The holographic c-theorem and infinite-dimensional Lie algebras

Abstract

We discuss a non-dynamical theory of gravity in three dimensions which is based on an infinite-dimensional Lie algebra that is closely related to an infinite-dimensional extended AdS algebra. We find an intriguing connection between on the one hand higher-derivative gravity theories that are consistent with the holographic c-theorem and on the other hand truncations of this infinite-dimensional Lie algebra that violate the Lie algebra structure. We show that in three dimensions different truncations reproduce, up to terms that do not contribute to the c-theorem, Chern-Simons-like gravity models describing extended 3D massive gravity theories. Performing the same procedure with similar truncations in dimensions larger than or equal to four reproduces higher derivative gravity models that are known in the literature to be consistent with the c-theorem but do not have an obvious connection to massive gravity like in three dimensions.

Lie algebra expansions, non-relativistic matter multiplets and actions

“…an infinite-dimensional algebra [77]. When this expansion is generalized to include supersymmetry, the D-dimensional Poincaré superalgebra brings the following two commutators for the SUSY generators…”

Higher-curvature Gravities from Braneworlds and the Holographic c-theorem

“…This vanishes for n > d because the totally antisymmetric product of Kronecker deltas is identically zero in that case, but it has been shown that a simple limiting procedure 11 can be applied to P (n) , which gives non-trivial densities for additional orders and dimensions [43] (see also [44,45]). One may also systematically consider all the densities of a given curvature order for fixed d, with arbitrary relative coefficients, and identify the combinations that satisfy the aforementioned conditions.…”

“…In addition, it has been found that certain theories that satisfy the holographic c-theorem-some of which involve explicit covariant derivatives-are equivalent to Chern-Simons gravities [49]. More recently, theories of this kind have been related to truncations of certain infinitedimensional Lie algebras [45]. It has also been shown that theories of this kind never propagate the scalar mode that is present in the linearized spectrum of generic higher-curvature theories [46].…”

Non-Relativistic and Ultra-Relativistic Scaling Limits of Multimetric Gravity

“…On the other hand, this procedure was the main tool to construct the action principle for Newtonian gravity in first-order formulation (see [6,7] for its second order formulation and the relevant 1/c 2 expansion) as well as establishing new, extended, two and three-dimensional non/ultra-relativistic gravity models . In fact, the massive gravity models of [5] arise as scaling limits of ghost-free bi-gravity models [35,36] which later lead to the discovery that there exist trajectories in the parameter space of bi-gravity theories that connect the central charges of bulk/boundary unitary three-dimensional bi-gravity models to non-unitary massive gravity theories by a continuous change of scaling parameter [37][38][39]. It is, thus, a natural question whether one can unify the Lie algebra expansion and the scaling limit together to define a non/ultra-relativistic limit for bimetric and multimetric models of gravity to establish similar connections between physical quantities.…”