We review what has been learnt and what remains unknown about the physics of hot enhancons following studies in supergravity. We recall a rather general family of static, spherically symmetric, non-extremal enhancon solutions describing D4 branes wrapped on K3 and discuss physical aspects of the solutions. We embed these solutions in the six dimensional supergravity describing type IIA strings on K3 and generalize them to have arbitrary charge vector. This allows us to demonstrate the equivalence with a known family of hot fractional D0 brane solutions, to widen the class of solutions of this second type and to carry much of the discussion across from the D4 brane analysis. In particular we argue for the existence of a horizon branch for these branes.
Surprisingly, when the system is near extremality and the asymptotic volume of the K3 is large, the first two terms combine into a dominant, negative, contribution. Thus the shell branch violates the WEC. It was argued @cite_7 that the shell branch should therefore be regarded as unphysical. Accordingly, the horizon branch should be considered the dominant, valid, supergravity solution for non-BPS , for the range of parameters admitting it. For the region of parameter space in which no horizon branch exists, other solutions, more general than those yet considered, might be valid @cite_7 .
@cite_7
2471626564 2079936263 326163211 2332339252
We study the supergravity solutions describing nonextremal enhan c c ons. There are two branches of solutions: a shell branch'' connected to the extremal solution, and a horizon branch'' which connects to the Schwarzschild black hole at large mass. We show that the shell branch solutions violate the weak energy condition, and are hence unphysical. We investigate linearized perturbations of the horizon branch and the extremal solution numerically, completing an investigation initiated in a previous paper. We show that these solutions are stable against the perturbations we consider. This provides further evidence that these latter supergravity solutions are capturing some of the true physics of the enhan c c on. The enhancon mechanism removes a family of time-like singularities from certain supergravity spacetimes by forming a shell of branes on which the exterior geometry terminates. The problematic interior geometry is replaced by a new spacetime, which in the prototype extremal case is simply flat. We show that this excision process, made inevitable by stringy phenomena such as enhanced gauge symmetry and the vanishing of certain D-branes' tension at the shell, is also consistent at the purely gravitational level. The source introduced at the excision surface between the interior and exterior geometries behaves exactly as a shell of wrapped D6-branes, and in particular, the tension vanishes at precisely the enhancon radius. These observations can be generalised, and we present the case for non-extremal generalisations of the geometry, showing that the procedure allows for the possibility that the interior geometry contains an horizon. Further knowledge of the dynamics of the enhancon shell itself is needed to determine the precise position of the horizon, and to uncover a complete physical interpretation of the solutions. We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a|<M, arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards. The aim of this paper is to show that spatial coupling can be viewed not only as a means to build better graphical models, but also as a tool to better understand uncoupled models. The starting point is the observation that some asymptotic properties of graphical models are easier to prove in the case of spatial coupling. In such cases, one can then use the so-called interpolation method to transfer known results for the spatially coupled case to the uncoupled one. Our main use of this framework is for Low-density parity check (LDPC) codes, where we use interpolation to show that the average entropy of the codeword conditioned on the observation is asymptotically the same for spatially coupled as for uncoupled ensembles. We give three applications of this result for a large class of LDPC ensembles. The first one is a proof of the so-called Maxwell construction stating that the MAP threshold is equal to the area threshold of the BP GEXIT curve. The second is a proof of the equality between the BP and MAP GEXIT curves above the MAP threshold. The third application is the intimately related fact that the replica symmetric formula for the conditional entropy in the infinite block length limit is exact.