On the Shape of Things: From holography to elastica

TL;DR

This paper investigates the shapes of manifolds as extremals of curvature-dependent functionals, deriving shape equations, applying them to holography and elastica, and revealing new entangling curves with implications for higher curvature gravity and CFTs.

Contribution

It introduces a detailed derivation of shape equations for curvature-based functionals, explores their applications in holography and elastica, and uncovers new entangling curves in higher curvature gravity.

Findings

New classes of entangling curves in holography.

Non-geodesic curves have lower entropy in New Massive Gravity.

Geodesics provide the correct entanglement entropy in dual logarithmic CFTs.

Abstract

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.