Affine Gauge Theory of Gravity (AGT)

• Affine Gauge Theory of Gravity is a framework where gravity is formulated via gauge connections and bundle fields rather than solely using a metric.
• It employs independent translational and linear sectors, using torsion, curvature, and nonmetricity as core field strengths in its gauge-geometric approach.
• The theory recovers Einstein gravity through gauge fixing, reduction processes, and dynamical mass-scale emergence while extending gravitational wave and perturbation analyses.

Affine Gauge Theory of Gravity (AGT) denotes a family of gravitational gauge theories in which the primary variables are connections and associated bundle fields tied to spacetime symmetry groups, rather than a metric alone. In the strict metric-affine sense, AGT is based on the affine group $A(4,\mathbb R)=GL(4,\mathbb R)\ltimes \mathbb R^4$, with independent translational and linear sectors and with torsion, curvature, and nonmetricity as the characteristic field strengths (Sobreiro et al., 2010, Jiménez-Cano et al., 2020). In a broader usage present across the literature, closely related Poincaré, conformal, Cartan, teleparallel, and conformal-affine constructions are also treated as AGT-type theories whenever the coframe, spin or linear connection, and sometimes the metric arise from a principal-bundle gauge structure and Einstein gravity appears as a reduction, gauge fixing, or special regime of a larger gauge system (Herfray et al., 2020, Tjandra et al., 30 Sep 2025).

1. Gauge groups and symmetry content

In the strict affine formulation, the gauge group is the affine group

$A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$

with Lie-algebra splitting into a linear part and a translational part. The translational sector is Abelian, the full affine group is non-semisimple, and the decomposition

$GL(d,\mathbb R)=K(d)\oplus O(d)$

plays a central role in later reductions to orthogonal geometry (Sobreiro et al., 2010, Sobreiro et al., 2010). This structure is what makes AGT broader than purely Lorentz or purely Poincaré gauge gravity: the linear connection need not be metric compatible, and the translational sector is retained explicitly.

The papers do not adopt a single universal symmetry choice. Some formulations work with the full affine or metric-affine group $GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$ (Jiménez-Cano et al., 2020), while others enlarge or specialize the symmetry to $ISO(1,3)$, $ISO(n)$, $SO(n+1)$, $SO(4,1)$, $SO(3,2)$, a parabolic subgroup of $SO(p+1,q+1)$, the conformal-affine group $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$0, or Maxwell-affine extensions with tensorial generators $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$1 (Santos, 2019, Kerr, 2014, Herfray et al., 2020, Cebecioğlu et al., 2015, Capozziello et al., 2014, Thibaut et al., 2024). Taken together, these works show that AGT is best understood as a gauge-geometric program rather than a single fixed model.

A recurrent point in this literature is that translational gauge symmetry is not merely auxiliary. In Poincaré and affine models, the coframe or tetrad is the gauge potential for translations, while the spin or linear connection gauges Lorentz or $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$2 transformations (Santos, 2019, Sobreiro et al., 2010). In conformal and tractor formulations, additional compensator or tractor fields reduce a larger internal symmetry to the gravitational subgroup relevant after gauge fixing (Herfray et al., 2020).

2. Bundle geometry and dynamical variables

A central AGT claim is that gravity should be formulated on a principal bundle endowed with an ordinary connection. One recent formulation argues that the affine bundle is “the most natural principal fibre bundle for a gauge theory of gravity,” with affine connection $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$3 as the basic field. Its local representative is

$A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$4

where $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$5 is the local representative of the frame-bundle connection and $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$6 is the local representative of the canonical 1-form. The corresponding affine curvature is

$A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$7

and in the teleparallel regime $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$8 one has $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$9 (Tjandra et al., 30 Sep 2025). This construction is explicitly designed to reproduce teleparallel variables while keeping a standard principal-bundle gauge structure.

A related clarification is provided by the affine-vector-bundle approach. There the appropriate carrier space for the affine group is not an ordinary associated vector bundle but an associated affine-vector bundle

$GL(d,\mathbb R)=K(d)\oplus O(d)$0

whose fibers contain both an affine reference point and an ordinary vector. In this setting the local covariant derivative splits naturally into an affine part and a vector part, and the coframe-like object arises as

$GL(d,\mathbb R)=K(d)\oplus O(d)$1

This is the bundle-theoretic origin of the parallel

$GL(d,\mathbb R)=K(d)\oplus O(d)$2

so that torsion becomes the translational holonomy of the affine bundle and curvature the linear holonomy (Chen et al., 2020).

