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For a full mathematical definition of 'orbifold' summarized above, check A. Adem and M. Klaus. String-theory compactifications on N-dimensional orbifolds is attractive, and essential in some cases. For N = 6, it allows the full determination of the emergent four-dimensional effective supergravity theory, including the gauge group and matter content, the superpotential and Kähler potential, as well as the gauge kinetic function, and yields the four-dimensional space-time supersymmetry-RT.

Before proceeding, what is in an equation like ...

where integration on a global quotient $\chi = \left[ {M/G} \right]$ is defined by

and $\omega \in {\Omega ^p}\left( M \right)$ is a G-invariant differential form, where $I\chi$ is the inertia stack of $\chi$, an orbifold groupoid, yeilding the Poincaré pairing on $I\chi$ defined as the direct sum of the pairings

with

The key to absorbing this post, following my last two, is to appreciate that the Standard Model 'particles'/fields can be interpreted as low-lying Kaluza-Klein excitations of Randall-Sundrum bulk fields (see header photo). To carry out a Horava-Witten decomposition, flesh out the bulk-field as

with ${y_n}\left( \phi \right)$ satisfying

and the action is then given by

After an integration by parts and substituting-in ${e^{ - 2\sigma \left( \phi \right)}}$, Eq. 1 is reduces to

Now, as in typical Kaluza-Klein compactifications, the bulk field $\Phi \left( {x,\phi } \right)$