**3. Rarita–Schwinger Lagrangian in 3D**

Commonly, *supergravity* [15, 19] with one supersymmetry generator, i.e. N =1, represents the simplest consistent coupling of a Rarita–Schwinger (RS) spin– <sup>3</sup> <sup>2</sup> field [35] to gravity.

[23] reads

The corresponding *manifestly Hermitian* RS type Lagrangian three–form of Howe and Tucker

S-Duality in Topological Supergravity 41

2

which is nothing more than the gauge covariant derivative of a spinor-valued one-form Ψ.

As in the case of the Rarita-Schwinger Lagrangian *L*RS, it is manifestly Hermitian when the additional quadratic derivative terms carry *s*<sup>1</sup> and *s*<sup>2</sup> as dimensionless coupling constants. In order to supersymmetrize this action, it will be coupled to topological gravity later on.

 + *i*

*L*<sup>Ψ</sup> = *L*RS + *s*<sup>1</sup> *D*Ψ ∧ <sup>∗</sup>(*D*Ψ) + *s*<sup>2</sup> *D*Ψ ∧ *γ* ∧ <sup>∗</sup>(*γ* ∧ *D*Ψ). (22)

<sup>+</sup> *<sup>D</sup> <sup>∂</sup>L*<sup>Ψ</sup>

*<sup>∂</sup>*<sup>Ψ</sup> <sup>−</sup> (*eα*�*D*Ψ) <sup>∧</sup>

*<sup>D</sup>*<sup>Ψ</sup> <sup>∧</sup> *<sup>e</sup>α*�∗(*D*Ψ) <sup>−</sup> (*eα*�*D*Ψ) <sup>∧</sup> <sup>∗</sup>

*∂L*<sup>Ψ</sup>

*D*Ψ ∧ *γ* ∧ *γ* 

<sup>4</sup> *<sup>m</sup>*<sup>Ψ</sup> <sup>∧</sup> *<sup>γ</sup>* <sup>∧</sup> <sup>Ψ</sup>, (20)

*<sup>∂</sup>T<sup>α</sup>* , (23)

*∂L*<sup>Ψ</sup> *∂D*Ψ, (24)

. (25)

*<sup>∂</sup>D*<sup>Ψ</sup> <sup>−</sup> (*eα*�*D*Ψ) <sup>∧</sup>

*D*Ψ 

. (26)

*γα*<sup>Γ</sup>*<sup>α</sup>* <sup>∧</sup> <sup>Ψ</sup>, (21)

Ψ ∧ *D*Ψ − Ψ ∧ *D*Ψ

including, however, a mass term. Here minimal coupling to gravity is achieved via

*<sup>D</sup>*<sup>Ψ</sup> <sup>=</sup> *<sup>d</sup>*<sup>Ψ</sup> <sup>−</sup> <sup>1</sup>

Only in 3D, however, there exists a generalization given by the following expression

By definition, the energy-momentum current two-form Σ*<sup>α</sup>* of matter is given by

*δϑ<sup>α</sup>* <sup>=</sup> *<sup>∂</sup>L*<sup>Ψ</sup> *∂ϑα*

where the second term accounts for the possibility of a non-minimal coupling to torsion via Pauli type terms, cf. Eq. (5.1.8) of Ref. [22]. According to the Noether theorem, the energy-momentum current two-form of matter Σ*<sup>α</sup>* without Pauli terms can be rewritten as

*∂L*<sup>Ψ</sup>

see Eq. (5.4.11) of Ref. [22] for details. This equivalent equation often is more convenient, since it involves only partial derivatives of the matter fields and avoids the intricate treatment of a possible dependence of the matter Lagrangian on the Hodge dual. Taking into account

<sup>Σ</sup>*<sup>α</sup>* :<sup>=</sup> *<sup>δ</sup>L*<sup>Ψ</sup>

*<sup>∂</sup>*<sup>Ψ</sup> <sup>−</sup> (*eα*�Ψ) <sup>∧</sup>

*<sup>D</sup>*<sup>Ψ</sup> <sup>∧</sup> *γα* <sup>∧</sup> <sup>∗</sup>(*<sup>γ</sup>* <sup>∧</sup> *<sup>D</sup>*Ψ) <sup>−</sup> (*eα*�*D*Ψ) <sup>∧</sup> <sup>∗</sup>

on the coframe *ϑα*, they provides no contribution to the energy-momentum current.

