Local existence proofs for the boundary value problem for static spherically symmetric Einstein–Yang–Mills fields with compact gauge groups

Given the local observables in the vacuum sector fulfilling a few basic principles of local quantum theory, we show that the superselection structure, intrinsically determined a priori, can always be described by a unique compact global gauge group acting on a field algebra generated by field operators which commute or anticommute at spacelike separations. The field algebra and the gauge group are constructed simultaneously from the local observables. There will be sectors obeying parastatistics, an intrinsic notion derived from the observables, if and only if the gauge group is non-Abelian. Topological charges would manifest themselves in field operators associated with spacelike cones but not localizable in bounded regions of Minkowski space. No assumption on the particle spectrum or even on the covariance of the theory is made. However the notion of superselection sector is tailored to theories without massless particles. When translation or Poincare covariance of the vacuum sector is assumed, our construction leads to a covariant field algebra describing all covariant sectors.

In this paper, we shall describe some recent results, ([6,7]), concerning the existence of smooth, globally defined static solutions of the Einstein /Yang-Mills (EYM) equations. The EYM equations extend Einstein’s celebrated gravitational field equations, to include other “force” fields: electromagnet ism, and the weak and strong nuclear force fields. It turns out that solutions of such coupled systems of equations, in n = 4 space-time dimensions, have some very interesting mathematical and physical properties. Thus for the non-abelian gauge group G = SU (2), we can prove the existence of infinitely many non-singular globally defined solutions - a result that fails to hold if G = U (1), (the coupling of gravity to electromagnetism). Since the (classical) Yang-Mills equations with G = SU (2) correspond to the weak nuclear force, our result indicates that coupling gravity to the weak nuclear force can prevent singularity formation in space-time. Furthermore, these solutions give rise to constants which correspond to masses. In other words, numbers representing masses are a direct consequence of our theory, and need not be prescribed in an ad-hoc manner - as is usually done in physical theories.

By employing the higher (N⩾5)-dimensional version of the Wu–Yang ansatz we obtain black hole solutions in the spherically symmetric Einstein–Yang–Mills (EYM) theory. Although these solutions were found recently by other means, our method provides an alternative way in which one identifies the contribution from the Yang–Mills (YM) charge. Our method has the advantage to be carried out analytically as well. We discuss some interesting features of the black hole solutions obtained.

We consider the standard Yang–Mills (YM) invariant raised to the power q, i.e., (Fμν(a)F(a)μν)q as the source of our geometry and investigate the possible black hole solutions. How does this parameter q modify the black holes in Einstein–Yang–Mills (EYM) and its extensions such as Gauss–Bonnet (GB) and the third order Lovelock theories? The advantage of such a power q (or a set of superposed members of the YM hierarchies) if any, may be tested even in a free YM theory in flat spacetime. Our choice of the YM field is purely magnetic in any higher-dimensions so that duality makes no sense. In analogy with the Einstein-power-Maxwell theory, the conformal invariance provides further reduction, albeit in a spacetime for dimensions of multiples of 4.

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified. The talk reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren, and the author.

In the present work, we propose a scheme for the digital formulation of lattice gauge theories with dynamical fermions in 3 + 1 dimensions. All interactions are obtained as a stroboscopic sequence of two-body interactions with an auxiliary system. This enables quantum simulations of lattice gauge theories where the magnetic four-body interactions arising in two and more spatial dimensions are obtained without the use of perturbation theory, thus resulting in stronger interactions compared with analogue approaches. The simulation scheme is applicable to lattice gauge theories with either compact or finite gauge groups. The required bounds on the digitization errors in lattice gauge theories, due to the sequential nature of the stroboscopic time evolution, are provided. Furthermore, an implementation of a lattice gauge theory with a non-abelian gauge group, the dihedral group D3, is proposed employing the aforementioned simulation scheme using ultracold atoms in optical lattices.

Yang-Mills theory in D = 7 with arbitrary gauge group G is considered. By use of the octonionic algebra we reduce D = 7 Yang-Mills equations to the nonlinear ordinary differential equations (Ward equations). As an example, we write out the explicit form of some solutions, which are the seven-dimensional analogues of the d = 4 instanton.

It is shown that infrared phenomena in the gauge theories are guided by certain classical solutions of the Yang-Mills equations. The existence of such solutions can lead to a finite correlation length which stops infrared catastrophe. In the present paper we deal only with theories with a compact but abelian gauge group. In this case the problems of correlation length and charge confinement are completely solved.

