Regge Action in Discrete Gravity

Updated 22 May 2026

Regge action is a discrete gravitational action defined on triangulated manifolds that encodes curvature via deficit angles at codimension-2 hinges.
The formulation in length and area Regge calculus reveals distinct implications for degrees of freedom, gauge invariance, and the continuum limit of general relativity.
It underpins state-sum models in quantum gravity, influencing discrete diffeomorphism invariance and the behavior under Pachner moves.

The Regge action is a discretized gravitational action fundamental to piecewise-flat formulations of general relativity and numerous approaches to quantum gravity, including spin-foam models, state-sum quantization, and discrete numerical relativity. It is constructed on a triangulated (simplicial) manifold by encoding curvature as deficit angles concentrated at codimension-2 simplices—”hinges”—with the action expressed as a sum over these hinges of the product of their geometric measures (e.g., triangle area in 4d) and associated deficits. Variants of the Regge action use as fundamental variables either edge lengths (length Regge calculus) or face areas (area Regge calculus), each with distinct implications for degrees of freedom, gauge invariance, and compatibility with continuum limit general relativity.

1. Formal Definition and Geometric Content

Let $\mathcal T$ be a triangulation of a $d$ -dimensional piecewise-flat manifold, with $L_e$ the edge-lengths for each 1-simplex $e$ . In $4$d, curvature is localized on triangles $t$ (2-simplices). The standard (length-based) Regge action is

$S_{\rm Regge}[L] = \sum_{t}A_t(L)\,\delta_t(L),$

where $A_t(L)$ is the area of triangle $t$ (determined by the edge-lengths via the Cayley–Menger determinant), and the deficit angle at $t$ ,

$d$ 0

measures the curvature concentrated at $d$ 1 by summing the dihedral angles $d$ 2 at $d$ 3 within each incident 4-simplex $d$ 4 (Mikovic, 2022, Mikovic, 2014).

In the area Regge action, the areas $d$ 5 are taken as independent variables, and for each 4-simplex, the dihedral angles $d$ 6 are re-expressed as functions of the areas in $d$ 7. The action is then

$d$ 8

with deficit $d$ 9 (Asante et al., 2018).

2. Equations of Motion and Flatness Conditions

Varying the length Regge action with respect to $L_e$ 0 yields

$L_e$ 1

which, for generic data, allows for curvature solutions ( $L_e$ 2) compatible with discrete Bianchi identities (Dittrich, 2021).

For the area Regge action, variation with respect to each area $L_e$ 3 yields, via the Schläfli identity,

$L_e$ 4

and, after summing and using $L_e$ 5, implies everywhere-vanishing deficit angles: $L_e$ 6 for all $L_e$ 7 (Asante et al., 2018, Neiman, 2013). This enforces local flatness of each simplex but does not guarantee a globally flat geometry due to potential non-metric identifications at gluing faces.

3. Gauge Symmetry, Diffeomorphisms, and Non-metric Degrees of Freedom

Length Regge calculus on a flat background supports a discrete analogue of diffeomorphism invariance: moving an interior vertex does not alter deficit angles (vertex translation symmetry) (Asante et al., 2018). However, the presence of non-metric (“twisted”) area assignments in the area Regge action breaks this symmetry.

In the area formulation, a 4-simplex can be uniquely described by 10 areas or 10 lengths, but as soon as two or more 4-simplices are glued, the number of area variables exceeds the number of lengths (e.g., 16 areas for 14 edges in two glued simplices). This mismatch allows non-metric area data, which cannot arise from any consistent set of edge-lengths: glued triangles may have matched areas but differ in shape (their boundary dihedral angles), giving rise to additional, non-metric degrees of freedom (Asante et al., 2018).

Non-metric modes are detected by considering the Jacobian $L_e$ 8 on a metric background. Null vectors of $L_e$ 9, $e$ 0, span the space of area variations not generated by any infinitesimal change in $e$ 1—these constitute the twisted, non-metric sector.

Numerically, translation mode Hessian eigenvalues vanish for metric data but pick up a quadratic dependence on non-metricity parameters (e.g., differences of certain boundary dihedral angles) as non-metricity is introduced (Asante et al., 2018).

