6.1 Wilson loops

As a first example of the use of classical solutions to brane effective actions to compute the expectation values of gauge invariant operators at strong coupling, I will review the prescription put forward in [367Jump To The Next Citation Point, 433Jump To The Next Citation Point] for Wilson loop operators in š’© = 4 SYM.

Wilson loop operators [494] in SU (N ) Yang–Mills theories are non-local gauge invariant operators

1-- iāˆ® A W (š’ž) = N TrPe š’ž , (429 )
depending on a closed loop in spacetime š’ž and where the trace is over the fundamental representation of the gauge group. This operator allows one to extract the energy E (L) of a quark-antiquark pair separated a distance L. Indeed, consider a rectangular closed loop in which the pair evolves in Euclidean time T. In the limit T → ∞, the expectation value of this rectangular Wilson loop equals
−TE(L) āŸØW (š’ž )āŸ© = A(L )e . (430 )

To understand the prescription in [367Jump To The Next Citation Point, 433Jump To The Next Citation Point], one must first introduce massive quarks in the theory. This is achieved by breaking the original gauge symmetry of the original š’© = 4 SYM according to

U (N + 1) → U(N ) × U (1). (431 )
The massive W-bosons generated by this process have a mass proportional to the norm of the Higgs field expectation value responsible for the symmetry breaking āƒ— (|Φ |) and transform in the fundamental representation of the U (N ) gauge symmetry, as required. Furthermore at energy scales much lower than |āƒ—Φ |, the U(N ) theory decouples from the U (1) theory.

In this set-up, the massive W-boson interacts with the U (N ) gauge fields, including the scalar adjoint fields I X [367Jump To The Next Citation Point], leading to the insertion of the operator

1 iāˆ® ds[Aμ(σ)Ė™σμ+šœƒI(s)XI (σ)√σĖ™2] W (š’ž) = N-TrPe š’ž . (432 )
The contour š’ž is parameterised by σμ(s) whereas the vector āƒ—šœƒ(s) maps each point on the loop to a point on the five-sphere.

The proposal made in [367, 433] to compute the expectation value of Eq. (432View Equation) was

−Sstring āŸØW (š’ž)āŸ© ∼ e . (433 )
This holds in the large gsN limit and Sstring stands for the proper area of a fundamental string describing the loop š’ž at the boundary of AdS5 and lying along šœƒI(s ) on S5. Notice that a quantum mechanical calculation at strong coupling reduces to determining a minimal worldsheet surface in AdS5, i.e., solving the worldsheet equations of motion with appropriate boundary conditions, and then solving for the worldsheet energy as a function of the separation L between the quark-antiquark. After subtracting the regularised mass of the W-boson one obtains the quark-antiquark potential energy
4π2(2g2 N )1āˆ•2 E (L) = − ------Y1M4------, (434 ) Γ (4) L
which differs from the linear perturbative dependence on g2 N YM.

If one considers multiply-wrapped Wilson loops, the many coincident strings will suffer from self-interactions. This may suggest that a more appropriate description of the system is in terms of a D3-brane with non-trivial world volume electric flux accounting for the fundamental strings. This is the approach followed in [189], where it was shown that for linear and circular loops the D3-brane action agreed with the string worldsheet result at weak coupling, but captures all the higher-genus corrections at leading order in ′ α.

  Go to previous page Go up Go to next page