Gauge Orbit Space Research Articles

Gauge and scalar fields on CP2 : A gauge-invariant analysis. II. The measure for gauge fields and a 4d WZW theory

We consider the volume of the gauge orbit space for gauge fields on four-dimensional complex projective space. The analysis uses a parametrization of gauge fields where gauge transformations act homogeneously on the fields, facilitating a manifestly gauge-invariant analysis. The volume element contains a four-dimensional Wess-Zumino-Witten (WZW) action for a hermitian matrix-valued field. There is also a mass-like term for certain components of the gauge field. We discuss how the mass term could be related to results from lattice simulations as well as Schwinger-Dyson equations. We argue for a kinematic regime where the Yang-Mills theory can be approximated by the 4d-WZW theory. The result is suggestive of the instanton liquid picture of QCD. Further it is also indicative of the mechanism for confinement being similar to what happens in two dimensions.

Fermions, mass-gap and Landau levels: gauge invariant Hamiltonian for QCD in D = 2+1

A gauge-invariant reformulation of QCD in three spacetime dimensions is presented within a Hamiltonian formalism, extending previous work to include fermion fields in the adjoint and fundamental representations. A priori there are several ways to define the gauge-invariant versions of the fermions; a consistent prescription for choosing the fermionic variables is presented. The fermionic contribution to the volume element of the gauge orbit space and the gluonic mass-gap is computed exactly and this contribution is shown to be closely related to the mechanism for induction of Chern–Simons terms by parity-odd fermions. The consistency of the Hamiltonian scheme with known results on index theorems, Landau levels and renormalization of Chern–Simons level numbers is shown in detail. We also comment on the fermionic contribution to the volume element in relation to issues of confinement and screening.

Remarks on the thermodynamics and the vacuum energy of a quantum Maxwell gas on compact and closed manifolds

The quantum Maxwell theory at finite temperature at equilibrium is studied on compact and closed manifolds in both the functional integral and Hamiltonian formalism. The aim is to shed some light onto the interrelation between the topology of the spatial background and the thermodynamic properties of the system. The quantization is not unique and gives rise to inequivalent quantum theories which are classified by θ-vacua. Based on explicit parametrizations of the gauge orbit space in the functional integral approach and of the physical phase space in the canonical quantization scheme, the Gribov problem is resolved and the equivalence of both quantization schemes is elucidated. Using zeta-function regularization the free energy is determined and the effect of the topology of the spatial manifold on the vacuum energy and on the thermal gauge field excitations is clarified. The general results are then applied to a quantum Maxwell gas on an n-dimensional torus providing explicit formulae for the main thermodynamic functions in the low- and high-temperature regimes, respectively.

Supersymmetry and mass gap in2+1dimensions: A gauge invariant Hamiltonian analysis

A Hamiltonian formulation of Yang-Mills-Chern-Simons theories with $0\leq N\leq 4$ supersymmetry in terms of gauge-invariant variables is presented, generalizing earlier work on nonsupersymmetric gauge theories. Special attention is paid to the volume measure of integration (over the gauge orbit space of the fields) which occurs in the inner product for the wave functions and arguments relating it to the renormalization of the Chern-Simons level number and to mass-gaps in the spectrum of the Hamiltonians are presented. The expression for the integration measure is consistent with the absence of mass gap for theories with extended supersymmetry (in the absence of additional matter hypermultiplets and/or Chern-Simons couplings), while for the minimally supersymmetric case, there is a mass-gap, the scale of which is set by a renormalized level number, in agreement with indications from existing literature. The realization of the supersymmetry algebra and the Hamiltonian in terms of the gauge invariant variables is also presented.

Properties of the generalN-Higgs-doublet model. I. The orbit space

We study the scalar sector of the general N-Higgs-doublet model via geometric constructions in the space of gauge orbits. We give a detailed description of the shape of the orbit space both for general N and, in more detail, for N=3. We also comment on remarkable analogies between NHDM and quantum information theory.

Properties of the general NHDM. II. Higgs potential and its symmetries

We continue our analysis of the general N-Higgs-doublet model and focus of the Higgs potential description in the space of gauge orbits. We develop a geometric technique that allows us to study the global minimum of the potential without explicitly finding its position. We discuss symmetry patterns of the NHDM potential, and illustrate the general discussion with various specific variants of the three-Higgs-doublet model.