4. Invariance under Triangulation Moves and Pachner Transformations

Pachner moves locally replace a cluster of $e$ 2 4-simplices by $e$ 3 4-simplices sharing the same boundary, serving as canonical tests of action invariance under triangulation change:

5–1 move: Both length and area Regge actions are invariant.
4–2 move: The length Regge action remains invariant for arbitrary boundary data, but the area Regge action is only invariant if non-metric (twist) modes vanish, as two non-metric boundary modes (differences of boundary dihedrals) generally produce an action mismatch.
3–3 move: In length Regge calculus, invariance is lost generically due to curvature at a single interior triangle; in area Regge calculus, flatness is enforced via bulk equations, but non-metric twists generically induce triangulation dependence unless they vanish (Asante et al., 2018).

Invariance under Pachner moves is thus sharply sensitive to non-metricity, making these moves critical diagnostics for physical viability and for the analysis of quantum spin-foam dynamics.

5. Relation to Quantum Gravity, Spin Foams, and BF Theory

The Regge action underlies state-sum models, including spin foams and simplicial quantum gravity. In particular:

Spin Foam Models: Barrett–Crane-type amplitudes reduce to area Regge actions in the large-spin (semiclassical) limit, carrying twisted boundary geometries. The EPRL/FK model, by contrast, does not admit generic non-metric twist configurations. In the semiclassical analysis of SU(2) spin-network invariants (e.g., 15j symbols), the Regge action governs the oscillatory asymptotics, with the emergence of non-metric twist modes linked to the existence of angle-matched (but not fully shape-matched) twisted geometries (Donà et al., 2017, Asante et al., 2018, Dittrich, 2021).
BF Theory Correspondence: Regge calculus emerges as the Hamilton–Jacobi function of $e$ 4 theory on manifolds with defect loci at the hinges, with edge lengths encoded as boundary data and the action derived as a defect term from flat $e$ 5 solutions (Kisielowski, 2017).
Path Integral and Measures: The Regge action admits quantum path-integral formulations, with the choice of measure (exponential damping, power-law, or discrete length spectra) deeply affecting the semiclassical regime, the emergence of a cosmological constant, and perturbative expansions (Mikovic, 2014, Mikovic, 2022, Khatsymovsky, 2017).

6. Physical Implications and Limitations

Several key implications follow from the structure of the Regge action:

Continuum Limit: To recover 4d general relativity, both the curvature per simplex and the magnitude of twist/non-metricity must vanish in the continuum/refinement limit. Semiclassical expansions and correct propagator spectra require a sufficiently fine discretization and the effective suppression of non-metric modes (Asante et al., 2018, Dittrich, 2021).
Non-metricity and Unphysical Modes: Area Regge discretizations admitting generic non-metricity propagate extraneous degrees of freedom (unphysical from the perspective of GR). The persistence of such modes manifests as breakdowns in discrete diffeomorphism invariance and failure of action invariance under standard Pachner moves (Neiman, 2013, Asante et al., 2018).
Compatibility with General Relativity: In the classical sector where area and length data are related by shape-matching (metric sector), the area Regge action becomes equivalent to the length Regge action, and the correct discrete Einstein equations are reproduced (Dittrich, 2021). However, in the presence of general non-metric twist, the area formulation fails to approximate vacuum general relativity, especially in the Euclidean and standard Lorentzian (all-spacelike) sectors (Neiman, 2013).

7. Generalizations, Extensions, and Discrete Palatini Forms

The Regge action admits several formal generalizations:

Affine (Palatini-like) Discretizations: By analogy with the continuum Palatini action, the Regge action can be recast in terms of discrete metric and independent connection variables (on 3-simplices), with the standard length Regge action recovered upon imposing the discrete metric-compatibility equations (Khatsymovsky, 2015, Khatsymovsky, 2016, Khatsymovsky, 2017). This approach clarifies gauge invariances and allows for the introduction of parity-violating (Holst-Immirzi) terms.
Torsionful (Teleparallel) Generalizations: In three dimensions, the Regge action admits torsional extensions via discrete Burgers vectors, yielding teleparallel gravity analogues. The semiclassical limit of the Ponzano–Regge state-sum is governed by the corresponding Regge or teleparallel action (Vargas, 2013).
Polytopal and High-dimensional Extensions: In cosmological and higher-dimensional settings, the Regge action can be formulated for discrete models based on regular polytopes and Schläfli-symbol combinatorics, retaining time-reparameterization invariance and yielding discrete analogues of the ADM Hamiltonian constraint and evolution equations. New dynamical phenomena—including oscillatory (recollapse) behavior in higher- $e$ 6 Regge–FLRW models—emerge in this context (Tsuda et al., 2021).