Supersymmetric Lorentz Chern–Simons

• Supersymmetric Lorentz Chern–Simons theories are gauge theories in odd-dimensional spacetimes that integrate supersymmetry with Chern–Simons actions and non-Riemannian geometric structures.
• They employ a superconnection to cohesively combine bosonic and fermionic fields, using deformed Poincaré algebras and teleparallel approaches to realize off-shell supersymmetry without conventional gravitinos.
• These frameworks enable novel mass generation mechanisms and offer versatile applications in topological quantum field theory, alternative gravity models, and even condensed matter physics.

Supersymmetric Lorentz Chern–Simons theories are gauge theories in odd-dimensional spacetimes whose actions are given by Chern–Simons forms, extended to include supersymmetry and constructed over Lorentz or generalized (super)algebras. These theories intertwine topological, geometric, and algebraic structures, providing frameworks for both quantum field theories and the geometric formulation of supergravity. They exhibit a range of distinctive features: supersymmetry realized via superconnections, topologically massless gauge dynamics, the absence (in some cases) of conventional gravitational or matter multiplets, and the emergence of non-Riemannian geometries in certain constructions.

1. Superalgebraic Foundations and Chern–Simons Construction

Supersymmetric Lorentz Chern–Simons (CS) theories are formulated by promoting the gauge group to a supergroup that extends the Lorentz group (e.g., OSp(2|2), SU(2,1|2)), or, as in teleparallel constructions, by deforming the Poincaré algebra to include input from the cosmological constant and torsion. The action is constructed as a Chern–Simons form for a superconnection $\mathcal{A}$ that packages all dynamical fields—bosonic (e.g., gauge fields, spin connections) and fermionic (e.g., Majorana or Dirac spinors)—as components transforming in the adjoint representation of the superalgebra (Alvarez et al., 2011, Alvarez et al., 2015, Caroca et al., 2021):

$$S_{\text{CS}}[\mathcal{A}] = \kappa\, \mathrm{STr}\left(\mathcal{A} \wedge d\mathcal{A} + \frac{2}{3}\mathcal{A}^3\right)$$

For $d=3$, representative cases include:

• $$\mathcal{A} = A K + \bar{Q}\psi + \psi Q + \omega^a J_a$$ for OSp(2|2),
• $$\mathcal{A} = \omega^a\mathcal{J}_a + A^I\mathcal{T}_I + \cdots$$ for SU(2,1|2) (Alvarez et al., 2015).

Gauge transformations are of the form $$\delta\mathcal{A} = d\Lambda + [\mathcal{A}, \Lambda]$$, allowing gauge parameters (including those generating supersymmetry) to intertwine bosonic and fermionic degrees of freedom. All fields transform in the adjoint representation and supersymmetry is realized off-shell as gauge symmetry; the metric/dreibein can be inert under supersymmetry, eliminating the gravitino (spin-3/2) as in (Alvarez et al., 2011), which sets such models apart from conventional supergravity.

2. Geometric and Dynamical Structure: Torsion and Non-Riemannian Geometry

In teleparallel and certain Lorentz-deformed Chern–Simons supergravities, the geometry is characterized by vanishing Lorentz curvature but non-vanishing torsion (Caroca et al., 2021):

$$R^a = d\omega^a + \frac{1}{2} \varepsilon^a_{\,\,bc} \omega^b \wedge \omega^c = 0,\quad T^a - \frac{1}{\ell} \varepsilon^a_{\,\,bc} e^b \wedge e^c = 0$$

The supersymmetric extension involves "super-torsion" terms:

$$T^a + \frac{1}{2}\bar{\psi} \gamma^a \psi = \frac{1}{\ell}\varepsilon^a_{\,\,bc} e^b \wedge e^c$$

Here, the cosmological constant ($\Lambda \propto -1/\ell^2$) appears explicitly in the algebra and serves as a source for torsion, in contrast to the conventional (Riemannian) geometric picture. The vanishing of curvature places the dynamical emphasis on torsion and its fermionic partners, leading to a non-Riemannian background geometry even in the presence of supersymmetry.

