**5. Towards supersymmetric S–duality**

There exists a continuous deformation [or a field redefinition (FR)] of the (Lorentz-) rotational connection by adding a tensor–valued one–form, similarly as in Eq. (3.11.1) of Ref. [22]. In 3D, the particular deformation

$$
\tilde{\Gamma}\_{\alpha}^{\star} = \Gamma\_{\alpha}^{\star} - (-1)^{s} \frac{\varepsilon}{2\ell} \theta\_{\alpha \prime} \tag{43}
$$

where *ε* is a continuous parameter, is involving the Lie dual Γ *<sup>α</sup>* = <sup>1</sup> <sup>2</sup> *ηαβγ*Γ*βγ* of the connection. In view of the definitions (2) and (3) of torsion and curvature, respectively, this FR implies

$$
\tilde{T}\_{\mathfrak{a}} = T\_{\mathfrak{a}} - \frac{\varepsilon}{\ell} \eta\_{a\nu} \qquad \tilde{R}\_{\mathfrak{a}}^{\star} = R\_{\mathfrak{a}}^{\star} - (-1)^{s} \frac{\varepsilon}{2\ell} T\_{\mathfrak{a}} + (-1)^{s} \frac{\varepsilon^{2}}{4\ell^{2}} \eta\_{a} \tag{44}
$$

for the deformed torsion and curvature, respectively. In particular, there can arise two subcases: Riemannian spacetime with deformed torsion *T<sup>α</sup>* = 0, or deformed *teleparallelism* in the gauge <sup>Γ</sup> *α* ∗ = 0, equivalent to the covariant constraint of vanishing modified RC curvature, i.e., *<sup>R</sup> <sup>α</sup>* = 0.

In the latter case, coframe and connection are *Lie dual* to each other, i.e.,

$$
\Gamma^\star\_a = (-1)^s \frac{\varepsilon}{2\ell} \theta\_a \qquad \Leftrightarrow \qquad \theta\_a = (-1)^s \frac{2\ell}{\varepsilon} \Gamma^\star\_a. \tag{45}
$$

Observe the inversion of the parameter *ε*, i.e., a small deformation *ε* of the connection will induce a large coframe proportional to 1/*ε* and vice versa, resembling strong/weak duality. Such a duality of the *strong/weak* coupling regime of gauge fields, is the so-called *S–duality*. For Chern-Simons (super-)gravity, some of its aspects have also been discussed in Ref. [16, 20].

There could also arise the seemingly trivial case of a completely *flat* deformed spacetime, i.e., *<sup>T</sup><sup>α</sup>* <sup>=</sup> 0 and *<sup>R</sup> <sup>α</sup>* = 0. This would correspond to configurations with constant axial torsion and constant RC curvature as originally envision by E. Cartan, i.e.,

$$T\_{\mathfrak{a}} = \frac{\varepsilon}{\ell} \,\eta\_{\mathfrak{a}\prime} \qquad R\_{\mathfrak{a}}^{\star} = \frac{\rho}{\ell^2} \,\eta\_{\mathfrak{a}\prime} \tag{46}$$

where *<sup>ρ</sup>* = (−1)*sε*2/4 depends quadratically on the deformation parameter *<sup>ε</sup>*.

Let us extend such ideas to supergravity in 3D: Generalizing the peculiar dynamical symmetry of BMH [2], identified as *S*–duality in Ref. [31], we try the following Ansatz

$$
\theta\_{\mathfrak{A}} = (-1)^{s} \ell \, \Gamma\_{\mathfrak{A}}^{\star} + \overline{\sigma} \, \gamma\_{\mathfrak{A}} \, \mathbb{Y}\_{\prime} \tag{47}
$$

8 Will-be-set-by-IN-TECH

<sup>2</sup> *<sup>m</sup>*(2*<sup>m</sup>* <sup>+</sup> <sup>1</sup>), *<sup>θ</sup>*<sup>L</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>θ</sup>*TL

