MacDowell-Mansouri Connection

• The MacDowell-Mansouri connection unifies the spin connection and vierbein into a single (A)dS-valued gauge connection within a larger Lie algebra.
• It constructs a curvature-squared action that, after symmetry breaking, reproduces Einstein-Cartan or Palatini gravity with additional topological invariants.
• Recent extensions incorporate AKSZ theory, higher gauge and supersymmetric models, highlighting its versatility in reformulating gravitational dynamics.

Searching arXiv for recent and foundational papers on the MacDowell-Mansouri connection to ground the article in current literature. arXiv search query: "MacDowell-Mansouri gravity connection BF supergravity" The MacDowell-Mansouri connection is a gauge-theoretic unification of the spin connection and the coframe or vierbein into a single connection valued in a larger Lie algebra containing the Lorentz algebra, typically $SO(1,4)$, $SO(2,3)$, or related super- or higher-gauge extensions. In its standard four-dimensional form, the construction embeds $\omega^{ab}$ and $e^a$ into an $(A)dS$-valued connection, whose curvature simultaneously contains the Lorentz curvature, torsion, and cosmological-constant term. The resulting action is quadratic in curvature but, through explicit or induced symmetry breaking to the Lorentz subgroup, reduces to Einstein-Cartan or Palatini gravity with cosmological constant, together with topological invariants such as the Euler, Pontryagin, and Nieh-Yan terms (Díaz-Saldaña et al., 2020, Durka, 2012). More recent work has shown that the same structure can arise identically from a Manin deformation of a four-dimensional AKSZ theory, where the MacDowell-Mansouri formulation of Einstein gravity with cosmological constant is recovered at the level of the action (Borsten et al., 2024).

1. Definition and basic geometric form

The defining feature of the MacDowell-Mansouri construction is the replacement of separate gravitational variables by a single connection for a larger gauge group. In the $SO(1,4)$ or $SO(2,3)$ formulations, the Lie algebra splits reductively as

$$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$

so that the gauge field decomposes into Lorentz and translational parts. One standard parametrization is

$$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$

while another equivalent convention writes

$$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$

In these conventions, $SO(2,3)$0 or $SO(2,3)$1 is a length scale related to the cosmological constant (Díaz-Saldaña et al., 2020, Durka, 2012).

The curvature of the MacDowell-Mansouri connection is

$SO(2,3)$2

and its decomposition is one of the central structural identities of the formalism. In the $SO(2,3)$3 case,

$SO(2,3)$4

with

$SO(2,3)$5

Equivalent formulas appear in the $SO(2,3)$6 and $SO(2,3)$7 conventions, differing only by sign choices associated with AdS or dS signature (Díaz-Saldaña et al., 2020, Durka, 2012, López-Domínguez et al., 2014).

A later reformulation emphasizes the connection in Cartan-geometric language. There the MacDowell-Mansouri connection is written

$SO(2,3)$8

with $SO(2,3)$9 the spin connection and $\omega^{ab}$0 the coframe. Its curvature becomes

$\omega^{ab}$1

which makes explicit that points in spacetime are infinitesimally modeled by a homogeneous space of radius $\omega^{ab}$2 (Reid, 5 Aug 2025).

2. Symmetry breaking and recovery of Einstein gravity

The MacDowell-Mansouri mechanism does not use the full $\omega^{ab}$3 symmetry as an off-shell invariance of the gravitational action. Instead, the action contains a structure that projects onto the Lorentz subalgebra. In the Stelle-West variant, this role is played by an $\omega^{ab}$4-valued vector field $\omega^{ab}$5 satisfying a norm constraint: $\omega^{ab}$6 The symmetry breaking condition

$\omega^{ab}$7

selects a preferred internal direction and reduces $\omega^{ab}$8. Under this condition the action becomes

$\omega^{ab}$9

so that the first term is Euler, the second is the Palatini sector, and the third is the cosmological term, with $e^a$0 (Díaz-Saldaña et al., 2020).

A closely related formulation uses a “director field” $e^a$1 and an internal dual operator

$e^a$2

so that the action takes the form

$e^a$3

Gauge-fixing $e^a$4 yields the standard Lorentz-projected form and reproduces Einstein-Palatini gravity plus cosmological constant and Euler density (Langenscheidt, 2019).

