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     

Geometry and topology of chiral anomalies in gauge theories

Geometrical and topological aspects of chiral anomalies in gauge theories are reviewed. Geometrical and topological concepts and results for chiral anomalies in gauge theories are considered, including differential forms, Lie groups, homotopy, homology, cohomology, Riemannian manifolds, fibre bundles, characteristic classes, index theorems and spectral flow. Gauge theories and their formulation in terms of differential forms and fibre bundles are described. The quantisation of gauge theories is performed using path integrals, and the orbit space, BRST symmetries and ? vacuum are discussed. Gauge theories with fermions are formulated, including realistic models of strong and weak interactions. Chiral anomalies and related issues such as the existence of Schwinger terms, their origins in terms of differential forms, cohomology, the orbit space, BRST identities, Hamiltonian systems and relations to index theorems are analysed. Constraints on models for particle physics from chiral anomalies and theories involving spontaneously broken chiral symmetry described by effective Lagrangians are also mentioned.     

Six-Dimensional Superconformal Field Theories from Principal 3-Bundles over Twistor Space

We construct manifestly superconformal field theories in six dimensions which contain a non-Abelian tensor multiplet. In particular, we show how principal 3-bundles over a suitable twistor space encode solutions to these self-dual tensor field theories via a Penrose–Ward transform. The resulting higher or categorified gauge theories significantly generalise those obtained previously from principal 2-bundles in that the so-called Peiffer identity is relaxed in a systematic fashion. This transform also exposes various unexplored structures of higher gauge theories modelled on principal 3-bundles such as the relevant gauge transformations. We thus arrive at the non-Abelian differential cohomology that describes principal 3-bundles with connective structure.     

Nonperturbative BRS invariance

In the context of lattice gauge theories the standard requirement of local gauge invariance is replaced by BRS invariance. The most general BRS invariant lagrangian is shown to contain as many parameters as the gauge invariant one. This is done by explicitly solving the lattice BRS cohomology problem.     

Higher bundle gerbes and cohomology classes in gauge theories

The notion of a higher bundle gerbe is introduced to give a geometric realisation of the highder degree integral cohomology of certain manifolds. We consider examples using the infinite-dimensional spaces arising in gauge theories.     

Are nonrenormalizable gauge theories renormalizable?

We raise the issue whether gauge theories, that are not renormalizable in the usual powercounting sense, are nevertheless renormalizable in the modern sense that all divergences can be cancelled by renormalization of the infinite number of terms in the bare action. We find that a theory is renormalizable in this sense if the a priori constraints that we impose on the form of the bare action correspond to the cohomology of the BRST-transformations generated by the action. Recent cohomology theorems of Bamich, Brandt, and Henneaux are used to show that conventionally nomenormalizable theories of Yang-Mills fields (such as quantum chromodynamics with heavy quarks integrated out) and/or gravitation are renormalizable in the modern sense.     

Propagators and renormalization transformations for lattice gauge theories. I

Lattice gauge theories may be looked at as perturbations of the theory of a vector field with a Gaussian action. We study this theory here and in following papers obtaining crucial results for understanding the renormalization group method in more complicated non-Abelian gauge field theories.Research supported in part by the National Science Foundation under Grant PHY-82-03669     

Characteristic cohomology ofp-form gauge theories

The characteristic cohomologyH ^k _char(d) for an arbitrary set of freep-form gauge fields is explicitly worked out in all form degreesk < n — 1, wheren is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interactingp-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned.     

Results on BRS cohomologies in gauge theory

We compute the cohomology of the Becchi-Rouet-Stora operator in gauge theory over general space-time M without boundary, with structure group G, for class of polynomial functions of the field. We show that the problem reduces to a standard problem for the finite dimensional group G. As a consequence, we prove that, within this class of polynomials, all anomalies and Schwinger terms are obtained from invariants of G.     

Reducible gauge algebra of BRST-invariant constraints

We show that it is possible to formulate the most general first-class gauge algebra of the operator formalism by only using BRST-invariant constraints. In particular, we extend a previous construction for irreducible gauge algebras to the reducible case. The gauge algebra induces two nilpotent, Grassmann-odd, mutually anti-commuting BRST operators that bear structural similarities with BRST/anti-BRST theories but with shifted ghost number assignments. In both cases we show how the extended BRST algebra can be encoded into an operator master equation. A unitarizing Hamiltonian that respects the two BRST symmetries is constructed with the help of a gauge-fixing boson. Abelian reducible theories are shown explicitly in full detail, while non-Abelian theories are worked out for the lowest reducibility stages and ghost momentum ranks.     

