MacDowell-Mansouri Formulation Overview

• MacDowell–Mansouri formulation is a gauge-theoretic approach that interprets Einstein gravity with a cosmological constant as a symmetry-broken (anti-)de Sitter theory.
• Its action decomposes into topological, Einstein–Hilbert, and cosmological volume terms through an algebraic projection that breaks the full gauge symmetry to Lorentz invariance.
• The formulation links gravity with topological field theory, supergravity, and higher-spin models, offering a unified framework for geometrical and quantum analyses.

The MacDowell–Mansouri (MM) formulation is a pivotal geometric approach to gravity that realizes Einstein’s theory, with cosmological constant and topological terms, as a symmetry-broken gauge theory for the (anti-)de Sitter group in four spacetime dimensions. This construction provides a unified algebraic and variational framework, connecting General Relativity (GR), supergravity, topological field theory, higher spin extensions, and categorical/higher gauge structures. The MM action's structure elegantly encodes both the dynamical gravitational field and the underlying topological content, with an explicit gauge-theoretic origin for all geometrical variables.

1. Algebraic Foundations and Geometric Structure

The MacDowell–Mansouri approach begins by enlarging the internal gauge symmetry of gravity to the de Sitter or anti-de Sitter group, $G=\mathrm{SO}(4,1)$ (dS), $G=\mathrm{SO}(3,2)$ (AdS), or $G=\mathrm{SO}(2,3)$ (Euclidean AdS), depending on the cosmological constant and signature. The Lie algebra admits a reductive decomposition: $$\mathfrak{g} \cong \mathfrak{so}(1,3) \oplus \mathbb{R}^{1,3}$$ The connection 1-form is

$A = \omega^{ab} J_{ab} + e^a P_a,$

where $J_{ab}$ generate Lorentz transformations and $P_a$ the “translational” part. The curvature splits as: $$F^{ab} = R^{ab}(\omega) - \frac{1}{\ell^2} e^a \wedge e^b, \qquad F^{a4} = \frac{1}{\ell} T^a(\omega, e)$$ with $R^{ab}$ the Lorentz curvature and $T^a$ the torsion. A key step is the invariance-breaking projection onto the Lorentz sector, implemented either by hand or dynamically (e.g., with an internal vector/Higgs field) (Díaz-Saldaña et al., 2020, Addazi, 22 Dec 2025). This projection is equivalent to “breaking” $G$ to its Lorentz subgroup.

2. Action Principle and Symmetry Breaking Mechanisms

The core MM action in four dimensions is: $$S_{MM} = \frac{3}{2\Lambda} \int \epsilon_{abcd}\, F^{ab} \wedge F^{cd}$$ where $\epsilon_{abcd}$ is the Levi–Civita tensor, and the connection/curvature are as above. Expanding the curvature, the action decomposes as: $$S_{MM}= \frac{3}{2\Lambda} \int \left[ \epsilon_{abcd} R^{ab} \wedge R^{cd} - \frac{2}{\ell^2}\epsilon_{abcd} R^{ab} \wedge e^c \wedge e^d + \frac{1}{\ell^4} \epsilon_{abcd} e^a \wedge e^b \wedge e^c \wedge e^d \right]$$ The $R \wedge R$ term is (in D=4) the Euler/Gauss–Bonnet invariant; the $R \wedge e \wedge e$ term gives the Einstein–Hilbert Lagrangian; the $e^4$ term is the cosmological constant volume form (Freidel et al., 2012, López-Domínguez et al., 2014, Díaz-Saldaña et al., 2020). The projection is achieved, in the original MM and in BF-like variants, via a fixed internal vector $V^A$ or a suitable choice of invariant bilinear form (Díaz-Saldaña et al., 2020, Addazi, 22 Dec 2025). In the pure-connection formalism, algebraic constraints on auxiliary fields enforce the correct sector (Quintero, 2023).

