantum gravity on discrete spaces (simplicial complexes) is proposed. A theory of gravitation interacting with Dirac field is considered. This theory is shown to be free of reparametrization anomaly. The problem of axial gauge anomaly and the associated problem of the doubling of fermion states on a lattice are discussed.     

Anomalous thermal expansion of intermediate valence compounds

A low temperature anomaly in the thermal expansion is predicted for all rare earth compounds with an intermediate valence. The anomaly is caused by a logarithmic variation of the f-electron self-energy with temperature. The sign of the anomaly depends on the Hund's rule correlations of the rare earth ions and is predicted unequivocally by the theory.     

Trace anomalies and λφ4 theory in curved space

It is shown for a conformally invariant λφ⁴ theory in a weakly curved background, how to extend previous results to obtain full information about the trace anomaly in perturbation theory, including the “topological” term in the gravitational part of the anomaly. There is a strong connection among renormalisability of the curved space theory, finiteness of the energy-momentum tensor, and the role of normal products. Combined with a renormalisation-group analysis this provides an efficient means of calculating some terms in the anomaly to high orders of perturbation theory. In particular, the first λ-dependent coefficient of the topological part of the anomaly appears at O(λ⁴) and can be deduced from simple flat-space results without the calculation of any further Feynman diagrams. Some techniques based on an absorptive-part argument are developed in order to compute other anomalous coefficients, and a direct 5-loop calculation confirms the indirect renormalisation-group derivation of a non-vanishing R² anomaly at O(λ⁵). All the essential information can be obtained from the massless theory. The underlying ideas are applicable to other theories, and similar results for massless QED are obtained in a subsequent paper.     

BRS anomaly in the fully quantized non-abelian gauge theory

The form of a possible anomalous term in Slavnov-Taylor identities of Yang-Mills theory is determined by making use of a cohomological exact sequence and the same method is applied to the theory with an anti-symmetric tensor gauge field. In both cases the unique anomalous term has the same form as the conventional non-abelian anomaly. The effect of the anomaly on the nilpotency of the BRS transformation is discussed also.     

Generalized Chern–Simons modified gravity in first-order formalism

We propose a generalization of Chern–Simons (CS) modified gravity in first-order formalism. CS modified gravity action has a term that comes from the chiral anomaly which is Pontryagin invariant. First-order CS modified gravity is a torsional theory and in a space-time with torsion the chiral anomaly includes a torsional topological term called Nieh-Yan invariant. We generalize the CS modified gravity by adding the Nieh-Yan term to the action and find the effective theory. We compare the generalized theory with the first-order CS modified gravity and comment on the similarities and differences.     

Anomalous inequivalence of phenomenological theories

When the transformation between bases in a phenomenological theory is anomalous, the theories are not equivalent, but can be made so by the addition of an appropriate anomaly functional. This functional can be given a simple form which allows the derivation of a sufficient condition for the impossibility of anomaly cancellation.     

Finite-mode regularization of the fermion functional integral (II)

Using the finite-mode regularization introduced in a previous paper, we define the functional integral for a theory of Weyl fermions. We check this definition by making sure the resulting triangle anomaly satisfies the Wess-Zumino consistency conditions. We compare our result with others found in the literature. We apply the finite-mode regularization to compute the axial anomaly in any space-time dimension and to find the explicit expression of anomalous currents in terms of the gauge fields. We illustrate the phenomenon of the infrared compensation of the chiral anomaly.     

Cancellation of quadratically divergent mass corrections in globally supersymmetric spontaneously broken gauge theories

The one-loop quadratically divergent mass corrections in globally supersymmetric gauge theories with spontaneously broken abelian and non-abelian gauge symmetry are studied. Quadratically divergent mass corrections are found to persist in an abelian model with an ABJ anomaly. However, additional supermultiplets necessary to cancel the ABJ anomaly, turn out to be sufficient to eliminate the quadratic divergences as well, rendering the theory natural. Quadratic divergences are shown to vanish also in the case of an anomaly free model with spontaneously broken non-abelian gauge symmetry.     

The Global Gravitational Anomaly of the Self-dual Field Theory

We derive a formula for the global gravitational anomaly of the self-dual field theory on an arbitrary compact oriented Riemannian manifold. Along the way, we uncover interesting links between the theory of determinant line bundles of Dirac operators, Siegel theta functions and a functor constructed by Hopkins and Singer. We apply our result to type IIB supergravity and show that in the naive approximation where the Ramond-Ramond fields are treated as differential cohomology classes, the global gravitational anomaly vanishes on all 10-dimensional spin manifolds. We sketch a few other important physical applications.     

Conformal anomalies in curved space-time

The general form of the conformal anomaly in a dimensionally regularized theory of massless fermions in a background metric is shown to be determined by the first few terms of weak field perturbation theory.     

Introduction to string theory and conformal field theory

A concise survey of noncritical string theory and two-dimensional conformal field theory is presented. A detailed derivation of a conformal anomaly and the definition and general properties of conformal field theory are given. Minimal string theory, which is a special version of the theory, is considered. Expressions for the string susceptibility and gravitational dimensions are derived.     

The Anomaly Line Bundle of the Self-Dual Field Theory

In this work, we determine explicitly the anomaly line bundle of the abelian self-dual field theory over the space of metrics modulo diffeomorphisms, including its torsion part. Inspired by the work of Belov and Moore, we propose a non-covariant action principle for a pair of Euclidean self-dual fields on a generic oriented Riemannian manifold. The corresponding path integral allows one to study the global properties of the partition function over the space of metrics modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual fields differs from the determinant bundle of the Dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle-degree cohomology, and whose holonomies are determined explicitly. We briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self-dual field theory, that will appear in another paper.