# On the Geometric Approach to the Boundary Problem in Supergravity

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## Abstract

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## 1. Preamble

## 2. Geometric Approach in the Presence of a Non-Trivial Boundary of Spacetime

#### 2.1. The Principle of Rheonomy

- They are defined off-shell from their symmetry properties, as in (2), in terms of Lorentz and R-symmetry covariant exterior derivatives of the superfield 1-forms ${\mu}^{\mathcal{A}}$, and have to satisfy consistency constraints given by the closure of Bianchi identities (${\mathrm{d}}^{2}=0$).
- However, being 2-forms in superspace, they can also be expanded along the supervielbein basis $\{{V}^{a},{\psi}_{A}\}$ of superspace,$${R}^{\mathcal{A}}={{R}^{\mathcal{A}}}_{ab}(x,\theta ){V}^{a}\wedge {V}^{b}+{{R}^{\mathcal{A}}}_{a\phantom{\rule{0.166667em}{0ex}}\alpha A}(x,\theta ){V}^{a}\wedge {\psi}^{\alpha A}+{{R}^{\mathcal{A}}}_{\alpha A\phantom{\rule{0.166667em}{0ex}}\beta B}(x,\theta ){\psi}^{\alpha A}\wedge {\psi}^{\beta B}\phantom{\rule{0.166667em}{0ex}},$$

#### 2.2. Supersymmetry Invariance of the Action

## 3. Facing the Boundary Problem in AdS${}_{\mathbf{4}}$ Pure Supergravities

#### 3.1. $\mathcal{N}=1$ Case

#### 3.2. $\mathcal{N}=2$ Case

## 4. Supersymmetry Invariance of Flat Supergravity with Boundary

#### 4.1. Symmetry Structure of Asymptotically Flat Spacetimes

#### 4.2. Construction of the Flat Model

#### 4.3. Recovering Flat Supergravity with Boundary from Super-AdS${}_{4}$

- We introduce a torsionful spin connection$${\widehat{\omega}}^{ab}\equiv {\omega}^{ab}+\frac{1}{{\ell}^{2}}{A}^{ab}\phantom{\rule{0.166667em}{0ex}},$$$$\begin{array}{cc}\hfill {\mathcal{R}}^{ab}& \to {\widehat{\mathcal{R}}}^{ab}=\mathrm{d}{\omega}^{ab}+{{\omega}^{a}}_{c}{\omega}^{cb}+\frac{1}{{\ell}^{2}}{\mathcal{D}}_{\left(\omega \right)}{A}^{ab}+\frac{1}{{\ell}^{4}}{{A}^{a}}_{c}{A}^{cb}\equiv {\mathcal{R}}^{ab}+\frac{1}{{\ell}^{2}}{\mathbb{F}}^{ab}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {R}^{a}& \to {\widehat{R}}^{a}={\mathcal{D}}_{\left(\omega \right)}{V}^{a}+\frac{1}{{\ell}^{2}}{{A}^{a}}_{b}{V}^{b}-\frac{\mathrm{i}}{2}\overline{\psi}{\gamma}^{a}\psi \phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$${\mathbb{F}}^{ab}\equiv {\mathcal{D}}_{\left(\omega \right)}{A}^{ab}+\frac{1}{{\ell}^{2}}{{A}^{a}}_{c}{A}^{cb}\phantom{\rule{0.166667em}{0ex}}.$$
- We redefine the gravitino 1-form with the introduction of the new spinor 1-form $\chi $,$$\psi \to \psi +\frac{1}{\ell}\chi \phantom{\rule{0.166667em}{0ex}},$$$$\begin{array}{cc}\hfill {\widehat{R}}^{a}& \to {\mathfrak{R}}^{a}\equiv {\mathcal{D}}_{\left(\omega \right)}{V}^{a}-\frac{\mathrm{i}}{2}\overline{\psi}{\gamma}^{a}\psi +\frac{1}{{\ell}^{2}}{{A}^{a}}_{b}{V}^{b}-\frac{\mathrm{i}}{\ell}\overline{\psi}{\gamma}^{a}\chi -\frac{\mathrm{i}}{2{\ell}^{2}}\overline{\chi}{\gamma}^{a}\chi \phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill \rho & \to \widehat{\rho}={\mathcal{D}}_{\left(\omega \right)}\psi +\frac{1}{\ell}\left({\mathcal{D}}_{\left(\omega \right)}\chi +\frac{1}{4\ell}{A}^{ab}{\gamma}_{ab}\psi +\frac{1}{4{\ell}^{2}}{A}^{ab}{\gamma}_{ab}\chi \right)\equiv \rho +\frac{1}{\ell}\Phi \phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\Phi \equiv {\mathcal{D}}_{\left(\omega \right)}\chi +\frac{1}{4\ell}{A}^{ab}{\gamma}_{ab}\psi +\frac{1}{4{\ell}^{2}}{A}^{ab}{\gamma}_{ab}\chi \phantom{\rule{0.166667em}{0ex}}.$$$$\begin{array}{cc}\hfill {\mathcal{R}}^{ab}& \equiv \mathrm{d}{\omega}^{ab}+{{\omega}^{a}}_{c}{\omega}^{cb}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {\mathfrak{R}}^{a}& \equiv \mathcal{D}{V}^{a}-\frac{\mathrm{i}}{2}\overline{\psi}{\gamma}^{a}\psi +\frac{1}{{\ell}^{2}}{{A}^{a}}_{b}{V}^{b}-\frac{\mathrm{i}}{\ell}\overline{\psi}{\gamma}^{a}\chi -\frac{\mathrm{i}}{2{\ell}^{2}}\overline{\chi}{\gamma}^{a}\chi \phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill \rho & \equiv \mathcal{D}\psi \phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {\mathbb{F}}^{ab}& \equiv \mathcal{D}{A}^{ab}+\frac{1}{{\ell}^{2}}{{A}^{a}}_{c}{A}^{cb}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill \Phi & \equiv \mathcal{D}\chi +\frac{1}{4\ell}{A}^{ab}{\gamma}_{ab}\psi +\frac{1}{4{\ell}^{2}}{A}^{ab}{\gamma}_{ab}\chi \phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{bulk}}^{\ell}& =\frac{1}{4}{\u03f5}_{abcd}{R}^{ab}{V}^{c}{V}^{d}+\frac{1}{4{\ell}^{2}}{\u03f5}_{abcd}{\mathbb{F}}^{ab}{V}^{c}{V}^{d}-\overline{\psi}{\gamma}_{5}{\gamma}_{a}\rho {V}^{a}-\frac{1}{\ell}\overline{\psi}{\gamma}_{5}{\gamma}_{a}\Phi {V}^{a}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& -\frac{1}{{\ell}^{2}}\overline{\chi}{\gamma}_{5}{\gamma}_{a}\Phi {V}^{a}-\frac{1}{\ell}\overline{\chi}{\gamma}_{5}{\gamma}_{a}\rho {V}^{a}-\frac{1}{8{\ell}^{2}}{\u03f5}_{abcd}{V}^{a}{V}^{b}{V}^{c}{V}^{d}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& -\frac{\mathrm{i}}{2\ell}\overline{\psi}{\gamma}_{5}{\gamma}_{ab}\psi {V}^{a}{V}^{b}-\frac{\mathrm{i}}{{\ell}^{2}}\overline{\chi}{\gamma}_{5}{\gamma}_{ab}\psi {V}^{a}{V}^{b}-\frac{\mathrm{i}}{2{\ell}^{3}}\overline{\chi}{\gamma}_{5}{\gamma}_{ab}\chi {V}^{a}{V}^{b}\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$