The symmetry breaking has several incarnations:

• Explicit, by invariant tensor projection (original MM).
• Dynamical via spontaneous symmetry breaking by an internal vector $\phi^A$ (“Higgs”)—the Stelle–West and pre-geometric models (Addazi, 22 Dec 2025, Díaz-Saldaña et al., 2020).
• Via Lagrange-multiplier constraints imposing $V^A V_A = \ell^{-2}$ (Díaz-Saldaña et al., 2020).
• Through AKSZ-Manin deformation in topological field theory terms (Borsten et al., 14 Oct 2024).

3. Relation to BF Theory, Topological Sectors, and Prospects for Quantization

The MM model is a member of the class of constrained/topologically-deformed BF theories. In the BF formulation, the action is: $$S_{BF} = \int B_{IJ} \wedge F^{IJ} - \frac{\alpha}{4} \epsilon_{ijkl} B^{ij} \wedge B^{kl}$$ On-shell, $B$ can be integrated out (via its algebraic EOM), and the MM quadratic action in curvature is recovered (Freidel et al., 2012, Durka et al., 2022, Borsten et al., 14 Oct 2024). The mechanism by which gravity arises from a topological phase is the breaking of the $G$ gauge symmetry to the Lorentz subgroup—dynamically or algebraically.

BF formulations enable straightforward extensions:

• Inclusion of Holst, Nieh–Yan, Pontryagin, and Euler–Gauss–Bonnet classes (López-Domínguez et al., 2014, Durka et al., 2022).
• Promotion of couplings (e.g., Immirzi parameter or topological coefficients) to dynamical fields leading to torsional and axion-like modifications (López-Domínguez et al., 2014).
• Constructions with higher gauge structures, e.g., categorical 2-groups and actions for higher-form equations (as in Kalb–Ramond coupling) (Oliveira, 2022).
• The presence of topological terms guarantees that, for asymptotically (A)dS spaces, the action and all associated physical charges are finite, without requiring ad hoc counterterms; the Gauss–Bonnet and Pontryagin invariants are built in (Durka et al., 2022).

4. Supersymmetric, Higher-Spin, and Non-Relativistic Extensions

The MM construction generalizes naturally:

• Supergravity: The principle applies by extending the algebra to supergroups (e.g., OSp($N|4$)), with the invariant constructed from the super-curvature. Both $N=1$ and $N=2$ AdS supergravity, and further $SU(2,2|2)$ and AdS-Lorentz supergravity, admit MM-like Lagrangians. The supersymmetric cosmological constant, boundary super-Chern–Simons terms, and the relation to the Barbero–Immirzi parameter arise from the curvature structure and invariant projectors (Concha et al., 2015, Alvarez et al., 2021, Eder et al., 2021, Peñafiel et al., 2018).
• Higher Spin: The MM action is the archetype for the “broken topological” formulation of Vasiliev’s higher spin theories. Here, the infinite-dimensional higher-spin algebra is broken to the Lorentz subalgebra; local propagating fields emerge from the coset. The first-class constraint structure of the Hamiltonian directly parallels that of the MM model (Doroud et al., 2011).
• Non-Relativistic Theories: S-expansion and appropriate gauge algebra choices yield non-relativistic MM gravity formulations, including Newton–Hooke and Newtonian/MOND-like models. The MM action is adapted to these algebras, encoding generalized non-relativistic “Einstein equations” (Concha et al., 2022).

5. Connections to Cartan Geometry, Conformal Extensions, and Categorical/Higher Gauge Structures

The MM connection is naturally interpreted within Cartan geometry: a Cartan connection encodes local “rolling” of a model Klein geometry (A)dS/Lorentz onto the spacetime manifold. The fundamental MM length parameter $\ell$ is the radius of curvature of the model space; promoting it to a scalar field, one passes to conformal extensions of gravity with corresponding Cartan-geometric interpretation. In such models, the conformal Einstein field equations arise as the equations of motion (Reid, 5 Aug 2025). The MM framework also generalizes to strict 2-groups and 2-BF theory, supporting the coupling to Kalb–Ramond fields as components of a higher gauge structure and thereby encoding the local and topological content of gravitationally coupled theories (Oliveira, 2022).

