A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance

We derive an asymptotic formula for operator product expansion coefficients of heavy operators in two dimensional conformal field theory. This follows from modular invariance of the genus two partition function, and generalises the asymptotic formula for the density of states from torus modular invariance. The resulting formula is universal, depending only on the central charge, but involves the asymptotic behaviour of genus two conformal blocks. We use monodromy techniques to compute the asymptotics of the relevant blocks at large central charge to determine the behaviour explicitly.Comment: 32 pages, 2 figures, 1 appendix, 2 moose, a bear and an o

Counting states in a model of replica wormholes

We study the Hilbert space of a system of $n$ black holes with an inner product induced by replica wormholes. This takes the form of a sum over permutations, which we interpret in terms of a gauge symmetry. The resulting inner product is degenerate, with null states lying in representations corresponding to Young diagrams with too many rows. We count the remaining states in a large $n$ limit, which is governed by an emergent collective Coulomb gas description describing the shape of typical Young diagrams. This exhibits a third-order phase transition when the null states become numerous. We find that the dimension of the black hole Hilbert space accords with a microscopic interpretation of Bekenstein-Hawking entropy.Comment: 27pp + appendices, 4 figure

Bit models of replica wormholes

We define bit models of evaporating black holes which incorporate the effects of replica wormholes. These enter as non-perturbative corrections to the gravitational inner product, arising from spacetime wormholes that connect replica black holes. The resulting models have exactly unitary evolution, predict measurements on Hawking radiation in accord with a Page curve for entropy, and are compatible with a smooth horizon for infalling observers

Phase transitions in 3D gravity and fractal dimension

We show that for three dimensional gravity with higher genus boundary conditions, if the theory possesses a sufficiently light scalar, there is a second order phase transition where the scalar field condenses. This three dimensional version of the holographic superconducting phase transition occurs even though the pure gravity solutions are locally AdS$_3$. This is in addition to the first order Hawking-Page-like phase transitions between different locally AdS$_3$ handlebodies. This implies that the R\'enyi entropies of holographic CFTs will undergo phase transitions as the R\'enyi parameter is varied, as long as the theory possesses a scalar operator which is lighter than a certain critical dimension. We show that this critical dimension has an elegant mathematical interpretation as the Hausdorff dimension of the limit set of a quotient group of AdS$_3$, and use this to compute it, analytically near the boundary of moduli space and numerically in the interior of moduli space. We compare this to a CFT computation generalizing recent work of Belin, Keller and Zadeh, bounding the critical dimension using higher genus conformal blocks, and find a surprisingly good match

The geometry and topology of quantum entanglement in holography

In this thesis I explore the connection between geometry and quantum entanglement, in the context of holographic duality, where entanglement entropies in a quantum field theory are associated with the areas of surfaces in a dual gravitational theory. The first chapter looks at a phase transition in such systems in finite size and at finite temperature, associated with the properties of minimal surfaces in a static black hole background. This is followed by the related problem of extremal surfaces in a spacetime describing the dynamical process of black hole formation, with a view towards understanding the connections between bulk locality and various field theory observables including entanglement entropy. The third chapter looks at the simple case of pure gravity in three spacetime dimensions, where I show how evaluating the entanglement entropy can be reduced to a simple algebraic calculation, and apply it to some interesting examples. Finally, the role played by topology of surfaces in a proposed derivation of a holographic entanglement entropy formula is investigated. This makes it clear what assumptions are required in order to reproduce the ‘homology constraint’, a topological condition necessary for consistency with field theory

Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information

In the 1980's, work by Coleman and by Giddings and Strominger linked the physics of spacetime wormholes to `baby universes' and an ensemble of theories. We revisit such ideas, using features associated with a negative cosmological constant and asymptotically AdS boundaries to strengthen the results, introduce a change in perspective, and connect with recent replica wormhole discussions of the Page curve. A key new feature is an emphasis on the role of null states. We explore this structure in detail in simple topological models of the bulk that allow us to compute the full spectrum of associated boundary theories. The dimension of the asymptotically AdS Hilbert space turns out to become a random variable $Z$, whose value can be less than the naive number $k$ of independent states in the theory. For $k>Z$, consistency arises from an exact degeneracy in the inner product defined by the gravitational path integral, so that many a priori independent states differ only by a null state. We argue that a similar property must hold in any consistent gravitational path integral. We also comment on other aspects of extrapolations to more complicated models, and on possible implications for the black hole information problem in the individual members of the above ensemble