Gravitational Higgs Mechanism and Massive Gravity

In arXiv:hep-th/0011245, in the context of domain wall backgrounds, it was shown that spontaneous breaking of diffeomorphism invariance results in gravitational Higgs mechanism. Recently in arXiv:0708.3184 't Hooft discussed gravitational Higgs mechanism in the context of obtaining massive gravity directly in four dimensions, and pointed out some subtleties with unitarity. We attribute these subtleties to breaking time-like diffeomorphisms, and discuss gravitational Higgs mechanism with all but time-like diffeomorphisms spontaneously broken. The resulting background is no longer flat but exhibits expansion, which is linear in time. For space-time dimensions D <= 10 the background is stable and has no non-unitary propagating modes. The absence of non-unitary modes is due to the unbroken time-like diffeomorphism invariance. The physical states correspond to those of a massive graviton. The effective mass squared of the graviton is positive for D < 10, and vanishes for D = 10. For D > 10 the graviton modes become effectively tachyonic. The special value of D=10, which coincides with the critical dimension of superstring theory, arises in our setup completely classically.Comment: 11 pages; minor misprints corrected, references adde

Geometry of Orientifolds with NS-NS B-flux

We discuss geometry underlying orientifolds with non-trivial NS-NS B-flux. If D-branes wrap a torus with B-flux the rank of the gauge group is reduced due to non-commuting Wilson lines whose presence is implied by the B-flux. In the case of D-branes transverse to a torus with B-flux the rank reduction is due to a smaller number of D-branes required by tadpole cancellation conditions in the presence of B-flux as some of the orientifold planes now have the opposite orientifold projection. We point out that T-duality in the presence of B-flux is more subtle than in the case with trivial B-flux, and it is precisely consistent with the qualitative difference between the aforementioned two setups. In the case where both types of branes are present, the states in the mixed (e.g., 59) open string sectors come with a non-trivial multiplicity, which we relate to a discrete gauge symmetry due to non-zero B-flux, and construct vertex operators for the the mixed sector states. Using these results we revisit K3 orientifolds with B-flux (where K3 is a T^4/Z_M orbifold) and point out various subtleties arising in some of these models. For instance, in the Z_2 case the conformal field theory orbifold does not appear to be the consistent background for the corresponding orientifolds with B-flux. This is related to the fact that non-zero B-flux requires the presence of both O5^- as well as O5^+ planes at various Z_2 orbifold fixed points, which appears to be inconsistent with the presence of the twisted B-flux in the conformal field theory orbifold. We also consider four dimensional N=2 and N=1 supersymmetric orientifolds. We construct consistent four dimensional models with B-flux which do not suffer from difficulties encountered in the K3 cases.Comment: 79 pages, revte