In metric-affine gravity, the dynamical variables are typically the metric $GL(d,\mathbb R)=K(d)\oplus O(d)$3, coframe $GL(d,\mathbb R)=K(d)\oplus O(d)$4, and independent linear connection $GL(d,\mathbb R)=K(d)\oplus O(d)$5. Equivalently, in tensor notation one writes

$GL(d,\mathbb R)=K(d)\oplus O(d)$6

with torsion

$GL(d,\mathbb R)=K(d)\oplus O(d)$7

and nonmetricity

$GL(d,\mathbb R)=K(d)\oplus O(d)$8

This independent-connection structure is the standard post-Riemannian content of AGT (Aoki et al., 2023). By contrast, some affine-gauge approaches initially treat the vielbein-like field $GL(d,\mathbb R)=K(d)\oplus O(d)$9 and premetric $GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$0 as matter fields on the gauge bundle, postponing their identification with the spacetime coframe and metric until a later reduction or mass-scale emergence (Sobreiro et al., 2010, Sobreiro et al., 2010).

3. Variational principles and recovery of Einstein gravity

A major line of AGT research asks how Einstein gravity emerges from genuinely gauge-theoretic variables without imposing metricity or torsion-freeness by hand. A prominent example is the conformal-tractor reformulation of Einstein gravity. There the basic fields are an abstract tractor bundle $GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$1 with tractor metric $GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$2, a distinguished null line generated by the position tractor $GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$3, a tractor connection $GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$4, a generalized conformal frame $GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$5, and a non-degenerate tractor $GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$6. The conformal metric is built as

$GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$7

with $GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$8. The field equations are first order and include

$GA(4,\mathbb R)=T^4\rtimes GL(4,\mathbb R)$9

which is the tractor form of the Einstein equation; the scalar curvature is encoded by

$ISO(1,3)$0

In this formulation torsion-freeness is not imposed externally but follows from the variational equations through the normality conditions on the tractor connection (Herfray et al., 2020).

A second route is the covariant canonical gauge theory of gravity. In that framework the metric $ISO(1,3)$1 and connection $ISO(1,3)$2 are independent, with conjugate momenta $ISO(1,3)$3 and $ISO(1,3)$4. The connection momentum couples directly to curvature, and the gauge-covariant action contains the term

$ISO(1,3)$5

The crucial criterion is that if the dynamical Hamiltonian $ISO(1,3)$6 does not depend on $ISO(1,3)$7, then

$ISO(1,3)$8

so the independent connection becomes Levi-Civita and the first-order affine-Palatini system reduces to the second-order metric formalism (Benisty et al., 2018).

Other gauge models achieve the same reduction by gauge fixing or vacuum alignment. In the Pagels-type $ISO(1,3)$9 and $ISO(n)$0 constructions, the spin connection and frame field are assembled into a single gauge connection $ISO(n)$1; after fixing the scalar multiplet $ISO(n)$2, the action becomes the Sciama–Kibble first-order Einstein–Cartan or Palatini action, with or without cosmological constant, and matter couplings for spin $ISO(n)$3, $ISO(n)$4, $ISO(n)$5, and $ISO(n)$6 can be written in the same gauge-invariant language (Kerr, 2014). In the generalized gauge theory with higher-rank gauge components, the mixed tensor field $ISO(n)$7 is assumed to acquire the vacuum expectation value

$ISO(n)$8

which yields a term $ISO(n)$9 in the bosonic sector and thereby reproduces the Einstein–Hilbert term; a scalar truncation gives a $SO(n+1)$0 scalar-tensor sector (Nishida, 2012).