The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes induces a refined graphic expansion, by which any tree-level EYM amplitude can be expressed as a summation over all possible refined graphs. Each graph contributes a unique coefficient as well as a proper combination of color-ordered Yang-Mills (YM) amplitudes. This expansion allows one to evaluate EYM amplitudes through YM amplitudes, the latter have much simpler structures in four dimensions than the former. In this paper, we classify the refined graphs for the expansion of EYM amplitudes into N k MHV sectors. Amplitudes in four dimensions, which involve k + 2 negative-helicity particles, at most get non-vanishing contribution from graphs in N k′ (k′ ≤ k) MHV sectors. By the help of this classification, we evaluate the non-vanishing amplitudes with two negative-helicity particles in four dimensions. We establish a correspondence between the refined graphs for single-trace amplitudes with left({g}_i^{-},{g}_j^{-}right) or left({h}_i^{-},{g}_j^{-}right) configuration and the spanning forests of the known Hodges determinant form. Inspired by this correspondence, we further propose a symmetric formula of double-trace amplitudes with left({g}_i^{-},{g}_j^{-}right) configuration. By analyzing the cancellation between refined graphs in four dimensions, we prove that any other tree amplitude with two negative-helicity particles has to vanish.

Globally regular (i.e., asymptotically flat and regular interior) spherically symmetric and localized (“particlelike”) solutions of the coupled Einstein Yang–Mills (EYM) equations with gauge group SU(2) have been known for more than 20 years, yet their properties are still not well understood. Spherically symmetric Yang–Mills fields are classified by a choice of isotropy generator and SO(5) is distinguished as the simplest gauge group having a model with a non-Abelian residual (little) group, SU(2)×U(1), which admits globally regular particlelike solutions. We exhibit an algebraic gauge condition which normalizes the residual gauge freedom to a finite number of discrete symmetries. This generalizes the well-known reduction to the real magnetic potential w(r,t) in the original SU(2) YM model. Reformulating using gauge-invariant polynomials dramatically simplifies the system and makes numerical search techniques feasible. We find three families of embedded SU(2) EYM equations within the SO(5) system, one of which was first detected only within the gauge-invariant polynomial reduced system. Numerical solutions representing mixtures of the three SU(2) subsystems are found, classified by a pair of positive integers.

I would like to explain some applications of quantum field theory methods to Donaldson theory. But first, perhaps, I should explain what Donaldson theory is. We start with an oriented four-manifold M,and a compact gauge group, say G = SU (2) We pick a principal G bundle P over M, and let A denote a connection on P The space of such connections will be called A.

We summarize the algebraic structure of spherically symmetric Yang–Mills potentials for a general compact gauge group, and investigate the particular case of gauge groups with Lie algebra su(n) in detail. We develop techniques that lead to a complete classification of the possible spherical symmetry ansätze, including descriptions of the reduced gauge group 𝒵, the space of magnetic potentials ℋ, and for those ansätze that admit extensions across the symmetry axis, a description of the space of vacuum potentials H0 and its little group Z0. These results are illustrated by listing all irreducible models for su(n), n⩽6.

Non-perturbative quasi-classical model in a gauge theory with the Yang-Mills (YM) field is developed. The self-consistent solutions of the Dirac equation in the SU(N) gauge field, which is in the eikonal approximation, and the Yang-Mills (YM) equations containing the external fermion current are solved. It shown that the developed model has the self-consistent solutions of the Dirac and Yang-Mills equations at N ≥ 3. In this way, the solutions take place provided that the fermion and gauge fields exist simultaneously, so that the fermion current completely compensates the current generated by the gauge field due to self-interaction of it.\n

We compute various averages over bulk geometries of quantum states prepared by the Chern-Simons path integral, for any level k and compact simple gauge group G. We do so by carefully summing over all topologically distinct bulk geometries which have n disjoint boundary tori and a decomposition into space×time of fixed spatial topology. We find that to leading order in the complexity of the state, the typical state contains many types of multiparty entanglement, proving a conjecture of Balasubramanian Combined with the fact that periodic link states are Greenberger–Horne–Zeilinger (GHZ)-like, this result implies that a generic fibered link state contains non-GHZ entanglement if and only if its link complement is hyperbolic. Additionally, we compute an averaged wave function which captures the leading order statistics of boundary observables in the n torus Chern-Simons Hilbert space.

We demonstrate that the five vortex equations recently introduced by Manton arise as symmetry reductions of the anti-self-dual Yang–Mills equations in four dimensions. In particular the Jackiw–Pi vortex and the Ambjørn–Olesen vortex correspond to the gauge group , and respectively the Euclidean or the symmetry groups acting with two-dimensional orbits. We show how to obtain vortices with higher vortex numbers, by superposing vortex equations of different types. Finally we use the kinetic energy of the Yang–Mills theory in 4 + 1 dimensions to construct a metric on vortex moduli spaces. This metric is not positive-definite in cases of non-compact gauge groups.