2D Horndeski Theories & KGB Equivalence

• Two-dimensional Horndeski theories are the most general 2D scalar–tensor models with second-order field equations, parameterized by arbitrary functions F2, F3, and F4.
• They recast curvature couplings into a Kinetic Gravity Braiding form via special 2D identities and disformal mappings, linking them to Jackiw–Teitelboim gravity and 4D reductions.
• An integrable sector emerges with static, Schwarzschild gauge metrics, highlighting connections to higher-dimensional pure gravity and emphasizing the role of boundary terms in 2D.

Two-dimensional Horndeski theories are the most general two-dimensional scalar–tensor, or dilaton, theories with second-order field equations. In two spacetime dimensions, their structure is sharply constrained by special identities: the curvature-coupled Horndeski sector can be traded, up to total derivatives, for a Kinetic Gravity Braiding form, so the physically relevant theory space is most naturally characterized either by a Horndeski-like action with $F_2,F_3,F_4$ or by an equivalent two-function KGB parametrization. Recent work places this class in a broader network of equivalences and embeddings: Jackiw–Teitelboim gravity generates it through invertible disformal transformations, spherical reduction of four-dimensional Horndeski yields a two-dimensional bi-scalar theory whose single-scalar sector is exactly the known two-dimensional Horndeski theory up to a boundary term, and all inequivalent two-dimensional Horndeski theories arise from $d\ge 4$ pure gravities once sufficiently general higher-dimensional actions are allowed (Nejati et al., 2023, Nejati et al., 2024, Borissova, 6 Mar 2026).

1. Canonical formulations and the two-dimensional theory space

A standard starting point is the general two-dimensional dilaton gravity action

$$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$

The two-dimensional Horndeski-type extension allows dependence on second derivatives of $\phi$ while still yielding second-order Euler–Lagrange equations. One convenient form is

$$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$

with $F_{4X}:=\partial F_4/\partial X$. Horndeski’s own two-dimensional representation is

$$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$

In the terminology adopted in the recent literature, the Kinetic Gravity Braiding subclass is therefore not separate from two-dimensional Horndeski: it is an equivalent parametrization of the same second-order theory, up to total derivatives (Nejati et al., 2023).

The two-dimensional character of the theory is essential. In $2$D, the final Horndeski family present in $4$D disappears because the Einstein tensor vanishes identically and the cubic Hessian combination also vanishes identically in two dimensions. This is why “Horndeski” in $2$D means precisely the scalar–tensor theory with arbitrary $d\ge 4$0, arranged so that the equations of motion for both $d\ge 4$1 and $d\ge 4$2 remain second order (Nejati et al., 2023).

An equivalent description, useful for dimensional reduction and integrability, is written in terms of

$d\ge 4$3

with

$d\ge 4$4

In this formulation, the physically meaningful data reduce to two functions $d\ge 4$5 and $d\ge 4$6,

$d\ge 4$7

where

$d\ge 4$8

and

$d\ge 4$9

Accordingly, “all $$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$0D Horndeski theories” means all inequivalent theories modulo integrations by parts, that is, all possible pairs $$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$1 realizable by such actions (Borissova, 6 Mar 2026).

2. Special two-dimensional identities and equivalence with KGB and dilaton frames

The key simplification in $$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$2D is the identity

$$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$3

where

$$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$4

This identity is special to two dimensions. It shows that the entire

$$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$5

sector is equivalent, up to a total derivative, to a redefinition of the $$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$6 and $$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$7 terms. Specifically,

$$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$8

Thus every $$S=\int d^2x\,\sqrt{g}\,\Big(F(\phi)R+G(\phi)X+V(\phi)\Big), \qquad X:=-\frac12\,\partial_\mu\phi\,\partial^\mu\phi.$$9 theory is equivalent, at the level of bulk equations of motion, to a KGB theory $\phi$0. Conversely, by choosing an arbitrary $\phi$1, one can trade part of the KGB structure back into an $\phi$2 term (Nejati et al., 2023).

The same mechanism clarifies the status of generalized dilaton gravities. The Grumiller–Ruzziconi–Zwikel action

$\phi$3

is a special case of $\phi$4 with

$\phi$5

Using the identity with $\phi$6, it can also be rewritten as a KGB theory: $\phi$7 because then

$\phi$8

In this sense, GRZ, KGB, and Horndeski-like representations are different frames of the same second-order two-dimensional theory space (Nejati et al., 2023).

A closely related result appears in the reduction of four-dimensional Horndeski to two-dimensional bi-scalar gravity. There the single-scalar precursor of the identity is

$\phi$9

and the restricted single-scalar theory obeys

$$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$0

The resulting Lagrangian

$$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$1

is therefore identical to the two-dimensional Horndeski theory up to a boundary term (Nejati et al., 2024).

