A functional CLT for fields of commuting transformations via martingale approximation

We consider a field $f\circ T^{i_1}_1\circ\dots\circ T_d^{i_d}$, where $T_1,\dots,T_d$ are completely commuting transformations in the sense of Gordin. If one of these transformations is ergodic, we give sufficient conditions in the spirit of Hannan under which the partial sum process indexed by quadrants converges in distribution to a Brownian sheet. The proof combines a martingale approximation approach with a recent CLT for martingale random fields due to Volný. We apply our results to completely commuting endomorphisms of the $m$-torus. In that case, the conditions can be expressed in terms of the $L^2$-modulus of continuity of $f$.