**4. Topological supersymmetry in 3D**

Let us consider the first order topological Lagrangian

$$L\_{\infty} = L\_{\infty}(\boldsymbol{\theta}^{\boldsymbol{\mu}}, \boldsymbol{\Gamma}\_{\boldsymbol{\mu}\prime}^{\boldsymbol{\star}} \boldsymbol{\Psi}) = L\_{\text{MB}} + L\_{\boldsymbol{\Psi}} \tag{30}$$

and verify if it is supersymmetric or not: The variation of its independent variables (*ϑα*, Γ *<sup>α</sup>*, Ψ) yields

$$
\delta L = \delta \theta^{\mathfrak{a}} \wedge \frac{\delta L}{\delta \theta^{\mathfrak{a}}} + \delta \Gamma\_{\mathfrak{a}}^{\star} \wedge \frac{\delta L}{\delta \Gamma\_{\mathfrak{a}}^{\star}} + \delta \overline{\Psi} \wedge \frac{\delta L}{\delta \overline{\Psi}} \tag{31}
$$

where, for convenience, it suffices to vary only for the Dirac adjoint Ψ.

The supersymmetric transformation of Deser [13, 14] read in exterior form notation

$$\begin{aligned} \delta\_{\text{susy}} \theta^{\mathfrak{a}} &= i \overline{\sigma}^{\mathfrak{a}} \Psi \gamma^{\mathfrak{a}}, & \delta\_{\text{susy}} \Gamma^{\star}\_{\mathfrak{a}} &= i \overline{\sigma} \, \gamma^{\ast}\_{\mathfrak{a}} D \Psi + i c \overline{\sigma} \, (\gamma\_{\mathfrak{a}} \Psi + e\_{\mathfrak{a}} \rfloor^{\ast} \Psi), \\ \delta\_{\text{susy}} \Psi &= 2 D \sigma + c \gamma \sigma, & \tag{32} \end{aligned} \tag{32}$$

where *σ* stands in for a spinor valued zero form and *c* a real constant. Inserting this into Eq. (31) yields

$$\delta\_{\rm susy} L = i\overline{\sigma} \, \Psi \gamma^{\mu} \wedge \frac{\delta L}{\delta \theta^{\mu}} + \delta\_{\rm susy} \Gamma\_{\alpha}^{\star} \wedge \frac{\delta L}{\delta \Gamma\_{\alpha}^{\star}} + \{2\overline{D\sigma} + c\overline{\sigma}\gamma\} \wedge \frac{\delta L}{\delta \overline{\Psi}^{\prime}} \tag{33}$$

where we used *cγσ* = *cσγ* for the Dirac adjoint.

In the following, we assume that the second field equation *δL*/*δ*Γ *<sup>α</sup>* ∼= 0 is fulfilled "on shell", i.e., Eq. (9) of the 'mixed' MB model. Then, the SUSY transformation reduce to

$$\delta\_{\rm susy} L \cong \overline{\sigma} \left( \mathrm{i} \gamma^{\mu} \Psi \wedge \frac{\delta L}{\delta \theta^{\alpha}} - 2D \wedge \frac{\delta L}{\delta \overline{\Psi}} + c \gamma \wedge \frac{\delta L}{\delta \overline{\Psi}} \right) + 2d \left( \overline{\sigma} \wedge \frac{\delta L}{\delta \overline{\Psi}} \right) \tag{34}$$

6 Will-be-set-by-IN-TECH

It should be noted that for the pure Rarita-Schwinger Lagrangian with *s*<sup>1</sup> = *s*<sup>2</sup> = 0, the

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

and verify if it is supersymmetric or not: The variation of its independent variables (*ϑα*, Γ

*α* ∧ *δL δ*Γ *α*

*<sup>α</sup>* = *iσ γ*<sup>∗</sup>

where *σ* stands in for a spinor valued zero form and *c* a real constant. Inserting this into Eq.

