**2. Topological gravity with torsion**

In three spacetime dimensions, the basic gravitational variables in the Riemann-Cartan (RC) formalisms are the one–forms of the coframe and the Lie dual of the (Lorentz-) rotational connection Γ*βγ* = Γ*<sup>j</sup> βγdx<sup>j</sup>* , i.e.,

$$
\theta^a = e\_i{}^a d\mathbf{x}^i \quad \text{and} \quad \Gamma\_a^\star := \frac{1}{2} \eta\_{a\beta\gamma} \Gamma^{\beta\gamma}. \tag{1}
$$

The related field strengths are the two–forms of torsion

$$T^{\mathfrak{a}} := d\theta^{\mathfrak{a}} - (-1)^{s} \eta^{a\beta} \wedge \Gamma^{\star}\_{\beta} \tag{2}$$

and curvature

$$R\_a^\star = \frac{1}{2} \eta\_{a\beta\gamma} R^{\beta\gamma} := d\Gamma\_a^\star + \frac{(-1)^s}{2} \eta\_{a\beta\gamma} \Gamma\_\beta^\star \wedge \Gamma\_{\gamma\prime}^\star \tag{3}$$

respectively, cf. the Appendices of Ref. [31]. Table 1 contains a summary of these basic variables and their components in various dimensions. Observe that only for *n* = 3 all fields have the same number of components. After converting bivectors into vectors via the Lie dual, a linear combination of all variables could pave the way for a better understanding of topological models.

In 3D, the Einstein-Cartan (EC) Lagrangian

$$L\_{\rm EC} := -\frac{\chi}{\ell} \theta^a \wedge \mathsf{R}\_a^\star \equiv -\chi \mathsf{C}\_{\rm TL} - \frac{\chi}{\ell} \, d(\Gamma\_a^\star \wedge \theta^a) \tag{4}$$

merely gives rise to a locally trivial dynamics [38]. This is due to its equivalence to a 'mixed' Chern-Simons type term *C*TL plus a total divergence, as indicated above.

In this paper, we generalize this trivial dynamics by adding Chern-Simons (CS) type terms, following the lead of Witten [43]. By gauging the Poincaré group IR3 ⊂× *SO*(1, 2), we arrive at the Mielke and Baekler (MB) model [2, 28] which is at most *linear* in the field strengths. This is slightly modified here by replacing *L*EC via the 'mixed' Chern-Simons type term *C*TL, which is simulating, in 3D, to some extend Einstein's theory with 'cosmological' term, as is indicated above. Thereby, we are able to depart from a completely topological theory.

Allowing for arbitrary "vacuum angles" *θ*T, *θ*<sup>L</sup> and *θ*TL = −*χ*, the most general purely *topological* gravity Lagrangian in 3D, in first order formalism, takes the form

$$L\_{\rm MB}(\theta^{\mathfrak{a}}, \Gamma\_{\mathfrak{a}}^{\star}) = \theta\_{\rm T} \mathbb{C}\_{\rm T} + \theta\_{\rm L} \mathbb{C}\_{\rm L} + \theta\_{\rm TL} \mathbb{C}\_{\rm TL} \,\tag{5}$$

where

2 Will-be-set-by-IN-TECH

*ϑ<sup>α</sup>* vector 1 *n*<sup>2</sup> 16 9 1

flexural modes of corrugated surfaces (2D membranes) embedded in 3D spacetime, as recently

In three spacetime dimensions, the basic gravitational variables in the Riemann-Cartan (RC) formalisms are the one–forms of the coframe and the Lie dual of the (Lorentz-) rotational

*<sup>T</sup><sup>α</sup>* :<sup>=</sup> *<sup>d</sup>ϑ<sup>α</sup>* <sup>−</sup> (−1)*<sup>s</sup> <sup>η</sup>αβ* <sup>∧</sup> <sup>Γ</sup>

respectively, cf. the Appendices of Ref. [31]. Table 1 contains a summary of these basic variables and their components in various dimensions. Observe that only for *n* = 3 all fields have the same number of components. After converting bivectors into vectors via the Lie dual, a linear combination of all variables could pave the way for a better understanding of

