Supersymmetric Lorentz Chern–Simons

• Supersymmetric Lorentz Chern–Simons theories are gauge frameworks extending 3D supergravity, characterized by deformed Poincaré superalgebras and torsion-based cosmological encoding.
• They employ a Chern–Simons action with a connection combining the spin connection, dreibein, and gravitino, yielding a teleparallel (curvature-free) yet torsion-rich dynamics.
• The formulation shifts gravitational roles by replacing curvature with non-vanishing torsion, offering alternative insights and potential non-relativistic extensions in supergravity models.

Supersymmetric Lorentz Chern–Simons theories are extensions of Chern–Simons gauge frameworks that incorporate Lorentz symmetry and $\mathcal{N}$-extended supersymmetry within a non-Riemannian (teleparallel) geometric context, most notably in three spacetime dimensions. These theories are formulated with gauge algebras that arise as deformations of the Poincaré superalgebra, and possess a defining feature in which the cosmological constant is encoded through non-vanishing (super-)torsion instead of curvature, fundamentally altering the geometric and dynamical structure compared to standard Riemannian (super)gravity models (Caroca et al., 2021, Concha et al., 2021).

1. Algebraic Structure and Supersymmetric Extensions

The algebra underlying supersymmetric Lorentz Chern–Simons theories is a deformation of the three-dimensional Poincaré superalgebra, characterized by the following set of (anti)commutation relations for the generators $J_a$ (Lorentz rotations), $P_a$ (teleparallel translations), and $Q_\alpha$ (Majorana supercharges):

$$[J_a, J_b] = \epsilon_{abc} J^c,\quad [J_a, P_b] = \epsilon_{abc} P^c,\quad [P_a, P_b] = -\frac{2}{\ell} \epsilon_{abc} P^c$$

$$[J_a, Q_\alpha] = -\frac{1}{2} (\gamma_a)_\alpha{}^\beta Q_\beta,\quad \{ Q_\alpha, Q_\beta \} = -(\gamma^a C)_{\alpha\beta}(P_a + \frac{2}{\ell} J_a)$$

Here, $\ell$ encodes the (negative) cosmological constant $\Lambda = -1/\ell^2$ and $C$ is the charge-conjugation matrix. In the vanishing cosmological constant limit $\ell \to \infty$, the algebra reduces to the standard Poincaré superalgebra, eliminating the $J_a$0 deformation (Caroca et al., 2021, Concha et al., 2021).

Supersymmetry admits $J_a$1 extensions via additional supercharges $J_a$2 and $J_a$3, together with R-symmetry generators $J_a$4, $J_a$5, $J_a$6, $J_a$7, yielding more intricate structure constants and multiplet content.

2. Chern–Simons Gauge Theory Formulation

The construction is based on a Chern–Simons action functional with a gauge connection valued in the aforementioned deformed superalgebra:

$J_a$8

Here, $J_a$9 is the spin connection, $P_a$0 the dreibein, and $P_a$1 a Majorana gravitino one-form. The non-degenerate invariant bilinear $P_a$2 on the algebra, essential for defining the Chern–Simons action, assigns the only non-vanishing entries to pairs such as $P_a$3, $P_a$4, $P_a$5, and $P_a$6 with coefficients involving parameters $P_a$7, $P_a$8, and the metric $P_a$9. The general Chern–Simons action reads:

$Q_\alpha$0

Substitution yields a theory with distinct "exotic" ($Q_\alpha$1) and minimal supergravity ($Q_\alpha$2) sectors, where all structure constants reflect the (anti)commutators of the teleparallel superalgebra (Caroca et al., 2021).

3. Field Strengths, Supertorsion, and Equations of Motion

The gauge field strengths decompose into Lorentz curvature, torsion, and supertorsion terms. For the bosonic sector:

$Q_\alpha$3

The supertorsion two-form is

$Q_\alpha$4

The equations of motion entirely set the Lorentz curvature to zero (Weitzenböck geometry), while torsion and supertorsion remain non-vanishing and are sourced by the cosmological constant and fermionic bilinears. The gravitino satisfies a Killing spinor condition:

$Q_\alpha$5

This succinctly demonstrates the characteristic shift from Riemannian (torsion-free) to teleparallel (curvature-free) supersymmetric gravity (Caroca et al., 2021).