Since the kinetic terms in the Rarita-Schwinger type Lagrangian *L*RS do not depend explicitly

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

2

*δL*<sup>Ψ</sup> *δ*Γ *α*

*∂L*<sup>Ψ</sup>

*m*Ψ ∧ *γα*Ψ + *s*<sup>1</sup>

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

*<sup>L</sup>*RS <sup>=</sup> *<sup>i</sup>* 4 

**3.1 Energy-momentum and spin currents**

Σ*<sup>α</sup>* := *eα*�*L*<sup>Ψ</sup> − (*eα*�Ψ) ∧

the identities of Appendix B, we find

<sup>Σ</sup>*<sup>α</sup>* <sup>=</sup> <sup>−</sup> *<sup>i</sup>* 4

> +*s*<sup>2</sup>

The 3-dual of the spin current is defined by

The Rarita-Schwinger [35] type spinor-valued one-form1

$$\Psi = \Psi\_i d\mathbf{x}^i = \Psi\_a \theta^a \tag{13}$$

can be written holononically and anholononically. However, it does not depend on the coframe, inasmuch Ψ*<sup>α</sup>* := *eα*�Ψ involves the inverse tetrad. In 3D, we adhere to the conventions that the holonomic indices run from *i*, *j*, *k*,... = 0, 1, 2, whereas *α*, *β*,... = 0,ˆ 1, ˆ 2ˆ for the anholonomic indices.

We are going to provide a brief summary of the spinors that will be used in three dimensions: As well known, the covering group of the rotation group *SO*(3) is isomorphic to the unitary group *SU*(2). Since an element of *SU*(2) can be parameterized by three numbers, the most convenient basis of the Lie algebra are the familiar Pauli spin matrices:

$$
\sigma^1 = \begin{pmatrix} 0 \ 1 \\ 1 \ 0 \end{pmatrix}, \qquad \sigma^2 = \begin{pmatrix} 0 \ -i \\ i \ 0 \end{pmatrix}, \qquad \sigma^3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \tag{14}
$$

These matrices satisfy the following Lie algebra:

$$\mathbb{E}\left[\sigma^{\alpha}, \sigma^{\beta}\right] = 2i\eta^{a\beta\gamma}\sigma\_{\gamma}.\tag{15}$$

However, for Lorentzian signature *s* = 1, the covering group of *SO*(1, 2) is isomorphic to the real group *SL*(2, IR). Then the generators of *SL*(2, IR) may be realized by the matrices

$$
\gamma\_0 = i\sigma^2, \quad \gamma\_1 = \sigma^1, \quad \gamma\_2 = \sigma^3. \tag{16}
$$

These real matrices [27] satisfying

$$
\gamma\_{\mathfrak{a}} \gamma\_{\mathfrak{F}} = \mathfrak{g}\_{a\mathfrak{F}} + \eta\_{a\mathfrak{F}\mathfrak{v}} \gamma^{\nu} \tag{17}
$$

also provide a realization of the Clifford algebra

$$
\gamma\_{\mathfrak{a}}\gamma\_{\mathfrak{F}} + \gamma\_{\mathfrak{F}}\gamma\_{\mathfrak{a}} = \mathfrak{Z}g\_{\mathfrak{a}\mathfrak{F}}\tag{18}
$$

in 3D. In addition, the coframe basis *ϑ<sup>α</sup>* converts into one Clifford algebra value one-form

$$
\gamma = \gamma\_\mathfrak{a} \mathfrak{e}^\mathfrak{a} \tag{19}
$$

Then Ψ will become real two-component spinors, with the Dirac adjoint defined by Ψ := Ψ†*γ*0.