## 5. Applications of the Formalism to Asymptotic Boundaries

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | With the terminology “generalized cosmological constant” we mean, according with the literature on the subject, a modification of the standard volume 4-form in which a dependence on additional 1-form gauge fields explicitly appears. |

2 | The Maxwell algebra was firstly introduced to describe the symmetries of a particle moving in a background in the presence of a constant electromagnetic field [58]. |

3 | |

4 | For this to be possible, the Lagrangian 4-form has to be entirely expressed as wedge product of differential forms and their exterior derivatives, without the use of tensor densities such as the Levi-Civita symbol ${\u03f5}_{\mu \nu \rho \sigma}$. For this reason, the kinetic terms in particular have to be written at first-order, thus avoiding the Hodge dual of the field-strengths, which is defined in terms of the Levi-Civita symbol. |

5 | This restriction can be relaxed by including auxiliary fields in the theory, but this is however possible at the supergravity level only in few cases (in theories exhibiting up to 8 supercharges, and with the inclusion of an infinite number of auxiliary fields for the off-shell description of hypermultiplets). In those cases, the Bianchi identities are proper identities and the supersymmetry algebra closes off-shell. |

6 | Typically, besides the contributions to the equations of motion coming from ${\mathcal{L}}_{\mathrm{bdy}}$ we also have extra contributions from ${\mathcal{L}}_{\mathrm{bulk}}$, neglected in the absence of a boundary, originating from the total differentials obtained by partial integration. |

7 | Regarding our notation, with respect to [34] here we define the Lorentz spin connection and curvature with extra minus signs, that is ${\omega}^{ab}\to -{\omega}^{ab}$ and ${\mathcal{R}}^{ab}\to -{\mathcal{R}}^{ab}$, to better fit in the conventions more commonly adopted in the literature. |

8 | |

9 | In [34], the AdS${}_{4}$ radius and the cosmological constant are expressed in terms of the parameter $e=\frac{1}{2\ell}$. |

