Wilson loops in Five-Dimensional Super-Yang-Mills

TL;DR

This paper investigates the divergences of Wilson loops in five-dimensional super-Yang-Mills theory and their string/M-theory duals, revealing a logarithmic divergence linked to conformal anomalies and a scheme-dependent finite part.

Contribution

It demonstrates the origin of divergences in 5D Wilson loops through string, M-theory, and gauge theory analyses, connecting them to conformal anomalies and regularization schemes.

Findings

Logarithmic divergence in string worldsheet area in D4-brane geometry.

Matching divergence in M2-brane Wilson surface in AdS_7 x S^4.

One-loop diagrams in 5D SYM are finite, but divergences appear in higher dimensions.

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 AdS_7 x 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 for general smooth loops by summing the leading divergences in the analytic continuation of dimensional regularization of planar rainbow/ladder diagrams. These diagrams are finite in 5 - epsilon 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 greater than or equal to 6.