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 is defined by

and is a G-invariant differential form, where is the inertia stack of , an **orbifold** groupoid, yeilding the Poincaré pairing on defined as the direct sum of the pairings

with

Let us go one small mathematical step at a time, and here is a nice visualization of an orbifold

Take two D-branes, with their world-volumes and velocities , with transverse positions , . Generally, the potential between D-branes is given by cylinder vacuum amplitudes and can be interpreted as the Casimir energy stemming from open string vacuum fluctuations. The closed string amplitude

is a tree level propagation between two boundary states with two sectors, RR and NSNS, corresponding to periodicity and anti-periodicity of the fermionic fields around the cylinder. Statically, we have Neumann b.c. in **time** and Dirichlet b.c. in **space**, and the dynamics of the boundary state is given by boosting the static one with a negative rapidity

In the large distance limit only world-sheets with will contribute. Hence, the moving boundary states, Eq. 4 and 5:

can only carry space-time momentum in the combinations

and

If one integrates over the bosonic zero modes and factors-in momentum conservation

then the amplitude factorizes into a bosonic and a fermionic partition functions

where

and , and the commutation relations are

For the RR zero modes satisfying a Clifford algebraic structure and thus are proportional to -matrices

the creation annihilation operators are given by

thus satisfying the algebra:

Hence, for an orbifold rotation , we have

and

and for a boost of rapidity , we have

and

finally getting us

- An orbifold compactification can be obtained by identifying points in the compact part of spacetime which are connected by discrete rotations

on some of the compact pairs and to preserve at least one supersymmetry, one imposes .

Since 3-branes, in a double-geometric e-folding sense, figure essentially in my last few posts on the Randall-Sandrum/Klebanov-Strassler relation/correspondence, let me consider such a 3-brane configuration: in the static case, I shall take Neumann b.c. for time, Dirichlet b.c. for space and mixed b.c. for each pair of compact directions.

We get new b.c. for the compact directions

and

Let us also define a new spinor vacuum

such that

is the compact part of the boundary state. Notice though that

the boundary state is not invariant under orbifold rotations, under which the modes of the fields transform as in eq. (3)

as well as the new spinor vacuum as

However, this was to be expected since a rotation mixes two directions with different b.c, and …

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