10 | In writing the expression above we are skipping some subtleties related to the fact that the extension of the action integral to superspace requires, properly speaking, that the 4-form Lagrangian in superspace should be written at first-order, thus avoiding use of the Hodge-dual which is not easily defined in superspace (unless one uses the formalism of integral forms in superspace, see [71,72]). However, these subtleties do not affect the arguments reviewed here. |

11 | Here let us emphasize that, in complete analogy to what has been observed in [26] for the case of AdS${}_{4}$ gravity, the action (33) is not invariant under local $\mathrm{OSp}\left(1\right|4)$ transformations, even though the supercurvatures (34) are covariant with respect to $\mathrm{OSp}\left(1\right|4)$. |

12 | Here, as for the $\mathcal{N}=1$ case, we shall adopt the notation of [34], but performing the changes ${\omega}^{ab}\to -{\omega}^{ab}$, ${\mathcal{R}}^{ab}\to -{\mathcal{R}}^{ab}$, and $A\to -\frac{1}{\sqrt{2}}A$. Our conventions on fermions can be found in Appendix A.2 of [74], where the notation of [75] was adopted. We generally use Majorana spinors, and redefine the constants appearing in [34] as ${L}^{0}=L=\frac{1}{\sqrt{2}}$ and $\frac{1}{\ell}=2e=\frac{P}{\sqrt{2}}=\sqrt{-\frac{\Lambda}{3}}$. |

13 | As we have done in the $\mathcal{N}=1$ case, we keep on adopting bold symbols to denote the $\mathrm{OSp}\left(2\right|4)$ super field-strengths, without changing, however, their name with respect to the $\mathcal{N}=1$ case, to lighten the notation. |

14 | The same remark as for the $\mathcal{N}=1$ case, regarding use of the Hodge symbol in superspace, holds here (see footnote). |

15 | Recall that here with the terminology “flat supergravity” we mean supergravity in the absence of any explicit internal scale in the Lagrangian. |

16 | We refer the reader to [88] for a meticulous definition of asymptotic symmetry group, quotient of the group of residual gauge transformations (or diffeomorphisms, for theories of gravity) modulo the group of trivial gauge transformations (or, again, diffeomorphisms, for theories of gravity). In particular, one should more properly state that the asymptotic symmetry group of a class of theories is the union of the asymptotic symmetry groups of each (equivalent) formulation of that theory, enforcing in this way the actual asymptotic symmetry group to be invariant under field redefinitions and gauge choices. |

17 | |

18 | In Strominger’s work [83] the BMS transformations acting on future null infinity (${\mathcal{I}}^{+}$) are denoted by BMS${}^{+}$. They are an infinite-dimensional set of “large” diffeomorphisms transforming one asymptotically flat solution of general relativity constraints at ${\mathcal{I}}^{+}$ to a new, physically inequivalent solution. There is then an isomorphic structure at past null infinity (${\mathcal{I}}^{-}$) acted on by a second copy of the group denoted BMS${}^{-}$. |

19 | Flat holography denotes the application of the holographic correspondence to the case of asymptotically flat spacetime. |

20 | |

21 | To be coherent with respect to the notation previously adopted, we multiply the flat supergravity Lagrangian considered in [51] by a factor $\frac{1}{4}$. |

22 | Notice that actually this is not a MacDowell-Mansouri Lagrangian, as the latter would be quadratic in the field-strengths, while here we have the wedge product of different 2-form curvatures. |

23 | |

24 | Here again we adapt the conventions in such a way to be consistent with the ones previously adopted in the present paper. |

25 | |

26 | Note that the field ${\chi}^{\left(\mathrm{AVZ}\right)}$ is a Dirac spinor, not to be confused with the 1-form field $\chi $ of Section 4. |

27 | The holographic gauge fixing consists in using the freedom on local parameters to fix the Lagrange multipliers associated with the radial components of the fields when doing the holographic analysis. |

28 | Here we rename the two-dimensional indices $i,j,\cdots $ of [123] as $\mathfrak{a},\mathfrak{b},\dots $, in order to avoid confusion with other indices adopted in the present work. |

29 | Let us warn the reader that sometimes the terminology “non-relativistic” is used in the literature to denote both Galilean and Carrollian structures, none of them being relativistic, even if they emerge in totally different regimes. |

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Andrianopoli, L.; Ravera, L.
On the Geometric Approach to the Boundary Problem in Supergravity. *Universe* **2021**, *7*, 463.
https://doi.org/10.3390/universe7120463

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Andrianopoli L, Ravera L.
On the Geometric Approach to the Boundary Problem in Supergravity. *Universe*. 2021; 7(12):463.
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Andrianopoli, Laura, and Lucrezia Ravera.
2021. "On the Geometric Approach to the Boundary Problem in Supergravity" *Universe* 7, no. 12: 463.
https://doi.org/10.3390/universe7120463