10.1 Supersymmetric Large Extra Dimensions (SLED) Model

This is a supersymmetric version of the Einstein–Maxwell model [72Jump To The Next Citation Point] (see [57, 58, 59, 247] for reviews). The bosonic part of the action is given by [3, 4, 2]
∫ √ ---[ 1 ( ) 1 ] S = d6x − g ---2 R − ∂M ϕ∂M ϕ − -e−ϕFMN F MN − eϕΛ6 . (437 ) 2 κ6 4
There exists a solution where the dilaton ϕ is constant, ϕ = ϕ0. The solution for the 6D spacetime is given by
2 μ ν i j ds = ημνdx dx + γijdx dx . (438 )
The gauge field is taken to consist of magnetic flux threading the extra dimensional space so that the field strength takes the form
√ -- Fij = γB0 𝜖ij, (439 )
where B0 is a constant, γ is the determinant of γij and 𝜖ij is the antisymmetric tensor normalized as 𝜖12 = 1. All other components of Fab vanish. A static and stable solution is obtained by choosing the extra-dimensional space to be a two-sphere
i j 2 2 2 2 γijdx dx = a 0(d 𝜃 + sin 𝜃d φ ). (440 )
The magnetic field strength B0 and the radius a0 are fixed by the cosmological constant
M 4 B20 = 2Λ6e2ϕ0, a20 = ----6ϕ--. (441 ) 2Λ6e 0.
The constant value ϕ0 is determined by the condition that the potential for ϕ has minimum [153Jump To The Next Citation Point]
′ 1- 2 − ϕ0 ϕ0 V (ϕ0) = − 2B 0e + Λ6e = 0. (442 )
This is exactly the condition to have a flat geometry on the brane (see Equation (441View Equation))
2 − ϕ0 ϕ0 B0e = 2 Λ6e . (443 )
Thus without tuning the 6D cosmological constant, it is possible to obtain a flat 4D spacetime.

Now we add branes to this solution [72Jump To The Next Citation Point]. The brane action is given by

∫ 4 √ ---- S4 = − σ d x − γ. (444 )
The solution for the extra dimensions is now given by
γijdxidxj = a20(d𝜃2 + α2 sin2 𝜃dφ2), (445 )
σ M 4 α = 1 − -----2, a20 = ----6ϕ-. (446 ) 2πM 6 2Λ6e 0
The coordinate φ ranges from 0 to 2π. Thus the effect of the brane makes a deficit angle δ = 2π (1 − α) in the bulk (see Figure 27View Image). This is a 6D realization of the ADD model including the self-gravity of branes. An interesting property of this model is that regardless of the tension of the brane, the 4D specetime on the brane is flat. Thus this could solve the cosmological constant problem – any vacuum energy on the brane only changes a geometry of the extra-dimensions and does not curve the 4D spacetime. This idea of solving the cosmological constant problem is known as self-tuning.
View Image

Figure 27: Removing a wedge from a sphere and identifying opposite sides to obtain a rugby ball geometry. Two equal-tension branes with conical deficit angles are located at either pole; outside the branes there is constant spherical curvature. From [72].

We should note that there have been several objections to the idea of self-tuning [153, 419Jump To The Next Citation Point, 420Jump To The Next Citation Point]. Consider that a phase transition occurs and the tension of the brane changes from σ1 to σ2. Accordingly, α changes from α1 = 1 − σ1∕ (2 πM 46) to α2 = 1 − σ2∕(2πM 46). The magnetic flux is conserved as the gauge field strength is a closed form, dF = 0. Then the magnetic flux which is obtained by integrating the field strength over the extra dimensions should be conserved

ΦB = 4πα1B0,1 = 4π α2B0,2. (447 )
The relation between Λ6 and B0, Equation (441View Equation), that ensures the existence of Minkowski branes cannot be imposed both for B0 = B0,1 and B0 = B0,2 when α1 ⁄= α2 unless ϕ0 changes. Moreover, the quantization condition must be imposed on the flux ΦB. What happens is that a modulus, which is a combination of ϕ and the radion describing the size of extra-dimension, acquires a runaway potential and the 4D spacetime becomes non-static.

An unambiguous way to investigate this problem is to study the dynamical solutions directly in the 6D spacetime. However, once we consider the case where the tension becomes time dependent, we encounter a difficulty to deal with the branes [419Jump To The Next Citation Point]. This is because for co-dimension 2 branes, we encounter a divergence of metric near the brane if we put matter other than tension on a brane. Hence, without specifying how we regularize the branes, we cannot address the question what will happen if we change the tension. Is the self-tuning mechanism at work and does it lead to another static solution? Or do we get a dynamical solution driven by the runaway behaviour of the modulus field?

There was a negative conclusion on the self-tuning in this supersymmetric model for a particular kind of regularization [419Jump To The Next Citation Point, 420Jump To The Next Citation Point]. However, the answer could depend on the regularization of branes and the jury remains out. It is important to study the time-dependent dynamics in the 6D spacetime and the regularization of the branes in detail [410, 60, 62, 411, 61, 33, 110, 338, 350, 100].

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