Gauge Orbit Research Articles

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.

Weak gauge-invariance of dimension two condensate in Yang–Mills theory

We give a formal proof that the spacetime average of the vacuum condensate of mass dimension two, i.e., the vacuum expectation value of the squared potential Aμ2, is gauge invariant in the weak sense that it is independent of the gauge-fixing condition adopted in quantizing the Yang–Mills theory. This is shown at least for the small deformation from the generalized Lorentz and the modified maximal Abelian gauge in the naive continuum formulation neglecting Gribov copies. This suggests that the numerical value of the condensate could be the same no matter what gauge-fixing conditions for choosing the representative from the gauge orbit are adopted to measure it. Finally, we discuss how this argument should be modified when the Gribov copies exist.

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.

Quasi-bialgebras and dynamical [formula omitted]-matrices

We study the relationship between general dynamical Poisson groupoids and Lie quasi-bialgebras. For a class of Lie quasi-bialgebras G naturally compatible with a reductive decomposition, we extend the description of the moduli space of classical dynamical r-matrices of Etingof and Schiffmann. We construct, in each gauge orbit, an explicit analytic representative l can . We translate the notion of duality for dynamical Poisson groupoids into a duality for Lie quasi-bialgebras. It is shown that duality maps the dynamical Poisson groupoid for l can and G to the dynamical Poisson groupoid for l can and the dual quasi-bialgebra G ★ .

Non-Abelianizable First Class Constraints

We study the necessary and sufficient conditions on Abelianizable first class constraints. The necessary condition is derived from topological considerations on the structure of gauge group. The sufficient condition is obtained by applying the theorem of implicit function in calculus and studying the local structure of gauge orbits. Since the sufficient condition is necessary for existence of proper gauge fixing conditions, we conclude that in the case of a finite set of non-Abelianizable first class constraints, the Faddeev-Popov determinant is vanishing for any choice of subsidiary constraints. This result is explicitly examined for SO(3) gauge invariant model.

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.

Gauge-invariant coordinates on gauge-theory orbit space

A gauge-invariant field is found which describes physical configurations, i.e. gauge orbits, of non-Abelian gauge theories. This is accomplished with non-Abelian generalizations of the Poincare'-Hodge formula for one-forms. In a particular sense, the new field is dual to the gauge field. Using this field as a coordinate, the metric and intrinsic curvature are discussed for Yang-Mills orbit space for the (2+1)- and (3+1)-dimensional cases. The sectional, Ricci and scalar curvatures are all formally non-negative. An expression for the new field in terms of the Yang-Mills connection is found in 2+1 dimensions. The measure on Schroedinger wave functionals is found in both 2+1 and 3+1 dimensions; in the former case, it resembles Karabali, Kim and Nair's measure. We briefly discuss the form of the Hamiltonian in terms of the dual field and comment on how this is relevant to the mass gap for both the (2+1)- and (3+1)-dimensional cases.

Solving loop equations by Hitchin systems via holography in large-NQCD4

For (planar) closed self-avoiding loops we construct a "holographic" map from the loop equations of large-N QCD_4 to an effective action defined over infinite rank Hitchin bundles. The effective action is constructed densely embedding Hitchin systems into the functional integral of a partially quenched or twisted Eguchi-Kawai model, by means of the resolution of identity into the gauge orbits of the microcanonical ensemble and by changing variables from the moduli fields of Hitchin systems to the moduli of the corresponding holomorphic de Rham local systems. The key point is that the contour integral that occurs in the loop equations for the de Rham local systems can be reduced to the computation of a residue in a certain regularization. The outcome is that, for self-avoiding loops, the original loop equations are implied by the critical equation of an effective action computed in terms of the localisation determinant and of the Jacobian of the change of variables to the de Rham local systems. We check, at lowest order in powers of the moduli fields, that the localisation determinant reproduces exactly the first coefficient of the beta function.

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.

