Boundary Conditions for Topological Quantum Field Theories,Anomalies and Projective Modular Functors

We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level m, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for n-characters on ∞-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for n + 1-dimensional TQFTs produce n-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.     

Current algebra and Wess-Zumino model in two dimensions

We investigate quantum field theory in two dimensions invariant with respect to conformal (Virasoro) and non-abelian current (Kac-Moody) algebras. The Wess-Zumino model is related to the special case of the representations of these algebras, the conformal generators being quadratically expressed in terms of currents. The anomalous dimensions of the Wess-Zumino fields are found exactly, and the multipoint correlation functions are shown to satisfy linear differential equations. In particular, Witten's non-abelean bosonisation rules are proven.     

Global Gauge Anomalies in Two-Dimensional Bosonic Sigma Models

We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. Obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.     

Chiral anomalies in supersymmetric theories

The supersymmetric non-abelian chiral anomaly is computed in a theory of chiral scalar superfields coupled to external gauge superfields, both in a vector-current conserving scheme and in a left-right symmetric scheme. The corresponding supersymmetric Wess-Zumino term is discussed; in particular we give an explicit expression for the anomalous bosonic term.     

Gauge invariant theories of the generalized chiral Schwinger model

The gauge invariant theories of the generalized chiral Schwinger model are constructed in terms of two schemes with and without Wess-Zumino terms, respectively. Following the former scheme, we calculate the Wess-Zumino term which cancels the gauge anomaly, and then constitute the gauge invariant theory by adding the Wess-Zumino term to the original Lagrangian of the model. According to the latter, we modify the original Hamiltonian by adding a term composed of constraints of the model. It is so designed that the theory described by the modified Hamiltonian and its corresponding first-order Lagrangian maintains gauge invariance. We show by the canonical Dirac method that each of the two gauge invariant theories has the same physical spectrum as that of the original gauge noninvariant formulation.     

Gauge invariant theories of the generalized chiral Schwinger model

The gauge invariant theories of the generalized chiral Schwinger model are constructed in terms of two schemes with and without Wess-Zumino terms, respectively. Following the former scheme, we calculate the Wess-Zumino term which cancels the gauge anomaly, and then constitute the gauge invariant theory by adding the Wess-Zumino term to the original Lagrangian of the model. According to the latter, we modify the original Hamiltonian by adding a term composed of constraints of the model. It is so designed that the theory described by the modified Hamiltonian and its corresponding first-order Lagrangian maintains gauge invariance. We show by the canonical Dirac method that each of the two gauge invariant theories has the same physical spectrum as that of the original gauge noninvariant formulation.     

Chiral anomalies, effective lagrangians and differential geometry

Using the differential geometric approach developed by Zumino and others, we study the relation between chiral anomalies and the effective lagrangian. We present a simple formula for the non-abelian (Bardeen) anomaly in any even dimensions. This is used to introduce gauge couplings to the Wess-Zumino effective action, as suggested by Witten.     

Structural Aspects of the Fermion-Boson Mapping in Two-Dimensional Gauge and Anomalous Gauge Theories with Massive Fermions

Using a synthesis of the functional integral and operator approaches we discuss the fermion-boson mapping and the role played by the Bose field algebra in the Hilbert space of two-dimensional gauge and anomalous gauge field theories with massive fermions. In QED₂ with quartic self-interaction among massive fermions, the use of an auxiliary vector field introduces a redundant Bose field algebra that should not be considered as an element of the intrinsic algebraic structure defining the model. In anomalous chiral QED₂ with massive fermions the effect of the chiral anomaly leads to the appearance in the mass operator of a spurious Bose field combination. This phase factor carries no fermion selection rule and the expected absence of Θ-vacuum in the anomalous model is displayed from the operator solution. Even in the anomalous model with massive Fermi fields, the introduction of the Wess-Zumino field replicates the theory, changing neither its algebraic content nor its physical content.     

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.     

Unoriented WZW Models and Holonomy of Bundle Gerbes

The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models. manche meinen lechts und rinks kann man nicht velwechsern werch ein illtum Ernst Jandl [Jan95] K.W. is supported with scholarships by the German Israeli Foundation (GIF) and by the Rudolf und Erika Koch–Stiftung.     

