Extended Supergravities

Abstract

Whereas in $\mathcal {N}=1$ supersymmetry scalar fields are only mapped to certain spin-1/2 fermions, the larger multiplets of extended supersymmetry may link scalars also to fields of spin greater than 1/2, in particular vector fields. As we will see in the following, this heavily constrains the structure and couplings of extended supergravities. The main reason for this is that in a theory without charged fields, the field equations of Abelian vector fields in four spacetime dimensions exhibit the phenomenon of electric–magnetic duality. In extended supersymmetry, this electric–magnetic duality structure of the vector field sector is then linked to the scalar field geometry by supersymmetry and constrains the scalar manifolds of most supermultiplets. Moreover, this same duality structure is going to be at the basis of the consistent construction of deformations that make some of the fields charged under the gauge group. We therefore first discuss in Sect. 8.1 the main features of electric–magnetic duality in preparation for our discussion of extended supergravity. In the subsequent sections of this chapter, we then explain how electric–magnetic duality and the structure of the R-symmetry group constrain the geometries of the corresponding scalar manifolds. In Sect. 8.2, this will lead us to special Kähler and quaternionic manifolds for the scalar fields in, respectively, the vector and hypermultiplets in $\mathcal {N}=2$ supergravity. The scalar manifolds for supergravity theories with $\mathcal {N}\geq 3$ supergravity are then discussed in Sect. 8.3. The appendix to this chapter contains computational details on the scalar field geometries in $\mathcal {N}=2$ supergravity.

Appendices

8.A Appendix: Details on and Origin of Local Special Kähler Geometry

In this appendix, we would like to understand the origin of the differences between rigid and local special Kähler geometry that are listed in Sect. 8.2.1.2. Our goal herein is not so much mathematical completeness or a full derivation of all terms and prefactors in the Lagrangian, but rather to emphasize the geometrical roots of these differences and to exhibit them as directly as possible and without introducing additional formalism [2, 4,5,6,7,8].

1.1 8.A.1 The Symplectic Section and Its Kähler Transformation

We begin with the necessity of the existence of a scalar field-dependent symplectic section, v. Just as for rigid special Kähler geometry, this necessity can be deduced from the requirement that the transformation laws of the gaugini should be symplectically invariant and covariant with respect to holomorphic scalar reparameterizations. More precisely, if we again consider the supersymmetry transformation with respect to the original $\mathcal {N}=1$ supersymmetry parameter 𝜖 ⁽¹⁾, the analogy with the rigid transformations (8.63), (8.64) suggests that in supergravity one has

(8.119)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta^{(1)}\lambda_L^{m(2)}& =&\displaystyle -\frac{1}{4}g^{m\overline{n}}\overline{f_n^I}(\mbox{Im}\mathcal{N}_{IJ})F_{\mu\nu}^{J-}\gamma^{\mu\nu}\epsilon_L^{(1)}.{} \end{array} \end{aligned} $$

(8.120)

Here, $\mathcal {N}_{IJ}(z,\overline {z})$ is the scalar field-dependent gauge kinetic matrix of the vector field Lagrangian (8.39), and $f_{n}^{I}(z,\overline {z})$ is a scalar field-dependent object that transforms as a holomorphic covector, $f_n^I\mapsto (\partial z^m/\partial \widetilde {z}^n)f_m^I$, under scalar field reparameterizations and forms the upper part of a symplectic vector

$$\displaystyle \begin{aligned} U_n(z,\overline{z})=\left(\begin{array}{c}f_n^I(z,\overline{z})\\h_{Jn}(z,\overline{z})\end{array}\right). \end{aligned} $$

(8.121)

To be most general as possible, we allow for a possible non-holomorphic z ^m-dependence, but, as in rigid supersymmetry, we do expect that also U _n will be related to a suitable derivative of another symplectic (and possibly non-holomorphic) section $V(z,\overline {z})$ with respect to z ⁿ. In order to ensure the symplectic invariance of (8.120), however, we only need to require

$$\displaystyle \begin{aligned} \overline{f_n^I}\mathcal{N}_{IJ}=\overline{h_{Jn}}, \end{aligned} $$

(8.122)

because then

$$\displaystyle \begin{aligned} \overline{f^I_n}\mbox{Im}\mathcal{N}_{IJ}F^{J-}_{\mu\nu}= \frac{1}{2i}\left(\overline{h_{Jn}}F_{\mu\nu}^{J-}-\overline{f^I_n}G_{\mu\nu I-}\right) = \frac{1}{2i}\langle \mathbb{F}_{\mu\nu}^{-},\overline{U_n}\rangle, \end{aligned} $$

(8.123)

which is manifestly symplectically invariant.

As mentioned above, we expect that U _n be related to a suitable derivative of a, possibly non-holomorphic, symplectic section, which we denote as

$$\displaystyle \begin{aligned} V(z,\overline{z})=\left(\begin{array}{c}L^I(z,\overline{z})\\M_J(z,\overline{z})\end{array}\right), \end{aligned} $$

(8.124)

and which will be closely related to the advertised holomorphic section $\mathcal {V}(z)$ mentioned in item 1.

In order to understand this better, it is necessary to address also item 2 in Sect. 8.2.1.2, namely, the claim that in supergravity the scalar field-dependent symplectic sections have to transform non-trivially under Kähler transformations (see (8.81) for $\mathcal {V}(z)$). The root of this behavior is the particular non-trivial Kähler transformation property of the fermions we already discovered in $\mathcal {N}=1$ supergravity. Repeating the arguments of Sect. 6.1.2 for each of the two supersymmetries, one now finds that the two gravitini $\psi _\mu ^{(1)}, \psi _\mu ^{(2)}$ have to transform under Kähler transformations as (cf. (6.18))

$$\displaystyle \begin{aligned} \psi_\mu^{(j)}\mapsto \exp\left[-\frac{i}{2}(\mbox{Im } h)\gamma_5\right]\psi_\mu^{(j)}, \qquad (j=1,2) \end{aligned} $$

(8.125)

which immediately implies

$$\displaystyle \begin{aligned} \epsilon^{(j)}\mapsto \exp \left[-\frac{i}{2}(\mbox{Im } h)\gamma_5\right]\epsilon^{(j)},\qquad (j=1,2) {} \end{aligned} $$

(8.126)

via $\delta \psi _{\mu }^{(j)}=\mathcal {D}_{\mu }\epsilon ^{(j)}+\ldots$, where 𝜖 ⁽ⁱ⁾ are the two Majorana spinor parameters.

This behavior is consistent with the SU(2)_R R-symmetry rotating the two $\psi _\mu ^{(j)}$ and the two 𝜖 ^(j) into each other and requires the introduction of Kähler covariant derivatives (cf. (6.16), (6.21)),

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathcal{D}_{[\nu}\psi_{\rho]}^{(j)}& \equiv&\displaystyle D_{[\nu}(\omega)\psi_{\rho]}^{(j)}+\frac{i}{2} Q_{[\nu}\gamma_5\psi_{\rho]}^{(j)} \end{array} \end{aligned} $$

(8.127)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathcal{D}_{\nu}\epsilon^{(j)}& \equiv&\displaystyle D_{\nu}(\omega)\epsilon^{(j)} + \frac{i}{2}Q_\nu\gamma_5\epsilon^{(j)} \end{array} \end{aligned} $$

(8.128)

with $Q_\nu (z,\overline {z})\equiv i/2[(\partial _{\overline {n}}K)\partial _{\nu }z^{\overline {n}}-(\partial _m K)\partial _{\nu }z^m]$.