3. Off-Shell Supersymmetry and Mass Spectrum

Supersymmetric Lorentz Chern–Simons theories, formulated as gauge theories for a superconnection, are characterized by off-shell invariance—that is, their invariance under supersymmetry transformations does not require the use of equations of motion (Alvarez et al., 2011). Unlike conventional supergravity, there is no requirement for supersymmetric multiplets to possess degenerate masses:

• The fermion (e.g., Dirac spinor $\psi$) can acquire a mass $m$ via coupling to a background with constant torsion: $m \propto T$.
• The bosonic gauge fields (U(1) or non-Abelian connections, spin connection) enter only through their Chern–Simons terms and thus remain massless; CS actions in 3D do not yield propagating local degrees of freedom for the connection.
• The absence of a dynamical gravitino or gaugino is a direct consequence of the superconnection content and the inertness of the dreibein/vielbein under supersymmetry.

This results in theories where fermion masses and boson masses are not related by supersymmetry, distinct from the multiplet structure of traditional supergravity (Alvarez et al., 2011, Alvarez et al., 2015).

4. Teleparallel Supersymmetric Chern–Simons Gravity

The teleparallel approach deforms the standard Poincaré algebra, introducing a cosmological parameter $\ell$ and modifying (anti)commutation relations:

• $$[J_a, Q_\alpha] = -\frac{1}{2} (\gamma_a)_\alpha^{\,\,\beta} Q_\beta$$
• $$\{Q_\alpha, Q_\beta\} = -(\gamma^a C)_{\alpha\beta} \left(\frac{2}{\ell} J_a + P_a\right)$$
• $[P_a, P_b] = -\frac{2}{\ell}\varepsilon_{abc} P^c$ (Caroca et al., 2021)

The Chern–Simons action constructed for this superalgebra yields a bosonic sector with vanishing curvature and shifted torsion constraints, while the fermionic sector features equations consistent with the superalgebraic deformation. In the $\ell \to \infty$ limit (vanishing cosmological constant), the superalgebra contracts to the standard Poincaré superalgebra, and one recovers the usual (torsionless) Poincaré supergravity.

Supersymmetric teleparallel CS actions generalize straightforwardly to higher $\mathcal{N}$ by extending the superalgebra to accommodate more fermionic charges and internal automorphism generators for so($p$) $\oplus$ so($q$) $\subset$ so($p+q$) symmetry, ensuring nondegenerate invariant bilinear forms in the action (Caroca et al., 2021).

5. Key Features, Extensions, and Applications

Feature	Description	Reference
Supersymmetry	Implemented as gauge symmetry via superconnection; off-shell closure	(Alvarez et al., 2011, Caroca et al., 2021)
Geometry	Non-Riemannian (zero curvature, nonzero torsion/super-torsion)	(Caroca et al., 2021)
Mass spectrum	No requirement for multiplet mass degeneracy; fermion mass via torsion	(Alvarez et al., 2011)
Gravitino	Typically absent due to inert dreibein and projections in superconnection	(Alvarez et al., 2011, Alvarez et al., 2015)
Limiting behavior	Poincaré supergravity recovered for vanishing cosmological constant	(Caroca et al., 2021)
Higher $\mathcal{N}$	Extension through additional supercharges and internal algebra generators	(Caroca et al., 2021)

Supersymmetric Lorentz Chern–Simons theories offer versatile frameworks with direct applications to:

• Topological quantum field theory and invariants in three and higher odd dimensions,
• Non-Riemannian gravity and alternative geometric phase spaces,
• The paper of mass generation via dynamical torsion,
• Supersymmetric models in condensed matter systems (e.g., effective CS theories for graphene and topological insulators (Alvarez et al., 2015)),
• Intrinsic connections to higher-spin superalgebras and extended supersymmetric gauge theories.

6. Distinctions from Supergravity and Outlook

A critical distinction between supersymmetric Lorentz Chern–Simons theories and traditional supergravity is the separation of the local Lorentz/supersymmetry structure (superalgebraic, CS formulation with inert geometry) from the dynamics of spacetime geometry (no propagating metric degrees of freedom in pure CS; torsion, not curvature, is fundamental). Supersymmetry is realized without requiring auxiliary or compensating (gravitino/gaugino) fields, and the spin structure is constrained by the construction of the superconnection.

Future directions involve systematic paper of these models' quantization, coupling to matter, holographic correspondences in higher dimensions, and extensions to more general superalgebraic contexts. The clarity provided by the Chern–Simons gauge perspective and superalgebraic formulation has established these theories as central to the ongoing exploration of topological, geometric, and supersymmetric field theory.