$$
\overline{\Psi} \wedge \Psi = 0, \quad \overline{\Psi} \wedge \gamma\_5 \gamma^a \Psi = 0, \quad \overline{\Psi} \wedge \gamma\_5 \Psi = 0
$$

<sup>1</sup> In four dimensions (4D), the Rarita–Schwinger field Ψ := Ψ*αϑ<sup>α</sup>* entering Eq. (13) is a *Majorana spinor* valued one-form. As it is well known [34], it satisfies the Majorana condition, i.e. Ψ = *C*Ψ*<sup>t</sup>* , where *C* is the charge conjugation matrix given by *<sup>C</sup>* <sup>=</sup> <sup>−</sup>*iγ*<sup>0</sup> satisfying *<sup>C</sup>*† <sup>=</sup> *<sup>C</sup>*−<sup>1</sup> , *<sup>C</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>*<sup>C</sup>* and *<sup>C</sup>*−1*γα<sup>C</sup>* <sup>=</sup> <sup>−</sup> (*γα*) *t* . Consequently,

For the real Majorana representation all *γ<sup>α</sup>* are purely imaginary and the components of the gravitino vector–spinor consequently are all real [30].

4 Will-be-set-by-IN-TECH

can be written holononically and anholononically. However, it does not depend on the coframe, inasmuch Ψ*<sup>α</sup>* := *eα*�Ψ involves the inverse tetrad. In 3D, we adhere to the conventions that the holonomic indices run from *i*, *j*, *k*,... = 0, 1, 2, whereas *α*, *β*,... = 0,

We are going to provide a brief summary of the spinors that will be used in three dimensions: As well known, the covering group of the rotation group *SO*(3) is isomorphic to the unitary group *SU*(2). Since an element of *SU*(2) can be parameterized by three numbers, the most

> <sup>0</sup> <sup>−</sup>*<sup>i</sup> i* 0

However, for Lorentzian signature *s* = 1, the covering group of *SO*(1, 2) is isomorphic to the

real group *SL*(2, IR). Then the generators of *SL*(2, IR) may be realized by the matrices

in 3D. In addition, the coframe basis *ϑ<sup>α</sup>* converts into one Clifford algebra value one-form

Then Ψ will become real two-component spinors, with the Dirac adjoint defined by Ψ :=

<sup>1</sup> In four dimensions (4D), the Rarita–Schwinger field Ψ := Ψ*αϑ<sup>α</sup>* entering Eq. (13) is a *Majorana spinor* valued one-form. As it is well known [34], it satisfies the Majorana condition, i.e. Ψ = *C*Ψ*<sup>t</sup>*

is the charge conjugation matrix given by *<sup>C</sup>* <sup>=</sup> <sup>−</sup>*iγ*<sup>0</sup> satisfying *<sup>C</sup>*† <sup>=</sup> *<sup>C</sup>*−<sup>1</sup> , *<sup>C</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>*<sup>C</sup>* and *<sup>C</sup>*−1*γα<sup>C</sup>* <sup>=</sup>

<sup>Ψ</sup> <sup>∧</sup> <sup>Ψ</sup> <sup>=</sup> 0 , <sup>Ψ</sup> <sup>∧</sup> *<sup>γ</sup>*5*γα*<sup>Ψ</sup> <sup>=</sup> 0 , <sup>Ψ</sup> <sup>∧</sup> *<sup>γ</sup>*5<sup>Ψ</sup> <sup>=</sup> <sup>0</sup> For the real Majorana representation all *γ<sup>α</sup>* are purely imaginary and the components of the gravitino