While this is completely analogous to $\mathcal {N}=1$ supergravity, an important difference occurs for the gaugini. If we again identify 𝜖 ⁽¹⁾ with the original $\mathcal {N}=1$ supersymmetry parameter and (λ ^m(1), λ ^m(2)) = (χ ^m, λ ^m∕2), we expect a supersymmetry transformation rule with respect to 𝜖 ⁽¹⁾ of the form (cf. (8.119), (8.120))

(8.129)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta \lambda^{m(2)}_L& =&\displaystyle -\frac{1}{4}g^{m\overline{n}}\overline{f_n^I}\mbox{Im}\mathcal{N}_{IJ}F^{J-}_{\mu\nu}\gamma^{\mu\nu}\epsilon_{L}^{(1)}. {} \end{array} \end{aligned} $$

(8.130)

From the first equation and (8.126), we deduce

$$\displaystyle \begin{aligned} \lambda_{L}^{m(1)}\mapsto \exp\left[+\frac{i}{2}(\mbox{Im }h)\right]\lambda^{m(1)}_L,{} \end{aligned} $$

(8.131)

which must then also hold for $\lambda ^{m(2)}_L$ due to the assumed SU(2)_R R-symmetry. This, however, seems to be in conflict with the second equation (8.130), which naively would imply a transformation with a minus sign in the exponent due to the appearance of $\epsilon ^{(1)}_L$ instead of $\epsilon ^{(1)}_R$. In $\mathcal {N}=1$ supergravity, we encountered a similar problem for the supersymmetry transformations of the fermions in the presence of a non-trivial superpotential (see the discussion below Eq. (6.30)). This problem could be solved by assuming that the superpotential transforms non-trivially under Kähler transformations. To solve the analogous problem in (8.130), one likewise requires that the field-dependent term in front of γ ^μν transform non-trivially under Kähler transformations. In this term, the inverse scalar field metric $g^{m\overline {n}}$ is by construction Kähler invariant, and a Kähler transformation of $\mathcal {N}_{IJ}$ and/or $F_{\mu \nu }^{J-}$ would lead to Kähler non-invariances in the gauge field sectors. This leaves $f_{n}^I$ as the only reasonable candidate for a non-trivial Kähler transformation, which then has to be of the form

$$\displaystyle \begin{aligned} f_n^I\mapsto\exp\left[-i(\mbox{Im} h)\right] f_n^I \quad \Longleftrightarrow\quad U_n \mapsto \exp\left[-i(\mbox{Im} h)\right] U_n {} \end{aligned} $$

(8.132)

in order to yield the correct Kähler transformation of the right-hand side of (8.130).

Equation (8.132) now also implies that U _n cannot be just the partial derivative of a symplectic section, V , as that would also produce derivatives of h in (8.132). Instead, U _n can at most be a Kähler covariant derivative,

$$\displaystyle \begin{aligned} U_n=\mathcal{D}_n V \equiv \left(\partial_n+\frac{1}{2}(\partial_n K)\right) V, \end{aligned} $$

(8.133)

where

$$\displaystyle \begin{aligned} V=V(z,\overline{z})=\left(\begin{array}{c}L^I(z,\overline{z})\\M_J(z,\overline{z})\end{array}\right){} \end{aligned} $$

(8.134)

is a symplectic section that transforms as

$$\displaystyle \begin{aligned} V\mapsto \exp\left[-i(\mbox{Im} h)\right]V \end{aligned} $$

(8.135)

under Kähler transformations. This transformation property of V  depends on h(z) and $\overline {h}(\overline {z})$, which implies that V  cannot depend holomorphically on the coordinates z ^m of the scalar manifold, as indicated in (8.134).

We can however define a modified symplectic section, v, that does transform holomorphically under Kähler transformations,

$$\displaystyle \begin{aligned} \mathcal{V}(z)= \left(\begin{array}{c}X^I(z)\\F_J(z)\end{array}\right):= e^{-\frac{K(z,\overline{z})}{2}}V(z,\overline{z}) \quad \Rightarrow\quad \mathcal{V}(z)\mapsto \exp\left[-h(z)\right]\mathcal{V}(z) {} \end{aligned} $$

(8.136)

and hence may sensibly be restricted to have a holomorphic coordinate dependence, as indicated. We thus have

$$\displaystyle \begin{aligned} U_n=\mathcal{D}_n\left(e^{\frac{K}{2}}\mathcal{V}\right)\equiv \Big(\partial_n+\frac{1}{2}(\partial_n K)\Big) \Big(e^{\frac{K}{2}}v\Big)= e^{\frac{K}{2}}\Big(\partial_n+(\partial_n K)\Big)v \equiv e^{\frac{K}{2}}\mathcal{D}_n \mathcal{V} \end{aligned} $$

(8.137)

so that, in particular,

$$\displaystyle \begin{aligned} f_n^I=e^{\frac{K}{2}}\mathcal{D}_{n} X^I \equiv e^{\frac{K}{2}}\Big(\partial_n+(\partial_n K)\Big)X^I \end{aligned} $$

(8.138)

and similarly for h _Jn. We finally note that the holomorphicity of v, $\partial _{\overline {n}} v=0$, is equivalent to

$$\displaystyle \begin{aligned} \mathcal{D}_{\overline{n}}V\equiv \left(\partial_{\overline{n}}-\frac{1}{2}(\partial_{\overline{n}}K)\right)V=0. \end{aligned} $$

(8.139)

1.2 8.A.2 The Kähler Potential

After having discussed the necessity of a holomorphic symplectic section, $\mathcal {V}(z)$, we now come to the explicit form of the Kähler potential (8.82) mentioned in item 3 of Sect. 8.2.1.2. The analogy with Eq. (8.51) in rigid supersymmetry suggests that the Kähler potential should be related to the real symplectic invariant $i\langle \overline {\mathcal {V}}, \mathcal {V}\rangle = i(\overline {X^I}F_I-X^I\overline {F_I})$. As opposed to the rigid case, however, we now have the behavior (8.136) under Kähler transformations, which implies

$$\displaystyle \begin{aligned} i\langle\overline{\mathcal{V}},\mathcal{V}\rangle\longrightarrow e^{-(h+\overline{h})} i\langle \overline{\mathcal{V}},\mathcal{V}\rangle. \end{aligned} $$

(8.140)

Hence, $i\langle \overline {\mathcal {V}},\mathcal {V}\rangle$ does not transform the right way to serve as the Kähler potential, but the logarithm of it would do the job. We are thus led to the logarithmic expression (8.82).

