Nonlinear Realisation in 3D Supergravity

• Nonlinear realisation approach is a geometric framework that constructs extended 3D supergravity theories by treating spacetime translations and supersymmetry as spontaneously broken symmetries.
• It employs a coset construction to derive composite gravitational and fermionic fields, which recover canonical AdS and Poincaré supergravity actions via unitary gauge fixing.
• The method introduces Stückelberg compensators and reveals dual local supersymmetry, enhancing the formulation of Lorentz Chern–Simons and other supergravity extensions.

The nonlinear realisation approach provides a systematic and geometric framework to construct $\mathcal{N}$-extended supergravity theories in three dimensions by interpreting both spacetime translations and supersymmetry transformations as spontaneously broken symmetries. Goldstone modes associated to these transformations—the physical spacetime coordinates and “Goldstini”—parameterise a coset space obtained by quotienting the extended super-Poincaré group by the Lorentz and $R$-symmetry subgroups. The resulting formulation yields Stückelberg-type extensions of known supergravity actions, and reveals a dual local supersymmetry structure underpinning both (anti-)de Sitter and Poincaré supergravities, as well as their supersymmetric Lorentz Chern–Simons extensions.

1. Nonlinear Realisation and Coset Construction

In the nonlinear realisation scheme, the extended super-Poincaré group $\mathcal{P}(3|\mathcal{N})$ is the fundamental symmetry group. Group elements are factorised as

$g = s(b,\Theta) \, h(M, \mathcal{R}),$

with $s(b,\Theta)$ in the coset space (parameterized by translation parameters $b^a$ and Goldstini $\Theta_{(I)}$), and $h(M, \mathcal{R})$ in the stability subgroup $SL(2,\mathbb{R}) \times SO(p)\times SO(q)$. Accordingly, the coset is

$$\mathcal{M}^{3|2\mathcal{N}} = \mathcal{P}(3|\mathcal{N}) / \big[SL(2,\mathbb{R}) \times SO(p)\times SO(q)\big],$$

which defines the three-dimensional $(p,q)$ (with $p+q=\mathcal{N}$, $p\geq q$) superspace.

Within this formulation, the supergravity fields—composite dreibein $E^a$ (“gravitational vielbein”) and a set of gravitini $\psi_{(I)}$—are constructed via

$$\Omega = \mathcal{S}^{-1} \mathcal{A} \mathcal{S} + \mathcal{S}^{-1} d \mathcal{S},$$

with $\mathcal{S}$ the coset representative and $\mathcal{A}$ the gauge connection one-form. These composite fields transform covariantly under the residual Lorentz and $R$-symmetry ($SO(p)\times SO(q)$).

Gauge fixing by setting the Goldstone fields $X^a=0$, $\Theta_{(I)}=0$ (“unitary gauge”) brings the action back to the canonical formulations of $(p,q)$ AdS and Poincaré supergravities, as originally constructed by Achúcarro-Townsend and Marcus-Schwarz, respectively.

2. Stückelberg Extensions in AdS and Poincaré Supergravity

In this approach, the Goldstone modes associated with the spontaneously broken spacetime and supersymmetry generators function as Stückelberg compensators. Their inclusion restores full invariance under nonlinearly realised symmetries, rendering the supergravity action a Stückelberg-type extension.

For $(p,q)$ AdS supergravities,

$$S_{AdS(p,q)} = S_{SG} + S_{\text{super-cosm}} + S_{VCS},$$

where \begin{align*} S_{SG} &= \frac{1}{2}\int E^a\wedge R^{bc} - 2i\int \psi_I\wedge \mathcal{D}\psi_I, \ S_{\text{super-cosm}} &= m^2 S_{\text{cosm}} - i m_{IJ} S_{\text{mass}},\ S_{VCS} &= \frac{1}{2 m^2} m_{IJ} \int \left[dr_{IK}\wedge r_{KJ} - \frac{2}{3} r_{IK} \wedge r_{KL} \wedge r_{LJ} \right], \end{align*} with $$m_{IJ} = m \, \mathrm{diag}(1,\ldots,1,-1,\ldots,-1)$$ ($p$ entries $1$, $q$ entries $-1$). Unitary gauge recovers the conventional AdS (or flat-space) supergravity action.