格点规范理论(Ⅰ)

本文介绍了由Wilson等人发展起来的处理粒子间强相互作用的格点规范理论。由于这个理论是建立在点阵上的规范理论,故首先讨论了点阵上体系的场论性质和统计物理性质之间的联系,介绍了处理粒子禁闭问题的Wilson判据,点阵的哈密顿形式。然后讨论了各种具体模型的计算方法,如规范场的点阵模型、紧致QED模型、费米子模型、阿贝尔Higgs模型等。在此基础上,总结出Wilson定理。本文也讨论了格点规范理论中的实空间重正化群方法,介绍了Heisenberg平面模型的重正化群分析,一维的二维的复现关系及Migdal近似。最后评介了近年来对于Wilson回路算子的一些研究,内容包括’t Hooft代数和Wilson回路算子方程等。     

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.     

On the gauge variance of action functions under transformations on space-time

The problem of the gauge variance or invariance of action functions in classical mechanics is discussed from a group and path-theoretic viewpoint. By using the elementary theory of the cohomology of groups, criteria are introduced which enable one to decide when action functions gauge variant under a kinematical group are equivalent to action functions invariant under the transformations of the group. The criteria are applied to action functions gauge variant under Lorentz and Galilei transformations, where we deduce that any action function gauge variant under the Lorentz group is equivalent to an action function invariant under Lorentz transformations, whilst action functions gauge variant under the Galilei group are not necessarily equivalent to Galilei-invariant action functions. It is also shown that any action function gauge variant in a more restricted fashion which we define in the text, is necessarily equivalent to a kinetic-energy action.     

BRS cohomology of the supertranslations inD=4

Supersymmetry transformations are a kind of square root of spacetime translations. The corresponding Lie superalgebra always contains the supertranslation operator . We find that the cohomology of this operator depends on a spin-orbit coupling in anSU(2) group and has a quite complicated structure. This spin-orbit type coupling will turn out to be basic in the cohomology of supersymmetric field theories in general.     

Topology of the gauge condition and new confinement phases in non-abelian gauge theories

The gauge-fixing constraint in a gauge field theory is crucial for understanding both short-distance and long-distance behavior of non-abelian gauge field theories. We define what we call “non-propagating” gauge conditions such as the unitary gauge and “approximately non-propagating” or renormalizable gauge conditions, and study their topological properties. By first fixing the non-abelian part of the gauge ambiguity we find that SU(N) gauge theories can be written in the form of abelian gauge theories with N ? 1 fold multiplicity enriched with magnetic monopoles with certain magnetic charge combinations. Their electric chargesare governed by the instanton angle θ.If θ is continuously varied from 0 to 2π and a confinement mode is assumed for some θ, then at least one phase-transition must occur. We speculate on the possibility of new phases: e.g., “oblique confinement,” where $\text{θ ? π}$, and explain some peculiar features of this mode. In principle there may be infinitely many such modes, all separated by phase transition boundaries.     

Wigner’s little group as a generator of gauge transformations

The role of Wigner’s little group, as an Abelian gauge generator in usual and topologically massive gauge theories, is studied.     

Inductive Construction of the Loop Transform for Abelian Gauge Theories

We construct the loop transform in the case of Abelian gauge theories as a unitary operator given by the inductive limit of Fourier transforms on tori. We also show that its range, i.e. the space of kinematical states of the quantum loop representation, is the Hilbert space of square integrable complex valued functions on the group of hoops.     

Quantum Cohomology and the Periodic Toda Lattice

We describe a relation between the periodic one-dimensional Toda lattice and the quantum cohomology of the periodic flag manifold (an infinite-dimensional K?hler manifold). This generalizes a result of Givental and Kim relating the open Toda lattice and the quantum cohomology of the finite-dimensional flag manifold. We derive a simple and explicit “differential operator formula” for the necessary quantum products, which applies both to the finite-dimensional and to the infinite-dimensional situations. Received: 20 April 1999 / Accepted: 12 April 2000