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

There exists a continuous deformation [or a field redefinition (FR)] of the (Lorentz-) rotational connection by adding a tensor–valued one–form, similarly as in Eq. (3.11.1) of Ref. [22]. In

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

In view of the definitions (2) and (3) of torsion and curvature, respectively, this FR implies

for the deformed torsion and curvature, respectively. In particular, there can arise two subcases: Riemannian spacetime with deformed torsion *T<sup>α</sup>* = 0, or deformed *teleparallelism* in

Observe the inversion of the parameter *ε*, i.e., a small deformation *ε* of the connection will induce a large coframe proportional to 1/*ε* and vice versa, resembling strong/weak duality. Such a duality of the *strong/weak* coupling regime of gauge fields, is the so-called *S–duality*. For Chern-Simons (super-)gravity, some of its aspects have also been discussed in Ref. [16, 20]. There could also arise the seemingly trivial case of a completely *flat* deformed spacetime, i.e.,

*ηα*, *<sup>R</sup>*

Let us extend such ideas to supergravity in 3D: Generalizing the peculiar dynamical symmetry

Γ

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

2

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

= 0, equivalent to the covariant constraint of vanishing modified RC curvature,

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

*<sup>α</sup>* = 0. This would correspond to configurations with constant axial torsion and

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

2

*<sup>α</sup>* = <sup>1</sup>

*<sup>T</sup><sup>α</sup>* + (−1)*<sup>s</sup> <sup>ε</sup>*<sup>2</sup>

*ε* Γ

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

*ηα*, *<sup>R</sup>*

In the latter case, coframe and connection are *Lie dual* to each other, i.e.,

2

*<sup>T</sup><sup>α</sup>* <sup>=</sup> *<sup>ε</sup>*

where *<sup>ρ</sup>* = (−1)*sε*2/4 depends quadratically on the deformation parameter *<sup>ε</sup>*.

of BMH [2], identified as *S*–duality in Ref. [31], we try the following Ansatz

*ϑα* = (−1)*<sup>s</sup>*

where *ε* is a continuous parameter, is involving the Lie dual Γ

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

constant RC curvature as originally envision by E. Cartan, i.e.,

*<sup>α</sup>* of the RS field will not contribute, again due to Fierz rearrangement (37). This

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

*ϑα*, (43)

<sup>2</sup> *ηαβγ*Γ*βγ* of the connection.

<sup>4</sup><sup>2</sup> *ηα* (44)

*<sup>α</sup>*. (45)

<sup>2</sup> *ηα*, (46)

*<sup>α</sup>* + *σ γα*Ψ , (47)

<sup>2</sup> *<sup>m</sup>*(2*<sup>m</sup>* <sup>+</sup> <sup>1</sup>) (42)

the dual spin *τ*

supersymmetry.

the gauge <sup>Γ</sup>

*<sup>α</sup>* = 0.

*<sup>T</sup><sup>α</sup>* <sup>=</sup> 0 and *<sup>R</sup>*

i.e., *<sup>R</sup>*

*α* ∗

finally leads to the "on shell" conditions

**5. Towards supersymmetric S–duality**

*<sup>T</sup><sup>α</sup>* <sup>=</sup> *<sup>T</sup><sup>α</sup>* <sup>−</sup> *<sup>ε</sup>*

Γ

*<sup>θ</sup>*<sup>T</sup> <sup>=</sup> <sup>−</sup>*m*<sup>2</sup>2, *<sup>θ</sup>*TL <sup>=</sup> (−1)*<sup>s</sup>*

3D, the particular deformation

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

By exterior differentiation, we find

$$d\theta\_{\mathfrak{a}} = (-1)^{s} \ell d\Gamma\_{\mathfrak{a}}^{\star} + d(\overline{\sigma}\gamma\_{\mathfrak{a}}\Psi),\tag{48}$$

