Three-Dimensional SU(3) gauge theory and the Spatial String Tension of the (3+1)-Dimensional Finite Temperature SU(3) Gauge Theory

We establish a close relation between the spatial string tension of the (3+1)-dimensional $SU(3)$ gauge theory at finite temperature (${\sigma }_s$) and the string tension of the 3-dimensional $SU(3)$ gauge theory (${\sigma }_3$) which is similar to what has been found previously for $SU(2)$. We obtain $\sqrt{{\sigma }_3}=(0.554\pm 0.004)g_3^2$ and $\sqrt{\sigma_s} = (0.586 \pm 0.045)g^2(T) T$, respectively. For temperatures larger than twice the critical temperature results are consistent with a temperature dependent coupling running according to the two-loop $\beta$-function with $\Lambda_T = 0.118(36)T_c$.