, *σ*<sup>3</sup> =

*γ*<sup>0</sup> = *iσ*2, *γ*<sup>1</sup> = *σ*1, *γ*<sup>2</sup> = *σ*3. (16)

*γαγβ* = *gαβ* + *ηαβνγ<sup>ν</sup>* (17)

*γαγβ* + *γβγα* = 2*gαβ* (18)

*γ* = *γαϑ<sup>α</sup>* (19)

convenient basis of the Lie algebra are the familiar Pauli spin matrices:

, *σ*<sup>2</sup> =

 *σα*, *σ<sup>β</sup>* 

Ψ = Ψ*idx<sup>i</sup>* = Ψ*αϑ<sup>α</sup>* (13)

 1 0 0 −1

= 2*iηαβγσγ*. (15)

ˆ 1, ˆ 2ˆ

. (14)

, where *C*

The Rarita-Schwinger [35] type spinor-valued one-form1

for the anholonomic indices.

*σ*<sup>1</sup> =

These real matrices [27] satisfying

Ψ†*γ*0.

<sup>−</sup> (*γα*) *t*

. Consequently,

vector–spinor consequently are all real [30].

These matrices satisfy the following Lie algebra:

also provide a realization of the Clifford algebra

 0 1 1 0  The corresponding *manifestly Hermitian* RS type Lagrangian three–form of Howe and Tucker [23] reads

$$L\_{\rm RS} = \frac{i}{4} \left( \overline{\mathbf{Y}} \wedge D\mathbf{Y} - \mathbf{Y} \wedge \overline{D\mathbf{Y}} \right) + \frac{i}{4} m \overline{\mathbf{Y}} \wedge \gamma \wedge \mathbf{Y},\tag{20}$$

including, however, a mass term. Here minimal coupling to gravity is achieved via

$$d\Psi = d\Psi - \frac{1}{2}\gamma\_a \Gamma^{\star \alpha} \wedge \Psi,\tag{21}$$

which is nothing more than the gauge covariant derivative of a spinor-valued one-form Ψ.

Only in 3D, however, there exists a generalization given by the following expression

$$L\_{\overline{\mathbf{Y}}} = L\_{\overline{\mathbf{RS}}} + s\_1 \overline{D\mathbf{Y}} \wedge \, ^\*(D\mathbf{Y}) + s\_2 \, \overline{D\mathbf{Y}} \wedge \gamma \wedge \, ^\*(\gamma \wedge D\mathbf{Y}).\tag{22}$$

As in the case of the Rarita-Schwinger Lagrangian *L*RS, it is manifestly Hermitian when the additional quadratic derivative terms carry *s*<sup>1</sup> and *s*<sup>2</sup> as dimensionless coupling constants.

In order to supersymmetrize this action, it will be coupled to topological gravity later on.

### **3.1 Energy-momentum and spin currents**

By definition, the energy-momentum current two-form Σ*<sup>α</sup>* of matter is given by

$$\Sigma\_{\mathfrak{n}} := \frac{\delta L\_{\Psi}}{\delta \theta^{\mathfrak{a}}} = \frac{\partial L\_{\Psi}}{\partial \theta\_{\mathfrak{a}}} + D \frac{\partial L\_{\Psi}}{\partial T^{\mathfrak{a}}},\tag{23}$$

where the second term accounts for the possibility of a non-minimal coupling to torsion via Pauli type terms, cf. Eq. (5.1.8) of Ref. [22]. According to the Noether theorem, the energy-momentum current two-form of matter Σ*<sup>α</sup>* without Pauli terms can be rewritten as