Let us summarize what we have shown so far. The scalar manifolds, $\mathcal {M}_{\mathrm {vec}}$, of vector multiplets in $\mathcal {N}=2$ supergravity are Kähler manifolds that allow for a symplectic section, $V(z,\overline {z})=e^{K(z,\overline {z})/2}\mathcal {V}(z)$, that satisfies the following properties:

• $\partial _{\overline {m}} v=0 \quad \Longleftrightarrow \quad \mathcal {D}_{\overline {m}}V =0$

• $K=-\log i\langle \overline {\mathcal {V}},\mathcal {V}\rangle \quad \Longleftrightarrow \quad \langle V,\overline {V}\rangle =i$

• $\mathcal {V}(z)\mapsto e^{-h(z)}\mathcal {V}(z)$ under Kähler transformations,

where it is understood that $\mathcal {V}(z)$ must be such that the resulting metric $g_{m\overline {n}}=\partial _m\partial _{\overline {n}} K$ is positive definite for the z ^m-domain of interest.

Geometrically, the above Kähler transformation property of $\mathcal {V}(z)$ means that $\mathcal {V}(z)$ is a section not only in an Sp$(2(n_V+1),\mathbb {R})$ bundle but, similar to the superpotential in $\mathcal {N}=1$ supergravity, also in a holomorphic line bundle. This line bundle obeys a topological restriction such that the (special) Kähler manifold must actually be a (special) Hodge-Kähler manifold, as we discussed for $\mathcal {N}=1$ supergravity in Sect. 6.2.

1.3 8.A.3 The Existence of a Prepotential

The properties listed at the end of the previous subsection are still not sufficient to allow for a consistent embedding into $\mathcal {N}=2$ supergravity and have to be supplemented by the additional requirement

$$\displaystyle \begin{aligned} \langle \mathcal{D}_m \mathcal{V},\mathcal{D}_n \mathcal{V}\rangle =0,{} \end{aligned} $$

(8.141)

where $\mathcal {D}_{n} \mathcal {V} =(\partial _n +(\partial _n K))\mathcal {V}$. Due to the antisymmetry of the symplectic product, this condition is antisymmetric in m and n, and hence non-trivial only for n _V > 1, where (8.141) is necessary for the symmetry of the gauge kinetic matrix $\mathcal {N}_{IJ}$ and ensures its uniqueness, as we will discuss in Appendix 8.A.4. Moreover, for n _V > 1, (8.141) implies that there is always a symplectic transformation that brings $\mathcal {V}(z)$ into the familiar prepotential form

$$\displaystyle \begin{aligned} \mathcal{V}=\left(\begin{array}{c}X^I\\F_J\end{array}\right), \qquad \mbox{with} \quad F_J=\frac{\partial F(X)}{\partial X^J} {} \end{aligned} $$

(8.142)

for a suitable holomorphic function F(X) that exists at least locally and, unlike in rigid supersymmetry, is also restricted to be homogeneous of degree two. A simple way to understand the homogeneity requirement is provided by the behavior X ^I↦e ^−h X ^I and F _J↦e ^−h F _J, which is consistent only if the F _J = ∂F∕∂X ^J are homogeneous functions of the X ^I of degree one, implying homogeneity of degree two for F(X).

The generic necessity of performing a suitable symplectic rotation in supergravity before the prepotential form (8.142) is achieved is an important difference to rigid special Kähler geometry, where the section is of a prepotential form in any symplectic duality frame. This difference is ultimately related to the fact that in supergravity the (n _V + 1) components X ^I cannot be swapped for the n _V independent coordinates z ^m. This is best illustrated by a simple example [7]. Choosing, e.g., n _V = 1 with F(X) = −iX ⁰ X ¹ and special coordinates X ⁰ = 1, X ¹ = z, the section $\mathcal {V}(z)$ in the form (8.142) is

$$\displaystyle \begin{aligned} \mathcal{V}(z)=\left(\begin{array}{c}1\\z\\-iz\\-i\end{array}\right) \end{aligned} $$

(8.143)

which leads to the metric $g_{z\overline {z}}=(z+\overline {z})^{-2}$ corresponding to the manifold $\mathcal {M}_{\mathrm {vec}}=SU(1,1)/U(1)$. Performing now a symplectic transformation of the form

$$\displaystyle \begin{aligned} \mathcal{V} \mapsto S \mathcal{V}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0&0&0&-1\\ 0&0&1&0\\ 0&1&0&0 \end{array}\right) \mathcal{V}= \left(\begin{array}{c}1\\ i\\ -iz\\ z\end{array}\right), \end{aligned} $$

(8.144)

we see that the transformed section can no longer be of the form (8.142), as the lower two components cannot be functions of the upper two components. The prepotential form thus in general only holds in some duality frames but not in all of them. If one does not impose (8.141), on the other hand, one could even write down sections $\mathcal {V}(z)$ that can never be rotated into a prepotential form (8.142), no matter what symplectic transformation one uses. The proof that (8.141) ensures the existence of a prepotential basis is a bit technical [2, 4] and will therefore not be repeated here. Instead we content ourselves with showing the opposite direction, namely, that in a basis (8.142) with second-degree prepotential, condition (8.141) follows. As (8.141) is automatically true for n _V = 1, we assume n _V > 1 unless stated otherwise. We start by showing that (8.141) is implied by⁵

$$\displaystyle \begin{aligned} \langle \mathcal{V},\partial_m \mathcal{V}\rangle =0. {} \end{aligned} $$

(8.145)

To see this, we first note that $\langle \mathcal {V}, \mathcal {V} \rangle =0$ due to antisymmetry of the symplectic product, so that (8.145) can be equivalently written as

$$\displaystyle \begin{aligned} \langle \mathcal{V},\mathcal{D}_{m} \mathcal{V} \rangle \equiv \langle \mathcal{V},(\partial_m + (\partial_m K))\mathcal{V}\rangle=0. \end{aligned} $$

(8.146)

Acting with $\mathcal {D}_{n}$ and antisymmetrizing in n and m and using (8.145) and $\langle \mathcal {V}, \mathcal {V} \rangle =0$ then imply (8.141). Using now the prepotential form (8.142), one finds

$$\displaystyle \begin{aligned} \begin{array}{rcl} \langle \mathcal{V},\partial_m \mathcal{V}\rangle& =&\displaystyle X^I (\partial_{m} F_I)- F_I (\partial_m X^I) = X^I\left(\frac{\partial X^J}{\partial z^m} \partial_J\partial_I F\right)-\partial_m F \\ & =&\displaystyle \frac{\partial X^J}{\partial z^m} X^I\partial_I(\partial_J F)-\partial_m F= \frac{\partial X^J}{\partial z^m}\partial_J F-\partial_m F=0, \end{array} \end{aligned} $$

(8.147)

where we used that ∂ _J F = X ^I ∂ _I ∂ _J F is homogeneous of degree one. This then implies (8.141).

Before we come to the discussion of the gauge kinetic matrix and its relation to (8.141), we comment on the case n _V = 1. In that case, (8.141) is empty and hence does not imply the existence of a prepotential. If one uses instead the (for n _V = 1 non-equivalent) condition (8.145), on the other hand, the existence of a prepotential basis can be proven in a similar way as for n _V > 1 [2]. However, as the condition (8.145) is no longer implied by the required symmetry of the gauge kinetic matrix $\mathcal {N}_{IJ}$ (see below), there seems to be no physical reason that enforces (8.145). And indeed, in [4], models without an action and one vector multiplet were constructed where (8.145) does not hold.

