Spatial 't Hooft loop in hot SUSY theories at weak coupling

Abstract

The spatial 't Hooft loop measures the colour electric flux in SU(N)/Z(N) gauge theory. It is a closed loop of Dirac Z(N) flux and has strength k=1, 2,.., N-1. It is analyzed for generic k and small gauge coupling in the high temperature phase of {\cal N} =1, 2 and 4 SUSY theory up and including cubic order. In one loop order no qualitative difference with gluodynamics shows. However the two loop order shows a logarithmic divergence appear in the centre of the wall. This is because the gluinos become bosonic in the centre and acquire long wavelength excitations. We discuss the cure for this divergence, due to a gluino zero mode. The physics of Casimir scaling is explained. The cubic order can be obtained from gluodynamics by simple rescalings of the Debye masses and the Casimir parameter $k/N_c$.