$$\Sigma\_{\mathfrak{a}} := e\_{\mathfrak{a}} \rfloor L\mathfrak{y} - (e\_{\mathfrak{a}} \rfloor \overline{\mathfrak{Y}}) \wedge \frac{\partial L\overline{\mathfrak{Y}}}{\partial \overline{\mathfrak{Y}}} - (e\_{\mathfrak{a}} \rfloor \overline{\overline{\mathfrak{Y}}}) \wedge \frac{\partial L\overline{\mathfrak{Y}}}{\partial \overline{\overline{\mathfrak{Y}}}} - (e\_{\mathfrak{a}} \rfloor D\overline{\mathfrak{Y}}) \wedge \frac{\partial L\overline{\mathfrak{Y}}}{\partial D\overline{\mathfrak{Y}}} - (e\_{\mathfrak{a}} \rfloor D\overline{\overline{\mathfrak{Y}}}) \wedge \frac{\partial L\overline{\mathfrak{Y}}}{\partial D\overline{\overline{\mathfrak{Y}}}} \tag{24}$$

see Eq. (5.4.11) of Ref. [22] for details. This equivalent equation often is more convenient, since it involves only partial derivatives of the matter fields and avoids the intricate treatment of a possible dependence of the matter Lagrangian on the Hodge dual. Taking into account the identities of Appendix B, we find

$$\Sigma\_{\mathfrak{a}} = -\frac{i}{4}m\overline{\Psi} \wedge \gamma\_{\mathfrak{a}}\Psi + \mathrm{s}\_{1}\left\{ \overline{D\Psi} \wedge \mathrm{e}\_{\mathfrak{a}} \right\}^{\*} (D\Psi) - (\mathrm{e}\_{\mathfrak{a}} \rfloor D\Psi) \wedge \,^{\*}\left(\overline{D\Psi}\right) \rangle$$

$$+ \mathrm{s}\_{2}\left[ \overline{D\Psi} \wedge \gamma\_{\mathfrak{a}} \wedge \,^{\*}(\gamma \wedge D\Psi) - (\mathrm{e}\_{\mathfrak{a}} \rfloor D\Psi) \wedge \,^{\*}\left(\overline{D\Psi} \wedge \gamma\right) \wedge \gamma\right].\tag{25}$$

Since the kinetic terms in the Rarita-Schwinger type Lagrangian *L*RS do not depend explicitly on the coframe *ϑα*, they provides no contribution to the energy-momentum current.

The 3-dual of the spin current is defined by

$$
\tau\_a^\star := \frac{1}{2} \eta\_{a\beta\gamma} \tau^{\beta\gamma} = \frac{(-1)^s}{2} \frac{\delta L\_{\overline{\Gamma}}}{\delta \Gamma\_a^\star} \,. \tag{26}
$$

Let us restrict for the moment to the usual Rarita-Schwinger Lagrangian *L*RS, or equivalently

S-Duality in Topological Supergravity 43

1 2

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

+ *γα*Ψ

 <sup>−</sup> *iR*

+ *cγ* ∧

1 4  *i* 2 *D*Ψ + *i* 4

*m*Ψ*γα*Ψ

*αγ<sup>α</sup>*<sup>Ψ</sup> <sup>−</sup> *<sup>i</sup>*

2

*<sup>γ</sup>α*<sup>Ψ</sup> <sup>∧</sup> <sup>Ψ</sup>*γα*<sup>Ψ</sup> <sup>=</sup> 0, (37)

*c* = −*m*, (38)

*ηα* + *τ α* 

*γα*Ψ = 0, (41)

*<sup>γ</sup>* <sup>∧</sup> *<sup>γ</sup>* <sup>=</sup> <sup>−</sup>2*γαηα*, (39)

*m*2 <sup>2</sup> *ηα*  *mTαγα*Ψ +

*mγ* ∧ Ψ ∼= 0 (35)

*mγ* ∧ Ψ

*i* 2

*γα*Ψ = 0, (40)

*mγ* ∧ *D*Ψ

(36)

*<sup>α</sup>* , are

to *L*<sup>Ψ</sup> with *s*<sup>1</sup> = *s*<sup>2</sup> = 0. Then the Rarita-Schwinger equation

