The Polyakov loop of an operator in the antisymmetric representation in $\mathcal{N}=4$ supersymmetric Yang-Mills theory on spacial ${\mathbb{R}}^3$ is calculated, to leading order in $1/N$ and at large ’t Hooft coupling, by solving the saddle point equations of the corresponding quantum impurity model. Agreement is found with previous results from the supergravity dual, which is given by a $D5$-brane in an asymptotically ${\mathrm{AdS}}_5\times S^5$ black brane background. It is shown that the azimuth angle, at which the dual $D5$-brane wraps the $S^5$, is related to the spectral asymmetry angle in the spectral density associated with the Green’s function of the impurity fermions. Much of the calculation also applies to the Polyakov loop on spacial $S^3$ or $H^3$.

• Received 25 January 2011

DOI:https://doi.org/10.1103/PhysRevD.83.066006