1.4 8.A.4 The Gauge Kinetic Matrix

We finally come to the discussion of the gauge kinetic matrix and its relation to condition (8.141). More precisely, we would like to show that (8.141) is necessary for a sensible gauge kinetic matrix, $\mathcal {N}_{IJ}$, for n _V > 1, and that, in a prepotential basis, this matrix can be written as announced in (8.84). We already have derived the constraint

$$\displaystyle \begin{aligned} \overline{f^I_m}\mathcal{N}_{IJ}=\overline{h_{Jm}}, {} \end{aligned} $$

(8.148)

which follows from the requirement that the gaugino transformation law (8.120) be symplectically invariant and covariant with respect to scalar reparameterizations. As we will now show, the gravitino transformation law imposes the additional condition

$$\displaystyle \begin{aligned} L^I\mathcal{N}_{IJ}=M_J.{} \end{aligned} $$

(8.149)

Note that there is no complex conjugation involved in this equation, as opposed to (8.148). In order to motivate this equation, we recall the rigid supersymmetry transformations of the $\mathcal {N}=1$ gravitino multiplet discussed in Sect. 2.2.2,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta \psi_{\mu L}& =&\displaystyle - \gamma^\rho\epsilon_R\, F_{\mu\rho}^{-} \end{array} \end{aligned} $$

(8.150)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta A_{\mu}& =&\displaystyle \overline{\epsilon} \psi_\mu, \end{array} \end{aligned} $$

(8.151)

where we used γ _μνρ = i𝜖 _μνρσ γ ^σ γ ₅ to write the gravitino transformation in terms of the anti-self-dual field strength.

These transformations leave invariant the Lagrangian

$$\displaystyle \begin{aligned} \mathcal{L}=-\frac{1}{2}\overline{\psi}_{\mu}\gamma^{\mu\nu\rho}\partial_{\nu}\psi_{\rho}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}. \end{aligned} $$

(8.152)

This theory should be embeddable in pure $\mathcal {N}=2$ supergravity by making the supersymmetry parameter x-dependent and by coupling it to the $\mathcal {N}=1$ supergravity multiplet in the usual way. This will introduce a derivative of 𝜖 as an additional term into the gravitino transformation law and require spacetime covariantization, but as there are no scalar fields involved, there is not much room for a modification of the above existing terms from rigid $\mathcal {N}=1$ supersymmetry. And indeed, one can write the above transformation laws and the Lagrangian formally in terms of local special Kähler geometry for the special case n _V = 0. To this end, we use the prepotential $F(X^0)=-\frac {i}{2}(X^0)^2$, which leads to

$$\displaystyle \begin{aligned} \mathcal{V}=\left(\begin{array}{c}X^0\\F_0\end{array}\right)=\left(\begin{array}{c}X^0\\-iX^0\end{array}\right). \end{aligned} $$

(8.153)

This implies via (8.82) and (8.84)

$$\displaystyle \begin{aligned} K=-\log |X^0|{}^2, \qquad \mathcal{N}_{00}=-i \end{aligned} $$

(8.154)

so that, after choosing X ⁰ = 1, one has V = (L ⁰, M ₀)^T = (1, −i)^T and hence

$$\displaystyle \begin{aligned} \delta \psi_{\mu L}=L^0\mbox{Im} \mathcal{N}_{00}F_{\mu\rho}^{0-}\gamma^\rho\epsilon_R. \end{aligned} $$

(8.155)

This expression is symplectically invariant, because

$$\displaystyle \begin{aligned} L^0\mathcal{N}_{00}= -i=M_0 {} \end{aligned} $$

(8.156)

implies $L^0\mbox{Im}\mathcal {N}_{00}F_{\mu \rho }^{0-}=(1/2i)(M_0F_{\mu \rho }^{0-}-L^0G_{\mu \rho 0-})=(1/2i)\langle \mathbb {F}_{\mu \nu },V\rangle$.

For $\mathcal {N}=2$ supergravity coupled to n _V vector multiplets, the gravitino transformation law should then contain the analogous term

$$\displaystyle \begin{aligned} \delta\psi_{\mu L}^{(1)} \propto L^I\mbox{Im}\mathcal{N}_{IJ}F^{J-}_{\mu\rho}\gamma^\rho\epsilon_{R}^{(2)}, {} \end{aligned} $$

(8.157)

because this ensures that both sides transform with $e^{-\frac {i}{2}(\mbox{Im} h)}$ and are symplectic invariant provided the analogue of equation (8.156) holds, which is just (8.149).

The gauge kinetic matrix $\mathcal {N}_{IJ}$ thus has to obey the two constraints (8.148) and (8.149). In fact, these conditions determine $\mathcal {N}_{IJ}$ uniquely. To see this, we observe from the gaugino transformation (8.130) and the gravitino transformation (8.157) with respect to 𝜖 ⁽²⁾ that the n _V independent gaugini $\lambda ^{m(1)}_{L}g_{m\overline {n}}$ transform into $\overline {f^I_n}\mbox{Im}\mathcal {N}_{IJ}F_{\mu \nu }^{J-}$, whereas the gravitino $\psi ^{(1)}_\mu$ transforms into the linear combination $L^I\mbox{Im}\mathcal {N}_{IJ}F_{\mu \nu }^{I-}$. For any fixed scalar field value z ^m, these must therefore be (n _V + 1) linearly independent combination of vector field strengths. This implies that the following ((n _V + 1) × (n _V + 1))-matrix, A, must be invertible:

$$\displaystyle \begin{aligned} A:=(\overline{f^I_m},L^I), \end{aligned} $$

(8.158)

where I is the row index and m labels the first n _V columns, whereas L ^I is the last column. Defining similarly

$$\displaystyle \begin{aligned} B:=(\overline{h_{Im}},M_I), \end{aligned} $$

(8.159)

Eqs. (8.148) and (8.149) can be written as

$$\displaystyle \begin{aligned} A^T\cdot \mathcal{N}=B^T, \end{aligned} $$

(8.160)

so that, with the invertibility of A, one has a unique expression for the gauge kinetic matrix

$$\displaystyle \begin{aligned} \mathcal{N}=A^{T-1}\cdot B^T. \end{aligned} $$

(8.161)

This has to be symmetric, $\mathcal {N}-\mathcal {N}^T=0$, which is then equivalent to

$$\displaystyle \begin{aligned} B^TA-A^TB=0\Longleftrightarrow \left(\begin{array}{cc}\langle\overline{U_m},\overline{U_n}\rangle &\langle \overline{U_n},V\rangle\\\langle V,\overline{U_n} \rangle &\langle V,V\rangle\end{array}\right)=0. \end{aligned} $$

(8.162)

This implies

$$\displaystyle \begin{aligned} \langle U_m,U_n\rangle=0,\qquad \langle V,\overline{U_n}\rangle=0. \end{aligned} $$

(8.163)

The second of these equations simply follows from taking a covariant derivative of $\langle V,\overline {V}\rangle =i$, whereas the first one is equivalent to (8.141), which implies the existence of a basis with a prepotential for n _V > 1, as discussed earlier.