Defining 〈A2〉 in the finite volume Hamiltonian formalism

It is shown how in principle for non-Abelian gauge theories it is possible in the finite volume Hamiltonian framework to make sense of calculating the expectation value of ‖A‖2=∫d3x(Aai(x→))2. Gauge invariance requires one to replace ‖A‖2 by its minimum over the gauge orbit, which makes it a highly non-local quantity. We comment on the difficulty of finding an expression for ‖A‖2min analogous to that found for the Abelian case, and the relation of this question to Gribov copies. We deal with these issues by implementing the Hamiltonian on the so-called fundamental domain, with appropriate boundary conditions in field space, essential to correctly represent the physics of the problem.

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.

Towards an explicit expression of the Seiberg-Witten map at all orders

The Seiberg-Witten map links noncommutative gauge theories to ordinary gauge theories, and allows to express the noncommutative variables in terms of the commutative ones. Its explicit form can be found order by order in the noncommutative parameter theta and the gauge potential A by the requirement that gauge orbits are mapped on gauge orbits. This of course leaves ambiguities, corresponding to gauge transformations, and there is an infinity of solutions. Is there one better, clearer than the others ? In the abelian case, we were able to find a solution, linked by a gauge transformation to already known formulas, which has the property of admitting a recursive formulation, uncovering some pattern in the map. In the special case of a pure gauge, both abelian and non-abelian, these expressions can be summed up, and the transformation is expressed using the parametrisation in terms of the gauge group.

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.

Partial ordering of gauge orbit types for SU n-gauge theories

The natural partial ordering of the orbit types of the action of the group of local gauge transformations on the space of connections in space–time dimension d≤4 is investigated. For that purpose, a description of orbit types in terms of cohomology elements of space–time, derived earlier, is used. It is shown that on the level of these cohomology elements, the partial ordering relation is characterized by a system of algebraic equations. Moreover, operations to generate direct successors and direct predecessors are formulated. The latter allow to successively reconstruct the set of orbit types, starting from the principal type.

Bulk quantization of gauge theories: confined and Higgs phases

We deepen the understanding of the quantization of the Yang-Mills field by showing that the concept of gauge fixing in 4 dimensions is replaced in the 5-dimensional formulation by a procedure that amounts to an $A$-dependent gauge transformation. The 5-dimensional formulation implements the restriction of the physical 4-dimensional gluon field to the Gribov region, while being a local description that is under control of BRST symmetries both of topological and gauge type. The ghosts decouple so the Euclidean probability density is everywhere positive, in contradistinction to the Faddeev-Popov method for which the determinant changes sign outside the Gribov region. We include in our discussion the coupling of the gauge theory to a Higgs field, including the case of spontaneously symmetry breaking. We introduce a minimizing functional on the gauge orbit that could be of interest for numerical gauge fixing in the simulations of spontaneously broken lattice gauge theories. Other new results are displayed, such as the identification of the Schwinger-Dyson equation of the five dimensional formulation in the (singular) Landau gauge with that of the ordinary Faddeev-Popov formulation, order by order in perturbation theory.

QUANTUM AND CLASSICAL GAUGE SYMMETRIES IN A MODIFIED QUANTIZATION SCHEME

The use of the mass term as a gauge fixing term has been studied by Zwanziger, Parrinello and Jona-Lasinio, which is related to the nonlinear gauge [Formula: see text] of Dirac and Nambu in the large mass limit. We have recently shown that this modified quantization scheme is in fact identical to the conventional local Faddeev–Popov formula without taking the large mass limit, if one takes into account the variation of the gauge field along the entire gauge orbit and if the Gribov complications can be ignored. This suggests that the classical massive vector theory, for example, is interpreted in a more flexible manner either as a gauge invariant theory with a gauge fixing term added, or as a conventional massive nongauge theory. As for massive gauge particles, the Higgs mechanics, where the mass term is gauge-invariant, has a more intrinsic meaning. It is suggested that we extend the notion of quantum gauge symmetry (BRST symmetry) not only to classical gauge theory but also to a wider class of theories whose gauge symmetry is broken by some extra terms in the classical action. We comment on the implications of this extended notion of quantum gauge symmetry.

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.