Quasi-Topological Gravities

• Quasi-topological gravities are higher-curvature extensions of Einstein gravity distinguished by static, spherically symmetric black-hole solutions governed by a single algebraic metric function.
• They are classified into three types based on the order of field equations and align with Birkhoff’s theorem, bridging Lovelock theories with generalized quasi-topological models.
• These theories maintain analytic control on maximally symmetric backgrounds, propagate only a massless graviton, and influence holographic dualities and black hole thermodynamics.

Quasi-topological gravities are higher-curvature extensions of Einstein gravity that are especially special in spherical symmetry. In the standard higher-dimensional setting they are most naturally studied in spacetime dimension $D\ge 5$, where one can add nontrivial curvature corrections beyond the Einstein-Hilbert term without reducing immediately to topological densities. Their defining hallmark is the existence of static and spherically symmetric black-hole solutions determined by a single metric function $f(r)$, with $-g_{tt}=g_{rr}^{-1}\equiv f(r)$, and with $f(r)$ satisfying an algebraic equation rather than a generic higher-derivative differential equation (Bueno et al., 29 Oct 2025). In the broader classification of higher-curvature theories, quasi-topological gravities form the algebraic black-hole subclass of generalized quasi-topological gravities (GQTGs), whose corresponding single-function equation is at most second order but not necessarily algebraic (Bueno et al., 2022).

1. Definition, scope, and place within higher-curvature gravity

The modern notion of quasi-topological gravity is tied to the behavior of the field equations on symmetry-reduced backgrounds rather than to topological invariance on arbitrary metrics. In the higher-curvature action

$$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$

the quasi-topological condition singles out those combinations of curvature invariants for which static, spherically symmetric black holes remain controlled by one function $f(r)$, and in the quasi-topological subclass the resulting equation for $f(r)$ is algebraic (Bueno et al., 2022).

This distinguishes quasi-topological gravities from both Lovelock gravity and generic higher-derivative gravity. Lovelock gravity yields second-order equations for arbitrary metrics; generic higher-curvature theories typically generate higher-derivative equations and additional propagating modes. Quasi-topological gravities relax the Lovelock requirement globally while preserving a Lovelock-like algebraic structure on the static, spherically symmetric sector. In that sense they sit between Lovelock theories and generic higher-derivative gravities (Fierro et al., 2020).

The standard static ansätze are

$$ds^2_{N,f}=-N(r)^2f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{(D-2)}^2,$$

and, after fixing the Schwarzschild-like gauge,

$$ds^2_f=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{(D-2)}^2,\qquad f(r)\equiv -g_{tt}=g_{rr}^{-1}.$$

A curvature density is a GQTG density if the Euler-Lagrange equation of $f(r)$ from the reduced action vanishes identically for arbitrary $f(r)$0,

$f(r)$1

equivalently if the reduced Lagrangian is a total derivative. Quasi-topological gravities are the special subclass for which the integrated equation for $f(r)$2 is algebraic rather than differential (Bueno et al., 2019).

The motivation for studying these theories is correspondingly broad. Higher-curvature corrections arise as effective field theory terms at high energies, in stringy $f(r)$3-expansions, in cosmology and inflation, and as controlled models for black holes and holography. What makes quasi-topological gravities distinctive in that landscape is the persistence of analytic control in the spherical sector (Bueno et al., 29 Oct 2025).

2. Definitions and the three-type classification

Different notions of quasi-topological gravity appeared in the literature over time, but they can be reduced to three inequivalent notions. The unifying formulation distinguishes type I, type II, and type III quasi-topological gravities (Bueno et al., 29 Oct 2025).

Type I quasi-topological gravities are those for which the field equations evaluated on the single-function static, spherically symmetric ansatz

$f(r)$4

are second order. Two earlier definitions collapse into this class. One is that the on-shell reduced equations on the single-function ansatz have a total-derivative structure; the other is that the entropy tensor is divergenceless on that ansatz,

$f(r)$5

These two notions are equivalent (Bueno et al., 29 Oct 2025).