3. Jackiw–Teitelboim gravity as a generator and the role of disformal frames

The central result of the disformal approach begins from the Jackiw–Teitelboim action

$$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$2

A general disformal transformation,

$$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$3

with

$$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$4

maps the JT action into Horndeski form, up to a total derivative: $$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$5 The generated curvature coupling is

$$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$6

In the authors’ interpretation, the two arbitrary disformal functions $$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$7 and $$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$8 match the two arbitrary functions $$S_{\rm H_1} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X) + F_3(\phi,X)\Box\phi + F_4(\phi,X)R + F_{4X}\big[(\Box\phi)^2-(\nabla_\mu\nabla_\nu\phi)^2\big] \Big),$$9 and $F_{4X}:=\partial F_4/\partial X$0 of the general two-dimensional second-order scalar–tensor theory, so that JT is not merely a simple reference model but, up to boundary terms, a generator of the entire class (Nejati et al., 2023).

Invertibility is essential. The number of degrees of freedom is guaranteed to be unchanged only if the disformal map is invertible, meaning that one can solve $F_{4X}:=\partial F_4/\partial X$1 for $F_{4X}:=\partial F_4/\partial X$2. The explicit condition is

$F_{4X}:=\partial F_4/\partial X$3

The paper also imposes $F_{4X}:=\partial F_4/\partial X$4 and $F_{4X}:=\partial F_4/\partial X$5, so that the transformed metric has a well-defined inverse and volume form and does not flip the sign of the time component. Under invertible, nonsingular disformal maps satisfying these conditions, the number of degrees of freedom is unchanged, although singular cases remain subtle (Nejati et al., 2023).

The same work proves closure. If one starts from a two-dimensional Horndeski/KGB theory and performs a generic disformal transformation with arbitrary $F_{4X}:=\partial F_4/\partial X$6 and $F_{4X}:=\partial F_4/\partial X$7, one remains within the same family, again up to a boundary term. This is unlike the four-dimensional case, where $F_{4X}:=\partial F_4/\partial X$8-dependent disformal transformations typically take Horndeski into the broader DHOST class. In $F_{4X}:=\partial F_4/\partial X$9D, because of the special identity that collapses the $$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$0-sector into KGB form, no enlargement of the theory space is needed. The family is closed (Nejati et al., 2023).

The same framework also yields explicit special cases. For the one-parameter family

$$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$1

one has constant $$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$2, and the transformed JT coefficients simplify to

$$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$3

For $$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$4, one reproduces the ordinary $$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$5D dilaton gravity family

$$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$6

while the identity transformation $$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$7, $$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$8 returns pure JT gravity itself (Nejati et al., 2023).

4. Higher-dimensional origin: from four-dimensional Horndeski to $$S_{\rm H_2} = \int d^2x\,\sqrt{g}\, \Big( F_2(\phi,X)+F_3(\phi,X)\Box\phi \Big).$$9 pure gravities

A direct reduction from four-dimensional Horndeski starts from the spherically symmetric ansatz

$2$0

so the reduced $2$1D theory contains the $2$2D metric $2$3, the original scalar $2$4, and the sphere radius $2$5, organized as

$2$6

After integrating over the sphere,

$2$7

with reduced bi-scalar Lagrangian

$2$8

where

$2$9

This is already a $4$0D bi-scalar theory with second-order equations. Under the single-scalar restriction $4$1, it becomes

$4$2

which is exactly the known two-dimensional Horndeski theory up to a boundary term (Nejati et al., 2024).

A broader result shows that all inequivalent two-dimensional Horndeski theories arise from $4$3 generally covariant pure gravities. The higher-dimensional warped-product metric is

$4$4

with compact constant-curvature transverse space and $4$5. Reduction gives

$4$6

and by symmetric criticality the reduced equations are equivalent to the higher-dimensional equations evaluated on the ansatz. The decisive distinction is between curvature-only actions and actions that also contain curvature-derivative invariants (Borissova, 6 Mar 2026).

Higher-dimensional class	Two-dimensional outcome
Polynomial curvature quasi-topological gravities	a narrow integrable subclass of $4$7D Horndeski theories, parameterized by one analytic function $4$8
Non-polynomial curvature quasi-topological gravities	a larger pure-curvature subclass, parameterized by two functions $4$9, $2$0
Generic quasi-topological gravities with curvature derivatives	all $2$1D Horndeski theories, i.e. arbitrary $2$2, hence arbitrary inequivalent pairs $2$3

For pure-curvature theories,

$2$4

the reduced scalar building blocks are

$2$5

so the reduced Horndeski functions depend on $2$6 only through

$2$7

To obtain generic dependence on $2$8, one must go beyond pure curvature invariants and introduce the additional reduced scalar

$2$9

Since $d\ge 4$00, the pair $d\ge 4$01 reconstructs $d\ge 4$02 and $d\ge 4$03 separately, and this is the precise sense in which all $d\ge 4$04D Horndeski theories arise from $d\ge 4$05 pure gravities (Borissova, 6 Mar 2026).