It remains to verify that in a basis with a prepotential $\mathcal {N}_{IJ}$ takes the form (8.84). To see this, we use

$$\displaystyle \begin{aligned} \begin{array}{rcl} \langle \overline{\mathcal{D}_{m}V}, \mathcal{V} \rangle& =&\displaystyle 0{} \end{array} \end{aligned} $$

(8.164)

$$\displaystyle \begin{aligned} \begin{array}{rcl} (\partial_m X^I)F_{IK}& =&\displaystyle \partial_m F_K{} \end{array} \end{aligned} $$

(8.165)

$$\displaystyle \begin{aligned} \begin{array}{rcl} X^IF_{IK}& =&\displaystyle F_K{} \end{array} \end{aligned} $$

(8.166)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \overline{f^I_n F_{IK}}& =&\displaystyle \overline{h_{Km}}. {} \end{array} \end{aligned} $$

(8.167)

The first of these equations is valid in any symplectic frame and equivalent to $\langle \overline {\mathcal {D}_{m}V},V\rangle =0$, which follows from $\langle \overline {V},V\rangle =-i$ and $\mathcal {D}_{\overline {m}}V=0$. The second uses the prepotential relations F _IK = ∂ _J ∂ _K F and F _K = ∂ _K F, which also imply the third equation due to the homogeneity of F of degree two. The fourth equation finally follows from the second and third upon using $f_{n}^{I}=e^{K/2}(\partial _{m}+(\partial _m K))X^K$ and h _Km = e ^K∕2(∂ _m + (∂ _m K))F _K.

We know that $\mathcal {N}_{IJ}$ is uniquely specified by (8.148) and (8.149), so it suffices to show that (8.84) satisfies these two equations in a prepotential basis. We begin with (8.148) and first show that

$$\displaystyle \begin{aligned} \begin{array}{rcl} 2i\overline{f_{m}^{I}}(\mbox{Im} F_{IK})X^K& =&\displaystyle \overline{f_{n}^{I}}F_{IK}X^K -\overline{f_{n}^{I} F_{IK}}X^K \end{array} \end{aligned} $$

(8.168)

(8.169)

The second term on the right-hand side of (8.84) thus does not contribute to (8.148), which then follows from $\overline {f^{I}_{m}}\mathcal {N}_{IK}=\overline {f^I_m F_{IK}}=\overline {h_{Km}}$.

Equation (8.149), finally, follows from (L ^I, M _K) = e ^K∕2(X ^I, F _K) and (8.166), with the non-antiholomorphic piece in $\mathcal {N}_{IJ}$ now playing a crucial role. This term is thus the unique extension of $\overline {F_{IJ}}$ that ensures (8.149) (i.e., ultimately the symplectic invariance of the gravitino transformation law) without destroying (8.148) (i.e., the symplectic invariance of the gaugino transformation law).

8.B Appendix: Quaternionic-Kähler vs. Hyper-Kähler Manifolds of Hypermultiplets

In this appendix we show in more detail how the invariance of the action implies a non-trivial SU(2) curvature for hypermultiplets in $\mathcal {N}=2$ supergravity and hence leads to the difference between hyper-Kähler geometry and quaternionic-Kähler geometry [9].

1.1 8.B.1 Sp(n _H) × SU(2)-Adapted Vielbein

To begin with, we consider the real 4n _H-dimensional scalar manifold, $\mathcal {M}_{\mathrm {hyper}}$, of n _H $\mathcal {N}=2$ hypermultiplets and parameterize it locally by 4n _H real scalar fields q ^X (X, Y, … = 1, …, 4n _H). Denoting an ordinary vielbein on $\mathcal {M}_{\mathrm {hyper}}$ by $\mathfrak {f}_{X}^{\varGamma }(q)$ with flat SO(4n _H)-indices Γ, Δ, … = 1, …4n _H, the metric components, h _XY(q), can be expressed as

$$\displaystyle \begin{aligned} h_{XY}(q)=\mathfrak{f}_{X}^{\varGamma}(q)\delta_{\varGamma\varDelta}\mathfrak{f}_{Y}^{\varDelta}(q). \end{aligned} $$

(8.170)

We now fix a particular point $p\in \mathcal {M}_{\mathrm {hyper}}$ and switch to a new basis of the vielbein at that point p. To this end, we split the vielbein components into two sets,

$$\displaystyle \begin{aligned} (\mathfrak{f}_X^\varLambda)=\left(\begin{array}{c}\mathfrak{f}_{X}^{1A}\\\mathfrak{f}_{X}^{2A}\end{array}\right), \qquad (A,B,\ldots =1,\ldots,2n_{H}). \end{aligned} $$

(8.171)

We then make a basis change with the non-orthogonal matrix (cf. (8.106) for the definition of E)

$$\displaystyle \begin{aligned} M:=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}i{\mathbf{1}}_{2n_H}& -iE\\ {\mathbf{1}}_{2n_{H}} & E\end{array}\right) \Leftrightarrow M^{-1}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-i {\mathbf{1}}_{2n_H}& {\mathbf{1}}_{2n_{H}}\\-iE & -E\end{array}\right), \end{aligned} $$

(8.172)

which satisfies

$$\displaystyle \begin{aligned} M^T M=e \otimes E=\left(\begin{array}{cc}0 & E\\ -E & 0\end{array}\right), \qquad (M^{-1})^{\ast}=(e\otimes E) M^{-1}.{} \end{aligned} $$

(8.173)

More precisely, we define a new vielbein at p,

$$\displaystyle \begin{aligned} f_{X}=\left(\begin{array}{c}f_{X}^{1A}\\f_{X}^{2A}\end{array}\right), \end{aligned} $$

(8.174)

by

$$\displaystyle \begin{aligned} f_X=M^{-1} \mathfrak{f}_{X}\leftrightarrow\begin{array}{l}f_{X}^{1}=\frac{1}{\sqrt{2}}(-i\mathfrak{f}_{X}^1 + \mathfrak{f}_{X}^{2})\\ {} f_{X}^{2}=\frac{1}{\sqrt{2}}(-iE\mathfrak{f}_{X}^{1}-E\mathfrak{f}_{X}^{2})\end{array}{} \end{aligned} $$

(8.175)

In terms of this new vielbein, we then have, with Eq. (8.173),

$$\displaystyle \begin{aligned} h_{XY}=\mathfrak{f}_{X}^{T}\mathfrak{f}_{Y}=f_{X}^{T}M^T M f_{Y}=f_X^T(e\otimes E)f_Y=f_X^{iA}e_{ij}E_{AB}f_{y}^{jB},{} \end{aligned} $$

(8.176)

as well as the reality condition

$$\displaystyle \begin{aligned} f_X^{\ast}=\left(\begin{array}{c}f_{X}^{1A}\\f_X^{2A}\end{array}\right)^{\ast}=M^{-1 \ast}\mathfrak{f}_{X}=(e\otimes E)M^{-1}\mathfrak{f}_{X}=(e\otimes E)f_X \end{aligned} $$

(8.177)

i.e.,

$$\displaystyle \begin{aligned} (f_{X}^{iA})^{\ast}=f_{X}^{jB}e_{ji}E_{BA}=:f_{X i A}.{} \end{aligned} $$

(8.178)

Equation (8.176) tells us that the new vielbein $f_{X}^{iA}$ is adapted to an Sp(n _H) × SU(2) split of the tangent space at the point p. In Sect. 8.2.2, we used the linearized supersymmetry transformations (8.102), (8.103) to argue that the tangent space group of the scalar manifold, $\mathcal {M}_{\mathrm {hyper}}$, of n _H $\mathcal {N}=2$ hypermultiplets should be contained in Sp(n _H) × SU(2) ∈ SO(4n _H). In other words, the above Sp(n _H) × SU(2) covariant form of the vielbein at p can actually be extended smoothly across all of $\mathcal {M}_{\mathrm {hyper}}$. We can thus consistently use a vielbein of the adapted form $f_{X}^{iA}$ for all points.

