$SU(N)$ gauge theories, extended with adjoint fermions having periodic boundary conditions, are confining at high temperature for sufficiently light fermion mass $m$. In the high-temperature confining region, the one-loop effective potential for Polyakov loops has a $Z(N)$-symmetric confining minimum. String tensions associated with Polyakov loops are calculable in perturbation theory, and display a novel scaling behavior in which higher representations have smaller string tensions than the fundamental representation. In the magnetic sector, the Polyakov loop plays a role similar to a Higgs field, leading to an apparent breaking of $SU(N)$ to $U(1{)}^{N-1}$. This in turn yields a dual effective theory where magnetic monopoles give rise to string tensions for spatial Wilson loops. The spatial string tensions arise semiclassically from kink solutions of the dual system. We prove that the spatial string tensions ${\sigma }_k^{(s)}$ associated with each $N$-ality $k$ are constrained by a rigorous upper bound. This bound is saturated for $N=2$ and 3, but is insufficient to determine the spatial string tension scaling law for $N\ge 4$. Lattice simulations indicate that the high-temperature confining region is smoothly connected to the confining region of low-temperature pure $SU(N)$ gauge theory. However, our results show that the string tension scaling behavior of the low-temperature region does not hold for the electric sector in the high-temperature region, and may not hold in the magnetic sector. The predicted change in the behavior of the electric sector should be readily distinguishable in lattice simulations.

• Received 1 June 2009

DOI:https://doi.org/10.1103/PhysRevD.81.025012