Confinement in a three-dimensional Yang–Mills theory

We show that, starting from known exact classical solutions of the Yang–Mills theory in three dimensions, the string tension is obtained and the potential is consistent with a marginally confining theory. The potential we obtain agrees fairly well with preceding findings in the literature but here we derive it analytically from the theory without further assumptions. The string tension is in strict agreement with lattice results and the well-known theoretical result by Karabali–Kim–Nair analysis. Classical solutions depend on a dimensionless numerical factor arising from integration. This factor enters into the determination of the spectrum and has been arbitrarily introduced in some theoretical models. We derive it directly from the solutions of the theory and is now fully justified. The agreement obtained with the lattice results for the ground state of the theory is well below 1% at any value of the degree of the group.     

The exact decomposition of gauge variables in lattice Yang–Mills theory

In this Letter, we consider lattice versions of the decomposition of the Yang–Mills field a la Cho–Faddeev–Niemi, which was extended by Kondo, Shinohara and Murakami in the continuum formulation. For the SU(N)$\mathit{SU}(N)$ gauge group, we propose a set of defining equations for specifying the decomposition of the gauge link variable and solve them exactly without using the ansatz adopted in the previous studies for SU(2)$\mathit{SU}(2)$ and SU(3)$\mathit{SU}(3)$. As a result, we obtain the general form of the decomposition for SU(N)$\mathit{SU}(N)$ gauge link variables and confirm the previous results obtained for SU(2)$\mathit{SU}(2)$ and SU(3)$\mathit{SU}(3)$.     

Slavnov–Taylor identities in Coulomb gauge Yang–Mills theory

The Slavnov–Taylor identities of Coulomb gauge Yang–Mills theory are derived from the (standard, second order) functional formalism. It is shown how these identities form closed sets from which one can in principle fully determine the Green’s functions involving the temporal component of the gauge field without approximation, given appropriate input.     

Mathematical quantum Yang–Mills theory revisited

A mathematically rigorous relativistic quantum Yang–Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is nonperturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers Dynin ([11] and [12]). The principal result is established anew: due to the quartic self-interaction term in the Yang–Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang–Mills energy spectrum has a positive mass gap. Furthermore, the quantum Yang–Mills Hamiltonian has a countable orthogonal eigenbasis in a Fock space, so that the quantum Yang–Mills spectrum is point and countable. In addition, a fine structure of the spectrum is elucidated.     

Quantum criticality and Yang–Mills gauge theory

We present a family of nonrelativistic Yang–Mills gauge theories in D+1$D+1$ dimensions whose free-field limit exhibits quantum critical behavior with gapless excitations and dynamical critical exponent z=2$z=2$. The ground state wavefunction is intimately related to the partition function of relativistic Yang–Mills in D   dimensions. The gauge couplings exhibit logarithmic scaling and asymptotic freedom in the upper critical spacetime dimension, equal to 4+1$4+1$. The theories can be deformed in the infrared by a relevant operator that restores Poincaré invariance as an accidental symmetry. In the large-N limit, our nonrelativistic gauge theories can be expected to have weakly curved gravity duals.     

Super Yang–Mills theory from nonlinear supersymmetry

The relation between a nonlinear supersymmetric (NLSUSY) theory and a SUSY Yang–Mills (SYM) theory is studied for N=3$N=3$ SUSY in two-dimensional space–time. We explicitly show the NL/L SUSY relation for the (pure) SYM theory by means of cancellations among Nambu–Goldstone fermion self-interaction terms.     

Equal-time two-point correlation functions in Coulomb gauge Yang–Mills theory

We apply a functional perturbative approach to the calculation of the equal-time two-point correlation functions and the potential between static color charges to one-loop order in Coulomb gauge Yang–Mills theory. The functional approach proceeds through a solution of the Schrödinger equation for the vacuum wave functional to order g²$g^2$ and derives the equal-time correlation functions from a functional integral representation via new diagrammatic rules. We show that the results coincide with those obtained from the usual Lagrangian functional integral approach, extract the beta function, and determine the anomalous dimensions of the equal-time gluon and ghost two-point functions and the static potential under the assumption of multiplicative renormalizability to all orders.     