*δL*

 *θ*TL *<sup>R</sup> <sup>α</sup>* + *θ*T <sup>2</sup> *<sup>T</sup><sup>α</sup>* <sup>+</sup> <sup>Σ</sup>*<sup>α</sup>*

 *θ*TL *<sup>R</sup> <sup>α</sup>* + *θ*T <sup>2</sup> *<sup>T</sup><sup>α</sup>* 

 *i* 2 *D*Ψ + *i* 4

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

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

 *R <sup>α</sup>* + *θ*<sup>T</sup> <sup>2</sup> <sup>−</sup> *<sup>m</sup>* 2 *T<sup>α</sup>* +

<sup>−</sup> <sup>1</sup>

in order that our Lagrangian becomes supersymmetric.

Chern-Simons term proportional to *θ*T. In this insertion

(−1)*<sup>s</sup> <sup>θ</sup>*TL

 <sup>+</sup> *θ*T <sup>2</sup> <sup>−</sup> *<sup>m</sup>* 2 *T<sup>α</sup>* + 1 2 *θ*<sup>T</sup> <sup>2</sup> <sup>+</sup> *<sup>m</sup>*<sup>2</sup>

*i θ*TL

 *R <sup>α</sup>* + 

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

supersymmetric transformations reads

∼= *iγα*Ψ

−*D iD*Ψ +

= *iγα*Ψ

+*cγ* ∧

*δL δϑ<sup>α</sup>* <sup>+</sup> *<sup>c</sup><sup>γ</sup>* <sup>∧</sup>

By a Fierz rearrangement, i.e.,

vanishing.

*i <sup>θ</sup>*<sup>L</sup> <sup>+</sup>

*θ*TL <sup>−</sup> <sup>1</sup>

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

2 *i δL*

*<sup>δ</sup>*<sup>Ψ</sup> <sup>−</sup> <sup>2</sup>*<sup>D</sup> <sup>δ</sup><sup>L</sup>*

*δ*Ψ

*mγ* ∧ Ψ

*mγ* ∧ Ψ

terms arising from the energy-momentum current Σ*α*, or likewise from the dual spin *τ*

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

*i* 2

*<sup>δ</sup>*<sup>Ψ</sup> <sup>=</sup> *<sup>D</sup>*<sup>Ψ</sup> <sup>+</sup>

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

$$\tau\_a^\* = \frac{(-1)^s}{2} \left\{ \frac{i}{4} \overline{\Psi} \gamma\_a \Psi + \frac{s\_1}{2} \left[ \overline{\Psi} \gamma\_a \wedge \,^\*(D\Psi) + \gamma\_a \Psi \wedge \,^\*\left(\overline{D\Psi}\right) \right] \right. $$

$$+ \frac{s\_2}{2} \left[ \overline{\Psi} \gamma\_a \wedge \,\gamma \wedge \,^\*(\gamma \wedge D\Psi) + \gamma\_a \Psi \wedge \gamma \wedge \,^\*(\gamma \wedge D\Psi) \right] \}. \tag{27}$$

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

$$\tau\_a^\star = \frac{(-1)^s}{2} \left[ \frac{i}{4} \overline{\mathbf{y}} \wedge \gamma\_a \mathbf{y} + s\_1 \overline{\mathbf{y}} \gamma\_a \wedge \,^\*(D\mathbf{y}) + s\_2 \overline{\mathbf{y}} \gamma\_a \wedge \gamma \wedge \,^\*(\gamma \wedge D\mathbf{y}) \right],\tag{28}$$

cf. the identities of Appendix C.

It should be noted that for the pure Rarita-Schwinger Lagrangian with *s*<sup>1</sup> = *s*<sup>2</sup> = 0, the energy-momentum current is proportional to its dual spin, i.e.

$$
\Sigma\_{\mathfrak{a}} = -(-1)^{s} 2m \tau\_{\mathfrak{a}}^{\star}.\tag{29}
$$
