Non-supersymmetric Wilson loop in N=4 SYM and defect 1d CFT

Abstract

Following Polchinski and Sully (arXiv:1104.5077), we consider a generalized Wilson loop operator containing a constant parameter $ζ$ in front of the scalar coupling term, so that $ζ=0$ corresponds to the standard Wilson loop, while $ζ=1$ to the locally supersymmetric one. We compute the expectation value of this operator for circular loop as a function of $ζ$ to second order in the planar weak coupling expansion in N=4 SYM theory. We then explain the relation of the expansion near the two conformal points $ζ=0$ and $ζ=1$ to the correlators of scalar operators inserted on the loop. We also discuss the $AdS_5\times S^5$ string 1-loop correction to the strong-coupling expansion of the standard circular Wilson loop, as well as its generalization to the case of mixed boundary conditions on the five-sphere coordinates, corresponding to general $ζ$. From the point of view of the defect 1d CFT defined on the Wilson line, the $ζ$-dependent term can be seen as a perturbation driving a RG flow from the standard Wilson loop in the UV to the supersymmetric Wilson loop in the IR. Both at weak and strong coupling we find that the logarithm of the expectation value of the standard Wilson loop for the circular contour is larger than that of the supersymmetric one, which appears to be in agreement with the 1d analog of the F-theorem.

Figures (3)

• Figure 1: Gauge field exchange diagram contributing the standard Wilson loop at the leading order. In the Wilson-Maldacena loop case there is an additional scalar exchange contribution.
• Figure 2: Order $\lambda^2$ contributions to the standard Wilson loop. The middle diagram contains the full self-energy 1-loop correction in SYM theory (with vector, ghost, scalar and fermion fields in the loop). For the Wilson-Maldacena loop there are additional diagrams with scalar propagators instead of some of the vector ones.
• Figure 3: Two of planar diagrams of ladder type $W_{2,1}=W^{(a)}_{2,1} +W^{(b)}_{2,1}$ with path-ordered four points $\tau_{1}, \dots, \tau_{4}$ in the WL ($\zeta=0$) case. For general $\zeta$ one needs also to add similar diagrams with scalar propagators.