Guillem Pérez-Nadal

A bstract: The Euclidean formulation of quantum gravity can be interpreted in terms of a probability distribution over Riemannian manifolds. In the context of de Sitter gravity, the statistics of the total volume according to this distribution is encoded in the dependence of the partition function on the cosmological constant. We use this observation to obtain a probability distribution for the volume from known results and proposals for the de Sitter partition function, in several levels of approximation: saddle point, one loop, an all-loop and a non-perturbative proposal in 3 dimensions, and an exact result in 2 dimensions, in the context of Liouville theory. In all cases we find a reasonable behavior: in the classical limit the distribution concentrates around the classical volume, and it spreads as quantum effects are turned on. We also find as a common trend that, as quantum effects are increased, the probability distribution favors increasingly smaller universes.

We compute the entanglement entropy of an interval for a chiral scalar on a circle at an arbitrary temperature. We use the resolvent method, which involves expressing the entropy in terms of the resolvent of a certain operator, and we compute that resolvent by solving a problem that entails finding an analytic function on the complex torus with certain jump conditions at the interval. The resolvent is relevant by itself, since it can be used to compute any function of the reduced density matrix. We illustrate that by also computing all the Rényi entropies for the model.

We compute the Rényi entropies of the massless Dirac field on the Euclidean torus (the Lorentzian cylinder at nonzero temperature) for arbitrary spatial regions. We do it by the resolvent method, i.e., we express the entropies in terms of the resolvent of a certain operator and then use the explicit form of that resolvent, which was obtained recently. Our results are different in appearance from those already existing in the literature (obtained via the replica trick), but they agree perfectly, as we show numerically for noninteger order and analytically for integer order. We also compute the Rényi mutual information, and find that, for appropriate choices of the parameters, it is nonpositive and nonmonotonic. This behavior is expected, but it cannot be seen with the simplest known Rényi entropies in quantum field theory because they are proportional to the entanglement entropy.

We consider the problem of defining spacelike-supported boundary-to-bulk propagators in AdS _d+1 down to the unitary bound ∆ = (d − 2)/2. That is to say, we construct the ‘smearing functions’ K of HKLL but with different boundary conditions where both dimensions ∆₊ and ∆ ₋ are taken into account. More precisely, we impose Robin boundary conditions, which interpolate between Dirichlet and Neumann boundary conditions and we give explicit expressions for the distributional kernel K with spacelike support. This flow between boundary conditions is known to be captured in the boundary by adding a double-trace deformation to the CFT. Indeed, we explicitly show that using K there is a consistent and explicit map from a Wightman function of the boundary QFT to a Wightman function of the bulk theory. In order to accomplish this we have to study first the microlocal properties of the boundary two-point function of the perturbed CFT and prove its wavefront set satisfies the microlocal spectrum condition. This permits to assert that K and the boundary two-point function can be multiplied as distributions.

In this paper we present the detailed calculation of a new modular Hamiltonian, namely that of a chiral fermion on a circle at non-zero temperature. We provide explicit results for an arbitrary collection of intervals, which we discuss at length by checking against known results in different limits and by computing the associated modular flow. We also compute the entanglement entropy, and we obtain a simple expression for it which appears to be more manageable than those already existing in the literature.

In this paper we present the detailed calculation of a new modular Hamiltonian, namely that of a chiral fermion on a circle at non-zero temperature. We provide explicit results for an arbitrary collection of intervals, which we discuss at length by checking against known results in different limits and by computing the associated modular flow. We also compute the entanglement entropy, and we obtain a simple expression for it which appears to be more manageable than those already existing in the literature.