Asymptotic freedom of Yang–Mills theory with gravity

We study the behaviour of Yang–Mills theory under the inclusion of gravity. In the weak-gravity limit, the running gauge coupling receives no contribution from the gravitational sector, if all symmetries are preserved. This holds true with and without cosmological constant. We also show that asymptotic freedom persists in general field-theory-based gravity scenarios including gravitational shielding as well as asymptotically safe gravity.     

Yang–Mills Algebra

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang–Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Koszul of global dimension 3 and Gorenstein but except for s=1 (i.e. in the two-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin–Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincaré series of this algebra and for the dimension in degree n of the graded Lie algebra of which is the universal enveloping algebra. In the four-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.     

Loop Expansion in Yang–Mills Thermodynamics

We argue that a self-consistent spatial coarse graining, which involves interacting (anti)calorons of unit topological charge modulus, implies that real-time loop expansions of thermodynamical quantities in the deconfining phase of SU(2) and SU(3) Yang–Mills thermodynamics are, modulo one-particle irreducible resummations, determined by a finite number of connected bubble diagrams.     

Study of Yang–Mills–Chern–Simons theory in presence of the Gribov horizon

The two-point gauge correlation function in Yang–Mills–Chern–Simons theory in three dimensional Euclidean space is analysed by taking into account the non-perturbative effects of the Gribov horizon. In this way, we are able to describe the confinement and de-confinement regimes, which naturally depend on the topological mass and on the gauge coupling constant of the theory.     

Homological perspective on edge modes in linear Yang–Mills and Chern–Simons theory

Letters in Mathematical Physics - We provide an elegant homological construction of the extended phase space for linear Yang–Mills theory on an oriented and time-oriented Lorentzian manifold...     

Quark-confinement mechanism for SU(2) Yang–Mills theory in abelian gauge

By dimensional reduction in the sense of Parisi and Sourlas (PS), the gauge fixing term in the abelian gauge of the SU(2) Yang–Mills field is reduced to a two-dimensional O(3) nonlinear model. The confinement potential is obtained from magnetic monopoles and frame fluctuations. But the source of quark confinement is frame fluctuations and not magnetic monopoles. Because the frame cannot be regarded as a fixed one, the abelian projected SU(2) Yang–Mills field turns into a gauge field – one group element being with fixed frame , another group gauging the frame . The nonperturbative part becomes a dynamical gauge field in two dimensions, giving rise to the short range linear potential. Received: 4 September 2000 / Published online: 23 February 2001     

A BRST gauge-fixing procedure for Yang–Mills theory on sphere

A gauge-fixing procedure for the Yang–Mills theory on an n  -dimensional sphere (or a hypersphere) is discussed in a systematic manner. We claim that Adler's gauge-fixing condition used in massless Euclidean QED on a hypersphere is not conventional because of the presence of an extra free index, and hence is unfavorable for the gauge-fixing procedure based on the BRST invariance principle (or simply BRST gauge-fixing procedure). Choosing a suitable gauge condition, which is proved to be equivalent to a generalization of Adler's condition, we apply the BRST gauge-fixing procedure to the Yang–Mills theory on a hypersphere to obtain consistent results. Field equations for the Yang–Mills field and associated fields are derived in manifestly O(n+1)$\mathrm{O}(n+1)$ covariant or invariant forms. In the large radius limit, these equations reproduce the corresponding field equations defined on the n-dimensional flat space.     

Lorentz-violating Yang–Mills theory: discussing the Chern–Simons-like term generation

We analyze the Chern–Simons-like term generation in the CPT-odd Lorentz-violating Yang–Mills theory interacting with fermions. Moreover, we study the anomalies of this model as well as its quantum stability. The whole analysis is performed within the algebraic renormalization theory, which is independent of the renormalization scheme. In addition, all results are valid to all orders in perturbation theory. We find that the Chern–Simons-like term is not generated by radiative corrections, just like its Abelian version. Additionally, the model is also free of gauge anomalies and quantum stable.