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

$$T\_{\mathfrak{a}} - (-1)^{s} \eta\_{a\mathfrak{f}} \wedge \Gamma^{\star \mathfrak{f}} = (-1)^{s} \ell R\_{\mathfrak{a}}^{\star} - \frac{\ell}{2} \eta\_{a\mathfrak{f}\gamma} \Gamma^{\star \mathfrak{f}} \wedge \Gamma^{\star \gamma} + d(\overline{\sigma} \wedge \gamma\_{a} \Psi) \tag{49}$$

Let us reconstitute our Ansatz (47) in order to replace all the connection terms Γ*<sup>β</sup>* . Then, using also the fundamental relation (18) for a Clifford algebra, we obtain

$$\begin{split} \Gamma\_{\mathfrak{a}} &+ \frac{2}{\ell} \eta\_{\mathfrak{a}} + \frac{1}{\ell} \eta\_{\mathfrak{a}\mathfrak{f}} \wedge \overline{\sigma} \gamma^{\mathfrak{f}} \Psi\\ &= (-1)^{\mathfrak{s}} \ell R\_{\mathfrak{a}}^{\star} - \frac{1}{2\ell} \eta\_{\mathfrak{a}\mathfrak{f}\gamma} (\mathfrak{e}^{\mathfrak{f}} - \overline{\sigma} \gamma^{\mathfrak{f}} \mathbf{Y}) \wedge (\mathfrak{e}^{\gamma} - \overline{\sigma} \gamma^{\gamma} \mathbf{Y}) + d(\overline{\sigma} \gamma\_{\mathfrak{a}} \mathbf{Y}). \end{split} \tag{50}$$

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

$$\frac{2}{A} \left[ (\theta\_{\rm TL} + (-1)^{s} \theta\_{\rm T}) \tau\_{\rm u}^{\star} - (\theta\_{\rm L} + (-1)^{s} \theta\_{\rm TL}) \ell \Sigma\_{\rm d} \right] \ell^{2} + (3 + 2\kappa - (-1)^{s} \rho) \eta\_{\rm d}$$

$$= -2\eta\_{\rm a\beta} \wedge \overline{\sigma} \gamma^{\beta} \mathbf{Y} - \frac{1}{2} \eta\_{\rm a\beta\mu} \overline{\sigma} \gamma^{\beta} \mathbf{Y} \wedge \overline{\sigma} \gamma^{\mu} \mathbf{Y} + \ell d(\overline{\sigma} \gamma\_{\rm a} \mathbf{Y}).\tag{51}$$

Together with (29), this leads to

$$\begin{split} \frac{B}{4A} \mathrm{i}\ell^{2} \overline{\Psi}\gamma\_{a}\Psi &+ \frac{\mathbb{C}}{A} \eta\_{a} \\ &= -2\eta\_{a\not\mathbb{B}} \wedge \overline{\sigma}\,\gamma^{\beta}\Psi - \frac{1}{2}\eta\_{a\not\mathbb{B}\mu} \overline{\sigma}\gamma^{\beta}\Psi \wedge \overline{\sigma}\gamma^{\mu}\Psi + \ell D(\overline{\sigma}\gamma\_{a}\Psi) \end{split} \tag{52}$$

as a condition for S-duality, where

$$B = \theta\_{\rm T} + (-1)^{s} \theta\_{\rm TL} + 2m\ell[\theta\_{\rm L} + (-1)^{s} \theta\_{\rm TL}] \tag{53}$$

and

$$\mathbf{C} = \mathbf{3}A + \theta\_{\mathrm{T}}[\theta\_{\mathrm{TL}} + (-1)^{s}\theta\_{\mathrm{T}}].\tag{54}$$

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 satisfied.

## **6. Membranes with torsion defects**

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

there follows the identity

*Rβ*

*<sup>α</sup>* <sup>∧</sup> *<sup>ϑ</sup><sup>β</sup>* <sup>=</sup> *<sup>ε</sup>*<sup>2</sup>

to the Pointrjagin type term *d*(A ∧ *d*A) from the axial torsion.

a spinning cosmic string exhibiting a *torsion line defect*.