Quark and gluon confinement within the variational approach to Yang-Mills theory in Coulomb gauge

The Yang-Mills Schrodinger equation is variationally solved in Coulomb gauge for the vacuum sector using a trial wave functional, which uses a Gaussian for the radial wave functional. The resulting Schwinger-Dyson equations are, for the first time, solved in a fully self-consistent way. We find the absence of gluons in the infrared and also a confining quark potential. Confinement is shown to be a consequence of the curvature of the space of gauge orbits.

Quark and gluon confinement in Yang-Mills theory in Coulomb gauge with the variational approach

The Yang-Mills Schrodinger equation is variationally solved in Coulomb gauge for the vacuum sector using a trial wave functional, which uses a Gaussian for the radial wave functional. The resulting Schwinger-Dyson equations are, for the first time, solved in a fully self-consistent way. We find the absence of gluons in the infrared and also a confining quark potential. Confinement is shown to be a consequence of the curvature of the space of gauge orbits.

Variational solution of the Yang-Mills Schrödinger equation in Coulomb gauge

The Yang-Mills Schr\"odinger equation is solved in Coulomb gauge for the vacuum by the variational principle using an Ansatz for the wave functional, which is strongly peaked at the Gribov horizon. A coupled set of Schwinger-Dyson equations for the gluon and ghost propagators in the Yang-Mills vacuum as well as for the curvature of gauge orbit space is derived and solved in one-loop approximation. We find an infrared suppressed gluon propagator, an infrared singular ghost propagator, and an almost linearly rising confinement potential.

How to study the physical relevance of gauge orbit space singularities?

Motivated by gauge theory, a strategy to quantize a system whose classical configuration space is the orbit space of a Lie group action is proposed. Special emphasis is placed on how the natural stratification of this space, induced by orbit types, is encoded in the quantum theory.

On the gauge orbit space stratification: a review

First, we review the basic mathematical structures and results concerning the gauge orbit space stratification. This includes general properties of the gauge group action, fibre bundle structures induced by this action, basic properties of the stratification and the natural Riemannian structures of the strata. In the second part, we study the stratification for theories with gauge group $\rmSU(n)$ in space time dimension 4. We develop a general method for determining the orbit types and their partial ordering, based on the 1-1 correspondence between orbit types and holonomy-induced Howe subbundles of the underlying principal $\rmSU(n)$-bundle. We show that the orbit types are classified by certain cohomology elements of space time satisfying two relations and that the partial ordering is characterized by a system of algebraic equations. Moreover, operations for generating direct successors and direct predecessors are formulated, which allow one to construct the set of orbit types, starting from the principal one. Finally, we discuss an application to nodal configurations in topological Chern-Simons theory.

Classification of gauge orbit types for SU n gauge theories

A method for determining the orbit types of the action of the group of gauge transformations on the space of connections for gauge theories with gauge group SU(n) in spacetime dimension d≤4 is presented. The method is based on the one-to-one correspondence between orbit types and holonomy-induced reductions of the underlying principal SU(n)-bundle. It is shown that the orbit types are labelled by certain cohomology elements of spacetime satisfying two relations. Thus, for every principal SU(n)-bundle the corresponding stratification of the gauge orbit space can be explicitly determined. As an application, a criterion characterizing kinematical nodes for physical states in Yang–Mills theory with the Chern–Simons term proposed by Asorey et al. is discussed.

Extremal curves in [formula omitted]-dimensional Yang–Mills theory

We examine the structure of the potential energy of (2+1)-dimensional Yang–Mills theory on a torus with gauge group SU(2). We use a standard definition of distance on the space of gauge orbits. The curves of extremal potential energy in orbit space satisfy a certain partial differential equation. We argue that the energy spectrum is gapped because the extremal curves are of finite length. Though classical gluon waves satisfy our differential equation, they are not extremal curves. We construct examples of extremal curves and find how the length of these curves depends on the dimensions of the torus. The intersections with the Gribov horizon are determined explicitly. The results are discussed in the context of Feynman's ideas about the origin of the mass gap.

Quantizing Yang-Mills theory: From Parisi-Wu stochastic quantization to a global path integral

Based on a generalization of the stochastic quantization scheme we recently proposed a generalized, globally defined Faddeev-Popov path integral density for the quantization of Yang-Mills theory. In this talk first our approach on the whole space of gauge potentials is shortly reviewed; in the following we discuss the corresponding global path integral on the gauge orbit space relating it to the original Parisi-Wu stochastic quantization scheme.