The reality condition (8.178), on the other hand, indicates that it is natural to denote complex conjugation by lowering the SU(2) indices, i, j, …, with e _ij and the Sp(n _H) indices, A, B, …, by E _AB according to the convention

$$\displaystyle \begin{aligned} V^i=e^{ij}V_{j}, \qquad V_{i} = V^{j}e_{ji}; \qquad \qquad V^A=E^{AB}V_B, \qquad V_A=V^B E_{BA} \end{aligned} $$

(8.179)

where $e^{ij}e_{ik}=\delta _{k}^{j}$ and $E^{AB}E_{AC}=\delta _C^B$.

Before we come to the supersymmetry transformation laws, we note two additional contraction identities,

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_X^{iA}f_{Y jA}+f_Y^{iA}f_{XjA}& =&\displaystyle h_{XY}\delta^i_j{} \end{array} \end{aligned} $$

(8.180)

$$\displaystyle \begin{aligned} \begin{array}{rcl} h^{XY}f_{X}^{iA}f_{YjB}& =&\displaystyle \delta^{i}_{j}\delta^{A}_{B}{} \end{array} \end{aligned} $$

(8.181)

The first of these is equivalent to $f^{iT}_{X}f^{j\ast }_{Y}+f^{iT}_{Y}f^{j\ast }_{X}=h_{XY}\delta ^{i}_{j}$, which may be verified by using (8.175), the reality of $\mathfrak {f}_X$, and $\mathfrak {f}_{X}^{1T}\mathfrak {f}^{1}_{Y}+\mathfrak {f}_{X}^{2T}\mathfrak {f}_{Y}^{2}=\mathfrak {f}_{X}^{T}\mathfrak {f}_{Y}=h_{XY}$. The second identity is equivalent to $h^{XY}f_X f_Y^{\dagger }={\mathbf {1}}_{4n_{H}}=\mathbf {1}{{ }_2}\otimes {\mathbf {1}}_{2n_{H}}$, which follows from (8.175), the reality of $\mathfrak {f}_{X}$, and the relations $h^{XY}\mathfrak {f}_{X}\mathfrak {f}_{Y}^{T}={\mathbf {1}}_{4n_{H}}$ and $M^{-1}M^{-1 \dagger }={\mathbf {1}}_{4n_{H}}$.

Remark: In the literature, one often also finds the statement that the vielbein $f_{X}^{iA}$ satisfies the additional relation

$$\displaystyle \begin{aligned} f_{X}^{iA}f_{YiB}+f_{Y}^{iA}f_{XiB}=\frac{1}{n_{H}}h_{XY}\delta^A_B. \qquad \mbox{(False!)}{} \end{aligned} $$

(8.182)

This equation, however, is false, except for the special case n _H = 1. This can be verified in many ways, e.g., contracting (8.182) with $f^{X}_{kC}f^{YlD}$, which, using (8.181), results in

$$\displaystyle \begin{aligned} \delta_{C}^{A}\delta_{B}^{D}+E^{AD}E_{BC}=\frac{1}{n_{H}}\delta_{C}^{D}\delta_{B}^{A}. \end{aligned} $$

(8.183)

Choosing A = C ≠ D = B such that E ^AD = 0 (e.g., C = 1, D = B = 4 for $E={\mathbb 1}_{n_H}\otimes e$), one finds a contradiction. Alternatively, one can contract (8.182) with $f_{Z}^{kB}$ and use (8.180), (8.182), and again (8.180) to arrive at

$$\displaystyle \begin{aligned} \left(1-\frac{1}{n_{H}}\right)h_{YZ}f_{X}^{kA} +\left(1-\frac{1}{n_{H}}\right)h_{XZ}f_{Y}^{kA} +\left(1-\frac{1}{n_{H}}\right)h_{XY}f_{Z}^{kA} =0 \end{aligned} $$

(8.184)

At the origin of Riemannian normal coordinates with h _XY = δ _XY, one can then take X = Y ≠ Z to infer that, for n _H > 1, $f_{Z}^{kA}=0$ for all Z ≠ X = Y . Contracting this with $f_{Z'kA}$ and setting Z = Z′ then would give the contradiction h _ZZ = δ _ZZ = 0. Yet another way to show that (8.182) cannot hold for n _H > 1 is to go to the origin of Riemannian normal coordinates with h _XY = δ _XY and $\mathfrak {f}_{X}^{\varGamma }=\delta _{X}^{\varGamma }$ and to use directly (8.175) for, e.g., X = Y = 1. This would lead (using $E={\mathbf {1}}_{n_H}\otimes e$) to $\delta _{1}^A\delta _1^B-\delta ^{AC}E_{C1}E_{1D}\delta ^{DB}=\delta _{1}^{A}\delta _{1}^{B}+\delta _{2}^{A}\delta _{2}^{B} =(1/n_{H})\delta ^{A}_{B}$, which is not a valid identity for n _H > 1.

1.2 8.B.2 Holonomy and Curvature

Up to now, we have shown that the 4n _H-dimensional scalar manifold, $\mathcal {M}_{\mathrm {hyper}}$, of n _H hypermultiplets in $\mathcal {N}=2$ supersymmetry admits a vielbein, $f_{X}^{iA}(q)$, that is adapted to the Sp(n _H) × SU(2) structure of $\mathcal {M}_{\mathrm {Hyper}}$ and that satisfies (8.176), (8.178), (8.180), and (8.181). We also saw that the reality condition (8.178) motivates the convention that complex conjugation raises or lowers the SU(2) index i and the Sp(n _H)-index A. According to our discussion around (8.102) and (8.103), the left-handed chiral fermions of hypermultiplets and the left-handed supersymmetry parameter transform in the fundamental representations of, respectively, Sp(n _H) and SU(2), so that we likewise use the analogous convention that charge conjugation lowers the corresponding indices, i.e., we set

$$\displaystyle \begin{aligned} \begin{array}{rcl} \zeta^A:=\chi^{A}_{L}, & &\displaystyle \qquad \zeta_{A}:=\chi^{\overline{A}}_{R} \end{array} \end{aligned} $$

(8.185)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \epsilon^i:=\epsilon^{(i)}_{L}, & &\displaystyle \qquad \epsilon_{i}:=\epsilon^{(i)}_{R}, \end{array} \end{aligned} $$

(8.186)

where, in contrast to the vielbein convention (8.178), the lower indices should not be thought of as arising from lowering the indices with e _ij or E _AB, as that cannot change the chirality.

