**A: Variations of Chern–Simons terms**

Gauging the Poincaré group in (2+1) dimensions, local translations and (Lorentz-) rotations give rise to two type of gauge potentials, the coframe *ϑ<sup>γ</sup>* and the dual of Lorentz-connection Γ *<sup>α</sup>*. Then the two Bianchi identities of Riemann-Cartan geometry can be rewritten as

$$DT^{\alpha} \equiv (-1)^{s} \eta^{a\beta} \wedge R^{\star}\_{\beta'} \tag{62}$$

$$DR\_{\mathfrak{a}}^{\star} \equiv 0.\tag{63}$$

In 3D the corresponding Chern–Simons three–forms of gauge type *C* = Tr{*A* ∧ *F*}, are

$$\mathbb{C}\_{\rm T} \coloneqq \frac{1}{2\ell^2} \theta^{\underline{a}} \wedge T\_{\underline{a}} = -\frac{(-1)^s}{\ell^2} \eta^{\underline{a}} \wedge \mathbb{K}\_{\underline{a}\prime}^{\star} \quad \mathbb{C}\_{\rm L} \coloneqq (-1)^s \Gamma^{\star \underline{a}} \wedge \mathbb{R}\_{\underline{a}}^{\star} - \frac{1}{3!} \eta\_{\underline{a}\not\rhd \gamma} \Gamma^{\star \underline{a}} \wedge \Gamma^{\star \underline{b}} \wedge \Gamma^{\star \underline{\gamma}}.\tag{64}$$

and

$$\mathbb{C}\_{\text{TL}} := \frac{1}{\ell} \left( \Gamma^{\star a} \wedge T\_a - \frac{(-1)^s}{2} \eta\_{a\beta\gamma} \Gamma^{\star a} \wedge \Gamma^{\star \beta} \wedge \mathfrak{o}^{\gamma} \right) \,. \tag{65}$$

The variational derivatives of these terms lead us to the following expressions

$$\frac{\delta \mathbf{C}\_{\rm T}}{\delta \theta^{a}} = \frac{1}{\ell^{2}} \, T\_{\rm d} \qquad \frac{\delta \mathbf{C}\_{\rm T}}{\delta \Gamma^{\star a}} = \frac{(-1)^{s}}{\ell^{2}} \, \eta\_{\rm a} \, \tag{66}$$

$$\frac{\delta \mathbb{C}\_{\mathcal{L}}}{\delta \theta^{\mathfrak{a}}} = 0, \qquad \frac{\delta \mathbb{C}\_{\mathcal{L}}}{\delta \Gamma^{\star \mathfrak{a}}} = (-1)^{s} 2 \mathcal{R}\_{\mathfrak{a} \prime}^{\star} \tag{67}$$

$$\frac{\delta \mathcal{C}\_{\rm TL}}{\delta \theta^{a}} = \frac{1}{\ell} R\_{a}^{\star} \qquad \frac{\delta \mathcal{C}\_{\rm TL}}{\delta \Gamma^{\star a}} = \frac{1}{\ell} T\_{a \prime} \tag{68}$$

respectively. Note that these three–forms are uniquely related to the torsion *Tα*, the curvature *R <sup>α</sup>*, and the cosmological term *ηα*, as developed in much more detail in Ref. [21].

## **B: The** *η***–basis for exterior forms in 3D**

The symbol ∧ denotes the exterior product of forms, the symbol � the interior product of a vector with a form and ∗ the Hodge star (or left dual) operator which maps a p–form into a (3 − *p*)–form. It has the property that

$$\Phi^{\*\;\*}\Phi^{(p)} = (-1)^{p(\mathfrak{J}-p)+s}\Phi^{(p)},\tag{69}$$

where *p* is the degree of the form Φ and *s* denotes the number of negative *eigenvalues* of the metric, i.e., the signature of spacetime.