**7. Outlook: Graphene and supersymmetry**

indications of dislocations [9] related to torsion.

developed to some extent in this paper.

**8. Acknowledgments**

Gérard Erik, for encouragement.

boost to the usual conical metric of a defect simulated by a cosmic string:

Recalling that *<sup>N</sup><sup>α</sup>* <sup>=</sup> *<sup>n</sup>*�*ϑ<sup>α</sup>* is the lapse and shift vector in the (2+1)–decomposition a la ADM, the corresponding coframe and connection can now explicitly be obtained by applying a finite

S-Duality in Topological Supergravity 47

*<sup>ϑ</sup>*0ˆ <sup>=</sup> *dt* <sup>+</sup> <sup>2</sup>*σρ*∗2[<sup>1</sup> <sup>−</sup> cos(*ρ*/*ρ*∗)]*d<sup>φ</sup> <sup>ϑ</sup>*1ˆ = *<sup>d</sup><sup>ρ</sup>* , *<sup>ϑ</sup>*2ˆ = *<sup>ρ</sup>*<sup>∗</sup> sin(*ρ*/*ρ*∗)*d<sup>φ</sup>* ,

From the Cartan type relation (11) and the identities (57) we can infer that the *axial* torsion

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

of such a membrane defect is a non-vanishing constant. Thus, in 3D there is no contribution

Moreover, the Nieh–Yan term *dC*<sup>T</sup> proportional to *d* ∗A *vanishes identically* for this example of

Fundamental interactions are rather successful formulated in terms of Yang-Mills theories with large gauge groups, stipulating that symmetry breaking is occurring in the ground state. The idea of supersymmetry or supergravity, anticipated to some extent already by Hermann Weyl [42], goes in the same direction but so far lacks empirical support in particle physics. Recently, graphene [33] as a new material has attracted a lot of attention because its charge carriers can be described by massless Dirac fields, cf. Ref. [41], whereas the flexural models of the 2D membrane of graphene have been tentatively considered as membranes, cf. Ref. [25], evolving in 2 + 1 dimensional curved, but conformally flat spacetime [24]. There are also

A related topological framework with a coupling to Dirac fields in 3D has been considered before by Lemke and Mielke [27]. It seems to be feasible to enlarge the dynamical framework of the theory by including supersymmetry, cf. Ref. [17] and apply the topological ideas

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

<sup>Γ</sup>1ˆ2ˆ <sup>=</sup> cos(*ρ*/*ρ*∗)*d<sup>φ</sup>* <sup>=</sup> <sup>−</sup>Γ2ˆ1ˆ

<sup>2</sup> (*x<sup>α</sup> <sup>Y</sup>* <sup>∧</sup> *<sup>X</sup>* <sup>∧</sup> *<sup>Y</sup>* <sup>−</sup> *<sup>y</sup><sup>α</sup> <sup>X</sup>* <sup>∧</sup> *<sup>X</sup>* <sup>∧</sup> *<sup>Y</sup>*) = 0 . (59)

. (60)

<sup>2</sup> (61)

Fig. 1. 'Screw' dislocation with singular torsion in a cubic lattice. (The Cartan circuit is indicate in blue, cf. Ref. [26].]

Let us assume that the 2D membrane of a corrugated graphene is evolving in an *intrinsic* three-dimensional spacetime, suppressing for the moment the embedding of a real graphene into *flat* 4D Minkowski spacetime. Then we may adopt the convention that *x<sup>α</sup>* together with *y<sup>α</sup>* are spacelike orthogonal vectors which span the (*x*, *y*)–plane perpendicular to the time coordinate *t*, which itself is orthogonal to the world sheet of the graphene. The corresponding one–forms [29] are denoted by capital letters, i.e.

$$X := \mathfrak{x}\_{\mathfrak{a}} \mathfrak{e}^{\mathfrak{a}} , \qquad Y := y\_{\mathfrak{a}} \mathfrak{e}^{\mathfrak{a}} . \tag{55}$$

Moreover, the vector *n<sup>α</sup>* is a timelike unit vector normal to the hypersurface with *n<sup>α</sup> n<sup>α</sup>* = *s*, the signature *s* of our 3D spacetime.

