${\cal N}=4$ supersymmetric Yang-Mills thermodynamics from effective field theory

The free energy density of ${\cal N}=4$ supersymmetric Yang-Mills theory in four space-time dimensions is derived through second order in the 't Hooft coupling $\lambda$ at finite temperature using effective-field theory methods. The contributions to the free energy density at this order come from the hard scale $T$ and the soft scale $\sqrt{\lambda} T$. The effects of the scale $T$ are encoded in the coefficients of an effective three-dimensional field theory that is obtained by dimensional reduction at finite temperature. The effects of the electric scale $\sqrt{\lambda} T$ are taken into account by perturbative calculations in the effective theory.