Type II quasi-topological gravities are those whose field equations on general static and spherically symmetric backgrounds are second order. The relevant ansatz is

$f(r)$6

Type II unifies three earlier conditions: that the reduced Lagrangian is a total derivative up to a boundary term, that the entropy tensor is divergenceless on general static spherical backgrounds,

$f(r)$7

and that the field equations on general static spherical backgrounds are second order. These are equivalent, and they further imply that the equations on general spherical backgrounds are second order (Bueno et al., 29 Oct 2025).

Type III quasi-topological gravities are those for which the trace of the field equations on a general background is second order. This is a weaker and more global condition than the type I and type II requirements, since it does not require the full reduced equations to be second order on the spherical sector (Bueno et al., 29 Oct 2025).

The inclusion relations are strict. Type II is a subset of type I. Type III is a subset of type II modulo pure Weyl invariants, specifically analytic functions of

$f(r)$8

A type III theory can therefore be written as a type II quasi-topological gravity plus an analytic function of these Weyl invariants (Bueno et al., 29 Oct 2025).

This classification sharpens the older distinction between quasi-topological gravities and proper GQTGs. In the all-orders GQTG classification, the defining structural distinction is that in a genuine GQTG the integrated equation depends on $f(r)$9, $-g_{tt}=g_{rr}^{-1}\equiv f(r)$0, and $-g_{tt}=g_{rr}^{-1}\equiv f(r)$1, whereas in a quasi-topological gravity it depends only on $-g_{tt}=g_{rr}^{-1}\equiv f(r)$2 (Moreno et al., 2023).

3. Relation to generalized quasi-topological gravities and all-order classification

Generalized quasi-topological gravities are higher-order extensions of Einstein gravity whose defining properties include second-order linearized equations around maximally symmetric backgrounds and non-hairy generalizations of the Schwarzschild black hole characterized by a single function $-g_{tt}=g_{rr}^{-1}\equiv f(r)$3 satisfying a second-order differential equation. Quasi-topological gravities are the special subclass for which that equation is algebraic (Bueno et al., 2022).

The all-orders classification in $-g_{tt}=g_{rr}^{-1}\equiv f(r)$4 is particularly rigid. At each curvature order $-g_{tt}=g_{rr}^{-1}\equiv f(r)$5, there are exactly $-g_{tt}=g_{rr}^{-1}\equiv f(r)$6 inequivalent GQTGs. Of these, one is a quasi-topological gravity and the remaining $-g_{tt}=g_{rr}^{-1}\equiv f(r)$7 are proper GQTGs. Equivalently, there is one unique quasi-topological gravity class at each $-g_{tt}=g_{rr}^{-1}\equiv f(r)$8, together with $-g_{tt}=g_{rr}^{-1}\equiv f(r)$9 inequivalent proper GQTG classes (Moreno et al., 2023).

A compact way to state the $f(r)$0 result is the following.

Curvature order $f(r)$1	Inequivalent classes in $f(r)$2	Quasi-topological content
$f(r)$3	$f(r)$4 GQTGs	1 QTG, $f(r)$5 proper GQTGs

The uniqueness of the quasi-topological sector can be expressed directly at the level of the reduced black-hole equation. In the all-orders construction, the quasi-topological class is singled out by an additional condition that kills the $f(r)$6 contribution on the static, spherically symmetric ansatz, so that the field equation becomes algebraic. The resulting black-hole equation is uniquely

$f(r)$7

which is algebraic in $f(r)$8 (Bueno et al., 2022).

The same literature also provides explicit covariant representatives. For $f(r)$9, the unique quasi-topological class at order $$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$0 is denoted $$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$1, while the proper GQTGs are denoted $$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$2, $$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$3. A technical advance enabling this classification is a dictionary that uplifts expressions evaluated on the single-function static, spherically symmetric ansatz to fully covariant densities (Moreno et al., 2023).

Four dimensions are exceptional. In $$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$4, there is strong evidence, and in later work a rigorous proof for the polynomial case, that there is one and only one proper inequivalent GQTG at each curvature order $$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$5, and that this four-dimensional family is not of the quasi-topological kind. In that sense, there are no nontrivial polynomial quasi-topological gravities in four dimensions, even though proper generalized quasi-topological gravities do exist there (Moreno et al., 2023).