The volume three–form is defined by

$$\eta := \frac{1}{3!} \,\eta\_{a\beta\gamma} \,\theta^{\mu} \wedge \theta^{\beta} \wedge \theta^{\gamma},\tag{70}$$

where *ηαβγ* := | det *gμν*| *�αβγ*, and *�αβγ* is the Levi–Civita symbol. The forms {*η*, *ηα*, *ηαβ*, *ηαβγ*} span a *dual basis* for the algebra of arbitrary p–forms in 3D, where

$$\begin{aligned} \eta\_{\boldsymbol{\alpha}} &:= e\_{\boldsymbol{\alpha}} \rfloor \eta = \frac{1}{2} \eta\_{\boldsymbol{\alpha}\boldsymbol{\beta}\gamma} \theta^{\beta} \wedge \theta^{\gamma} = \,^\*\theta\_{\boldsymbol{\alpha}}, \\ \eta\_{\boldsymbol{\alpha}\boldsymbol{\beta}} &:= e\_{\boldsymbol{\beta}} \rfloor \eta\_{\boldsymbol{\alpha}} = \eta\_{\boldsymbol{\alpha}\boldsymbol{\beta}\gamma} \theta^{\gamma} = \,^\*(\theta\_{\boldsymbol{\alpha}} \wedge \theta\_{\boldsymbol{\beta}}), \\ \eta\_{\boldsymbol{\alpha}\boldsymbol{\beta}\gamma} &:= e\_{\boldsymbol{\gamma}} \rfloor \eta\_{\boldsymbol{\alpha}\boldsymbol{\beta}}. \end{aligned} \tag{71}$$

In 3D, the following relations for the *η*–basis hold:

$$\begin{aligned} \eta^{a\beta\gamma}\eta\_{a\beta\gamma} &= (-1)^{s} 3! \\ \eta^{a\beta\gamma}\eta\_{a\beta\nu} &= (-1)^{s} 2\delta\_{\nu}^{\gamma} \\ \eta^{a\beta\gamma}\eta\_{a\mu\nu} &= (-1)^{s} \delta\_{\mu}^{\beta}\delta\_{\nu}^{\gamma} = (-1)^{s} 2\delta\_{[\mu}^{\beta}\delta\_{\nu]}^{\gamma} \\ \eta^{a\beta\gamma}\eta\_{\rho\mu\nu} &= (-1)^{s} \delta\_{\rho\mu\nu}^{a\beta\gamma} \end{aligned} \tag{72}$$

and

12 Will-be-set-by-IN-TECH

Gauging the Poincaré group in (2+1) dimensions, local translations and (Lorentz-) rotations give rise to two type of gauge potentials, the coframe *ϑ<sup>γ</sup>* and the dual of Lorentz-connection

*DT<sup>α</sup>* <sup>≡</sup> (−1)*<sup>s</sup> <sup>η</sup>αβ* <sup>∧</sup> *<sup>R</sup>*

*<sup>α</sup>*, *<sup>C</sup>*<sup>L</sup> := (−1)*<sup>s</sup>* <sup>Γ</sup>*<sup>α</sup>* <sup>∧</sup> *<sup>R</sup>*

*δC*<sup>T</sup>

*<sup>α</sup>* , *<sup>δ</sup>C*TL

respectively. Note that these three–forms are uniquely related to the torsion *Tα*, the curvature

The symbol ∧ denotes the exterior product of forms, the symbol � the interior product of a vector with a form and ∗ the Hodge star (or left dual) operator which maps a p–form into a

where *p* is the degree of the form Φ and *s* denotes the number of negative *eigenvalues* of the

∗ ∗Φ(*p*) = (−1)*p*(3−*p*)+*<sup>s</sup>*

*<sup>η</sup>* :<sup>=</sup> <sup>1</sup>

*<sup>δ</sup>*<sup>Γ</sup>*<sup>α</sup>* <sup>=</sup> (−1)*<sup>s</sup>*

*<sup>δ</sup>*<sup>Γ</sup>*<sup>α</sup>* = (−1)*<sup>s</sup>* <sup>2</sup>*<sup>R</sup>*

*<sup>δ</sup>*<sup>Γ</sup>*<sup>α</sup>* <sup>=</sup> <sup>1</sup> 

Φ(*p*)

