Generalized quasi-topological gravities: the whole shebang

TL;DR

This paper classifies and analyzes generalized quasi-topological gravities (GQTGs) in higher dimensions, revealing multiple classes at each order and exploring their thermodynamic properties, with implications for black hole solutions.

Contribution

It proves the existence of multiple inequivalent classes of GQTGs at each order in curvature for dimensions D≥5 and characterizes their properties, including the unique algebraic quasi-topological density.

Findings

Existence of (n-1) inequivalent classes of order-n GQTGs for D≥5.

Identification of a unique quasi-topological density with algebraic equations for f(r).

Verification that thermodynamic charges satisfy the first law and relate to the embedding function.

Abstract

Generalized quasi-topological gravities (GQTGs) are higher-curvature extensions of Einstein gravity in $D$-dimensions. Their defining properties include possessing second-order linearized equations of motion around maximally symmetric backgrounds as well as non-hairy generalizations of Schwarzschild's black hole characterized by a single function, $f(r)\equiv - g_{tt}=g_{rr}^{-1}$, which satisfies a second-order differential equation. In arXiv:1909.07983 GQTGs were shown to exist at all orders in curvature and for general $D$. In this paper we prove that, in fact, $n-1$ inequivalent classes of order-$n$ GQTGs exist for $D\geq 5$. Amongst these, we show that one -- and only one -- type of densities is of the Quasi-topological kind, namely, such that the equation for $f(r)$ is algebraic. Our arguments do not work for $D=4$, in which case there seems to be a single unique GQT density at each order which is not of the Quasi-topological kind. We compute the thermodynamic charges of the most general $D$-dimensional order-$n$ GQTG, verify that they satisfy the first law and provide evidence that they can be entirely written in terms of the embedding function which determines the maximally symmetric vacua of the theory.