Following Soleng [37], cf. Anandan [1, 3, 22], we assume that the two–forms Σ*<sup>α</sup>* and *τ <sup>α</sup>* of the energy–momentum and spin current, respectively, vanish outside of the graphene sheet, whereas "inside" they are *constant*, i.e.

$$
\Sigma\_{\mathfrak{a}} = \varepsilon \ge\_{\mathfrak{a}} X \wedge Y, \qquad \tau\_{\mathfrak{a}}^{\star} = \sigma \, y\_{\mathfrak{a}} \, X \wedge Y,\tag{56}
$$

which satisfy

$$
\vartheta^{\mathfrak{a}} \wedge \Sigma\_{\mathfrak{a}} = 0 \,, \qquad \vartheta^{\mathfrak{a}} \wedge \tau\_{\mathfrak{a}}^{\star} = 0 \,\tag{57}
$$

by construction. The constant parameters *ε* and *σ* of this *spinning string* type Ansatz are related to the exterior vacuum solution by appropriate matching conditions. For the related solution with *conical singularities* and torsion of Tod [40], we can infer that *ε* and *σ* are *delta distributions* [39] at the location of the defect, cf. Fig 1. From the specification (55) of the one–forms *X* and *Y* it can easily be inferred that the only nonzero components are Σ0ˆ �= 0 and *τ*1ˆ2ˆ = −*τ*2ˆ1ˆ �= 0.

Due to the identities (57), contractions of the second field equation (12) with *x<sup>α</sup>* and *y<sup>α</sup>* reveal that *<sup>x</sup>*[*αyβ*] *<sup>R</sup>αβ* <sup>=</sup> *<sup>R</sup>*1ˆ2ˆ <sup>=</sup> <sup>−</sup>*R*2ˆ1ˆ �<sup>=</sup> 0 are the only nonvanishing components of the RC curvature. From its covariant expression

$$\mathcal{R}^{a\beta} = \varepsilon \ell^2 \ge^{|a|} y^{\beta|} \ge \mathcal{N} \tag{58}$$

there follows the identity

10 Will-be-set-by-IN-TECH

Fig. 1. 'Screw' dislocation with singular torsion in a cubic lattice. (The Cartan circuit is

Let us assume that the 2D membrane of a corrugated graphene is evolving in an *intrinsic* three-dimensional spacetime, suppressing for the moment the embedding of a real graphene into *flat* 4D Minkowski spacetime. Then we may adopt the convention that *x<sup>α</sup>* together with *y<sup>α</sup>* are spacelike orthogonal vectors which span the (*x*, *y*)–plane perpendicular to the time coordinate *t*, which itself is orthogonal to the world sheet of the graphene. The corresponding

Moreover, the vector *n<sup>α</sup>* is a timelike unit vector normal to the hypersurface with *n<sup>α</sup> n<sup>α</sup>* = *s*,

Following Soleng [37], cf. Anandan [1, 3, 22], we assume that the two–forms Σ*<sup>α</sup>* and *τ*