3! *ηαβγ <sup>ϑ</sup><sup>α</sup>* <sup>∧</sup> *<sup>ϑ</sup><sup>β</sup>* <sup>∧</sup> *<sup>ϑ</sup>γ*, (70)

*DR*

In 3D the corresponding Chern–Simons three–forms of gauge type *C* = Tr{*A* ∧ *F*}, are

<sup>Γ</sup>*<sup>α</sup>* <sup>∧</sup> *<sup>T</sup><sup>α</sup>* <sup>−</sup> (−1)*<sup>s</sup>*

The variational derivatives of these terms lead us to the following expressions

<sup>2</sup> *<sup>T</sup><sup>α</sup>*

*δϑ<sup>α</sup>* <sup>=</sup> 0 , *<sup>δ</sup>C*<sup>L</sup>

*<sup>α</sup>*, and the cosmological term *ηα*, as developed in much more detail in Ref. [21].

<sup>2</sup> *<sup>η</sup><sup>α</sup>* <sup>∧</sup> *<sup>K</sup>*

*δC*<sup>T</sup> *δϑ<sup>α</sup>* <sup>=</sup> <sup>1</sup>

*δC*<sup>L</sup>

*δC*TL *δϑ<sup>α</sup>* <sup>=</sup> <sup>1</sup> *R*

*<sup>C</sup>*TL :<sup>=</sup> <sup>1</sup>  *<sup>β</sup>* , (62)

3! *ηαβγ* <sup>Γ</sup>*<sup>α</sup>* <sup>∧</sup> <sup>Γ</sup>*<sup>β</sup>* <sup>∧</sup> <sup>Γ</sup>*<sup>γ</sup>*. (64)

. (65)

*<sup>α</sup>* ≡ 0. (63)

<sup>2</sup> *ηα* , (66)

*<sup>α</sup>*, (67)

*Tα*, (68)

, (69)

*<sup>α</sup>* <sup>−</sup> <sup>1</sup>

<sup>2</sup> *ηαβγ* <sup>Γ</sup>*<sup>α</sup>* <sup>∧</sup> <sup>Γ</sup>*<sup>β</sup>* <sup>∧</sup> *<sup>ϑ</sup><sup>γ</sup>*

*<sup>α</sup>*. Then the two Bianchi identities of Riemann-Cartan geometry can be rewritten as

**9. Appendices**

Γ

*<sup>C</sup>*<sup>T</sup> :<sup>=</sup> <sup>1</sup>

and

*R*

**A: Variations of Chern–Simons terms**

<sup>2</sup><sup>2</sup> *<sup>ϑ</sup><sup>α</sup>* <sup>∧</sup> *<sup>T</sup><sup>α</sup>* <sup>=</sup> <sup>−</sup>(−1)*<sup>s</sup>*

**B: The** *η***–basis for exterior forms in 3D**

(3 − *p*)–form. It has the property that

metric, i.e., the signature of spacetime. The volume three–form is defined by

$$
\eta\_{\beta} \wedge \eta^{a\beta} = e\_{\beta} \rfloor (\eta \wedge \eta^{a\beta}) + \eta \wedge e\_{\beta} \rfloor \eta^{a\beta} \equiv 0 \tag{73}
$$

due to the antisymmetry of *<sup>η</sup>αβ* and the fact that *<sup>η</sup>* <sup>∧</sup> *<sup>η</sup>αβ* would already be a four-form in 3D.

### **C: Identities for spinor–valued forms**

Now some relations of special importance are presented which take care of the order of the forms in the exterior products and its Dirac adjoint: We would like to remind the reader that Φ is a *p*–form and Ψ a *q*–form with the spinor indices suppressed:

