Four-Dimensional $N=2$ Matter-Coupled Supergravity

Adopt the two-derivative four-dimensional $N=2$ component framework with one gravity multiplet, $n_V$ vector multiplets whose scalar manifold is special Kähler, and $n_H$ hypermultiplets whose scalar manifold is quaternionic-Kähler; fix a selected electric frame and the local symplectic section $L^{\mathrm{\Lambda }}$ and its covariant derivatives $f_i^{\mathrm{\Lambda }}$. A gauging is specified by Killing vectors on both scalar factors and the corresponding moment maps; the action is the on-shell matter-coupled supergravity action in the source convention. This item does not cover tensor multiplets, dyonic embedding-tensor gaugings, or a universal off-shell action.

Use the source normalization $M_P=1$ and denote the gauge coupling by $g$. In the electric frame, $L^{\mathrm{\Lambda }}$, $k_{\mathrm{\Lambda }}^i,k_{\mathrm{\Lambda }}^u$, and $P_{\mathrm{\Lambda }}^x$ denote the electric section components, scalar Killing fields, and quaternionic triplet moment maps of the local scalar-data convention above. The adopted gauging-dependent fermion shifts are $S_{AB}=\frac{i}{2}({\sigma }^x{)}_A{}^C{ϵ}_{BC}{P}_{\mathrm{\Lambda }}^x{L}^{\mathrm{\Lambda }},$ $W^{iAB}={ϵ}^{AB}{k}_{\mathrm{\Lambda }}^i\overline{L^{\mathrm{\Lambda }}}-i({\sigma }^x{)}_C{}^B{ϵ}^{CA}{P}_{\mathrm{\Lambda }}^x{g}^{i\bar{j}}{\bar{f}}_{\bar{j}}^{\mathrm{\Lambda }},N_{\alpha }{}^A=2{\mathcal{U}}^A{}_{u\alpha }{k}_{\mathrm{\Lambda }}^u\overline{L^{\mathrm{\Lambda }}}\mathrm{.}$ Here $A,B=1,2$ are $SU(2)$ indices, $\alpha$ is a hyperino-frame index, ${ϵ}_{AB}$ is the antisymmetric $SU(2)$ invariant tensor with ${ϵ}_{12}=+1$, and ${\sigma }^x$ are the Pauli matrices. The lowered hyperino frame is ${\mathcal{U}}^A{}_{u\alpha }=C_{\alpha \beta }{\mathcal{U}}_u^{A\beta }$ with conjugate ${\bar{\mathcal{U}}}^{\alpha }{}_{vB}=({\mathcal{U}}^B{}_{v\alpha }{)}^{\ast }$, using the scalar-data reality convention. The source raises and lowers $SU(2)$ indices by $v_A={ϵ}_{AB}{v}^B$ and $v^A={ϵ}^{BA}{v}_B$, with ${ϵ}^{12}={ϵ}_{12}=+1$; its hyperino symplectic indices obey $v_{\alpha }=C_{\alpha \beta }{v}^{\beta }$ and $v^{\alpha }=C^{\beta \alpha }{v}_{\beta }$, with $C^{\alpha \beta }$ numerically the same canonical antisymmetric matrix as $C_{\alpha \beta }$. These are the index-order rules of source equation (2.27), footnote 3. Transformations contain $gS,gW,gN$. These are adopted source component conventions, not consequences of scalar geometry alone.

Andrianopoli et al., §§4–9

Andrianopoli et al., §5

Andrianopoli et al., §2

Four-Dimensional $N=2$ Electric-Frame Scalar Data

The printed $N=2$ gaugino shift in hep-th/9605032v1, equation (8.23), has a relative-sign typo. The shift above uses the corrected sign stated in arXiv:1508.01474v2, §2, equations (2.31)–(2.32); the scalar-potential normalization is independently checked against hep-th/9605032v1, equation (9.49). The resulting local claim remains in the electric frame and $M_P=1$ source convention.

The convention input is Supergravity Conventions And Normalizations at the exact four-dimensional Einstein-frame and Planck normalization used above. The result remains conditional on its adopted supergravity framework.