Abstract
An infinite-dimensional family of analytic solutions in pure SU(2) Yang–Mills theory at finite density in (3+1) dimensions is constructed. It is labelled by two integeres (p and q) as well as by a two-dimensional free massless scalar field. The gauge field depends on all the 4 coordinates (to keep alive the topological charge) but in such a way to reduce the (3+1)-dimensional Yang–Mills field equations to the field equation of a 2D free massless scalar field. For each p and q, both the on-shell action and the energy-density reduce to the action and Hamiltonian of the corresponding 2D CFT. The topological charge density associated to the non-Abelian Chern–Simons current is non-zero. It is possible to define a non-linear composition within this family as if these configurations were “Lego blocks”. The non-linear effects of Yang–Mills theory manifest themselves since the topological charge density of the composition of two solutions is not the sum of the charge densities of the components. This leads to an upper bound on the amplitudes in order for the topological charge density to be well-defined. This suggests that if the temperature and/or the energy is/are high enough, the topological density of these configurations is not well-defined anymore. Semiclassically, one can show that (depending on whether the topological charge is even or odd) some of the operators appearing in the 2D CFT should be quantized as Fermions (despite the Bosonic nature of the classical field).
Highlights
A systematic tool to construct non-spherical hedgehog ansatz are suitable to describe finite density effects have been developed in [43–53] for the Skyrme model [54– 56]. Such strategy is quite effective in the Einstein–Yang–Mills case as well [64–66]. This approach will be adapted to the situation in which the (3+1)-dimensional Yang–Mills theory is defined within a flat region of finite spatial volume
If one is interested in non-spherical situations one should expect that a consistent ansatz should include, at least, two different profiles. This apparently obvious observation leads to the novel ansatz which is able to disclose an infinite-dimensional family of topologically non-trivial configuration in (3+1)-dimensional Yang–Mills theory at finite Baryon density
The main goal of the paper is to construct a formalism able to describe how topologically non-trivial configurations of Yang–Mills theory react when they are forced to live within a finite box
Summary
The Yang–Mills theory is defined by the action (here we will consider the SU (2) case but the present results can be extended to the SU (N ) case). Fμν = ∂μ Aν − ∂ν Aμ + [ Aμ, Aν ], A = Aμd xμ = Aμj t j d xμ, t j = i σ j. E is the Yang–Mills coupling constant and the matrices t j are the generators of the SU (2) group being σ j the Pauli matrices.
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