Loops, Knots, Gauge Theories and Quantum Gravity 2026, 27 - Wilson Loops and Confinement

In a YM theory coupled to fermions with quark-antiquark pair generation and annihilation, the Hamiltonian includes an interaction term that, neglecting vacuum polarization, a weight proportional to the connection holonomy along the closed path is expected. For quarks as separate final-state particles, the line in this process need to be well separable, if the points of creation are far enough apart (p. 125). Quark confinement is the analog to confinement of Wilson loops, as they confine the electric flux lines. Each confining regime is characterized by an order parameter. t'Hooft would introduce a YM-theory parameter, emergent in SU(N) YM in 2+1 dimensions (p. 126). An operator ϕ(x₀) materializing a singular gauge transformation that changes the winding number of the fields. Ω(x₀) is the gauge transformation at x₀ so that all (oriented) curves surrounding x₀ given the path parameter e^2πi/N. Curves surrounding a singularity will detect the multivaluedness induced by its winding-number operators. Curves surrounding several singularities will detect their combined winding number. ϕ functions as disorder parameter for the theory, as evident from its interaction with the Wilson loop. The Wilson loop acts as the creation operator for a domain inside of which ϕ has a different value (p. 127). The t'Hooft operator B in the commutation relation of the Wilson loop, can be shown to always commute with the Hamiltonian, which suggests that this YM theory is always in a confining phase (p. 130).