4. Teleparallel and Non-Riemannian Geometric Interpretation

The Weitzenböck or teleparallel geometry underlying these theories is defined by vanishing Lorentz (Riemann–Cartan) curvature but non-zero (super-)torsion. All gravitational dynamics are encoded in the torsion tensor, with the cosmological constant directly generating the torsion component. In the presence of supersymmetry, the gravitino bilinear further deforms the torsion structure. This geometric realization is fundamentally distinct from that in Riemannian CS supergravity, where both curvature and torsion vanish on-shell, yielding a locally AdS spacetime (Caroca et al., 2021, Concha et al., 2021).

5. Non-Relativistic and Extended Supergravity Generalizations

Teleparallel Chern–Simons (super)gravity admits non-relativistic limits via Lie algebra expansion techniques. By decomposing and expanding in suitable semigroups, one obtains non-relativistic teleparallel superalgebras with new bosonic and fermionic generators. The resulting non-relativistic theory maintains non-vanishing spatial (super-)torsion sourced by the cosmological constant (Concha et al., 2021). For $Q_\alpha$6 supersymmetries, one introduces additional supercharges and R-symmetry fields, with the gauge connection and invariant forms extended accordingly. In both cases, all torsion and supertorsion properties persist, with the algebraic structure naturally reducing to Poincaré CS supergravity as $Q_\alpha$7.

6. Comparison with Riemannian Chern–Simons Supergravity

The teleparallel Chern–Simons supergravity differs fundamentally from the standard (Riemannian) AdS$Q_\alpha$8 CS supergravity, which is constructed with the $Q_\alpha$9 superalgebra. In the Riemannian case, field equations impose vanishing of both curvature and torsion, resulting in a Riemannian geometry with local AdS structure. In contrast, the teleparallel (super)gravity features non-vanishing torsion (with curvature identically zero) and embeds the cosmological constant entirely in the torsion sector. The supersymmetry algebra forces a non-trivial interplay between the translation and Lorentz generators in both the field equations and the underlying geometry (Caroca et al., 2021, Concha et al., 2021).

7. Summary Table: Key Structural Elements

Feature	Teleparallel CS Supergravity	Riemannian CS Supergravity
Underlying algebra	Deformed Poincaré superalgebra	$$[J_a, J_b] = \epsilon_{abc} J^c,\quad [J_a, P_b] = \epsilon_{abc} P^c,\quad [P_a, P_b] = -\frac{2}{\ell} \epsilon_{abc} P^c$$0
On-shell curvature	$$[J_a, J_b] = \epsilon_{abc} J^c,\quad [J_a, P_b] = \epsilon_{abc} P^c,\quad [P_a, P_b] = -\frac{2}{\ell} \epsilon_{abc} P^c$$1 (Weitzenböck)	$$[J_a, J_b] = \epsilon_{abc} J^c,\quad [J_a, P_b] = \epsilon_{abc} P^c,\quad [P_a, P_b] = -\frac{2}{\ell} \epsilon_{abc} P^c$$2
On-shell torsion	$$[J_a, J_b] = \epsilon_{abc} J^c,\quad [J_a, P_b] = \epsilon_{abc} P^c,\quad [P_a, P_b] = -\frac{2}{\ell} \epsilon_{abc} P^c$$3 (cosm. const. source)	$$[J_a, J_b] = \epsilon_{abc} J^c,\quad [J_a, P_b] = \epsilon_{abc} P^c,\quad [P_a, P_b] = -\frac{2}{\ell} \epsilon_{abc} P^c$$4
Cosmological constant role	Sources torsion	Encoded in curvature
Supersymmetry effect	Introduces supertorsion, deforms torsion equation	No supertorsion

A plausible implication is that teleparallel CS supergravity realigns the assignment of cosmological constant effects from curvature to torsion, providing an alternative geometric and algebraic realisation of three-dimensional (super)gravity, with possible ramifications for non-Riemannian and non-relativistic gravitational models (Caroca et al., 2021, Concha et al., 2021).