### 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 [367, 433] for Wilson loop operators in SYM.
Wilson loop operators [494] in Yang–Mills theories are non-local gauge invariant operators

depending on a closed loop in spacetime and where the trace is over the fundamental representation of
the gauge group. This operator allows one to extract the energy of a quark-antiquark pair separated
a distance . Indeed, consider a rectangular closed loop in which the pair evolves in Euclidean
time . In the limit , the expectation value of this rectangular Wilson loop equals
To understand the prescription in [367, 433], one must first introduce massive quarks in the theory.
This is achieved by breaking the original gauge symmetry of the original SYM according to

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 and transform in the fundamental
representation of the gauge symmetry, as required. Furthermore at energy scales much lower than
, the theory decouples from the theory.
In this set-up, the massive W-boson interacts with the gauge fields, including the scalar adjoint
fields [367], leading to the insertion of the operator

The contour is parameterised by whereas the vector 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. (432) was

This holds in the large limit and stands for the proper area of a fundamental string
describing the loop at the boundary of AdS_{5} and lying along on S^{5}. Notice that a quantum
mechanical calculation at strong coupling reduces to determining a minimal worldsheet surface in AdS_{5}, 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 between the quark-antiquark. After
subtracting the regularised mass of the W-boson one obtains the quark-antiquark potential energy
which differs from the linear perturbative dependence on .
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 .