Burgess and his collaborators argue that one plausible reason is that space-time has additional dimensions, and total space-time is not Lorentz-invariant. In other words, it is a scenario with Branas in higher dimensions. In such a situation, the problematic quantum contributions, which normally, due to Lorentz-invariance, take the form of a cosmological constant, may not be noticeable in the Brana, which is where we live.
The example they give is that of a cosmic chain. If we calculate the metric that the string induces, we find that space is flat, but has a defect angle that depends on the tension of the string. The string itself is not affected by what it does to the background. The scenario that Burgess et al construct is basically a higher dimensional version, where our universe plays the role of the rope and creates a defect, but no curvature is induced in our own universe.
Specifically, they have two extra large dimensions. (There may be more than that, but if they are much smaller, their presence doesn't matter for the argument.) These extra dimensions have the topology of a sphere. At the two poles of the sphere, there is a Brana at each, one of which you can interpret as our universe. As with the cosmic string, the density of matter on the branes induces a defect angle to the sphere, creating a multiple that they call a "rugby ball". The radius of the sphere is flux stabilized, which leaves a free parameter (a combination of the radius and the dilaton field).
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