4. Spherical symmetry, Birkhoff’s theorem, and black-hole equations

The most powerful structural result in the recent literature is the identification of type II quasi-topological gravities with Birkhoff’s theorem, up to a zero-measure set of degenerate theories. A theory is said to satisfy Birkhoff’s theorem if all spherically symmetric solutions are static and uniquely characterized by a single continuous integration constant, plus possibly a discrete label. Any theory satisfying Birkhoff’s theorem is a type II quasi-topological gravity, and the reverse implication also holds up to a zero-measure set of theories (Bueno et al., 29 Oct 2025).

The relevant general spherically symmetric background allowing time dependence is

$$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$6

For type II theories, the field equations reduce to

$$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$7

For generic theories, $$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$8, and the equations imply that $$S=\frac{1}{16\pi G}\int d^Dx\,\sqrt{|g|}\left[\frac{(D-1)(D-2)}{L^2}+R+\sum_{n=2}\sum_{i_n}L^{2(n-1)}\mu_{i_n}^{(n)}\mathcal{R}^{(n)}_{i_n}\right],$$9 is time independent, $f(r)$0 is radially constant, and after a time redefinition one may set $f(r)$1. The most general spherical solution is therefore Schwarzschild-like and characterized by a single function satisfying an algebraic equation (Bueno et al., 29 Oct 2025).

The exceptional zero-measure set is explicitly identified: it consists of theories for which there exists a radius where

$f(r)$2

allowing a degenerate family of non-static spherical solutions of the form

$f(r)$3

Outside this nongeneric locus, Birkhoff’s theorem and type II quasi-topological structure are equivalent (Bueno et al., 29 Oct 2025).

This spherical algebraic structure generalizes the Wheeler-polynomial form familiar from Lovelock gravity. In five-dimensional cubic quasi-topological gravity, for example, the static blackening factor is determined by the cubic Wheeler-like polynomial

$f(r)$4

In quintic quasi-topological gravity in five dimensions, the black-hole equation becomes

$f(r)$5

These examples realize explicitly the quasi-topological principle that the metric function is determined algebraically even though the full theory is higher-curvature (Fierro et al., 2020, Cisterna et al., 2017).

A complementary and older route to the same single-function structure is the generalized quasi-topological condition

$f(r)$6

on the static, spherically symmetric ansatz, up to terms that vanish when setting $f(r)$7. In the cubic construction this ensures that the vacuum field equations collapse to a single ordinary differential equation for one metric function and that the remaining equation integrates once to a total derivative form (Hennigar et al., 2017).

5. Linearized spectrum, entropy tensor, and holographic structure

A recurring technical object is the entropy tensor

$f(r)$8

through which the field equations can be written as

$f(r)$9

The divergenceless conditions on $f(r)$0 under spherical symmetry provide the bridge between reduced second-order dynamics, algebraic black-hole equations, and Birkhoff-type uniqueness (Bueno et al., 29 Oct 2025).

A second hallmark is the behavior on maximally symmetric backgrounds. GQTGs, and in particular quasi-topological gravities, have second-order linearized equations of motion around maximally symmetric vacua. Therefore they propagate only the usual massless graviton on those backgrounds (Moreno et al., 2023). In quartic quasi-topological gravity in five dimensions, the linearized equations around AdS take the Einstein operator form multiplied by the factor

$f(r)$1

and the theory is ghost free around the AdS vacuum if

$f(r)$2

This same positivity condition matches positivity of the dual CFT central charge $f(r)$3 (Dehghani et al., 2013).

The quasi-topological simplification is not uniform across all perturbative sectors. Around AdS, the graviton equations can remain second order, while perturbations around black-hole backgrounds need not. In quartic quasi-topological gravity the linearized field equation around a black hole has fourth-order radial derivatives of the perturbation, and the tensor-channel causality constraint becomes nontrivial only when both cubic and quartic terms are present (Dehghani et al., 2013).

