Wilson loops in terms of color invariants

Abstract

We derive an expression for the vacuum expectation value (vev) of the 1/2 BPS circular Wilson loop of ${\cal N}=4$ super Yang Mills in terms of color invariants, valid for any representation R of any gauge group G. This expression allows us to discuss various exact relations among vevs in different representations. We also display the reduction of these color invariants to simpler ones, up to seventh order in perturbation theory, and verify that the resulting expression is considerably simpler for the logarithm of $\left<W\right>_R$ than for $\left<W\right>_R$ itself. We find that in the particular case of the symmetric and antisymmetric representations of SU(N), the logarithm of $\left<W\right>_R$ satisfies a quadratic Casimir factorization up to seventh order, and argue that this property holds to all orders. Finally, we derive the large N expansion of $\left<W\right>_R$ for an arbitrary, but fixed, representation of SU(N), up to order $1/\text{N}^2$.