A gauge-invariant Hamiltonian analysis for non-Abelian gauge theoreiesin (2+1) dimensions

Abstract

The Hamiltonian formulation for a non-Abelian gauge theory in two spatial dimensions is carried out in terms of a gauge-invariant matrix parametrization of the fields. The Jacobian for the relevant transformation of variables is given in terms of the WZW-action for a hermitian matrix field. Some gauge-invariant eigenstates of the kinetic term of the Hamiltonian are given; these have zero charge and exhibit a mass gap.

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

“…Further, the formalism has been extended [10] to the Yang-Mills-Chern-Simons theory [11]. Most importantly, at least as far as the focus of the present paper is concerned, the origin of the mass-gap in the purely gluonic theory can be understood in a geometric fashion as the effect of the volume measure on C, a fact made transparent in terms of manifestly gauge-invariant variables [1,12].…”

“…In this paper we present a reformulation of N = 1, 2 and 4 supersymmetric Yang-Mills theories (with and without Chern-Simons couplings) in three spacetime dimensions in terms of gauge-invariant variables, following the Hamiltonian approach in [1]. As part of this reformulation, we compute the volume measure on the physical gaugeinvariant configuration space C of the theories under consideration and study the interplay between dynamical mass-generation and supersymmetry in D = 2 + 1.…”

“…Keeping these broad motivations in mind it is interesting to note that, in the case of the purely gluonic theory, it has been possible to do ab-initio strong coupling computations 1 of certain physical quantities using the gauge-invariant Hamiltonian formalism [1]. For example, systematic computations of the string tension [6], screening effects [7] and the inclusion of the effects of non-trivial spatial geometries [8] have been studied within this framework.…”

“…The potent methods that enable these computations often use manifest supersymmetry in one way or another and are not readily generalizable to non-supersymmetric theories. A natural question to ask is if these two approaches, namely, the gauge-invariant Hamiltonian point of view due to KKN [1], which does not rely on supersymmetry in any way, and the powerful computational tools that have been so successful in the recent studies of supersymmetric gauge theories in three dimensions, can be fruitfully combined. In this paper we take a step towards answering this question.…”

“…For the pure glue theory, this measure is given in terms of a Wess-Zumino-Witten (WZW) functional (24) with a level number equal to 2 c A , where c A is the quadratic Casimir value for the adjoint representation of the gauge group (which we shall take as SU(N)) [19,20]. Ultimately, this level number determines the mass-gap in the spectrum of the theory [1], the basic argument for which is also outlined in this section. We also discuss the effect of the Chern-Simons coupling (when it is present), and its renormalization, on the measure and the massgap.…”