Wilson loop

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Introduction

A Wilson loop W_C\, is a defined by a path-ordered exponential of a gauge potential A = A_\mu dx^\mu\, (Lie algebra-valued oneform) along a closed contour:

W_C = \mathrm{Tr}\, \mathcal{P}\exp \oint_C A_\mu dx^\mu .

Here it is assumed that if the gauge group is SU(N) then A\, is anti-Hermitian. Under a gauge transformation \mathcal{P}e^{ \oint_C A_\mu dx^\mu} \to g(x) \mathcal{P}e^{\oint_C A_\mu dx^\mu} g^{-1}(x)\,, where x\, denotes the (arbitrary) initial point of the loop. By the cyclicity of the trace, it follows that W_C\, is gauge invariant.

Migdal-Makeenko loop equations

See also [1] [2] [3] [4] [5]

Zig-zag symmetry

References

Further reading: [6] [7]

  1. ↑ Y.M. Makeenko, A.A. Migdal, Exact equation for the loop average in multicolor QCD, Phys. Lett. B 88 (1979) 135
  2. ↑ Y.M. Makeenko, A.A. Migdal, Quantum chromodynamics as dynamics of loops, Nucl. Phys. B 188 (1981) 269
  3. ↑ R.A. Brandt, A. Gocksch, M.A. Sato and F. Neri, Loop Space, Phys. Rev. D 26 (1982) 3611
  4. ↑ V.A. Kazakov and I.K. Kostov, Nonlinear strings in two-dimensional U(1) gauge theory, Nucl. Phys. B 176 (1980) 199
  5. ↑ N. Drukker, A new type of loop equations, J. High Energy Phys 11 (1999) 006
  6. ↑ R. Giles, "Reconstruction of Gauge Potentials from Wilson loops", Phys. Rev. D 24, 2160 (1981)
  7. ↑ K. Wilson, "Confinement of quarks", Phys. Rev. D 10, 2445 (1974)
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