Wilson-Polyakov loop

Usually, a Wilson-Polyakov loop refers to a temporal Wilson loop taken over a circle in the Euclidean time direction,

$U(\mathbf{x}) = \mathcal{P} \exp \,{\int_0^\beta\!dt\,A_0(\mathbf{x})}\,$ ,

or to its trace, $\operatorname{tr}\,U(\mathbf{x})\,$ , taken in some representation of the gauge group. In pure gauge theory without matter, it is an order parameter for confinement ^[1]^[2]. In the confined phase, $\langle \operatorname{tr}_F\,U\rangle = 0\,$ in the fundamental representation. Intuitively, inserting $\operatorname{tr}_F\,U\,$ into the path integral corresponds to deforming the theory to include a single, static quark, so that $\left\langle \operatorname{tr}_F U\right\rangle \neq 0\,$ corresponds to a finite free energy. In the confining phase, this free energy is infinite, so that $\left\langle \operatorname{tr}_F U\right\rangle = 0\,$ .

References

↑ ^1.0 ^1.1 Polyakov, Alexander M. (1978). "Thermal Properties of Gauge Fields and Quark Liberation". Phys. Lett. B72: 477-480. DOI:10.1016/0370-2693(78)90737-2.
↑ ^2.0 ^2.1 Susskind, Leonard (1979). "Lattice Models of Quark Confinement at High Temperature". Phys. Rev. D20: 2610-2618. DOI:10.1103/PhysRevD.20.2610.
↑ Rey, Soo-Jong; Theisen, Stefan; Yee, Jung-Tay (1998). "Wilson-Polyakov loop at finite temperature in large N gauge theory and anti-de Sitter supergravity". Nucl. Phys. B527: 171-186. arXiv:hep-th/9803135. DOI:10.1016/s0550-3213(98)00471-4.