Wilson loops in $\mathcal{N}=4\mathrm{}$ super Yang-Mills theory: Rotation in ${\mathrm{S}}^5$

We study Wilson loops in $\mathcal{N}=4\mathrm{}$ super Yang-Mills (SYM) theory which are nonconstant in the scalar $\left({\mathrm{S}}^5\right)$ directions and open string solutions associated with them in the context of AdS conformal field theory correspondence. An interplay between Minkowskian and Euclidean pictures turns out to be nontrivial for time-dependent Wilson loops. We find that in the ${\mathrm{S}}^5$-rotating case there appears to be no direct open-string duals for the Minkowskian Wilson loops, and their expectation values should be obtained by analytic continuation from the Euclidean-space results. In the Euclidean case, we determine the dependence of the “quark-antiquark” potential on the rotation parameter $\nu ,$ both at weak coupling (i.e., in the 1-loop perturbative SYM theory) and at strong coupling (i.e., in the classical string theory in ${\mathrm{AdS}}_5\times {\mathrm{S}}^5).\mathrm{}$