3. Supersymmetric Lorentz Chern–Simons and Its Stückelberg Reformulation

The N-extended supersymmetric Lorentz Chern–Simons action receives a similar treatment, with the total action expressed schematically as

$S_{CSG} = S_{LCS} + S_{FCS} + S_{VCS},$

where $S_{LCS}$ is the Lorentz Chern–Simons term, $S_{FCS}$ is the fermionic (gravitino) Chern–Simons term, and $S_{VCS}$ encodes $R$-symmetry gauge dynamics. Goldstone fields again enter in a Stückelberg-like role. In the gauge $X^a=0, \Theta_{(I)}=0$, this construction reproduces the on-shell N-extended conformal supergravity actions of Lindström-Roček, Nishino-Gates, etc., but without invoking the superconformal algebra (e.g. no explicit special conformal or dilatation symmetries).

4. Dual Local $\mathcal{N}$-Extended Supersymmetry Structure

A principal feature is the emergence of two local $\mathcal{N}$-extended supersymmetries:

• The first supersymmetry is the canonical (or standard) one, emerging directly from the coset construction, under which the Goldstini transform nontrivially.
• The second (“inert Goldstino”) supersymmetry leaves the Goldstini invariant.

Explicitly, the operator of local supersymmetry that acts on Goldstini can be used as a gauge degree of freedom, enabling gauge fixing $X^a=\Theta_{(I)}=0$ (unitary gauge). The surviving inert supersymmetry is compatible with standard formulations in the literature. For the Lorentz Chern–Simons sector, only the standard supersymmetry is preserved; the inert structure does not extend fully.

This dual structure unifies and generalizes earlier coset-based constructions and reveals a deeper redundancy and flexibility in the local symmetry content of 3D extended supergravity theories.

5. Key Algebraic and Geometric Ingredient Summary

• The coset structure: $$\mathcal{M}^{3|2\mathcal{N}} = \mathcal{P}(3|\mathcal{N})/\big[SL(2,\mathbb{R})\times SO(p)\times SO(q)\big]$$.
• Mass matrix for AdS extensions: $$m_{IJ} = m \, \mathrm{diag}(1,\ldots,1,-1,\ldots,-1)$$.
• Composite connection: $$\Omega = \mathcal{S}^{-1} \mathcal{A} \mathcal{S} + \mathcal{S}^{-1} d\mathcal{S}$$.
• In unitary gauge, Stückelberg fields are eliminated, and the action reduces to canonical supergravity forms.

6. Relation to Established Formulations and Literature

The nonlinear realisation approach as developed in (Stirling, 3 Sep 2025) recovers, after gauge fixing, the supergravity actions of Achúcarro and Townsend for $(p,q)$ AdS supergravity, Marcus and Schwarz for N-extended Poincaré supergravity, and established conformal supergravity actions (Lindström-Roček, Nishino-Gates). Its central advance is exhibiting these well-known models as special gauges of a more general framework with dual local supersymmetry and an explicit Stückelberg field content arising from the spontaneous breaking viewpoint.

7. Implications and Perspective

This nonlinear realisation strategy offers not only a unified geometric derivation, but also an organizational principle for understanding how gauge and supersymmetry redundancies interlock in three-dimensional supergravity. The dual local supersymmetry structure may bear on partial spontaneous supersymmetry breaking phenomena, and the Stückelberg realisation clarifies the status of compensators and their role in rendering dynamics or topological sectors fully gauge invariant under both spacetime and internal symmetries.

The approach also suggests a natural path for future generalizations to higher $\mathcal{N}$, diverse dimensions, or other gauge and supergravity theories where local symmetry redundancy and its gauge fixing play a critical role in the structure of the theory.