---
title: Teleparallel Supersymmetric Chern–Simons Gravity
url: https://www.emergentmind.com/topics/teleparallel-supersymmetric-chern-simons-gravity
type: topic
---

# Teleparallel Supersymmetric Chern–Simons Gravity

Teleparallel Supersymmetric Chern–Simons Gravity is a gauge-invariant, three-dimensional supergravity framework constructed within the Chern–Simons (CS) formalism. Distinguished by its non-Riemannian geometry, this theory features vanishing curvature and non-vanishing torsion at the bosonic level, and, in the presence of supersymmetry, a non-vanishing super-torsion. The cosmological constant acts as a torsion source, and the theory reproduces standard Poincaré supergravity in the flat (vanishing cosmological constant) limit. The structure and dynamics are encoded by a supersymmetric extension of a specific deformation of the Poincaré algebra, allowing generalization to extended supersymmetry sectors with $\mathcal{N}=p+q$ supercharges [2103.06717].

## 1. Underlying Gauge Superalgebra

The minimal ($\mathcal{N}=1$) teleparallel superalgebra is generated by the Lorentz generators $J_a$, translation generators $P_a$, and a Majorana supercharge $Q_\alpha$. The non-vanishing commutators are:
\[
\begin{aligned}
[J_a, J_b] &= \epsilon_{abc} J^c, \\
[J_a, P_b] &= \epsilon_{abc} P^c, \\
[P_a, P_b] &= -\frac{2}{\ell} \epsilon_{abc} P^c, \\
[J_a, Q_\alpha] &= -\frac{1}{2} (\gamma_a)_\alpha{}^\beta Q_\beta, \\
\{Q_\alpha, Q_\beta\} &= -(\gamma^a C)_{\alpha\beta}(P_a + \frac{2}{\ell} J_a).
\end{aligned}
\]
Here, $\ell$ defines the (A)dS radius, and the cosmological constant is $\Lambda = -1/\ell^2$. In the limit $\ell \to \infty$, the algebra reduces to the standard Poincaré superalgebra.

For $\mathcal{N} = p+q$ extensions, additional supercharges $Q^i_\alpha$ and $Q^I_\alpha$, as well as internal and automorphism generators $T^{ij},\,T^{IJ},\,S^{ij},\,S^{IJ}$, are introduced. The full set of (anti)commutators includes additional non-abelian structure involving the internal and automorphism symmetries. After specific redefinitions, the algebra accommodates both the deformation parameter and supersymmetry [2103.06717].

## 2. Chern–Simons Action and Invariant Bilinear Forms

The dynamics are formulated via a Chern–Simons action,
\[
S_{\rm CS}[A] = k\int_{\mathcal{M}} \left\langle A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right\rangle,
\]
where $A$ is the gauge connection valued in the superalgebra and $k = 1/4G$ with $G$ the gravitational constant. The minimal super-connection is
\[
A = \omega^a J_a + e^a P_a + \bar\psi Q,
\]
with bilinear invariants specified by
\[
\begin{aligned}
\langle J_a J_b\rangle &= \alpha_0 \eta_{ab}, \qquad 
\langle J_a P_b\rangle = \alpha_1 \eta_{ab}, \\
\langle P_a P_b\rangle &= -\frac{2\alpha_1}{\ell} \eta_{ab}, \quad
\langle Q_\alpha Q_\beta\rangle = 2\left(\frac{2\alpha_0}{\ell} + \alpha_1\right) C_{\alpha\beta}.
\end{aligned}
\]
The component form of the action includes Lorentz and Einstein–Hilbert terms, cosmological and torsion contributions, as well as gravitino kinetic terms. For arbitrary $\mathcal{N}$, the action generalizes to include terms for the extended gauge fields and gravitini, with separate $\alpha_0$ (exotic) and $\alpha_1$ (kinetic/torsion) sectors reflecting the underlying algebraic structure [2103.06717].

