On Orbifold Compactifications, D/p-Branes and Strings

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

    \[\int_\chi \omega : = \frac{1}{{\left| G \right|}}\int_M \omega \]

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


    \[{\rm{e}}{{\rm{v}}_i}:{\not \bar {\rm M}_{0,n}}\left( {\chi ,\beta } \right) \to I\chi \]

Randall-Sundrum Compactification Models and Bulk-Field Actions

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

    \[\Phi \left( {x,\phi } \right) = \sum\limits_n {{\psi _n}} \frac{{{y_n}\left( \phi \right)}}{{\sqrt {{r_c}} }}\]

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

    \[\int\limits_{ - 1\pi }^\pi {d\phi {e^{ - 2\sigma (\phi )}}} {y_n}\left( \phi \right){y_m}\left( \phi \right) = {\delta _{nm}}\]

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)