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...
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...
Homework Statement
Forgive the awkward title, it was hard to think how to describe the problem in such a short space. I'm following the proof of Proposition 1.2 of Nakahara's "Geometry, Topology and Physics", and the following commutation relation has been established:
\partial_x^n e^{ikx} =...
Yeah, the source of the differential equation is indeed the Laplacian operator with factors of 1/(sin θ), and thus the poles have to be excluded. But is there a physical justification for excluding the poles? I won't pretend to know enough about it, but it seems to be just a mathematical...
I'm not so sure there is another way to obtain the same result; the (limited) workings in the book suggest a division by \sin^2\theta, so I can only assume there is a physical interpretation that justifies it. The only thing I can think of is that, going by the previous workings in the section...
:redface: The irony is that I thought twice about equating the two differentials but managed to convince myself that they were equal... Thank you ever so much, LawlQuals, I feel rather silly now!
I understand that this is more of a physics-related question and thus perhaps in the wrong forum...
(Apologies for not following the template for topic creation, but I wasn't sure how to adapt my problem to fit it). I'm following the derivation of the spherical harmonics in section 3.3 of Rae's "Quantum Mechanics", but have come across a step I can't quite understand. It seems like such a...