Wilson loops in five-dimensional Super-Yang-Mills

Abstract

We consider circular non-BPS Maldacena-Wilson loops in five-dimensional supersymmetric Yang-Mills theory (d = 5 SYM) both as macroscopic strings in the D4- brane geometry and directly in gauge theory. We find that in the Dp-brane geometries for increasing p, p = 4 is the last value for which the radius of the string worldsheet describing the Wilson loop is independent of the UV cut-off. It is also the last value for which the area of the worldsheet can be (at least partially) regularized by the standard Legendre transformation. The asymptotics of the string worldsheet allow the remaining divergence in the regularized area to be determined, and it is found to be logarithmic in the UV cut- off. We also consider the M2-brane in AdS₇ × S ⁴ which is the M-theory lift of the Wilson loop, and dual to a “Wilson surface” in the (2,0), d = 6 CFT. We find that it has exactly the same logarithmic divergence in its regularized action. The origin of the divergence has been previously understood in terms of a conformal anomaly for surface observables in the CFT. Turning to the gauge theory, a similar picture is found in d = 5 SYM. The divergence and its coefficient can be recovered by summing the leading divergences in the analytic continuation of dimensional regularization of planar rainbow/ladder diagrams. These diagrams are finite in 5 − ϵ dimensions. The interpretation is that the Wilson loop is renormalized by a factor containing this leading divergence of six-dimensional origin, and also subleading divergences, and that the remaining part of the Wilson loop expectation value is a finite, scheme-dependent quantity. We substantiate this claim by showing that the interacting diagrams at one loop are finite in our regularization scheme in d = 5 dimensions, but not for d ≥ 6.