5. Integrability, Birkhoff theorem, and algebraic black-hole sectors

In the reduced description, a distinguished subclass is defined by the extended Schwarzschild gauge

$d\ge 4$06

for which

$d\ge 4$07

The independent field equations are

$d\ge 4$08

$d\ge 4$09

Assuming $d\ge 4$10, one gets $d\ge 4$11, together with

$d\ge 4$12

The special condition for Schwarzschild gauge with $d\ge 4$13, that is $d\ge 4$14, is the integrability condition

$d\ge 4$15

Equivalently, there exists a characteristic function $d\ge 4$16 such that

$d\ge 4$17

and the remaining equation integrates to

$d\ge 4$18

This defines the integrable $d\ge 4$19D Horndeski theories: staticity, Schwarzschild gauge, and algebraic determination of $d\ge 4$20 (Borissova, 6 Mar 2026).

The higher-dimensional consequence is a Birkhoff-type theorem. If a generally covariant pure gravity in $d\ge 4$21 has second-order equations on the $d\ge 4$22 warped-product ansatz and its reduction yields an integrable $d\ge 4$23D Horndeski theory, then the corresponding vacuum solutions are necessarily static, satisfy

$d\ge 4$24

in Schwarzschild gauge, and $d\ge 4$25 is determined by an algebraic equation. This motivates the extended use of the term quasi-topological gravities for pure gravities whose reduction yields an integrable Horndeski theory (Borissova, 6 Mar 2026).

The same framework supports a reverse reconstruction theorem. Given any static, spherically symmetric, asymptotically flat metric

$d\ge 4$26

with invertible dependence on ADM mass,

$d\ge 4$27

one identifies

$d\ge 4$28

then

$d\ge 4$29

Because every $d\ge 4$30D Horndeski theory lifts to a $d\ge 4$31-dimensional pure gravity, the spacetime can be reconstructed as a vacuum solution of some generally covariant $d\ge 4$32-dimensional gravitational action (Borissova, 6 Mar 2026).

Examples illustrate the hierarchy. The Hayward metric fits the pure-curvature algebraic form with $d\ge 4$33, so it arises from non-polynomial CQTGs; the Dymnikova metric also belongs to non-polynomial CQTGs but is not obtainable from polynomial CQTGs; and the Bardeen metric is not of the pure-curvature form, so it cannot come from curvature-only CQTGs, polynomial or non-polynomial, but it can be obtained from the broader QTGs involving curvature-derivative invariants (Borissova, 6 Mar 2026).

6. Degrees of freedom, boundary terms, and non-Riemannian extensions

In two-dimensional dilaton gravity there are no local bulk propagating graviton modes; physical excitations arise as boundary modes. This is why boundary terms are unusually important. Many of the equivalences discussed above hold only up to total derivatives, and in $d\ge 4$34D these terms can affect the boundary theory. For the Horndeski-form action $d\ge 4$35, a boundary term that makes the variational principle well posed is

$d\ge 4$36

where

$d\ge 4$37

Here

$d\ge 4$38

This reinforces the statement that disformal transformations preserve bulk degrees of freedom under invertibility, but they do modify boundary structures (Nejati et al., 2023).

The reduced bi-scalar theory obtained from four-dimensional Horndeski also has nontrivial local dynamics. Around a static background

$d\ge 4$39

the quadratic perturbation action takes the form

$d\ge 4$40

The counting argument shows that there is only one propagating mode. The no-ghost and gradient-stability conditions require

$d\ge 4$41

and the propagation speed is

$d\ge 4$42

The paper further argues that $d\ge 4$43 and $d\ge 4$44, consistent with one propagating mode, and remarks that this speed matches the scalar-wave speed in the parent four-dimensional Horndeski theory because the Kaluza–Klein reduction keeps only the spherically symmetric $d\ge 4$45 mode (Nejati et al., 2024).

A non-Riemannian extension is provided by symmetric teleparallel Horndeski gravity. There the direct result is four-dimensional: Horndeski-like scalar–gravity theories can be formulated in a flat and torsionless geometry with gravity encoded in nonmetricity, the theory can be recast as a sum of the Riemannian Horndeski theory and new terms that are purely teleparallel, and one can formulate an extension of Kinetic Gravity Braiding in this setting. A cautious inference is that in lower dimensions, including $d\ge 4$46D, one should still expect a larger space of second-order scalar–tensor theories in the nonmetricity formulation than in the curvature formulation, because arbitrary functions of first-derivative nonmetricity invariants can be included in $d\ge 4$47-type sectors without automatically generating higher-order equations (Bahamonde et al., 2022).

Taken together, these developments imply that two-dimensional Horndeski theories are best understood not as isolated low-dimensional analogues of four-dimensional Horndeski, but as a tightly organized equivalence class of second-order metric–scalar theories. Their characteristic features are the collapse of the quartic sector into KGB form, the role of disformal frame changes with JT gravity as a generating representative, the existence of higher-dimensional pure-gravity and Horndeski uplifts, the distinguished integrable sector with $d\ge 4$48, and the persistent importance of boundary data in a dimension where bulk graviton propagation is absent (Nejati et al., 2023, Nejati et al., 2024, Borissova, 6 Mar 2026).