**8. Acknowledgments**

We would like to thank to Friedrich W. Hehl for constructive comments. Moreover, (E.W.M.) acknowledges the support of the SNI and thanks Noelia, Miryam Sophie Naomi and Markus Gérard Erik, for encouragement.

where *ηαβγ* :=

and

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

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

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


*ηαβγ* := *eγ*�*ηαβ*. (71)

2*δ β* [*μ δ γ ν*] ,

*ηβ* <sup>∧</sup> *<sup>η</sup>αβ* <sup>=</sup> *<sup>e</sup>β*�(*<sup>η</sup>* <sup>∧</sup> *<sup>η</sup>αβ*) + *<sup>η</sup>* <sup>∧</sup> *<sup>e</sup>β*�*ηαβ* <sup>≡</sup> <sup>0</sup> (73)

<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) <sup>Φ</sup>*<sup>p</sup>* <sup>∧</sup> <sup>Ψ</sup>*<sup>q</sup>* = (−1)*p*·*<sup>q</sup>* <sup>Ψ</sup>*<sup>q</sup>* <sup>∧</sup> <sup>Φ</sup>*p*, (75) <sup>Φ</sup>*<sup>p</sup>* <sup>∧</sup> <sup>∗</sup>Ψ*<sup>p</sup>* <sup>=</sup> <sup>Ψ</sup>*<sup>p</sup>* <sup>∧</sup> <sup>∗</sup>Φ*p*, (76) *<sup>e</sup>α*� (Φ*<sup>p</sup>* <sup>+</sup> <sup>Ψ</sup>*q*) <sup>=</sup> *<sup>e</sup>α*�Φ*<sup>p</sup>* <sup>+</sup> *<sup>e</sup>α*�Ψ*q*, (77)

*<sup>ϑ</sup><sup>α</sup>* <sup>∧</sup> (*eα*�Φ) = *<sup>p</sup>*<sup>Φ</sup> (79) <sup>∗</sup>(Φ ∧ *ϑα*) = *eα*� <sup>∗</sup>Φ, (80)

*γ* = *γ*. (81)

<sup>2</sup><sup>2</sup> *ηαβγϑ<sup>β</sup>* <sup>∧</sup> *<sup>ϑ</sup>γ*, (82)

*<sup>e</sup>α*�(Φ*<sup>p</sup>* <sup>∧</sup> <sup>Ψ</sup>*q*)=(*eα*�Φ*p*) <sup>∧</sup> <sup>Ψ</sup>*<sup>q</sup>* + (−1)*p*Φ*<sup>p</sup>* <sup>∧</sup> (*eα*�Ψ*q*), (78)

*ρμν* , (72)

*ηαβγϑ<sup>β</sup>* <sup>∧</sup> *<sup>ϑ</sup><sup>γ</sup>* <sup>=</sup> <sup>∗</sup>*ϑα*,

{*η*, *ηα*, *ηαβ*, *ηαβγ*} span a *dual basis* for the algebra of arbitrary p–forms in 3D, where

2

*ηαβ* :<sup>=</sup> *<sup>e</sup>β*�*ηα* <sup>=</sup> *ηαβγϑ<sup>γ</sup>* <sup>=</sup> <sup>∗</sup>(*ϑα* <sup>∧</sup> *ϑβ*),

S-Duality in Topological Supergravity 49

3!,

2*δ γ ν* ,

*δ β μδ γ <sup>ν</sup>* = (−1)*<sup>s</sup>*

*δ αβγ*

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.

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

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.

<sup>2</sup> *ηα* <sup>=</sup> *<sup>ρ</sup>*

*ηα* :<sup>=</sup> *<sup>e</sup>α*�*<sup>η</sup>* <sup>=</sup> <sup>1</sup>

*<sup>η</sup>αβγηαβγ* = (−1)*<sup>s</sup>*

*<sup>η</sup>αβγηαβν* = (−1)*<sup>s</sup>*

*<sup>η</sup>αβγηαμν* = (−1)*<sup>s</sup>*

*<sup>η</sup>αβγηρμν* = (−1)*<sup>s</sup>*

Φ is a *p*–form and Ψ a *q*–form with the spinor indices suppressed:

*R <sup>α</sup>* <sup>=</sup> *<sup>ρ</sup>*