4. Torsion, nonmetricity, and reduction to Riemann–Cartan geometry

One of the clearest structural results in AGT is the parallel between curvature and torsion. On the affine-vector bundle, the curvature operator decomposes into a vector part governed by $SO(n+1)$1 and an affine part governed by $SO(n+1)$2. In this language, curvature measures the nontrivial linear holonomy of transported vectors, while torsion measures the translational holonomy of transported affine origins. The coframe is therefore the gauge potential for the translational sector, and Trautman’s $SO(n+1)$3-field is reinterpreted as the choice of affine reference point rather than an ad hoc auxiliary variable (Chen et al., 2020).

A separate but related theme is the reduction of affine or metric-affine gauge gravity to orthogonal geometry. Both the bundle-theoretic and dynamical analyses show the contraction

$SO(n+1)$4

or, in $SO(n+1)$5 dimensions, $SO(n+1)$6. Because the translational sector $SO(n+1)$7 and the coset $SO(n+1)$8 are contractible, the affine connection can be decomposed as

$SO(n+1)$9

with $SO(4,1)$0 orthogonal, $SO(4,1)$1 the symmetric $SO(4,1)$2 sector, and $SO(4,1)$3 the translational sector. After reduction, the geometry becomes Riemann–Cartan, while the nonorthogonal pieces survive as matter fields rather than disappearing (Sobreiro et al., 2010, Sobreiro et al., 2010).

This matter reinterpretation is one of the most distinctive claims of AGT. After a dynamically generated mass scale $SO(4,1)$4 appears, one may set

$SO(4,1)$5

so that the spacetime metric becomes

$SO(4,1)$6

The residual fields $SO(4,1)$7 and $SO(4,1)$8 then encode the nonmetric sector of the original affine theory as post-Riemannian matter (Sobreiro et al., 2010).

At the restricted Poincaré-gauge level, the same logic yields Einstein–Cartan–Sciama–Kibble gravity. There the connection decomposes as

$SO(4,1)$9

with torsion sourced algebraically by the spin current density $SO(3,2)$0. In the minimal ECSK theory, torsion does not propagate in vacuum; it is a local algebraic response to spin density rather than an independent wave-like field (Santos, 2019). This restricted case is therefore an important subtheory of AGT, but not the full affine or metric-affine theory.

5. Waves, perturbations, and propagating post-Riemannian modes

Quadratic metric-affine gravity possesses a much richer wave sector than ordinary Riemannian gravity. In the general parity-even MAG action containing all linear and quadratic invariants built from torsion, nonmetricity, and curvature, exact plane-fronted gravitational-wave solutions exist. The theory admits the ordinary Riemannian pp-wave as the limit $SO(3,2)$1, $SO(3,2)$2, but also teleparallel waves with $SO(3,2)$3, symmetric teleparallel waves with $SO(3,2)$4, and fully general MAG waves in which curvature, torsion, and nonmetricity all propagate. In the general solution class, the field equations separate into parity-even and parity-odd sectors, with one massless graviton mode reproducing the GR-like wave and additional massive propagating modes beyond it (Jiménez-Cano et al., 2020).

Cosmological perturbation theory in metric-affine gravity makes this enlargement explicit. Around spatially curved FLRW backgrounds with independent affine connection, torsion and nonmetricity admit irreducible decompositions that generate helicity $SO(3,2)$5, $SO(3,2)$6, $SO(3,2)$7, and $SO(3,2)$8 sectors on top of the usual scalar, vector, and tensor metric perturbations. In particular, the helicity-2 modes of torsion and nonmetricity source the helicity-2 metric tensor perturbation at linear order, leading to gravitational-wave production by genuinely affine degrees of freedom (Aoki et al., 2023).

The most distinctive new mode is the helicity-3 excitation carried by the fully symmetric traceless nonmetricity component $SO(3,2)$9. Its quadratic action takes the form

$SO(p+1,q+1)$0

with ghost avoidance requiring

$SO(p+1,q+1)$1

In helicity space the dispersion relation becomes

$SO(p+1,q+1)$2

Because the background torsion amplitude $SO(p+1,q+1)$3 enters linearly, a parity-preserving action can still produce parity-violating propagation on a torsionful cosmological background (Aoki et al., 2023). A plausible implication is that AGT phenomenology is not exhausted by modified Friedmann dynamics; it extends to higher-spin and polarization-sensitive propagation sectors.