*α* ∧ *δL δ*Γ *α* +

*δL δ*Ψ

+ *cγ* ∧

*δL δ*Ψ + 2*d σ* ∧ *δL δ*Ψ 

*δ*susyΨ = 2*Dσ* + *cγσ*, (32)

+ *δ*Ψ ∧

*δL*

*<sup>α</sup>D*Ψ + *icσ* (*γα*Ψ + *eα*�∗Ψ),

2*Dσ* + *cσγ*

∧ *δL*

*L*<sup>∞</sup> = *L*∞(*ϑα*, Γ

*δL δϑ<sup>α</sup>* <sup>+</sup> *<sup>δ</sup>*<sup>Γ</sup>

The supersymmetric transformation of Deser [13, 14] read in exterior form notation

*δϑ<sup>α</sup>* <sup>+</sup> *<sup>δ</sup>*susy<sup>Γ</sup>

i.e., Eq. (9) of the 'mixed' MB model. Then, the SUSY transformation reduce to

Ψ*γα* ∧ *γ* ∧ <sup>∗</sup>(*γ* ∧ *D*Ψ) + *γα*Ψ ∧ *γ* ∧ <sup>∗</sup> (*γ* ∧ *D*Ψ)

Ψ ∧ *γα*Ψ + *s*<sup>1</sup> Ψ*γα* ∧ <sup>∗</sup>(*D*Ψ) + *s*<sup>2</sup> Ψ*γα* ∧ *γ* ∧ <sup>∗</sup>(*γ* ∧ *D*Ψ)

2*mτ*

<sup>Ψ</sup>*γα* <sup>∧</sup> <sup>∗</sup>(*D*Ψ) + *γα*<sup>Ψ</sup> <sup>∧</sup> <sup>∗</sup>

*D*Ψ 

*<sup>α</sup>* . (29)

*<sup>δ</sup>*<sup>Ψ</sup> (31)

*δ*Ψ, (33)

(34)

*<sup>α</sup>* ∼= 0 is fulfilled "on shell",

*<sup>α</sup>*, Ψ) = *L*MB + *L*<sup>Ψ</sup> (30)

. (27)

, (28)

*<sup>α</sup>*, Ψ)

In view of the definition (21) of the covariant derivative, we find

Ψ*γα*Ψ +

*s*1 2 

Using the Hermetian properties of the spinor-valued *p*–forms, we finally obtain

 *i* 4

energy-momentum current is proportional to its dual spin, i.e.

*<sup>δ</sup><sup>L</sup>* <sup>=</sup> *δϑ<sup>α</sup>* <sup>∧</sup>

*δ*susy*ϑ<sup>α</sup>* = *iσ* Ψ*γα*, *δ*susyΓ

*<sup>δ</sup>*susy*<sup>L</sup>* <sup>=</sup> *<sup>i</sup><sup>σ</sup>* <sup>Ψ</sup>*γ<sup>α</sup>* <sup>∧</sup>

*<sup>i</sup>γα*<sup>Ψ</sup> <sup>∧</sup>

where we used *cγσ* = *cσγ* for the Dirac adjoint.

*δ*susy*L* ∼= *σ*

where, for convenience, it suffices to vary only for the Dirac adjoint Ψ.

*δL*

*δL δϑ<sup>α</sup>* <sup>−</sup> <sup>2</sup>*<sup>D</sup>* <sup>∧</sup>

In the following, we assume that the second field equation *δL*/*δ*Γ

*τ*

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

cf. the identities of Appendix C.

**4. Topological supersymmetry in 3D**

Let us consider the first order topological Lagrangian

*τ*

yields

(31) yields

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

> + *s*2 2

> > *i* 4

Let us restrict for the moment to the usual Rarita-Schwinger Lagrangian *L*RS, or equivalently to *L*<sup>Ψ</sup> with *s*<sup>1</sup> = *s*<sup>2</sup> = 0. Then the Rarita-Schwinger equation

$$\frac{2}{i}\frac{\delta L}{\delta \Psi} = D\Psi + \frac{1}{2}m\gamma \wedge \Psi \cong 0\tag{35}$$

becomes massive. Moreover, in Eq. (34) the term in brackets following form the supersymmetric transformations reads

