Unitarity at Infinity and Topological Holography

Recently it has been suggested that non-gaussian inflationary perturbations can be usefully analysed in terms of a putative dual gauge theory defined on the future conformal infinity generated by an accelerating cosmology. The problem is that unitarity of this gauge theory implies a strong constraint [the “Strominger bound”] on the matter fields in the bulk. We argue that the bound is just a reflection of the equation of state of cosmological matter. The details motivate a discussion of the possible relevance of the “dS/CFT correspondence” to the resolution of the Big Bang singularity. It is argued that the correspondence may require the Universe to come into existence along a non-singular spacelike hypersurface, as in the theories of “creation from nothing” discussed by Firouzjahi, Sarangi, and Tye, and also by Ooguri et al. The argument makes use of the unusual properties of gauge theories defined on topologically non-trivial spaces. 1. dS/CFT: Limitations and Applications Efforts to connect the AdS/CFT correspondence [1] with cosmology lead naturally to the idea of a dS/CFT correspondence, in which the physics of accelerating spacetimes is related to a Euclidean CFT defined on spacelike conformal infinity [2][3]. In the form used by Maldacena [4][5][6], this version of holography has recently been revived [7] [see also [8]], in the hope of establishing a new understanding of the improved observational data. There has also been a revival of interest in the theoretical aspects of the correspondence. Thus for example Polchinski [9] has recently discussed the idea of emergent time in asymptotically de Sitter spacetimes, comparing it with “emergent gauge symmetries” of a possible gauge theory at future infinity. There are, however, some doubts as to whether a precise de Sitter analogue of the AdS/CFT correspondence can actually be constructed. It is generally agreed that a de Sitter phase can only be metastable in string theory [10][11]; generically it would decay to a cosmological spacetime with a Crunch; so the future boundary on which the dual theory is to be defined may not exist. This could simply mean that the dual theory can only be defined on some spacelike hypersurface at large but finite proper time, or that the dual theory can only be formulated in the Euclidean version of the spacetime [as in the work of Maldacena and Maoz [12]], or it may imply more serious limitations. A concrete proposal for understanding the limitations of dS/CFT was suggested by van der Schaar [6], who argued that dS/CFT is effective only at the level of a coarse-graining in which each lattice site of the field theory corresponds to an entire static patch in the de Sitter bulk. One way of stating the case would be to suggest that, on the bulk side of the correspondence, dS/CFT is primarily relevant to specifically “cosmological” features of spacetime and its matter content, not to their detailed structure. In this spirit, we wish to make an observation regarding a curious and apparently unphysical feature of the dS/CFT correspondence: it appears to impose a severe upper bound on the masses of particles. In detail, the limit of a massive p-form field amplitude [for a p-form field φ of mass mφ] in de Sitter spacetime defines a CFT two-point function at de Sitter conformal infinity. In the p = 0 case, the conformal weight corresponding to the boundary operator defined by φ is given, in four spacetime dimensions, by