Stratification of the Generalized Gauge Orbit Space

The action of Ashtekar's generalized gauge group $\Gb$ on the space $\Ab$ of generalized connections is investigated for compact structure groups $\LG$. First a stratum is defined to be the set of all connections of one and the same gauge orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice theorem is proven on $\Ab$. This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, $\Ab$ is topologically regularly stratified by $\Gb$. These results coincide with those of Kondracki and Rogulski for Sobolev connections. As a by-product, we prove that the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of $\LG$. Finally, we show that the set of all gauge orbits with maximal type has the full induced Haar measure 1.

Global path integral quantization of Yang–Mills theory

Based on a generalization of the stochastic quantization scheme recently a modified Faddeev–Popov path integral density for the quantization of Yang–Mills theory was derived, the modification consisting in the presence of specific finite contributions of the pure gauge degrees of freedom. Due to the Gribov problem the gauge fixing can be defined only locally and the whole space of gauge potentials has to be partitioned into patches. We propose a global path integral density for the Yang–Mills theory by summing over all patches, which can be proven to be manifestly independent of the specific local choices of patches and gauge fixing conditions, respectively. In addition to the formulation on the whole space of gauge potentials we discuss the corresponding global path integral on the gauge orbit space relating it to the original Parisi–Wu stochastic quantization scheme and to a proposal of Stora, respectively.

Maximal non-abelian gauges and topology of gauge orbit space

We introduce two maximal non-abelian gauge fixing conditions on the space of gauge orbits M for gauge theories over spaces with dimensions d ⩽ 3. The gauge fixings are complete in the sense that they describe an open dense set M 0 of the space of gauge orbits M and select one and only one gauge field per gauge orbit in M 0. There are no Gribov copies or ambiguities in these gauges. M 0 is a contractible manifold with trivial topology. The set of gauge orbits which are not described by the gauge conditions M / M 0 is the boundary of M 0 and encodes all non-trivial topological properties of the space of gauge orbits. The gauge field configurations of this boundary M / M 0 can be explicitly identified with non-abelian monopoles and they are shown to play a very relevant role in the non-perturbative behavior of gauge theories in one, two and three space dimensions. It is conjectured that their role is also crucial for quark confinement in 3+1-dimensional gauge theories.

Geometrical aspects of chiral anomalies in the overlap

The set of one dimensional lowest energy eigenspaces used to construct the overlap induces a two form on gauge orbit space which is the locally exact curl of Berry's connection. If anomalies do not cancel, examples of two dimensional closed sub-manifolds of orbit space are produced over which the integral of the above two form does not vanish. Based on these observations, a natural definition of covariant currents is obtained, a simple way to calculate chiral anomalies on the lattice is found, and indications for how to construct an ideal regularization of chiral gauge theories are seen to emerge.

Some remarks on the cohomology of SU(3) gauge orbit space

I compute the rational cohomology ring of the physical configuration space of gauge theories with structure group SU(3) over a simply connected four-manifold. The consequences of this computation are analyzed, in relation with gauge anomalies of the Dirac operator coupled with gauge field and with possible definition of SU(3)-polynomial invariants for smooth manifolds.

NONPERTURBATIVE QUANTUM GEOMETRY AND TORSION

The relevance of torsion in nonperturbative quantum geometry is studied here from the viewpoint of the equivalence of the loop space to the space of gauge potentials modulo gauge transformations satisfying Mandelstam constraints. The topological feature associated with the gauge orbit space of a non-Abelian gauge theory when the topological θ term is introduced in the Lagrangian corresponds to a vortex line and the gauge orbit space appears to be multiply connected in nature. This has an implication in the loop space formalism, in the sense that the latter involves nonlocality and there is no way we could arrive at a corresponding continuum limit. In the gravity without the metric formalism of Capovilla, Jacobson and Dell, the θ term in the Lagrangian corresponds to torsion. This suggests that in the construction of a solution of functionals annihilated by the Hamiltonian constraint, any regularization procedure which destroys the gauge invariance of the loop space variables also destroys the topology of the gauge orbit space and the continuum limit can be achieved only by removing the vortex line. Thus the constraint equations of canonical quantization of gravity can be achieved only in the limit of torsion tending to zero. This provides the link between covariant and canonical quantization of gravity and demonstrates explicitly the role of the arrow of time in nonperturbative quantum geometry also when we take the gauge-invariant holonomy variables as the fundamental entity.