$$\begin{split} i\gamma^{a}\Psi \wedge \frac{\delta L}{\delta\theta^{a}} + c\gamma \wedge \frac{\delta L}{\delta\overline{\Psi}} - 2D \frac{\delta L}{\delta\overline{\Psi}} \\ \cong i\gamma^{a}\Psi \left( \frac{\theta\_{\rm TL}}{\ell} R\_{a}^{\star} + \frac{\theta\_{\rm T}}{\ell^{2}} T\_{a} + \Sigma\_{a} \right) + c\gamma \wedge \left( \frac{i}{2} D\Psi + \frac{i}{4} m\gamma \wedge \Psi \right) \\ -D \left( iD\Psi + \frac{i}{2} m\gamma \wedge \Psi \right) \\ = i\gamma^{a}\Psi \left( \frac{\theta\_{\rm TL}}{\ell} R\_{a}^{\star} + \frac{\theta\_{\rm T}}{\ell^{2}} T\_{a} \right) + \gamma^{a}\Psi \left( \frac{i}{4} m\overline{\Psi}\gamma\_{a}\Psi \right) \\ + c\gamma \wedge \left( \frac{i}{2} D\Psi + \frac{i}{4} m\gamma \wedge \Psi \right) - iR\_{a}^{\star}\gamma^{a}\Psi - \frac{i}{2} mT\_{a}\gamma^{a}\Psi + \frac{i}{2} m\gamma \wedge D\Psi \end{split} (36)$$

By a Fierz rearrangement, i.e.,

$$
\gamma^a \Psi \wedge \overline{\Psi} \gamma\_a \Psi = 0,\tag{37}
$$

terms arising from the energy-momentum current Σ*α*, or likewise from the dual spin *τ <sup>α</sup>* , are vanishing.

Moreover, in our restricted model with *s*<sup>1</sup> = *s*<sup>2</sup> = 0 we have to put

$$
\mathcal{L} = -\mathfrak{m},
\tag{38}
$$

in order to eliminate the kinetic *γ* ∧ *D*Ψ terms. Then, using the formula

$$
\gamma \wedge \gamma = -2\gamma^{\mu}\eta\_{\mu\nu} \tag{39}
$$

of Howe and Tucker [23], we find from Eq. (36) the requirement

$$i\left[\left(\frac{\theta\_{\rm TL}}{\ell}-1\right)R\_a^\star + \left(\frac{\theta\_{\rm T}}{\ell^2} - \frac{m}{2}\right)T\_a + \frac{m^2}{2}\eta\_a\right]\gamma^a\Psi = 0,\tag{40}$$

in order that our Lagrangian becomes supersymmetric.

At first sight, it appears that there is no cosmological constant in order to compensate a similar one arising from the RS mass. However, one should compare the bracket with the second field equation (9) *inserted*, which indeed involves a cosmological term induced by the translational Chern-Simons term proportional to *θ*T. In this insertion

$$i\left[\left(\theta\_{\rm L} + \frac{\theta\_{\rm TL}}{\ell} - 1\right)R\_a^\star + \left((-1)^s \frac{\theta\_{\rm TL}}{\ell} + \frac{\theta\_{\rm T}}{\ell^2} - \frac{m}{2}\right)T\_a + \frac{1}{2}\left(\frac{\theta\_{\rm T}}{\ell^2} + m^2\right)\eta\_a + \tau\_a^\star\right]\gamma^a\Psi = 0,\tag{41}$$

where *σ* is again a spinor valued zero-form and a fundamental length.

or, after separating the covariant two-forms of torsion and curvature,

*ηαβ* <sup>∧</sup> <sup>Γ</sup>*<sup>β</sup>* = (−1)*<sup>s</sup>*

using also the fundamental relation (18) for a Clifford algebra, we obtain

Now we can eliminate torsion and RC curvature via (11) and (12) with the result

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

2

<sup>=</sup> <sup>−</sup>2*ηαβ* <sup>∧</sup> *σ γβ*<sup>Ψ</sup> <sup>−</sup> <sup>1</sup>

