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| Wilson loop | |
|---|---|
| Name | Wilson loop |
| Field | Theoretical physics |
| Introduced | 1974 |
| Introduced by | Kenneth G. Wilson |
Wilson loop is an observable in gauge theories defined by the trace of the path-ordered exponential of a gauge connection around a closed curve. It plays a central role in non-abelian gauge theory, lattice gauge theory, quantum chromodynamics, and the study of confinement, and connects to mathematical topics such as holonomy, loop space, and knot invariants.
A Wilson loop is constructed from a gauge connection A integrated along a closed curve C and then traced in a representation of a gauge group such as SU(3), SU(2), or U(1). The definition uses the path-ordered exponential to produce a group element in representations associated with Lie groups like SO(3), Spin groups, or exceptional groups such as E8. Key properties include gauge invariance under transformations generated by elements of groups like Gauge theory institutions (e.g., Yang–Mills theory), behavior under group center elements related to center symmetry, and dependence on the choice of representation, such as the fundamental or adjoint representation prominent in Quantum chromodynamics and Electroweak interaction studies.
In continuum Yang–Mills theory and Quantum field theory, Wilson loops serve as nonlocal operators sensitive to the topology of closed curves, linking to observables in Quantum chromodynamics used by collaborations like those at CERN and Fermilab for theoretical modeling. Their expectation values probe phases distinguished by concepts such as confinement and Deconfinement phase transition appearing in contexts studied at facilities like RHIC and LHC. The role of Wilson loops appears in theoretical frameworks developed by figures such as Gerard 't Hooft, Kenneth G. Wilson, and in approaches related to Polyakov action and work on nonperturbative aspects by groups at Princeton University and Harvard University.
Path ordering is necessary because the connection components do not commute in non-abelian groups like SU(N), a feature highlighted in analyses by mathematicians and physicists associated with institutions such as Institute for Advanced Study. The resulting holonomy around a loop is an element of the gauge group and relates to classical concepts studied by Élie Cartan, Évariste Galois (via group theory lineage), and differential geometers at places like Courant Institute. Holonomy connects to concepts in Differential geometry and the study of Principal bundles, and appears implicitly in constructions used in theorems by researchers at MPI (Max Planck Institute) and IHES.
On the lattice, Wilson introduced a discretized formulation used in many numerical studies by collaborations such as MILC Collaboration and CP-PACS. Wilson loops on a lattice become products of link variables around plaquettes and larger closed contours, central to Monte Carlo simulations pioneered at laboratories like SLAC and computing centers at CERN. The lattice formulation underpins calculations of string tension, heavy-quark potentials, and hadronic spectra studied by research groups at Brookhaven National Laboratory and DESY.
The behavior of large Wilson loops distinguishes confining and nonconfining phases: an area law behavior of expectation values signals confinement as argued in seminal work by Kenneth G. Wilson and elaborated by Gerard 't Hooft and Alexander Polyakov. Area law scaling yields a nonzero string tension used in modeling flux tubes between color charges in Quantum chromodynamics, a subject of investigation in experiments at CERN and theoretical programs at institutions like Perimeter Institute. In contrast, perimeter law behavior characterizes screened or Coulombic phases relevant to phenomena analyzed by groups at Stanford University and Cambridge University.
In supersymmetric contexts such as N=4 supersymmetric Yang–Mills theory, Wilson loops have exact results and dual descriptions via the AdS/CFT correspondence proposed by Juan Maldacena and studied at centers like KITP and IAS. Supersymmetric localization techniques developed by researchers associated with Harvard University and Princeton University allow exact evaluation of certain Wilson loop expectation values, while in conformal field theories Wilson loops connect to operator product expansions and to defect CFTs explored at institutes such as Perimeter Institute.
Mathematically, Wilson loops correspond to trace functions on moduli spaces of flat connections studied by scholars at IHES, MSRI, and Cambridge University. They relate to loop operators and holonomy in the theory of Principal bundles, to the representation theory of Kac–Moody algebras and Affine Lie algebras, and to knot invariants via relations explored in work tied to Vladimir Drinfeld and research programs at IAS and MPI. Loop space formulations and operators appear in contexts linked to the Atiyah–Bott framework and to investigations by mathematicians at Princeton University and University of Oxford.