<sup>Φ</sup>*<sup>p</sup>* <sup>∧</sup> <sup>Ψ</sup>*<sup>q</sup>* = (−1)*p*·*q*Ψ*<sup>q</sup>* <sup>∧</sup> <sup>Φ</sup>*p*, (74)

$$
\overline{\Phi^p \wedge \Psi^q} = (-1)^{p \cdot q} \overline{\Psi^q} \wedge \overline{\Phi^p} \, \, \, \tag{75}
$$

$$
\!\!\!\!\Phi^p \wedge \!\!\!^\*\Psi^p = \Psi^p \wedge \!\!\!^\*\Phi^p \,\!\!\Phi^p \,\!\!\tag{76}
$$

$$e\_{\mathfrak{a}}\rfloor\left(\Phi^{p} + \Psi^{q}\right) = e\_{\mathfrak{a}}\rfloor\Phi^{p} + e\_{\mathfrak{a}}\rfloor\Psi^{q} \,. \tag{77}$$

$$\left(\varepsilon\_{\mathfrak{a}}\right)\left(\Phi^{p}\wedge\Psi^{q}\right) = \left(\varepsilon\_{\mathfrak{a}}\left|\Phi^{p}\right>\wedge\Psi^{q} + (-1)^{p}\Phi^{p}\wedge\left(\varepsilon\_{\mathfrak{a}}\left|\Psi^{q}\right>\right),\tag{78}$$

$$
\theta^{\mathfrak{A}} \wedge (e\_{\mathfrak{A}} \rfloor \Phi) = p\Phi
\tag{79}
$$

$$\mathfrak{l}^\*(\Phi \wedge \theta\_\mathfrak{a}) = e\_\mathfrak{a} \lrcorner \mathfrak{d}\_\prime \tag{80}$$

$$
\overline{\gamma} = \gamma.\tag{81}
$$

### **D: No axial torsion restrictions in 3D**

Spaces of constant curvature deserve special attention in General Relativity, in particular in the cosmological context. In particular, when the RC curvature is constant as in Eq. (10), i.e.

$$R\_{\alpha}^{\star} = \frac{\rho}{\ell^2} \eta\_{\alpha} = \frac{\rho}{2\ell^2} \eta\_{\alpha\beta\gamma} \theta^{\beta} \wedge \theta^{\gamma} \,. \tag{82}$$

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the Bianchi identities (62) and (63) could lead to constraints on the admissible torsion *Tα*, as in 4D and higher dimensions. However, in 3D the situation is different: Using Appendix B, the first Bianchi identity yields

$$\begin{split} (-1)^{s} \eta^{a\beta} \wedge \mathcal{R}\_{\beta}^{\*} &= (-1)^{s} \frac{\rho}{\ell^{2}} \eta\_{\beta} \wedge \eta^{a\beta} = (-1)^{s} \frac{\rho}{2\ell^{2}} \left( \eta\_{\beta\mu\upsilon} \eta^{a\beta\gamma} \right) \theta\_{\gamma} \wedge \theta^{\mu} \wedge \mathcal{\theta}^{\upsilon} \\ &= -(-1)^{s} 2 \delta\_{[\mu}^{\mu} \delta\_{\upsilon]}^{\gamma} \theta\_{\gamma} \wedge \theta^{\mu} \wedge \mathcal{\theta}^{\upsilon} = 0. \end{split} \tag{83}$$

Furthermore, the exterior covariant derivative of Eq. (10) provides the identity

$$DT\_{\mathfrak{a}} = \frac{2\kappa}{\ell} D\eta\_{\mathfrak{a}} = \frac{2\kappa}{\ell} T^{\mathfrak{f}} \wedge \eta\_{a\mathfrak{f}} = \frac{4\kappa^2}{\ell^2} \eta\_{a\mathfrak{f}} \wedge \eta^{\mathfrak{f}} \equiv 0. \tag{84}$$

Thus the first Bianchi identity does not give any further information. The second Bianchi identity (63) yields

$$DR\_{\mathfrak{a}}^{\star} = \frac{\rho}{\ell^2} D\eta\_{\mathfrak{a}} = \frac{2\kappa\rho}{\ell^3} \eta\_{a\mathfrak{f}} \wedge \eta^{\mathfrak{f}} \equiv 0 \tag{85}$$

which is identically zero by a similar argument, or by employing Eq. (73). Consequently, the Bianchi identities impose *no* restrictions on the axial torsion given by (10) in 3D, a fact which has allowed us to construct something non-trivial from the MB model.