The same basic action can also be obtained by integrating out an auxiliary $e^a$5-field in deformed BF theory. In the $e^a$6 version,

$e^a$7

After solving algebraically for $e^a$8, one recovers a first-order gravitational action containing Einstein-Cartan, cosmological, Holst, and topological terms (Durka, 2012). This BF viewpoint is also central in the comparison with the Plebanski formulation in asymptotically AdS spacetime, where both theories give Einstein equations in the bulk but differ in edge charges (Durka et al., 2022).

3. Action principles, auxiliary fields, and topological sector

The curvature-squared character of the MacDowell-Mansouri action often gives it a Yang-Mills-like appearance, but the formalism differs from ordinary Yang-Mills theory in a structurally important way. The invariant used in the action is not the spacetime Hodge dual, but an internal or projected dual depending on the symmetry-breaking field. As a result, the action is sensitive to the decomposition of the larger gauge algebra into Lorentz and translational sectors (Langenscheidt, 2019, Díaz-Saldaña et al., 2020).

In the BF and deformed BF formulations, the MacDowell-Mansouri action naturally packages the standard first-order gravitational terms together with all parity-even local Lorentz- and diffeomorphism-invariant four-dimensional topological densities. A representative reduced action is

$e^a$9

Here the Einstein-Cartan, cosmological, Holst, Euler, Pontryagin, and Nieh-Yan sectors all appear explicitly (Durka, 2012).

The generalized MacDowell-Mansouri action studied in the presence of a pseudo-projector $(A)dS$0 introduces the Immirzi parameter $(A)dS$1 directly into the projected curvature,

$(A)dS$2

and after symmetry breaking yields a bulk sector equal to Einstein-Cartan gravity with cosmological constant, plus the Holst term and the topological Euler, Pontryagin, and Nieh-Yan classes (Díaz-Saldaña et al., 2020). The appearance of the Holst term in this gauge-theoretic framework is significant because the same parameter reappears in supergravity extensions and in the analysis of asymptotic and Noether charges (Eder et al., 2021, Durka et al., 2022).

Some later work considers promoting coefficients of the Holst, Nieh-Yan, and Pontryagin terms to spacetime fields. In that setting, torsion is no longer forced to vanish, and the resulting equations relate topological densities on shell. The modified Einstein equations then acquire torsional contributions determined by derivatives of the new fields (López-Domínguez et al., 2014).

4. Variants of the connection: AKSZ, pure-connection, and dynamical scale formulations

A notable 2024 result derives the MacDowell-Mansouri action identically from a four-dimensional AKSZ-Manin theory based on the graded Lie algebra

$(A)dS$3

Starting from the AKSZ $(A)dS$4 action

$(A)dS$5

one adds a Manin deformation determined by a Hodge structure $(A)dS$6,

$(A)dS$7

Decomposing the $(A)dS$8 connection by

$(A)dS$9

and integrating out the auxiliary $SO(1,4)$0-fields yields

$SO(1,4)$1

with $SO(1,4)$2 and $SO(1,4)$3. In this construction the cosmological constant is not added separately; it emerges directly from the algebraic splitting of the connection (Borsten et al., 2024).

A different generalization replaces the standard variable set by a pure connection $SO(1,4)$4 and algebraic constraints enforced by multiplier fields $SO(1,4)$5 and $SO(1,4)$6: $SO(1,4)$7 For $SO(1,4)$8, and in particular $SO(1,4)$9 or $SO(2,3)$0, suitable choices of trace and constraints recover the MacDowell-Mansouri and Stelle-West formulations as special cases. Under appropriate simplicity constraints, the resulting equations give torsionless conformally flat Einstein manifolds (Quintero, 2023).

Another variant promotes the fixed length scale $SO(2,3)$1 in the Cartan connection to a dynamical scalar field. The modified MacDowell-Mansouri connection becomes

$SO(2,3)$2

and the deformed BF action reduces on shell to

$SO(2,3)$3

After Palatini-type variation the field equations are equivalent to the conformal Einstein equations, and the scalar $SO(2,3)$4 acquires a Cartan-geometric interpretation as the field controlling the local radius of the homogeneous model space (Reid, 5 Aug 2025).

