Now we add branes to this solution [72]. The brane action is given by

The solution for the extra dimensions is now given by where The coordinate ranges from to . Thus the effect of the brane makes a deficit angle in the bulk (see Figure 27). 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.We should note that there have been several objections to the idea of self-tuning [153, 419, 420]. Consider that a phase transition occurs and the tension of the brane changes from to . Accordingly, changes from to . The magnetic flux is conserved as the gauge field strength is a closed form, . Then the magnetic flux which is obtained by integrating the field strength over the extra dimensions should be conserved

The relation between and , Equation (441), that ensures the existence of Minkowski branes cannot be imposed both for and when unless changes. Moreover, the quantization condition must be imposed on the flux . What happens is that a modulus, which is a combination of 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 [419]. 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 [419, 420]. 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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