Holographically, quasi-topological gravities exhibit both Lovelock-like universality and genuinely new effects. In five-dimensional quasi-topological gravity with Gauss-Bonnet and cubic quasi-topological terms, the planar AdS black hole satisfies

$f(r)$4

and the ratio $f(r)$5 is corrected by the higher-curvature couplings. The same model yields two butterfly velocity modes because the shock-wave equation factorizes into two second-order operators rather than one (Peng et al., 2018). By contrast, in the charged axion background of that theory, the thermoelectric DC conductivities coincide with those of Einstein and Gauss-Bonnet gravities (Peng et al., 2018).

6. Extensions, special dimensions, and restricted notions

The original quasi-topological program was developed mainly for $f(r)$6, but several later directions generalize or sharpen the notion.

One major branch is the study of generalized quasi-topological gravities in four dimensions. Polynomial quasi-topological gravities do not exist in $f(r)$7, but non-polynomial curvature quasi-topological gravities do. In that setting the cosmological equations on an FLRW ansatz collapse to a generalized Friedmann equation

$f(r)$8

with the infrared limit

$f(r)$9

The resulting non-polynomial theories admit three generic nonsingular cosmological scenarios with the correct infrared limit: a universe emerging from a de Sitter phase, a bouncing universe requiring a multi-valued Lagrangian, and an asymptotically Minkowski origin reminiscent of an eternally loitering universe (Borissova et al., 18 Mar 2026).

A second extension enlarges the quasi-topological framework to include covariant derivatives of the curvature. Such derivative GQTs can still preserve the hallmark single-function black-hole structure and second-order linearization on maximally symmetric backgrounds, but they do not belong to the original polynomial quasi-topological family. In four dimensions, the first derivative-based density that modifies the Schwarzschild solution appears at ten derivatives, and it changes the small-black-hole scaling from the polynomial-GQT behavior $$ds^2_{N,f}=-N(r)^2f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{(D-2)}^2,$$0 to $$ds^2_{N,f}=-N(r)^2f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{(D-2)}^2,$$1 (Aguilar-Gutierrez et al., 2023). At the same time, a separate no-go result shows that finite local truncations with explicit covariant derivatives of the Riemann tensor generically present ghosts in their spectrum, unless one allows contrived mixed-order tunings; this sharpens the special status of curvature-only Einsteinian and quasi-topological constructions (Edelstein et al., 2022).

A third direction studies more restrictive notions of quasi-topological behavior. One construction requires that the higher-curvature terms have no contribution to the equations of motion on the general static spherically symmetric metric

$$ds^2_{N,f}=-N(r)^2f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{(D-2)}^2,$$2

with $$ds^2_{N,f}=-N(r)^2f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{(D-2)}^2,$$3. Up to quintic order, such densities can be constructed so that they vanish identically on the most general non-stationary spherically symmetric metric. These terms therefore do not affect black-hole thermodynamics or background solutions in that symmetry class, but they can still modify perturbations and holographic transport coefficients such as shear viscosity (Chen, 2022).

Related variants include Ricci-polynomial quasi-topological gravities, where quasi-topological terms are classified according to whether they are trivial on the most general static metrics, on the special static metrics with $$ds^2_{N,f}=-N(r)^2f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{(D-2)}^2,$$4, or only at the linearized level on Einstein metrics (Li et al., 2017). There are also lower-dimensional analogues with matter support: in $$ds^2_{N,f}=-N(r)^2f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{(D-2)}^2,$$5 dimensions, coupling a scalar field in a magnetic configuration leads to electromagnetic quasi-topological and electromagnetic generalized quasi-topological gravities, whose black-hole solutions again involve a single metric function $$ds^2_{N,f}=-N(r)^2f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{(D-2)}^2,$$6 (Bueno et al., 2022).

Taken together, these developments suggest that “quasi-topological gravity” is no longer a single definition but a family of integrability principles centered on highly symmetric sectors. The recent three-type classification indicates that, among the older polynomial curvature notions in $$ds^2_{N,f}=-N(r)^2f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{(D-2)}^2,$$7, the physically meaningful core is the type II class, because it is essentially characterized by Birkhoff’s theorem itself (Bueno et al., 29 Oct 2025).