Counting paths with Schur transitions

In this work we explore the structure of the branching graph of the unitary group using Schur transitions. We find that these transitions suggest a new combinatorial expression for counting paths in the branching graph. This formula, which is valid for any rank of the unitary group, reproduces known asymptotic results. We proceed to establish the general validity of this expression by a formal proof. The form of this equation strongly hints towards a quantum generalization. Thus, we introduce a notion of quantum relative dimension and subject it to the appropriate consistency tests. This new quantity finds its natural environment in the context of RCFTs and fractional statistics; where the already established notion of quantum dimension has proven to be of great physical importance.Comment: 30 pages, 5 figure

Spinning probes and helices in AdS3_3

We study extremal curves associated with a functional which is linear in the curve's torsion. The functional in question is known to capture the properties of entanglement entropy for two-dimensional conformal field theories with chiral anomalies and has potential applications in elucidating the equilibrium shape of elastic linear structures. We derive the equations that determine the shape of its extremal curves in general ambient spaces in terms of geometric quantities. We show that the solutions to these shape equations correspond to a three-dimensional version of Mathisson's helical motions for the centers of mass of spinning probes. Thereafter, we focus on the case of maximally symmetric spaces, where solutions correspond to cylindrical helices and find that the Lancret ratio of these equals the relative speed between the Mathisson-Pirani and the Tulczyjew-Dixon observers. Finally, we construct all possible helical motions in three-dimensional manifolds with constant negative curvature. In particular, we discover a rich space of helices in AdS$_3$ which we explore in detail.Comment: 28 pages, 5 figure

On the Shape of Things: From holography to elastica

We explore the question of which shape a manifold is compelled to take when immersed in another one, provided it must be the extremum of some functional. We consider a family of functionals which depend quadratically on the extrinsic curvatures and on projections of the ambient curvatures. These functionals capture a number of physical setups ranging from holography to the study of membranes and elastica. We present a detailed derivation of the equations of motion, known as the shape equations, placing particular emphasis on the issue of gauge freedom in the choice of normal frame. We apply these equations to the particular case of holographic entanglement entropy for higher curvature three dimensional gravity and find new classes of entangling curves. In particular, we discuss the case of New Massive Gravity where we show that non-geodesic entangling curves have always a smaller on-shell value of the entropy functional. Then we apply this formalism to the computation of the entanglement entropy for dual logarithmic CFTs. Nevertheless, the correct value for the entanglement entropy is provided by geodesics. Then, we discuss the importance of these equations in the context of classical elastica and comment on terms that break gauge invariance.Comment: 54 pages, 8 figures. Significantly improved version, accepted for publication in Annals of Physics. New section on logarithmic CFTs. Detailed derivation of the shape equations added in appendix B. Typos corrected, clarifications adde

Graph duality as an instrument of Gauge-String correspondence

We explore an identity between two branching graphs and propose a physical meaning in the context of the gauge-gravity correspondence. From the mathematical point of view, the identity equates probabilities associated with $\mathbb{GT}$, the branching graph of the unitary groups, with probabilities associated with $\mathbb{Y}$, the branching graph of the symmetric groups. In order to furnish the identity with physical meaning, we exactly reproduce these probabilities as the square of three point functions involving certain hook-shaped backgrounds. We study these backgrounds in the context of LLM geometries and discover that they are domain walls interpolating two AdS spaces with different radii. We also find that, in certain cases, the probabilities match the eigenvalues of some observables, the embedding chain charges. We finally discuss a holographic interpretation of the mathematical identity through our results.Comment: 34 pages. version published in journa

Renormalized Entanglement Entropy for BPS Black Branes

We compute the renormalized entanglement entropy (REE) for BPS black solutions in ${\cal N}=2$, 4d gauged supergravity. We find that this quantity decreases monotonically with the size of the entangling region until it reaches a critical point, then increases and approaches the entropy density of the brane. This behavior can be understood as a consequence of the REE being driven by two competing factors, namely entanglement and the mixedness of the black brane. In the UV entanglement dominates, whereas in the IR the mixedness takes over.Comment: 6 pages, 4 figures; v2: Typos fixed, citation and clarifying text added, version accepted in Physical Review

Quantum corrections to extremal black brane solutions

We discuss quantum corrections to extremal black brane solutions in N=2 U(1) gauged supergravity in four dimensions. We consider modifications due to a certain class of higher-derivative terms as well as perturbative corrections to the prepotential. We use the entropy function formalism to assess the impact of these corrections on singular brane solutions and we give a few examples. We then use first-order flow equations to construct solutions that interpolate between quantum corrected fixed points of the associated potentials.Comment: 30 pages, 10 figures; v2: references added, as well as a discussion about how to obtain the attractor equations by extremizing an effective potential, coincides with published versio