The topological meaning of non-abelian anomalies

We show how the non-abelian anomaly for gauge fields coupled to Weyl fermions in 2n dimensions is related to the non-trivial topology of gauge orbit space. The form of the anomaly and its normalization are shown to follow from a familiar index theorem for a certain (2n + 2)-dimensional Dirac operator. We are thus able to recover and give topological meaning to a variety of results concerning anomalies in 4- and higher-dimensional theories.     

反常、Chern-Simons上链

本文提出一种联络空间上同调的概念,建立这种上同调群与Chern-Simons型示性类系列的同态关系,给出它们在反常、Wess-Zumino有效作用与Schwinger项等问题上的应用;从而概括了Faddeev,宋行长,Zumino等人最近关于这些问题的探讨。 关键词：     

The structure of gauge and gravitational anomalies

It is shown how the form of the gauge and gravitational anomalies in quantum field theories may be derived from classical index theorems. The gravitational anomaly in both Einstein and Lorentz form is considered and their equivalence is exhibited. The formalism of gauge and gravitational theories is reviewed using the language of differential geometry, and notions from the theory of characteristic classes necessary for understanding the classical index theorems are introduced. The treatment of known topological results includes a pedagogical derivation of the Wess-Zumino effective Lagrangian in arbitrary even dimension. The relation between various forms of the anomaly present in the literature is also clarified.     

THE GAUGE INVARIANT WESS-ZUMINO ACTION FUNCTIONAL,CHIRAL ANOMALIES AND TOPOLOGY

The gauge invariant Wess-Zumino action functional in 2n dimensional space is obtained by properly constructing the manifestly gauge invariant (2n+1)-forms which satisfy the Abelian anomalous Ward identity in 2n+2 dimensional space. Apart from its simplicity, directness, the construction clearlu embodies the connections between the gauge invariant Wess-Zumino action functional, cniral anomalies and topology.     

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.     

Covariant Hamiltonian Field Theory: Path Integral Quantization

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative quantum field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field systems are equivalent.     

A topological investigation of the Quantum Adiabatic Phase

Using algebraic topology, the appearance of the Quantum Adiabatic Phase over various parameter manifolds is investigated. The relation with nontrivial gauge bundles (both abelian and non-abelian) is studied and it is shown that the phase appears as a result of homotopically non-trivial mappings, induced by the Hamiltonian in the space of wave-functions. The cohomological picture is developed and some topological considerations concerning field theory anomalies in the Hamiltonian picture are presented. A proof of the Nielsen-Ninomiya theorem is given inspired from the notion of the adiabatic phase.Work supported in part by the U.S. Department of Energy under contract DEAC 03-81-ER 40050     

On the Gauge and BRST Invariance of the Chiral QED with Faddeevian Anomaly

Chiral Schwinger model with the Faddeevian anomaly is considered. It is found that imposing a chiral constraint this model can be expressed in terms of chiral boson. The model when expressed in terms of chiral boson remains anomalous and the Gauss law of which gives anomalous Poisson brackets between itself. In spite of that a systematic BRST quantization is possible. The Wess-Zumino term corresponding to this theory appears automatically during the process of quantization. A gauge invariant reformulation of this model is also constructed. Unlike the former one gauge invariance is done here without any extension of phase space. This gauge invariant version maps onto the vector Schwinger model. The gauge invariant version of the chiral Schwinger model for a=2 has a massive field with identical mass however gauge invariant version obtained here does not map on to that.     

Locally supersymmetric σ-model with Wess-Zumino term in two dimensions and critical dimensions for strings

We construct an N = 1 locally supersymmetric σ-model with a Wess-Zumino term coupled to supergravity in two dimensions. If one takes the σ-model manifold to be the product of d-dimensional Minkowski space M^d and a group manifold G, and if the radius of G is quantized in appropriate units of the string tension, then the model describes a Neveu-Schwarz-Ramond (NSR)-type string moving on M_d × G. (Our model generalizes earlier work of refs. [1,2] which do not contain a Wess-Zumino term and that of refs. [5,6] which is not locally supersymmetric.) The zweibein and the gravitino field equations yield constraints which generalize those of the NSR model to the case of a non-abelian group manifold. In particular, the fermionic constraint contains a new term trilinear in the fermionic fields. We quantize the theory in the light-cone gauge and derive the critical dimensions. We compute the mass spectrum of a closed string moving on M_d × G and show that massless fermions do not arise for non-abelian G for the spinning string, in agreement with the result of Friedan and Shenker [22].