Abstract
Two dimensional field theories invariant under the Bondi-Metzner-Sachs (BMS) group are conjectured to be dual to asymptotically flat spacetimes in three dimensions. In this paper, we continue our investigations of the modular properties of these field theories. In particular, we focus on the BMS torus one-point function. We use two different methods to arrive at expressions for asymptotic structure constants for general states in the theory utilising modular properties of the torus one-point function. We then concentrate on the BMS highest weight representation, and derive a host of new results, the most important of which is the BMS torus block. In a particular limit of large weights, we derive the leading and sub-leading pieces of the BMS torus block, which we then use to rederive an expression for the asymptotic structure constants for BMS primaries. Finally, we perform a bulk computation of a probe scalar in the background of a flatspace cosmological solution based on the geodesic approximation to reproduce our field theoretic results.
Highlights
1.1 Preliminaries: conformal symmetry The techniques of conformal field theory (CFT) [1, 2] have been central to our understanding of various diverse subjects including the theory of phase transitions in statistical physics, the understanding of string theory from worldsheet symmetries, the study of quantum gravity in AdS spacetimes through the celebrated gauge-gravity correspondence, as well as explorations in cosmology
Of particular importance is the procedure of conformal bootstrap [3,4,5,6], which aims to solve CFTs by imposing an infinite set of consistency conditions on the theory that stem from crossing symmetry of the four-point function
E|O|E, in the bulk side denotes the calculation of a one-point function of a light operator O, or a probe, in the background of a heavy state |E, which is given by a Flat Space Cosmologies (FSCs) solution
Summary
1.1 Preliminaries: conformal symmetry The techniques of conformal field theory (CFT) [1, 2] have been central to our understanding of various diverse subjects including the theory of phase transitions in statistical physics, the understanding of string theory from worldsheet symmetries, the study of quantum gravity in AdS (and dS) spacetimes through the celebrated gauge-gravity correspondence, as well as explorations in cosmology. One can generate other values of the central term by considering more general theories of gravity e.g. by adding gravitational Chern-Simons term to the Einstein-Hilbert action [17] It is natural, following lessons of AdS3, to attempt constructions of holography in 3d asymptotically flat spacetimes, using the algebra (1.1). As we will go on to describe, the BMS3 algebra can be obtained by an Inönü-Wigner contraction of two copies of the Virasoro algebra This contraction which is the infinite radius limit of AdS, manifests itself on the boundary theory as an ultra-relativistic or a Carrollian limit on the parent 2d CFT, where the speed of light goes to zero. Some other interesting advances in the study of holography for three dimensional asymptotic flat spacetimes include the construction of the flat limit of Liouville theory as an explicit putative boundary theory [28], matching of correlation functions between bulk and boundary for stress-energy tensors [29] and generic fields [30], aspects of entanglement entropy [30,31,32,33,34,35,36,37,38], understanding holographic reconstruction [39], and the flat version of the fluid-gravity correspondence [40]
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