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Compactification of M Theory on Smooth G2 Manifolds

  1. Jul 16, 2014 #1
    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.
  2. jcsd
  3. Jul 21, 2014 #2
    Never mind, I've answered my own question with the help of my supervisor. It appears that taking [itex]b_0=1[/itex] (as is the case for the [itex]G_2[/itex] manifolds under consideration) indeed results in a single 3-form gauge field [itex]A_3[/itex] in the effective 4D theory via this particular KK ansatz, and the terms containing [itex]A_3[/itex] consist only of a kinetic term of the form [itex]F_4 \wedge *F_4[/itex] for [itex]F_4 = dA_3[/itex]. This is because any interaction terms containing [itex]A_3[/itex] resulting from compactification of the Chern-Simons term in 11D will necessarily be a [itex]p[/itex]-form with [itex]p>4[/itex], 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.
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