Compactification of M Theory on Smooth G2 Manifolds

[d.hatch75]

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 $G_2$ 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 $b_1=0$ (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 $b_0$ 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 $b_0=0$ for $G_2$ 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.

 

[d.hatch75]

Never mind, I've answered my own question with the help of my supervisor. It appears that taking $b_0=1$ (as is the case for the $G_2$ manifolds under consideration) indeed results in a single 3-form gauge field $A_3$ in the effective 4D theory via this particular KK ansatz, and the terms containing $A_3$ consist only of a kinetic term of the form $F_4 \wedge *F_4$ for $F_4 = dA_3$. This is because any interaction terms containing $A_3$ resulting from compactification of the Chern-Simons term in 11D will necessarily be a $p$-form with $p>4$, and so they will vanish when integrated over the 4D spacetime. The kinetic term after integration contributes a cosmological constant, which is interesting in itself but not all that relevant to a field theory discussion, so I am satisfied that my query has been resolved.