Teleparallel Supersymmetric Chern–Simons Gravity

• The topic defines a three-dimensional supergravity theory using the Chern–Simons framework that integrates teleparallel geometry with vanishing curvature and non-vanishing torsion.
• It outlines the supersymmetric extension of the Poincaré algebra with explicit gauge connections and invariant bilinear forms to establish the action and field equations.
• The framework demonstrates how the cosmological constant acts as a torsion source, reducing to standard Poincaré supergravity in the flat limit.

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 (Caroca et al., 2021).

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 $\mathcal{N}=1$0 and $\mathcal{N}=1$1, as well as internal and automorphism generators $\mathcal{N}=1$2, 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 (Caroca et al., 2021).

2. Chern–Simons Action and Invariant Bilinear Forms

The dynamics are formulated via a Chern–Simons action,

$\mathcal{N}=1$3

where $\mathcal{N}=1$4 is the gauge connection valued in the superalgebra and $\mathcal{N}=1$5 with $\mathcal{N}=1$6 the gravitational constant. The minimal super-connection is

$\mathcal{N}=1$7

with bilinear invariants specified by

$\mathcal{N}=1$8

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}=1$9, the action generalizes to include terms for the extended gauge fields and gravitini, with separate $J_a$0 (exotic) and $J_a$1 (kinetic/torsion) sectors reflecting the underlying algebraic structure (Caroca et al., 2021).

3. Torsion, Super-Torsion, and Field Equations

Ordinary torsion is defined by

$J_a$2

while the “cosmological deformation” is

$J_a$3

Bosonic field equations enforce $J_a$4 and $J_a$5, implying

$J_a$6

Supersymmetry promotes torsion and curvature to superforms: $J_a$7 On-shell field equations for allowed couplings reduce to

$J_a$8

yielding non-vanishing super-torsion: $J_a$9 For $P_a$0 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 $P_a$1 explicitly sources torsion, as

$P_a$2

demonstrates. As $P_a$3, the theory flows to the standard Poincaré (super-)gravity limit: $P_a$4 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 (Caroca et al., 2021).

5. Supersymmetry Transformations and Closure

Supersymmetry transformations act as gauge variations of the super-connection $P_a$5: $P_a$6 which yield the explicit variations

$P_a$7

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 ($P_a$8) hold.

6. Extended ($P_a$9) Supersymmetry Generalization

The $Q_\alpha$0 generalization introduces additional gauge fields, gravitini, and internal automorphisms: $Q_\alpha$1 Pairings in the invariant tensor are specified for all new generators (see Eq.~(4.16)), and the CS action splits into $Q_\alpha$2 and $Q_\alpha$3 sectors similar to the minimal case but with couplings extending to the internal symmetry and automorphism fields. The field equations,

$Q_\alpha$4

yield generalized (super-)torsion conditions. Again, the flat limit recovers the Poincaré supergravity enlarged by internal $Q_\alpha$5 algebras, with vanishing (super-)torsion on-shell (Caroca et al., 2021).