*<sup>B</sup>* <sup>=</sup> *<sup>θ</sup>*<sup>T</sup> + (−1)*<sup>s</sup>*

*<sup>d</sup>ϑα* = (−1)*<sup>s</sup>*

*d*Γ

S-Duality in Topological Supergravity 45

*R <sup>α</sup>* <sup>−</sup> 2

Let us reconstitute our Ansatz (47) in order to replace all the connection terms Γ*<sup>β</sup>* . Then,

*<sup>α</sup>* + *d*(*σγα*Ψ), (48)

*ηαβγ*<sup>Γ</sup>*<sup>β</sup>* <sup>∧</sup> <sup>Γ</sup>*<sup>γ</sup>* <sup>+</sup> *<sup>d</sup>*(*<sup>σ</sup>* <sup>∧</sup> *γα*Ψ) (49)

*ηαβ* <sup>∧</sup> *σ γβ*<sup>Ψ</sup> (50)

<sup>2</sup> + (<sup>3</sup> <sup>+</sup> <sup>2</sup>*<sup>κ</sup>* <sup>−</sup> (−1)*<sup>s</sup>*

*ηαβμσγ<sup>β</sup>*<sup>Ψ</sup> <sup>∧</sup> *σγμ*<sup>Ψ</sup> <sup>+</sup> *<sup>d</sup>*(*σγα*Ψ). (51)

*ηαβμσγ<sup>β</sup>*<sup>Ψ</sup> <sup>∧</sup> *σγμ*<sup>Ψ</sup> <sup>+</sup> *<sup>D</sup>*(*σγα*Ψ) (52)

*ρ*)*ηα*

*θ*TL] (53)

*θ*T]. (54)

*ηαβγ*(*ϑ<sup>β</sup>* <sup>−</sup> *σγβ*Ψ) <sup>∧</sup> (*ϑ<sup>γ</sup>* <sup>−</sup> *σγγ*Ψ) + *<sup>d</sup>*(*σγα*Ψ).

*θ*TL)Σ*<sup>α</sup>*

2

*<sup>C</sup>* <sup>=</sup> <sup>3</sup>*<sup>A</sup>* <sup>+</sup> *<sup>θ</sup>*T[*θ*TL + (−1)*<sup>s</sup>*

In the case of vanishing *B* and *C* and in view of the massive Rarita-Schwinger equation (35), there remains a first order nonlinear differential equation for *σ* coupled to RS fields to be

As an example of a spacetime with torsion and/or curvature *defects* [9] or singularities, let us consider a *a planar graphene* solution within the 'mixed' MB model governed by the two

*<sup>θ</sup>*TL <sup>+</sup> <sup>2</sup>*<sup>m</sup>*[*θ*<sup>L</sup> + (−1)*<sup>s</sup>*

By exterior differentiation, we find

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

*T<sup>α</sup>* + 2 *ηα* + 1 

Together with (29), this leads to

*B*

as a condition for S-duality, where

**6. Membranes with torsion defects**

Einstein-Cartan type field equations (11) and (12).

and

satisfied.

2 *A*  = (−1)*<sup>s</sup>*

(*θ*TL + (−1)*<sup>s</sup>*

<sup>4</sup>*<sup>A</sup> <sup>i</sup>*<sup>2</sup>Ψ*γα*<sup>Ψ</sup> <sup>+</sup>

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

*θ*T)*τ*

<sup>=</sup> <sup>−</sup>2*ηαβ* <sup>∧</sup> *σ γβ*<sup>Ψ</sup> <sup>−</sup> <sup>1</sup>

*C <sup>A</sup> ηα*

the dual spin *τ <sup>α</sup>* of the RS field will not contribute, again due to Fierz rearrangement (37). This finally leads to the "on shell" conditions

$$\theta\_{\rm T} = -m^2 \ell^2, \qquad \theta\_{\rm TL} = \frac{(-1)^s}{2} m(2m+1)\ell, \qquad \theta\_{\rm L} = 1 - \frac{\theta\_{\rm TL}}{\ell} = 1 - \frac{(-1)^s}{2} m(2m+1) \tag{42}$$

for the coupling constants of the bosonic part of our Lagrangian *L*∞. Consequently, massless RS spinors do not require a translational nor a 'mixed' CS term in order to acquire supersymmetry.