*<sup>α</sup>* :<sup>=</sup> <sup>1</sup> 2

*<sup>α</sup>* <sup>+</sup> (−1)*<sup>s</sup>*

*<sup>α</sup>* ≡ −*<sup>χ</sup> <sup>C</sup>*TL <sup>−</sup> *<sup>χ</sup>*

merely gives rise to a locally trivial dynamics [38]. This is due to its equivalence to a 'mixed'

In this paper, we generalize this trivial dynamics by adding Chern-Simons (CS) type terms, following the lead of Witten [43]. By gauging the Poincaré group IR3 ⊂× *SO*(1, 2), we arrive at the Mielke and Baekler (MB) model [2, 28] which is at most *linear* in the field strengths. This is slightly modified here by replacing *L*EC via the 'mixed' Chern-Simons type term *C*TL, which is simulating, in 3D, to some extend Einstein's theory with 'cosmological' term, as is indicated

<sup>2</sup> *ηαβγ*<sup>Γ</sup>

*<sup>d</sup>*(<sup>Γ</sup>

*<sup>β</sup>* <sup>∧</sup> <sup>Γ</sup>

*ηαβγ*Γ*βγ*. (1)

*<sup>β</sup>* (2)

*<sup>γ</sup>*, (3)

*<sup>α</sup>* <sup>∧</sup> *<sup>ϑ</sup>α*) (4)

*<sup>α</sup>dx<sup>i</sup>* and Γ

*ηαβγRβγ* := *<sup>d</sup>*<sup>Γ</sup>

*<sup>α</sup>* vector 1 *<sup>n</sup>*<sup>2</sup> 16 9 1 *<sup>T</sup><sup>α</sup>* vector 2 *<sup>n</sup>*2(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)/2 24 9 2 *<sup>R</sup>αβ* bivector 2 *<sup>n</sup>*2(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)2/4 36 9 1 <sup>Σ</sup>*<sup>α</sup>* vector *<sup>n</sup>* <sup>−</sup> <sup>1</sup> *<sup>n</sup>*<sup>2</sup> 16 9 4 *ταβ* bivector *<sup>n</sup>* <sup>−</sup> <sup>1</sup> *<sup>n</sup>*2(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)/2 24 9 2 *ηα* vector *<sup>n</sup>* <sup>−</sup> <sup>1</sup> *<sup>n</sup>*<sup>2</sup> 16 9 4

Γ

Table 1. Geometrical objects and fields

**2. Topological gravity with torsion**

*βγdx<sup>j</sup>*

, i.e.,

The related field strengths are the two–forms of torsion

*R <sup>α</sup>* <sup>=</sup> <sup>1</sup> 2

In 3D, the Einstein-Cartan (EC) Lagrangian

*<sup>L</sup>*EC :<sup>=</sup> <sup>−</sup>*<sup>χ</sup>*

Chern-Simons type term *C*TL plus a total divergence, as indicated above.

above. Thereby, we are able to depart from a completely topological theory.

*<sup>ϑ</sup><sup>α</sup>* <sup>∧</sup> *<sup>R</sup>*

connection Γ*βγ* = Γ*<sup>j</sup>*

and curvature

topological models.

realized by the rather prospective new material called *graphene*.