5. Higher-gauge, supersymmetric, and algebra-expanded extensions

The MacDowell-Mansouri connection has been generalized in several directions that preserve its basic logic: a larger gauge symmetry, a curvature-squared or BF-type action, and a mechanism selecting a Lorentz or Lorentz-like sector.

In higher gauge theory, a categorical generalization based on a strict 2-group replaces the ordinary connection by a 2-connection $SO(2,3)$5, where $SO(2,3)$6 is $SO(2,3)$7-valued and $SO(2,3)$8 is an $SO(2,3)$9-valued 2-form. The corresponding curvatures are

$$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$0

which in the de Sitter 2-group reduce to

$$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$1

A higher Yang-Mills-type action and a constrained 2-BF action then produce Einstein-Cartan gravity coupled to Kalb-Ramond fields, together with Pontryagin, Nieh-Yan, and Euler terms (Oliveira, 2022).

Supersymmetric extensions replace $$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$2 by supergroups such as $$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$3, $$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$4, $$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$5, $$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$6, or expanded Maxwell and AdS-Lorentz superalgebras. In the $$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$7 model, the superconnection $$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$8 combines gravitational, internal $$\mathfrak{so}(1,4)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3} \qquad\text{or}\qquad \mathfrak{so}(2,3)=\mathfrak{so}(1,3)\oplus \mathbb{R}^{1,3},$$9, and fermionic sectors, and the action

$$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$0

uses an extended Hodge operator $$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$1 acting as an outer automorphism on superalgebra-valued 2-forms. Off shell, the preserved symmetry is $$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$2, while translation-type symmetries and supersymmetry are recovered only on the surface of integrability conditions including the Rarita-Schwinger equation and torsion-like constraints (Alvarez et al., 2021).

Other supergravity constructions use S-expanded superalgebras. A MacDowell-Mansouri-like action built from the minimal Maxwell superalgebra $$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$3 reproduces $$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$4, $$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$5 pure supergravity, with additional Maxwell fields contributing only boundary terms (Concha et al., 2014). A related construction based on a new AdS-Lorentz superalgebra produces a generalized supersymmetric cosmological term using the curvatures of the expanded algebra (Peñafiel et al., 2018). In $$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$6, a chiral supergravity action of the form

$$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$7

breaks $$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$8 gauge invariance to $$A_\mu^{AB}= \begin{pmatrix} \omega_\mu^{ab} & -\dfrac{1}{l} e_\mu^a \ \dfrac{1}{l} e_\mu^b & 0 \end{pmatrix},$$9, leaving half of the gauge supersymmetry manifest and allowing a self-dual projection of the spin connection (Castellani, 2017). In the presence of boundaries, the Holst-MacDowell-Mansouri formulation for $$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$0 and $$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$1 AdS supergravity yields boundary terms that become super Chern-Simons actions with gauge group $$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$2 in the chiral limit (Eder et al., 2021).

A further extension applies MacDowell-Mansouri-type symmetry breaking outside gravity. For the pair $$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$3, inserting a $$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$4-invariant matrix $$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$5 under the trace in

$$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$6

produces a variational problem whose critical points are constant scalar curvature almost-Kähler 4-manifolds, and under additional assumptions become Kähler-Einstein 4-manifolds (Alvarez et al., 23 Mar 2026). This suggests that the defining mechanism is not limited to relativistic gravity, but more generally to gauge-theoretic functionals built from Pontryagin density plus symmetry-breaking insertions.

6. Dynamics, charges, matter couplings, and open structural issues

Several developments concern not the derivation of Einstein gravity itself but the dynamical and boundary properties of the MacDowell-Mansouri connection. In the covariant De Donder-Weyl Hamiltonian formalism, the $$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$7 MacDowell-Mansouri action admits a graded Poisson-Gerstenhaber bracket description. There, symmetry breaking $$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$8 can be performed before or after variation without changing the resulting field equations, in contrast to the Lagrangian approach, and Einstein’s equations are recovered in the reduced theory (Berra-Montiel et al., 2017).

