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Register a new userLight-cone SU(2) Yang-Mills theory and conformal mechanics
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We examine the mechanical matrix model that can be derived from the SU(2) Yang-Mills light-cone field theory by restricting the gauge fields to depend on the light-cone time alone. We use Diracs generalized Hamiltonian approach. In contrast to its well-known instant-time counterpart the light-cone version of SU(2) Yang-Mills mechanics has in addition to the constraints, generating the SU(2) gauge transformations, the new first and second class constraints also. On account of all of these constraints a complete reduction in number of the degrees of freedom is performed. It is argued that the classical evolution of the unconstrained degrees of freedom is equivalent to a free one-dimensional particle dynamics. Considering the complex solutions to the second class constraints we show at this time that the unconstrained Hamiltonian system represents the well-known model of conformal mechanics with a ``strength of the inverse square interaction determined by the value of the gauge field spin.
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We study the classical dynamics of mechanical model obtained from the light-cone version of SU(2) Yang-Mills field theory under the supposition of gauge potential dependence only on ``time along the light-cone direction. The computer algebra system Maple was used strongly to compute and separate the complete set of constraints. In contrast to the instant form of Yang-Mills mechanics the constraints here represent a mixed form of first and second-class constraints and reduce the number of the physical degrees of freedom up to four canonical one.
By using the method of center projection the center vortex part of the gauge field is isolated and its propagator is evaluated in the center Landau gauge, which minimizes the open 3-dimensional Dirac volumes of non-trivial center links bounded by the closed 2-dimensional center vortex surfaces. The center field propagator is found to dominate the gluon propagator (in Landau gauge) in the low momentum regime and to give rise to an OPE correction to the latter of ${sqrt{sigma}}/{p^3}$.The screening mass of the center vortex field vanishes above the critical temperature of the deconfinement phase transition, which naturally explains the second order nature of this transition consistent with the vortex picture. Finally, the ghost propagator of maximal center gauge is found to be infrared finite and thus shows that the coset fields play no role for confinement.
This paper deals with various interrelations between strings and surfaces in three dimensional ambient space, two dimensional integrable models and two dimensional and four dimensional decomposed SU(2) Yang-Mills theories. Initially, a spinor version of the Frenet equation is introduced in order to describe the differential geometry of static three dimensional string-like structures. Then its relation to the structure of the su(2) Lie algebra valued Maurer-Cartan one-form is presented; while by introducing time evolution of the string a Lax pair is obtained, as an integrability condition. In addition, it is show how the Lax pair of the integrable nonlinear Schroedinger equation becomes embedded into the Lax pair of the time extended spinor Frenet equation and it is described how a spinor based projection operator formalism can be used to construct the conserved quantities, in the case of the nonlinear Schroedinger equation. Then the Lax pair structure of the time extended spinor Frenet equation is related to properties of flat connections in a two dimensional decomposed SU(2) Yang-Mills theory. In addition, the connection between the decomposed Yang-Mills and the Gauss-Godazzi equation that describes surfaces in three dimensional ambient space is presented. In that context the relation between isothermic surfaces and integrable models is discussed. Finally, the utility of the Cartan approach to differential geometry is considered. In particular, the similarities between the Cartan formalism and the structure of both two dimensional and four dimensional decomposed SU(2) Yang-Mills theories are discussed, while the description of two dimensional integrable models as embedded structures in the four dimensional decomposed SU(2) Yang-Mills theory are presented.
The analysis of the large-$N$ limit of $U(N)$ Yang-Mills theory on a surface proceeds in two stages: the analysis of the Wilson loop functional for a simple closed curve and the reduction of more general loops to a simple closed curve. In the case of the 2-sphere, the first stage has been treated rigorously in recent work of Dahlqvist and Norris, which shows that the large-$N$ limit of the Wilson loop functional for a simple closed curve in $S^{2}$ exists and that the associated variance goes to zero. We give a rigorous treatment of the second stage of analysis in the case of the 2-sphere. Dahlqvist and Norris independently performed such an analysis, using a similar but not identical method. Specifically, we establish the existence of the limit and the vanishing of the variance for arbitrary loops with (a finite number of) simple crossings. The proof is based on the Makeenko-Migdal equation for the Yang-Mills measure on surfaces, as established rigorously by Driver, Gabriel, Hall, and Kemp, together with an explicit procedure for reducing a general loop in $S^{2}$ to a simple closed curve. The methods used here also give a new proof of these results in the plane case, as a variant of the methods used by L{e}vy. We also consider loops on an arbitrary surface $Sigma$. We put forth two natural conjectures about the behavior of Wilson loop functionals for topologically trivial simple closed curves in $Sigma.$ Under the weaker of the conjectures, we establish the existence of the limit and the vanishing of the variance for topologically trivial loops with simple crossings that satisfy a smallness assumption. Under the stronger of the conjectures, we establish the same result without the smallness assumption.
We consider spatial coarse-graining in statistical ensembles of non-selfintersecting and one-fold selfintersecting center-vortex loops as they emerge in the confining phase of SU(2) Yang-Mills thermodynamics. This coarse-graining is due to a noisy environment and described by a curve shrinking flow of center-vortex loops locally embedded in a two-dimensional flat plane. The renormalization-group flow of an effective `action, which is defined in purely geometric terms, is driven by the curve shrinking evolution. In the case of non-selfintersecting center-vortex loops, we observe critical behavior of the effective `action as soon as the center-vortex loops vanish from the spectrum of the confining phase due to curve shrinking. This suggest the existence of an asymptotic mass gap. An entirely unexpected behavior in the ensemble of one-fold selfintersecting center-vortex loops is connected with the spontaneous emergence of order. We speculate that the physics of planar, one-fold selfintersecting center-vortex loops to be relevant for two-dimensional systems exhibiting high-temperature superconductivity.
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