## 3. Torsion, Super-Torsion, and Field Equations

Ordinary torsion is defined by
\[
T^a = d e^a + \epsilon^{a}{}_{bc} \omega^b \wedge e^c,
\]
while the “cosmological deformation” is
\[
\hat{T}^a = T^a - \frac{1}{\ell} \epsilon^{a}{}_{bc} e^b \wedge e^c.
\]
Bosonic field equations enforce $R^a=0$ and $\hat{T}^a=0$, implying
\[
T^a = \frac{1}{\ell} \epsilon^{a}{}_{bc} e^b e^c.
\]
Supersymmetry promotes torsion and curvature to superforms:
\[
\begin{aligned}
\mathcal{R}^a &= R^a + \frac{1}{\ell} \bar\psi \gamma^a \psi, \\
\mathcal{T}^a &= \hat{T}^a + \frac{1}{2} \bar\psi \gamma^a \psi.
\end{aligned}
\]
On-shell field equations for allowed couplings reduce to
\[
\mathcal{R}^a = 0, \quad \mathcal{T}^a = 0, \quad \nabla\psi = 0,
\]
yielding non-vanishing super-torsion:
\[
T^a + \frac{1}{2} \bar\psi \gamma^a \psi = \frac{1}{\ell} \epsilon^{a}{}_{bc} e^b e^c.
\]
For $\mathcal{N}=p+q$ extensions, the structure of super-torsion generalizes via additional gravitini, preserving the property that (super-)torsion is proportional to the cosmological deformation parameter.

## 4. Cosmological Constant as Torsion Source and Flat Limit

The cosmological constant $\Lambda = -1/\ell^2$ explicitly sources torsion, as 
\[
T^a = \frac{1}{\ell} \epsilon^a{}_{bc} e^b e^c + \dots,
\]
demonstrates. As $\ell \rightarrow \infty$, the theory flows to the standard Poincaré (super-)gravity limit:
\[
[P_a, P_b] \to 0,\quad T^a \to 0,\quad \mathcal T^a \to \frac{1}{2} \bar\psi \gamma^a \psi,
\]
with algebra, action, and field equations reducing correspondingly. This confirms the role of the cosmological constant as the unique source of (super-)torsion in teleparallel supergravity within this construction [2103.06717].

## 5. Supersymmetry Transformations and Closure

Supersymmetry transformations act as gauge variations of the super-connection $A$:
\[
\delta_\varepsilon A = D\varepsilon = d\varepsilon + [A, \varepsilon],\quad \varepsilon = \bar\varepsilon Q,
\]
which yield the explicit variations
\[
\begin{aligned}
\delta e^a &= \frac{1}{2} \bar\varepsilon \gamma^a \psi, \\
\delta\omega^a &= \frac{1}{2\ell} \bar\varepsilon \gamma^a \psi, \\
\delta\psi &= D\varepsilon.
\end{aligned}
\]
All additional fields remain inert under minimal supersymmetry. The algebra of supersymmetry transformations closes on-shell, up to a translation and Lorentz rotation, provided the field equations ($\mathcal{R}^a = \mathcal{T}^a = 0$) hold.

## 6. Extended ($\mathcal{N}=p+q$) Supersymmetry Generalization

The $\mathcal{N}=p+q$ generalization introduces additional gauge fields, gravitini, and internal automorphisms:
\[
A = \omega^a J_a + e^a P_a + \frac{1}{2} A^{ij} T_{ij} + \frac{1}{2}A^{IJ} T_{IJ} + \frac{1}{2}C^{ij} S_{ij} + \frac{1}{2}C^{IJ} S_{IJ} + \bar\psi^i Q^i + \bar\psi^I Q^I.
\]
Pairings in the invariant tensor are specified for all new generators (see Eq.~(4.16)), and the CS action splits into $\alpha_0$ and $\alpha_1$ sectors similar to the minimal case but with couplings extending to the internal symmetry and automorphism fields. The field equations,
\[
\mathcal R^a = 0,\quad \tilde{\mathcal T}^a = 0,\quad \tilde{F}^{ij} = \tilde{G}^{ij} = 0,\quad \nabla\psi^i = 0,\quad \nabla\psi^I = 0,
\]
yield generalized (super-)torsion conditions. Again, the flat limit recovers the Poincaré supergravity enlarged by internal $\mathfrak{so}(p)\oplus \mathfrak{so}(q)$ algebras, with vanishing (super-)torsion on-shell [2103.06717].

Source: https://www.emergentmind.com/topics/teleparallel-supersymmetric-chern-simons-gravity