The proper generalization of the linearized transformation laws (8.102)–(8.105) can then be written in the form

(8.187)

(8.188)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta q^{X}& =&\displaystyle -if_{iA}^{X}(q)\overline{\epsilon}^{i}\zeta^{A}+if^{XiA}(q)\overline{\epsilon}_{i}\zeta_{A}.{} \end{array} \end{aligned} $$

(8.189)

In fact, by setting

$$\displaystyle \begin{aligned} \begin{array}{rcl} (q^X)&=&\left(\begin{array}{c}q^{\widetilde{X}}\\q^{X'}\end{array}\right)=\left(\begin{array}{c}\sqrt{2}\delta_{m}^{\widetilde{X}}\mbox{Re }\phi^{m}\\\sqrt{2}\delta_{m}^{X'}\mbox{Im }\phi^{m}\end{array}\right) \end{array} \end{aligned} $$

(8.190)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \Longleftrightarrow\phi^{m}&=&\frac{1}{\sqrt{2}}(\delta_{\widetilde{X}}^{m}q^{\widetilde{X}}+i\delta_{X'}^{m}q^{X'}), \end{array} \end{aligned} $$

(8.191)

where $\widetilde {X},\widetilde {Y},\ldots =1,\ldots ,2n_{H}$, and X′, Y ′, … = 1, …, 2n _H, and by going to the origin of Riemann normal coordinates with the choice $\mathfrak {f}_{X}^{\varGamma }=\delta _{X}^{\varGamma }$, i.e., $\mathfrak {f}_{\widetilde {X}}^{1A}=\delta _{\widetilde {X}}^{A}$, $\mathfrak {f}_{X'}^{2A}=\delta _{X'}^{A}$ and $\mathfrak {f}_{\widetilde {X}}^{2A}=\mathfrak {f}_{X'}^{1A}=0$, the relation (8.175) brings (8.187)–(8.189) to the form (8.102)–(8.105).

Using (8.189), the commutator of two supersymmetry transformations of q ^X is, to lowest order in fermion fields,

$$\displaystyle \begin{aligned}{}[ \delta_{\eta},\delta_\epsilon ] q^X=\frac{1}{2}(f_{iA}^{X}f_{Y}^{jA}+f^{XjA}f_{YiA})(\overline{\epsilon}^{i}\gamma^{\mu}\eta_{j})\partial_{\mu}q^Y + c.c., \end{aligned} $$

(8.192)

which is equal to the desired result $\frac {1}{2}(\overline {\epsilon }^i\gamma ^{\mu }\eta _{i})\partial _{\mu }q^X+c.c.$ precisely when (8.180) holds. For the fermions ζ ^A, on the other hand, one uses the Fierz identities

$$\displaystyle \begin{aligned} \begin{array}{rcl} \epsilon_{i}\overline{\eta}^{j}& =&\displaystyle -\frac{1}{2}\overline{\eta}^{j}\gamma^{\nu}\epsilon_{i}\gamma_{\nu}P_{L} \end{array} \end{aligned} $$

(8.193)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \epsilon_{i}\overline{\eta}_j& =&\displaystyle -\frac{1}{2}\overline{\eta}_{j}\epsilon_{i}P_R+\frac{1}{4}\overline{\eta}_{j}\gamma_{\nu\rho}\epsilon_{i}\gamma^{\nu\rho}P_{R} \end{array} \end{aligned} $$

(8.194)

to obtain for the terms involving ∂ _μ ζ ^A,

$$\displaystyle \begin{aligned} \begin{array}{rcl} [ \delta_{\eta},\delta_{\epsilon} ]\zeta^{A}& =&\displaystyle -\frac{1}{4}f_X^{iA}f_{jB}^{X}(\overline{\eta}^{j}\gamma^{\nu}\epsilon_{i})\gamma^\mu\gamma_{\nu}\partial_{\mu}\zeta^{B}\\ & &\displaystyle +\frac{1}{4}f_{X}^{iA}f^{XjB}\left(\overline{\eta}_{j}\epsilon_{i}\gamma^{\mu}\partial_{\mu}\zeta_{B}-\frac{1}{2}\overline{\eta}_{j}\gamma_{\nu\rho}\epsilon_{i}\gamma^{\mu}\gamma^{\nu\rho}\partial_{\mu}\zeta_{B}\right) - (\epsilon \leftrightarrow \eta) \end{array} \end{aligned} $$

Using the linearized field equation and $\gamma ^\mu \gamma _{\nu }=-\gamma _{\nu }\gamma ^{\mu }+2\delta ^{\mu }_{\nu }$, this becomes

$$\displaystyle \begin{aligned} \begin{array}{rcl} [ \delta_{\eta},\delta_{\epsilon} ]\zeta^{A}& =&\displaystyle -\frac{1}{2}f_X^{iA}f_{jB}^{X}(\overline{\eta}^{j}\gamma^{\mu}\epsilon_{i} -\overline{\epsilon}^{j}\gamma^{\mu}\eta_{i} )\partial_{\mu}\zeta^{B}\\ & &\displaystyle -\frac{1}{8}f_{X}^{iA}f^{XjB}\left(\overline{\eta}_{j}\gamma_{\nu\rho}\epsilon_{i} -\overline{\epsilon}_{j}\gamma_{\nu\rho}\eta_{i}\right) \gamma^{\mu}\gamma^{\nu\rho}\partial_{\mu}\zeta_{B}. \end{array} \end{aligned} $$

The spinor bilinear in the second line is symmetric under exchange of i and j due to (1.44) and hence vanishes upon contraction with the vielbein terms due to (8.181), which also yields the desired result $\frac {1}{2}\overline {\epsilon }^{i}\gamma ^{\mu }\eta _{i}\partial _{\mu }\zeta ^{A} + c.c.$ for the first line. This shows that (8.180), (8.181) essentially ensure the closure of the supersymmetry algebra and that an equation of the (incorrect) form (8.182) is indeed not required.

We now come to the final part and discuss the invariance of the action. This will tell us that the connection compatible with the Sp(n _H) × SU(2) structure of $\mathcal {M}_{\mathrm {Hyper}}$ is actually the torsion-free Levi–Civita connection and that the SU(2) curvature must be non-trivial in supergravity.

Starting point is the kinetic term of the hypermultiplet fields,

(8.195)

where $\mathcal {D}_{\mu }$ is the spacetime and Sp(n _H) covariant derivative with respect to an Sp(n _H) connection compatible with the Sp(n _H) × SU(2) structure on $\mathcal {M}_{\mathrm {hyper}}$. In the following, we will in general denote by $\mathcal {D}_{\mu }$ the covariant derivative with respect to spacetime coordinate transformations, local Lorentz transformations, local composite Sp(n _H) × SU(2) transformations, and general field reparameterizations $q^X\rightarrow \widetilde {q}^{X}(q)$.

