  We investigate the minimal genus problem for the second homology of a right
angled Artin group (RAAG). Firstly, we present a lower bound for the minimal
genus of a second homology class, equal to half the rank of the corresponding
cap product matrix. We show that for complete graphs, trees, and complete
bipartite graphs, this bound is an equality, and furthermore in these cases the
minimal genus can always be realised by a disjoint union of tori. Additionally,
we give a full characterisation of classes that are representable by a single
torus. However, the minimal genus of a second homology class of a RAAG is not
always realised by a disjoint union of tori as an example we construct in the
pentagon shows.
