Finite action of solutions of Euclidean field equations in Yang-Mills theory saturates the bound determined by the topological charge of the homotopy group Z. Z is either the charactersitic group with index 3, for pure gauge theory, or with index 2 for the adjoint Higgs model. The former results in a mathematical construction of instantons, the latter for magnetic monopoles or dyons. In the deconfined phase, these instantons appear in their periodic incarnations, called calorons, responsible for the emergence of a thermal ground state and stable magnetic monopoles.
For SU(2), the construction of the simplest instanton (|k| = 1) on ℝ⁴ and |k| > 1 is derived from the topological current and the decomposition of the YM-action. The resulting closed-form solution is a gauge-transformed version with the nontrivial topology at the center of the action density. SU(2)Π₃(S₃) has topologically stabilized finite-action solutions. This can be shown using the gauge invariant current
where n are the Chern-Simons numbers. These are the charge of CS-current on the time slices at infinity and describe the behavior at, along with of the invariance of the topological charge (Pontryagin index) under smooth, spatially localized deformation away from the boundaries. The Euclidean YM-action can be decomposed into a topological invariant term and a positive definite, non-invariant term. The minimal action is reached for a vanishing non-invariant term. A regular gauge-field configuration has finite action if it approaches a pure gauge. For Pauli-matrices τ, unit quaternions σ and t'Hooft symbol η. η is self-dual, anti-self-dual and antisymmetric.
for |k| = 1, eight free parameters emerge, spanning an 8D Riemannian manifold called a moduli space. The solutions decay as |x| → ∞, so their topological charge is picked up by surface integrals of the CS-current over a 3-sphere with infinite radius.
Using the conjugation of the gauge fields as a gauge transformations introduces singularities at the origin, by which the invariance of the YM-action becomes visible. The saturation of the action at the |k| = 1 bound is fixed at the anti-self-dual gauge-field level. These can be narrowed down by using generalizing the boundary condition to higher topological charge moduli |k|.
Evaluated straight-forwardly, the configurations will turn out to be generalizable through multiplication with constant SO(3) matrices. The gauge-field configurations are then globally oriented in the Lie algebra, so that each |k| = 1 (anti-)instanton contributes one unit of topological charge with |k| = n, oriented the same way in su(2). These periodic configurations of the Euclidean formulation of the finite-temperature Yang-Mills theory will create calorons on the trivial holonomy. These configurations are generalizable to SU(N).
Assume a traceless, antihermitian gauge field. This will restore tⁿtᵐ = -1/2 δⁿᵐ, and make sure that the coefficients in the gauge field are real. It also eliminates the imaginary coefficients in the field strength tensor that would have decorated the commutator.
For k > 0 in SU(N), the self-dual gauge field emerges from the kernel of a 2k ⨉ (2k+N) matrix Δ⁺ = A + BX, where X = σᵢ⁺xᵢ and A, B (2k + N) ⨉ 2k matrices. Δ⁺Δ needs to be invertible and commute with the quaternions. Let V be the kernel of Δ⁺, then
A and B should be sufficient to generate all charge-k self-dual SU(N) configurations. The way to obtain all self-dual gauge-field configurations for some topological charge k at finite action is the ADHM construction.
The ADHM construction can itself be generalized to instantons on the 4-torus T4. For this, begin at SU(2), so to ignore gauge coupling, and the factor of -i into the gauge fields, so that the field potential ϕ follows the schema of the typical Lagrange density (up to a factor of -1/g²). Since ϕ ≠ 0 introduced a breaking of gauge-symmetry SU(2) → U(1), to consider Π₂(SU(2)/U(1)) = Π₂(S₂) = ℤ suggests topologically stabilized, static configurations with minimal energies for each topological sector. The field equation for the model become
Which can be simplified to the spatial dimensions if the solution is assumed to be static.
With some real scalar functions H, K which can be transitioned into solutions of the self-duality equations for the gauge-fields. This eventually results in differential equations of second order
This fixes the expectation at spatial infinity. An abelian spatial field strength should exist, which is associated with the magnetic charge of the static solitonic configuration. In the unitary gauge, this field strength tensor reduces to the usual U(1) expression, so fixing it to the magnetic field strength, This will derive the magnetic charge gₘ as a dual quantity to the gauge coupling g, along with the mass of the magnetic monopole, which vanishes as g diverges.
This is a sensible observation, seeing as the scenario would see a vanishing magnetic moment as well. This is in contrast to the Dirac monopole. This configuration has holonomy with its spatial Polyakov loop at infinity.
The BPS monopole with |k| = 1 represents the caloron configuration with trivial holonomy. New solutions are generated through symmetry of the Euclidean YM-action under spatial translation and global phase changes.
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