In this post, the mathematics applies to both, Randall-Sundrum-1and-2 models, hence I will not distinguish between them here. One of the most powerful aspects of M-theory’s braneworld scenarios is that the bosonic and fermionic fields of the Standard Model of physics can be interpreted as low-lying Kaluza-Klein excitations of Randall-Sundrum bulk fields, after extra dimensional modulus stabilization, and recalling that Randall-Sundrum bulk/brane interactions yield a very deep solution to the EW hierarchy problem. Start with the theory defined by the following action:

with the bulk field given by:

where generally, the bulk action, with worldsheet-uplift, is given by:

and satisfying:

with a Dirac-Born-Infeld brane interaction term:

which, after integration by parts and upon substituting  in our action, we get the Horava-Witten action variant:

Now, the bulk fields manifest themselves to 4-D ‘observers’ as infinite towers of scalars  with masses . After change of variables to:

our actions reduce to two interaction terms:

and:

where we have:

and the Bessel functions of order:

yield the standard Bertotti-Robinson-solutions. Hence, we have:

with  a normalization factor. That the differential operator on the LHS of:

is self-adjoint means that the derivative of  is continuous at the orbifold fixed points, giving us:

Four-dimensionally, these induce couplings between the Kaluza-Klein modes and so the exponential factor in:

where  are Lorentz coordinates on the four-dimensional surfaces of constant thus plays an essential role in determining the effective scale of the couplings. If the Planck scale sets the scale of the five-dimensional couplings, the low-lying Kaluza-Klein modes will have TeV-range self-interactions.

Now, a Klebanov-Strassler geometry naturally arises by considering string theory compactification on  where  is the Einstein manifold in five dimensions, with the interaction-Lagrangian of the massless Klebanov-Strassler field and the brane fields fermions is:

which, after integrating over the extra dimensional part, the effective 4-D Lagrangian reduces to:

with the fundamental Planck scale  and the 4-D Planck scale related as

Hence, in light of the Klebanov-Strassler/Randall-Sundrum throat-bulk isomorphism, this defines a background geometry given by:

with  and the induced metric on the hidden and visible brane-sectors,  the 5-D metric, with the 5-D Planck scale,  the cosmological ‘constant’,  the scalar field and the corresponding potential.

Working in the -warp-factor metric:

the corresponding 5-D Einstein and scalar field equations are:

and

with the index over the branes and our boundary-conditions of and  are given by:

To analytically solve in the backreacted Randall-Sundrum model-type, we use the quadratic/quartic bulk/brane dualized potential:

with:

Now we can derive solutions:

where is the scalar field on the Planck brane. Hence, and are given by:

and

We can now address the modulus stability of the braneworld. Substituting  into:

gives us the 4-D potential for the radion:

One then achieves inter-brane stabilization by minimizing the above potential with respect to the radion:

Hence, for the modulus field , the stabilization condition is:

Note now, in a backreacted RS model,

has no minima that is consistent with inflationary coupling-running. Thus, a quartic term of the bulk stabilising field potential must be coupled to the action factoring the mass of the scalar field:

Consider brane-fluctuations localized at stable inter-brane separation  as a function of brane-coordinates. The metric is hence:

The KK-modified brane warping is given by:

for the 5-D modulus brane angular coordinates. The Einstein-Hilbert action now is given by:

and by integrating over the 5th dimensional scalar field, we get

with kinetic sub-part:

where is the normalised radion field:

Thus, coupling the mass to the effective potential term and the inter brane measure  gives us:

and:

hence yielding the mass term:

Standard Model physics naturally arises now. One first derives the scalar radion field via interaction terms in the Standard Model, since the metric:

implies that the visible RS brane:

and:

couple to the Higgs sector of the SM via the action:

where is the Higgs field. Normalizing via:

thus reduces our action to:

where:

By an Euler-Lagrange derivation, we get the Higgs-field energy-momentum tensor:

thus coupling the radion to the Higgs via:

It is straightforward now to generalize to all fields of the Standard Model. Let be any field

By the above Higgs method, the corresponding RS-SM coupling term is:

Hence, yielding the coupled action:

Going back to the Klebanov-Strassler/Randall-Sundrum throat-bulk isomorphism, a stack of N D3-branes placed at the singularity  backreacts on the KS-geometry, creating a warped background with the following ten dimensional line element

with  the metric

and the warp factor is

and

### the deep part is that this AdS background is an explicit realization of the Randall-Sundrum scenario in string theory

that I discussed here and here. So in line with the AdS/CFT duality, the  geometry

### has a dual gauge theory interpretation

namely, an  gauge theory coupled to bifundamental chiral superfields, and adding  D5-branes wrapped over the  inside , then the gauge group becomes

giving a cascading gauge theory. The three-form flux induced by the wrapped D5-branes – fractional D3-branes – satisfies

and the Klebanov-Strassler warp-throat factor is

with

Thus from:

one derives the wave-function and the superposition-principle for every SM field from Kirchhoff ’s integral theorem. That gets very philosophically deep in the context of the Wheeler–DeWitt equation and the Hartle–Hawking wave function , since as one might expect, boundary conditions and the potential condition corresponding to the path-integral of:

are highly problematic, to say the least, though in upcoming posts, I will show how M-theory successfully deals with both via geometric surgery/quantum engineering methods in homological mirror symmetry.