*ϑ<sup>α</sup>* = *ei*

objects *p*-forms components n=4 3 2

$$\mathbb{C}\_{\Gamma} := \frac{1}{2\ell^2} \theta^{\mathfrak{a}} \wedge T\_{\mathfrak{a}\prime} \qquad \mathbb{C}\_{\mathcal{L}} := (-1)^s \, \Gamma^{\star \mathfrak{a}} \wedge \mathbb{R}\_{\mathfrak{a}}^{\star} - \frac{1}{3!} \, \eta\_{\mathfrak{a}\beta\gamma} \, \Gamma^{\star \mathfrak{a}} \wedge \Gamma^{\star \mathfrak{f}} \wedge \Gamma^{\star \gamma} \tag{6}$$

and

$$\mathbb{C}\_{\text{TL}} := \frac{1}{\ell} \left( \Gamma^{\star \mathfrak{a}} \wedge T\_{\mathfrak{a}} - \frac{(-1)^{s}}{2} \eta\_{a\beta\gamma} \Gamma^{\star \mathfrak{a}} \wedge \Gamma^{\star \mathfrak{b}} \wedge \mathfrak{o}^{\gamma} \right), \tag{7}$$

respectively, are the translational, rotational and 'mixed' Chern-Simons type three forms of gauge type *C* =Tr{*A* ∧ *F*} in RC spacetime [8, 12, 43]. The equation (5) is the known topological Lagrangian of the Mielke-Baekler (MB) *mix*-model [28, 31]. Since the translational term *C*<sup>T</sup> is covariant, it appears that the MB model is semi-topological, with interesting consequence on the degrees of propagating modes, cf. Ref. [4, 32, 36].

Varying the Lagrangian (5) with respects to *ϑ<sup>α</sup>* and Γ*<sup>α</sup>* and employing the results of Appendix A, yields the topological field equations

$$-\theta\_{\rm TL} \, R\_{\alpha}^{\star} - \frac{\theta\_{\rm T}}{\ell} \, T\_{\alpha} = \ell \, \Sigma\_{\alpha} \, \tag{8}$$

and

$$-\left(-1\right)^{s} \theta\_{\rm TL} \, T\_{\rm a} - \frac{\theta\_{\rm T}}{2\ell} \eta\_{\rm a} - \theta\_{\rm L} \ell \, R\_{\rm a}^{\star} = \ell \, \tau\_{\rm a}^{\star} \,. \tag{9}$$

cf. Eq. (6.9) of Ref. [2]. Observe that the translational CS term proportional to *θ*<sup>T</sup> induces in the second field equation a constant term, familiar in 4D from Einstein's equation with cosmological constant Λ.

Thereby, combining the *vacuum* field equations (9) and (8) yield for the torsion and the RC curvature the constrictions:

$$T\_{\mathfrak{a}} = \frac{2\kappa}{\ell} \eta\_{\mathfrak{a}} \,, \qquad R\_{\mathfrak{a}}^{\star} = \frac{\rho}{\ell^2} \eta\_{\mathfrak{a}} \,\tag{10}$$

where the contortional constants *<sup>κ</sup>* <sup>=</sup> *<sup>θ</sup>*TL*θ*T/2*<sup>A</sup>* and *<sup>ρ</sup>* <sup>=</sup> <sup>−</sup>*θ*<sup>2</sup> <sup>T</sup>/*A* are related to the vacuum angles. A singular case is exclude by assuming that *<sup>A</sup>* <sup>=</sup>: <sup>−</sup>(−1)*sθ*<sup>2</sup> TL + 2*θ*T*θ*<sup>L</sup> �= 0.

When including matter couplings, we explicitly find for the torsion

$$T\_{\mathfrak{a}} - \frac{2\kappa}{\ell} \eta\_{\mathfrak{a}} = \frac{2}{A} \ell \left( \theta\_{\text{TL}} \tau\_{\mathfrak{a}}^{\star} - \theta\_{\text{L}} \ell \Sigma\_{\mathfrak{a}} \right) \,, \tag{11}$$

and the RC curvature

$$R^\star\_{\mathfrak{a}} - \frac{\rho}{\ell^2} \eta\_{\mathfrak{a}} = \frac{2}{A} \left( \theta\_{\text{TL}} \ell \Sigma\_{\mathfrak{a}} - \theta\_{\text{T}} \tau^\star\_{\mathfrak{a}} \right) . \tag{12}$$

cf. Ref. [31].