6. Extensions and alternative affine constructions

Several papers enlarge the affine algebra itself. In Maxwell-affine gravity, the affine algebra is extended by antisymmetric tensorial generators $SO(p+1,q+1)$4 satisfying

$SO(p+1,q+1)$5

Gauging the resulting Maxwell-affine algebra introduces the connection

$SO(p+1,q+1)$6

and the curvatures

$SO(p+1,q+1)$7

Both first-order and second-order invariant actions can be written, and for a particular solution of the constraint equation the second-order theory reduces to generalized Bianchi identities (Cebecioğlu et al., 2015).

A different enlargement is the conformal-affine program based on $SO(p+1,q+1)$8 and its partial relation to a centrally extended $SO(p+1,q+1)$9. There the gauge symmetry includes Lorentz transformations, translations, special conformal transformations, nine shear generators, and one dilation generator. A nonlinear realization produces coset fields, Cartan forms, inverse Higgs constraints, and geometric Lagrangians of Born–Infeld or Yang–Mills type depending on the semisimple or semidirect version of the symmetry (Capozziello et al., 2014). Closely related topological gauge constructions derive MacDowell–Mansouri gravity, Holst, Euler, Pontrjagin, and Nieh–Yan terms from characteristic classes on Cartan geometries; in Lorentz $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$00 Weyl geometry the equations of motion contain a secondary source for curvature expressed in terms of spin density, torsion, and their variations (Thibaut et al., 2024).

There are also more heterodox proposals. One paper advocates representing gauge fields directly by affine connections rather than by principal-bundle connections and claims geometric origins for coupling constants, chiral asymmetry, PMNS mixing, CKM mixing, color confinement, and the avoidance of $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$01-type proton decay (Man, 2020). Another develops gravity as a gauge theory of translations derived from the isometry group of a maximally symmetric space, with a nonlinear realization leading to a first-order diagonal form for the dynamical variables and off-diagonal contributions only at higher orders (Julve et al., 2010). These proposals reinforce how wide the affine-gauge label has become in current usage.

7. Diffeomorphism invariance, teleparallel regimes, and common misconceptions

A persistent conceptual concern is whether AGT is genuinely diffeomorphism invariant or merely gauge-covariant on a fixed background. One recent affine-bundle formulation gives an explicit proof of diffeomorphism invariance by showing that the diffeomorphism-induced bundle automorphism preserves the canonical 1-form,

$A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$02

and preserves the flatness property of frame-bundle connections of the form

$A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$03

On that basis, both TEGR and AGT are argued to be strictly diffeomorphism invariant, and AGT is presented as background independent in the physically relevant sense, though the paper notes that background independence admits “varying degrees of strictness” (Tjandra et al., 30 Sep 2025).

A second misconception is that AGT is simply teleparallel gravity in different notation. The affine-bundle construction explicitly distinguishes the two. AGT is a gauge theory with affine curvature

$A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$04

whereas teleparallel gravity corresponds only to the special regime $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$05, so that $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$06 (Tjandra et al., 30 Sep 2025). Thus AGT may reproduce TEGR locally, but it is not exhausted by the teleparallel sector.

A third misconception is that every gauge theory of gravity is automatically a full affine gauge theory. The literature itself differentiates these cases. The pedagogical treatment of Einstein–Cartan–Sciama–Kibble gravity is explicitly described as a Poincaré gauge theory rather than a full affine theory, because it gauges local Lorentz transformations and translations but does not include the full $A(d,\mathbb R)=GL(d,\mathbb R)\ltimes \mathbb R^d,$07 or nonmetric sector (Santos, 2019). Likewise, reduction results show that passing from affine or metric-affine gravity to orthogonal or Riemann–Cartan geometry does not erase the affine content; the nonorthogonal pieces reappear as matter fields carrying the memory of the original affine symmetry (Sobreiro et al., 2010, Sobreiro et al., 2010).

These clarifications fix the place of AGT within gauge approaches to gravitation. AGT is neither a single model nor a synonym for any gauge formulation of gravity whatsoever. It is a family of bundle-based theories in which affine, metric-affine, or closely related enlarged spacetime symmetries generate the coframe, linear connection, and sometimes the metric as gauge variables, and in which torsion, curvature, and nonmetricity become the natural carriers of gravitational dynamics.