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.

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