Editing Wilson loop (section)

===String theory compactifications===

In [[compactification (physics)|compactified]] theories, zero mode gauge field states that are locally pure gauge configurations but are globally inequivalent to the vacuum are parameterized by closed Wilson lines in the compact direction. The presence of these on a compactified [[String (physics)|open]] [[string theory]] is equivalent under [[T-duality]] to a theory with non-coincident [[D-brane|D-branes]], whose separations are determined by the Wilson lines.<ref>{{cite book|last=Polchinski|first=J.|author-link=Joseph Polchinski|date=1998|title=String Theory Volume I: An Introduction to the Bosonic String|url=|doi=|location=|publisher=Cambridge University Press|chapter=8|pages=263–268|isbn=978-0143113799}}</ref> Wilson lines also play a role in [[orbifold]] compactifications where their presence leads to greater control of gauge [[symmetry breaking]], giving a better handle on the final unbroken gauge group and also providing a mechanism for controlling the number of matter multiplets left after compactification.<ref>{{cite journal|last1=Ibanez|first1=L.E.|authorlink1=|last2=Nilles|first2=H.P.|authorlink2=|last3=Quevedo|first3=F.|authorlink3=Fernando Quevedo|date=1986|title=Orbifolds and Wilson Lines|url=https://cds.cern.ch/record/173902|journal=Phys. Lett. B|volume=187|issue=1–2|pages=25–32|doi=10.1016/0370-2693(87)90066-9|pmid=|arxiv=|s2cid=|access-date=}}</ref> These properties make Wilson lines important in compactifications of superstring theories.<ref>{{cite book|last=Polchinski|first=J.|author-link=Joseph Polchinski|date=1998|title=String Theory Volume II: Superstring Theory and Beyond|url=|doi=|location=|publisher=Cambridge University Press|chapter=16|pages=288–290|isbn=978-1551439761}}</ref><ref>{{cite book|last1=Choi|first1=K.S.|author-link1=|last2=Kim|first2=J.E.|author-link2=|date=2020|title=Quarks and Leptons From Orbifolded Superstring|edition=2|url=|doi=|location=|publisher=|chapter=|page=|isbn=978-3030540043}}</ref>