I am currently reading the paper given here by Acharya+Gukov titled "M Theory and Singularities of Exceptional Holonomy Manifolds", and in particular right now am following section 4 where the field content of the effective 4-dimensional theory is derived by harmonic decomposition of the 11D bosonic fields compactified over a smooth [itex]G_2[/itex] manifold. I accept the reasoning behind the derivation of the scalar and gauge fields in terms of the Betti numbers for the manifold, and understand why [itex]b_1=0[/itex] (and so there's no need to take the ansatz of a term summing over harmonic 1-forms on the compact space). However, there is no mention of a possible term that sums over harmonic 0-forms on the compact space, which as I understand it would lead similarly to [itex]b_0[/itex] 3-form gauge fields in the N=1 theory in 4D, which to me doesn't sound like a trivial result that's not worth mentioning. Since in general we don't necessarily have [itex]b_0=0[/itex] for [itex]G_2[/itex] manifolds, is there a particular reason why one does not write this ansatz? I am most likely missing something embarrassingly obvious, but try as I might, I cannot see it.