The BF and AKSZ sigma-model descriptions—emphasized in categorical generalizations and the “sandwich” construction—clarify the role of higher-form symmetries and the connection to symmetry-protected phases, higher-form global symmetries (e.g., 1-form $\mathbb{Z}_2$ center symmetry), and their realization in topological boundary conditions (Borsten et al., 10 Sep 2025).

6. Physical and Mathematical Implications

A hallmark of the MM action is the emergence of the Einstein–Hilbert theory with cosmological constant (and Planck mass) not as fundamental input constants, but as dynamically generated scales determined by symmetry-breaking or coupling hierarchies (Addazi, 22 Dec 2025). The formalism robustly accounts for topological invariants (Euler/Gauss–Bonnet, Pontryagin, Nieh–Yan) and encodes them as either total derivatives or as non-dynamical terms, which, however, possess non-trivial global and quantum implications (e.g., Kodama state, Chern–Simons invariants, anomaly structure, $10^{120}$ vacuum energy scale) (Aydemir, 2017, López-Domínguez et al., 2014). In canonical (covariant De Donder–Weyl) quantization, the polysymplectic structure and graded Poisson-Gerstenhaber brackets uplift the classical MM theory to a setting amenable to manifestly covariant field-theoretic quantization (Berra-Montiel et al., 2017).

Table: Summary of Key Features

Aspect	MM Formulation	Reference(s)
Gauge Algebra	SO(4,1)/SO(3,2)/SO(2,3); broken to SO(1,3)	(Freidel et al., 2012, Díaz-Saldaña et al., 2020)
Action Structure	Quadratic in (A)dS curvature; symmetry-breaking	(López-Domínguez et al., 2014, Quintero, 2023)
Relation to BF Theory	Constrained/deformed BF with quadratic term	(Freidel et al., 2012, Durka et al., 2022)
Topological Terms	Euler, Pontryagin, Nieh–Yan, Holst	(López-Domínguez et al., 2014, Durka et al., 2022)
Supergravity/Higher Spin	Manifest generalization via enlarged superalgebra	(Concha et al., 2015, Alvarez et al., 2021, Doroud et al., 2011, Peñafiel et al., 2018)
Categorical/Higher Gauge	2-group structures, Kalb–Ramond extension	(Oliveira, 2022, Borsten et al., 10 Sep 2025)
Conformal & Scalar-Tensor	Promote $\ell\to\varphi(x)$, conformal gravity	(Reid, 5 Aug 2025)
Hamiltonian/Covariant Quant.	De Donder–Weyl, first-class constraint algebra	(Berra-Montiel et al., 2017)

7. Unified Perspective and Ongoing Research Directions

The MM formulation unifies several previously distinct themes in gravity: gauge symmetry breaking, topological field theory, Cartan geometry, higher gauge extensions, and quantum properties of the gravitational phase space. It provides a robust framework for extensions to supergravity, higher spins, conformal gravity, and gravity coupled to higher-form matter. The unique capability to build in topological invariants and encode the Einstein dynamics with cosmological constant via algebraic symmetry breaking underlies many advances in emergent and quantum gravity. Recent work continues to clarify the relation to AKSZ/BV quantization, boundary dualities, symmetry-protected topological structures, and dynamical symmetry breaking mechanisms (Borsten et al., 14 Oct 2024, Quintero, 2023, Borsten et al., 10 Sep 2025, Addazi, 22 Dec 2025).

The MM formalism persists as a central paradigm in research at the intersection of geometry, quantum field theory, and gravity, providing both a model for geometric unification and a concrete computational tool for gravitational, topological, and field-theoretic analyses.