For asymptotically AdS spacetimes, the MacDowell-Mansouri BF formulation has a distinctive boundary behavior. The on-shell $$A_\mu^{ab}=\omega_\mu^{ab},\qquad A_\mu^{a4}=\frac{1}{\ell}e_\mu^a.$$9-field is algebraically proportional to the AdS curvature, and the edge charge associated with a Killing vector $SO(2,3)$00 takes the form

$SO(2,3)$01

For AdS-Schwarzschild, AdS-Kerr, and AdS-Taub-NUT, the MacDowell-Mansouri charges are finite, whereas the corresponding Plebanski charges are divergent. More generally, the action, asymptotic charges, and symplectic form are finite for arbitrary asymptotically AdS spacetimes, so no counterterms are required (Durka et al., 2022).

The connection has also been adapted to non-relativistic gravity. Using a Newton-Hooke version of the Newtonian algebra, one defines a gauge connection containing $SO(2,3)$02, $SO(2,3)$03, $SO(2,3)$04, and additional fields, and constructs a MacDowell-Mansouri-type action quadratic in the associated curvatures. The resulting equations contain the Poisson equation with cosmological constant and, for a suitable ansatz, a MOND-like modified Poisson equation (Concha et al., 2022).

Matter coupling reveals a structural limitation of the Yang-Mills analogy. In the MacDowell-Mansouri-Stelle-West $SO(2,3)$05 framework, spin-$SO(2,3)$06 fields admit manifestly covariant couplings, but standard kinetic terms for scalar and spin-1 gauge fields require auxiliary fields if one insists on manifest $SO(2,3)$07 covariance. This shows that the construction does not put gravity and Yang-Mills theories on fully equal footing at the level of kinetic terms (Langenscheidt, 2019).

A related cosmological line of work treats the de Sitter Higgs or Goldstone field $SO(2,3)$08 as dynamical and identifies $SO(2,3)$09 with $SO(2,3)$10. In a polynomial version of Stelle-West gravity coupled to a modified perfect fluid, the field equation for $SO(2,3)$11 becomes nontrivial, enabling radiation and matter eras and accelerated expansion in a manifestly de Sitter-covariant formulation (Lu, 2021).

The topological sector has also motivated phenomenological speculation. One proposal links the Gauss-Bonnet term in the MacDowell-Mansouri action to a characteristic scale around $SO(2,3)$12 MeV or $SO(2,3)$13, identified through a large coefficient $SO(2,3)$14 in front of the Gauss-Bonnet density and compared with scenarios involving six compact extra dimensions (Aydemir, 2017). This suggests a possible physical role for the topological normalization of the MacDowell-Mansouri action, though the cited work presents it as a gravitational and cosmological interpretation rather than a derived prediction.

7. Conceptual status

Across its variants, the MacDowell-Mansouri connection consistently serves as the central object that recasts gravity as a broken gauge theory for a group larger than the Lorentz group. In the classical four-dimensional case, it packages $SO(2,3)$15 and $SO(2,3)$16 into an $SO(2,3)$17-valued Cartan connection, whose curvature unifies $SO(2,3)$18, $SO(2,3)$19, and the cosmological term. The action constructed from this curvature yields Einstein-Cartan or Palatini gravity with cosmological constant, while retaining topological densities that are classically nondynamical but important for boundary terms, asymptotic charges, and quantum-oriented extensions (Díaz-Saldaña et al., 2020, Durka, 2012, Durka et al., 2022).

The later literature broadens that picture rather than replacing it. AKSZ-Manin theory shows that the MacDowell-Mansouri action can arise identically from deformation of a topological field theory (Borsten et al., 2024). Higher gauge theory embeds the same idea in strict 2-groups and 2-BF theory (Oliveira, 2022). Supergravity generalizations reinterpret the symmetry-breaking projector as a superalgebraic dual or outer automorphism and connect boundary consistency to super Chern-Simons theory (Alvarez et al., 2021, Eder et al., 2021). Pure-connection, dynamical-scale, and non-relativistic versions indicate that the notion of a “MacDowell-Mansouri connection” is best understood not as a single formula but as a family of Cartan or gauge connections whose curvature-squared actions become gravitational only after a controlled reduction to a distinguished subgroup (Quintero, 2023, Reid, 5 Aug 2025, Concha et al., 2022).

A plausible implication is that the enduring significance of the MacDowell-Mansouri connection lies in the combination of three ingredients that recur across all these settings: enlargement of gauge symmetry, curvature-based action principles, and symmetry breaking or projection that extracts the gravitational sector. Within the literature surveyed here, that triad is the stable content of the construction.