Gauge Orbit Research Articles

COCYCLES IN TOPOLOGICAL YANG-MILLS THEORIES

We show that the cocycles arise in the Yang-Mills theories with topological terms added. They are found in the representation of the translation group in the configuration space or gauge orbit space, and are shown to be directly related to the U(1) functional connection in these spaces. In 3+1 dimensions, if we adopt a vector representation (i.e. vanishing 2-cocycle), we find the θ-parameter in the Pontryagin term to be 2π×integers. While, in 2+1 dimensions, insisting that finite translations be associative (i.e. vanishing 3-cocycle) leads to the known quantization of the topological mass. Some related aspects with anomalies are also discussed.

Induced angular momentum and gauge invariance.

We give a detailed analysis of the eduction of gauge invariance in our calculation of induced angular momentum (2+1)-dimensional QED. We present a detailed account of the trace-identity calculation of induced angular momentum. We give a general procedure for constructing gauge-covariant operators from those which are defined at only one point of the gauge orbit. Finally we give a critical analysis of other calculations of induced angular momentum and the controversy that has arisen over them.

Geometry of classical mechanics with non-Abelian gauge symmetry

The geometry of the gauge orbit bundle is considered in SO(3) Yang-Mills mechanics. It is shown that it can serve as a finite-dimensional model of the geometry of non-Abelian field theory with respect to properties such as the absence of a global gauge, the Gribov ambiguities, the convexity of the region within the corresponding horizon, and stratification. The connection and Lagrangian on the orbit space are obtained.

Geometry and anomalies in superconnection space

Using the concepts of gauge and supersymmetry orbits in superconnection space we give a geometrical derivation of the consistency condition for both gauge and supersymmetry anomalies. A relationship between supersymmetry anomalies in different approaches is established. An application is studied.

Cohomology of gauge fields and explicit construction of Wess-Zumino lagrangians in nonlinear sigma models over G/H

By generalizing the well-known Chern-Weil and Chern-Simons theorems, we are able to define a tower of cocycles in the entire space of gauge potentials. This construction not only induces (though not directly) the usual tower of cocycles on a gauge orbit, but also generates nontrivial cocycles, by appropriate projection, in the gauge-orbit space. As an application, using the Chern-Simons one-cocycle in this new tower, we explicitly construct the Wess-Zumino action in nonlinear sigma models over a homogeneous space G/H in any even dimensions.

Cohomology of the gauge orbit space and (2+1)-dimensional Yang-Mills theory with the Chern-Simons term

We consider Yang-Mills theory with the Chern-Simons term in 2+1 dimensions. We first formulate via a Dirac constraint analysis the non-degenerate classical hamiltonian system on the cotangent bundle of the gauge orbit space equipped with a non-degenerate symplectic structure which turns out to be different from the canonical one. We then explain the quantization in the canonical formalism, and point out the relationship of the quantization of the parameter involved in the Chern-Simons term and the second cohomology group H 2( M, Z ) of the gauge orbit space M .

Abelian gauge structure inside nonabelian gauge theories

We show that the inclusion of topological lagrangians in nonabelian gauge theories introduces certain topologically nontrivial abelian background fields in the configuration space of these theories. In particular, the θ-term and the topological mass term lead, respectively, to a vortex and a monopole in gauge orbit space in 3 + 1 and 2 + 1 dimensions. The identification of the theta-vacuum effect with a kind of Bohm-Aharonov effect in gauge-orbit space motivates a discussion on the possibility of solving the strong CP problem dynamically.

The topological meaning of non-abelian anomalies

We show how the non-abelian anomaly for gauge fields coupled to Weyl fermions in 2 n 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 (2 n + 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.

Path integral for gravity and supergravity

It is shown that the specification of the path integral variables in quantum gravity, which was previously formulated on the basis of the anomaly consideration and the BRS supersymmetry, can also be understood by the more familiar consideration of the unit jacobian approach. We describe how to derive the same prescription in the framework of the basic path integral formalism without recourse to the BRS supersymmetry, which is one of the consequences of the Faddeev-Popov effective action. A novel feature of quantum gravity (and the gauge theory in curved space-time in general) is that a proper choice of the invariant gauge orbit volume element becomes essential to ensure the gauge independence of the partition function. We also briefly note that the conformal (Weyl) symmetry is generally spoiled by the anomaly in quantum theory. The connection between the present “anomaly-free” prescription and the true gravitational anomaly by Alvarez-Gaumé and Witten is also clarified.

The action of the group of bundle-automorphisms on the space of connections and the geometry of gauge theories

Some aspects of the geometry of gauge theories are sketched in this review. We deal essentially with Yang-Mills theory, discussing the structure of the space of gauge orbits and the geometrical interpretation of ghosts and anomalies. Occasionally we deal also with classical ⪡gauge theories⪢ of gravitation and in particular we study the action of the group of diffeomorphisms on the space of linear connections. Finally we comment on the mathematical interpretation of anomalies in field theories.

An effective potential for classical Yang-Mills fields as outline for bifurcation on gauge orbit space

Classical Yang-Mills (Y.M.) equations with static external sources are formulated as a Hamiltonian system with gauge symmetry in A 0 = 0 gauge. Using the concept of a “momentum mapping” (J. Marsden and A. Weinstein, Rep. Math. Phys. 5 (1974), 121) on symplectic manifolds with symmetry, an analogue of centrifugal potential of a mass point in a spherically symmetric potential is derived. This gives rise to an effective potential V eff, whose critical points are rigorously proved to be in one-to-one correspondence with static Y.M. solutions. V eff additionally depends on the prescribed external source ϱ, which is as a constant of motion analogous to angular moment of the mass point. Thus bifurcation of static solutions is caused by bifurcation of critical points of V eff under variation of the external parameter ϱ. Some closing remarks on dynamics and stability on gauge orbit space are added.

An effective potential for classical Yang-Mills fields as outline for bifurcation on gauge orbit space

An effective potential for classical Yang-Mills fields as outline for bifurcation on gauge orbit space

On the bundle of connections and the gauge orbit manifold in Yang-Mills theory

In an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations (in Yang-Mills theory) is aC ∞ principal fibration. The underlying quotient space, the gauge orbit space, is seen explicitly to be aC ∞ manifold modelled on a Hilbert space.

Attraction to large gauge orbits

SU( N) gauge systems are attracted to large orbits of the global gauge group by an invariant and calculable potential. In models the wave function becomes increasingly localized near the maximal orbit as N → ∞, which explains the success of semiclassical methods. Picturing the links U i of the lattice as particles on an SU( N) group manifold, the effect favors large moments of inertia about U = 1. It thus opposes the magnetic interaction, and tends to destabilize the perturbative vacuum.

Regularized, continuum Yang-Mills process and Feynman-Kac functional integral

Giving an ultraviolet regularization and volume cut off we construct a nuclear Riemannian structure on the Hilbert manifold $$\mathfrak{M}$$ of gauge orbits. This permits us to define a regularized Laplace-Beltrami operator δ on $$\mathfrak{M}$$ and an associated global diffusion in $$\mathfrak{M}$$ governed by δ. This enables us to define, via a Feynman-Kac integral, a Euclidean, continuum regularized Yang-Mills process corresponding to a suitable regularization (of the kinetic term) of the classical Yang-Mills Lagrangian onT $$\mathfrak{M}$$ .