We first consider the supersymmetry variation of the scalar kinetic term with respect to the scalar fields,

(8.196)

where we used partial integration between the first and second line and $\mathcal {D}_{\mu }\partial ^{\mu }q^{Y}=\nabla _{\mu }\partial ^\mu q^Y + \varGamma _{XZ}^{Y}\partial _{\mu }q^X\partial ^{\mu }q^{Z}=\partial _{\mu } q^Y +\varGamma _{\mu \nu }^{\mu }\partial ^{\nu }q^Y+ \varGamma _{XZ}^{Y}\partial _{\mu }q^X\partial ^{\mu }q^{Z}$.

Now consider the variation of the fermionic term due to the variation of the fermions,

(8.197)

where we used partial integration and (1.44) in the first line. Using γ ^μ γ ^ν = γ ^μν + g ^μν and that $\mathcal {D}_{[\mu }\partial _{\nu ]}q^X=0$ due to the symmetry of the Christoffel symbols $\varGamma _{\mu \nu }^{\rho }$ and $\varGamma _{XY}^{Z}$, the second term becomes

$$\displaystyle \begin{aligned} if_{XiA}\overline{\zeta}^{A}\epsilon^{i}\mathcal{D}_{\mu}\partial^{\mu}q^X+c.c.=-h_{XZ}\delta q^Z \mathcal{D}_{\mu}\partial^{\mu}q^X =-\delta[e^{-1}\mathcal{L}_{\mathrm{scalar}}], \end{aligned} $$

(8.198)

and hence

(8.199)

with the supercurrent

(8.200)

The vanishing of the first term in (8.199) requires

$$\displaystyle \begin{aligned} \mathcal{D}_{X}f_{Y}^{iA}\equiv \partial_{X}f_{Y}^{iA}-\varGamma_{XY}^{Z}f_{Z}^{iA}+f_{Y}^{jA}\omega_{Xj}{}^{i}+f_{Y}^{iB}\omega_{XB}{}^{A}=0, \end{aligned} $$

(8.201)

where $\mathcal {D}_{X}$ contains the Sp(n _H) and SU(2) connections, ω _XA ^B(q) and ω _Xi ^j(q), and the Christoffel connection, $\varGamma _{XY}^{Z}(q)$, on $\mathcal {M}_{\mathrm {hyper}}$, as indicated. Just as for the spin connection (cf. (3.18) in Sect. 3.2), this equation is simply the statement that the Sp(n _H) × SU(2)-compatible connection is equivalent to the torsion-free Levi–Civita connection, and we have Sp(n _H) × SU(2) holonomy with respect to the Levi–Civita connection, as announced earlier.

To eliminate the second term in (8.199), we add the usual Noether term to the Lagrangian,

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{Noether}}=-\mathcal{J}_i^\mu\psi^i_{\mu}+c.c. \end{aligned} $$

(8.202)

for which the gravitino variation $\delta \psi _{\mu }^{i}=\mathcal {D}_{\mu }\epsilon ^i+\ldots$ just leads to the negative of the last term in (8.199). Just as in our discussion for $\mathcal {N}=1$ supergravity, the variation of the matter fields in $\mathcal {J}_{i}^{\mu }$ itself leads to terms involving the energy momentum tensor and is cancelled by variations of the metric but also to a term involving an antisymmetrized product of three gamma matrices that cannot be so absorbed. It is instead cancelled by the non-trivial composite connection terms (here the composite SU(2) connection) in the gravitino variation of the kinetic term of the gravitini. To see how this works in detail, we use again γ ^μ γ ^ν = γ ^μν + g ^μν and (1.44) to write

$$\displaystyle \begin{aligned} e^{-1}\mathcal{L}_{\mathrm{Noether}}= -if_{XiA}(\partial_{\nu}q^{X})\overline{\psi}^i_\mu(-\gamma^{\mu\nu}+g^{\mu\nu})\zeta^{A} +c.c.{} \end{aligned} $$

(8.203)

The variation of ζ ^A introduces one additional gamma matrix, so that a term involving γ ^μνρ can only arise from the γ ^μν-term in (8.203). We thus have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta_{\zeta}[e^{-1}\mathcal{L}_{\mathrm{Noether}}]|{}_{\gamma^{\mu\nu\rho}}& =&\displaystyle if_{XiA}(\partial_{\nu}q^{X})\overline{\psi}_{\mu}^{i}\gamma^{\mu\nu}\left(\frac{i}{2}f_{Y}^{jA}\gamma^{\rho}\partial_{\rho}q^Y\epsilon_{j}\right)|{}_{\gamma^{\mu\nu\rho}}+c.c.\\ & =&\displaystyle -\frac{1}{2}f_{XiA}f_{Y}^{jA}(\partial_{\nu}q^X)(\partial_{\rho}q^Y)\overline{\psi}^{i}_{\mu}\gamma^{\mu\nu\rho}\epsilon_{j}+c.c.{} \end{array} \end{aligned} $$

(8.204)

This is cancelled by the gravitino variation $\delta \psi _{\mu }^{i}=\mathcal {D}_{\mu }\epsilon ^i+\ldots$ in the kinetic term $\mathcal {L}_{\mathrm {gravitino}}=-\overline {\psi }^i_{\mu }\gamma ^{\mu \nu \rho }\mathcal {D}_{\nu }\psi _{\rho i}$,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta\mathcal{L}_{\mathrm{gravitino}}& =&\displaystyle -\overline{\psi}^i_{\mu}\gamma^{\mu\nu\rho}\mathcal{D}_{\nu}\mathcal{D}_{\rho}\epsilon_{i}+c.c. \end{array} \end{aligned} $$

(8.205)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & =&\displaystyle -\frac{1}{2}R_{XYi}{}^{j}(\partial_{\nu}q^X)(\partial_{\rho}q^Y)\overline{\psi}_{\mu}^i\gamma^{\mu\nu\rho}\epsilon_{j}, \end{array} \end{aligned} $$

(8.206)

where R _XYi ^j is the composite SU(2) curvature. Comparing with (8.204), we conclude that

$$\displaystyle \begin{aligned} R_{XYi}{}^{j}=M_{P}^{-2}f_{[XiA}f_{Y]}^{jA},{} \end{aligned} $$

(8.207)

where we have reinstalled the Planck mass, which originates from $\delta \psi _{\mu }^{i}\propto M_{P}$ and $\mathcal {L}_{\mathrm {Noether}}\propto M_{P}^{-1}$. Using the explicit relation (8.175) to the standard SO(4n _H) vielbein $\mathfrak {f}_{X}^{\varGamma }$, one can verify that the right-hand side of the above equation (8.207) is not identically zero and hence that the SU(2) curvature in $\mathcal {N}=2$ supergravity must be non-trivial. Obviously, the above reasoning becomes empty in the case of rigid supersymmetry, and the SU(2) curvature becomes flat, as also follows from (8.207) by formally sending M _P →∞.

Exercises

8.1

Construct the Kähler potential, the metric, and the Killing vectors and compute the action of the isometries on the symplectic sections for the STU model, i.e., for the prepotential

$$\displaystyle \begin{aligned}F(X) = \frac{X^1 X^2 X^3}{X^0}.\end{aligned}$$