*<sup>ϑ</sup><sup>α</sup>* <sup>∧</sup> <sup>Σ</sup>*<sup>α</sup>* <sup>=</sup> 0 , *<sup>ϑ</sup><sup>α</sup>* <sup>∧</sup> *<sup>τ</sup>*

by construction. The constant parameters *ε* and *σ* of this *spinning string* type Ansatz are related to the exterior vacuum solution by appropriate matching conditions. For the related solution with *conical singularities* and torsion of Tod [40], we can infer that *ε* and *σ* are *delta distributions* [39] at the location of the defect, cf. Fig 1. From the specification (55) of the one–forms *X* and *Y* it can easily be inferred that the only nonzero components are Σ0ˆ �= 0 and *τ*1ˆ2ˆ = −*τ*2ˆ1ˆ �= 0. Due to the identities (57), contractions of the second field equation (12) with *x<sup>α</sup>* and *y<sup>α</sup>* reveal that *<sup>x</sup>*[*αyβ*] *<sup>R</sup>αβ* <sup>=</sup> *<sup>R</sup>*1ˆ2ˆ <sup>=</sup> <sup>−</sup>*R*2ˆ1ˆ �<sup>=</sup> 0 are the only nonvanishing components of the RC curvature.

<sup>Σ</sup>*<sup>α</sup>* <sup>=</sup> *<sup>ε</sup> <sup>x</sup><sup>α</sup> <sup>X</sup>* <sup>∧</sup> *<sup>Y</sup>* , *<sup>τ</sup>*

the energy–momentum and spin current, respectively, vanish outside of the graphene sheet,

*X* := *x<sup>α</sup> ϑ<sup>α</sup>* , *Y* := *y<sup>α</sup> ϑ<sup>α</sup>* . (55)

*<sup>R</sup>αβ* <sup>=</sup> *<sup>ε</sup>*<sup>2</sup> *<sup>x</sup>*[*αyβ*] *<sup>X</sup>* <sup>∧</sup> *<sup>Y</sup>* (58)

*<sup>α</sup>* = *σ y<sup>α</sup> X* ∧ *Y* , (56)

*<sup>α</sup>* = 0 (57)

*<sup>α</sup>* of

indicate in blue, cf. Ref. [26].]

one–forms [29] are denoted by capital letters, i.e.

the signature *s* of our 3D spacetime.

whereas "inside" they are *constant*, i.e.

From its covariant expression

which satisfy

$$R\_{\hat{\mathcal{B}}}{}^{\mathfrak{a}} \wedge \mathfrak{d}^{\mathfrak{f}} = \frac{\varepsilon \ell^2}{2} (\mathfrak{x}^{\mathfrak{a}} \, Y \wedge X \wedge Y - \mathfrak{z}^{\mathfrak{a}} X \wedge X \wedge Y) = 0. \tag{59}$$

Recalling that *<sup>N</sup><sup>α</sup>* <sup>=</sup> *<sup>n</sup>*�*ϑ<sup>α</sup>* is the lapse and shift vector in the (2+1)–decomposition a la ADM, the corresponding coframe and connection can now explicitly be obtained by applying a finite boost to the usual conical metric of a defect simulated by a cosmic string:

$$\begin{aligned} \theta^{\bar{0}} &= dt + \ell^2 \sigma \rho^{\*2} [1 - \cos(\rho/\rho^\*)] d\phi \\ \theta^{\underline{1}} &= d\rho \, \, \, \qquad \theta^{\underline{2}} = \rho^\* \sin(\rho/\rho^\*) d\phi \, \, \end{aligned}$$

$$\Gamma^{\underline{12}} = \cos(\rho/\rho^\*) d\phi = -\Gamma^{\underline{21}} \, \, \, \tag{60}$$

From the Cartan type relation (11) and the identities (57) we can infer that the *axial* torsion

$$\mathcal{A} = \,^\*(\theta^a \wedge T\_a) = -(-1)^s \frac{2\kappa}{\ell^2} \tag{61}$$

of such a membrane defect is a non-vanishing constant. Thus, in 3D there is no contribution to the Pointrjagin type term *d*(A ∧ *d*A) from the axial torsion.

Moreover, the Nieh–Yan term *dC*<sup>T</sup> proportional to *d* ∗A *vanishes identically* for this example of a spinning cosmic string exhibiting a *torsion line defect*.
