# Positive Energy and Non-SUSY Flows in ISO(7) Gauged Supergravity

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. The Setup: ${\mathrm{G}}_{\mathbf{2}}$-Invariant $\mathrm{ISO}\left(\mathbf{7}\right)$ Gauged Maximal Supergravity

## 3. First-Order Flows: Bubbles, DWs, and Positive Energy

- The effective FRW cosmology on the bubble wall is of an expanding type;
- The asymptotic geometry is our (putative) AdS vacuum;
- The difference in Euclidean action $0<{S}_{\mathrm{E}}\left(\mathrm{Bubble}\right)-{S}_{\mathrm{E}}\left(\mathrm{AdS}\right)<\infty $.

## 4. Flat First-Order Flows: Separable HJ Treatment

## 5. Positive Energy Theorems for Non-SUSY Vacua

**Theorem**

**1.**

- $\left(i\right)$${\partial}_{i}{f|}_{{\varphi}_{0}}=0$,
- $\left(ii\right)$$V\left(\varphi \right)\ge -3f{\left(\varphi \right)}^{2}$, $\forall \varphi \in {\mathcal{M}}_{\mathrm{scalar}}$,

#### Interpolating Flat DW Solutions

## 6. Curved DW Geometries: A Non-Separable HJ Treatment

## 7. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Consistent Truncations of Massive Type IIA Supergravity on ${\mathit{S}}^{\mathbf{6}}$

## Notes

1 | From now on, we set ${\kappa}_{4}=1$. |

2 | From now on, we will fix $g=m$. |

3 | It is worth mentioning that, to obtain this reduced 1D Lagrangian, an integration by parts is required in order to eliminate the term with the second derivative ${A}^{\prime \prime}$. |

4 | We may set $B=0$ here, since it is a completely irrelevant overall factor. However, as pointed out earlier, the first-order flow equations are sensitive to the choice of B and may simplify significantly in particularly clever gauge choices. |

5 | In this general formalism, we collectively denote the scalar fields by $\varphi \equiv (\phi ,\chi )$. As a consequence, the kth derivative of a scalar function f is represented by a rank k tensor field. |

6 | We have fixed $m=\frac{2}{{5}^{7/12}}{\ell}^{-1}$ in such a way that the value of the cc in the $\mathrm{SO}\left(7\right)$ vacuum equals $-\frac{3}{{\ell}^{2}}$. |

7 | We still stick to the $B=0$ gauge for the sake of simplicity here. |

8 | Note that it correctly reproduces the flat result ${F}^{\left(0\right)}=\frac{{e}^{3A}}{\ell}$ in the limit where $L\gg \ell $. |

## References

- Denef, F.; Douglas, M.R. Distributions of flux vacua. J. High Energy Phys.
**2004**, 5, 072. [Google Scholar] [CrossRef] [Green Version] - Taylor, W.; Wang, Y.-N. The F-theory geometry with most flux vacua. J. High Energy Phys.
**2015**, 12, 164. [Google Scholar] [CrossRef] [Green Version] - Arkani-Hamed, N.; Motl, L.; Nicolis, A.; Vafa, C. The String landscape, black holes and gravity as the weakest force. J. High Energy Phys.
**2007**, 6, 060. [Google Scholar] [CrossRef] - Brennan, T.D.; Carta, F.; Vafa, C. The String Landscape, the Swampland, and the Missing Corner. arXiv
**2017**, arXiv:1711.00864. [Google Scholar] - Palti, E. The Swampland: Introduction and Review. Fortsch. Phys.
**2019**, 67, 1900037. [Google Scholar] [CrossRef] [Green Version] - Ooguri, H.; Vafa, C. Non-supersymmetric AdS and the Swampland. Adv. Theor. Math. Phys.
**2017**, 21, 1787–1801. [Google Scholar] [CrossRef] [Green Version] - Freivogel, B.; Kleban, M. Vacua Morghulis. arXiv
**2016**, arXiv:1610.04564. [Google Scholar] - Horowitz, G.T.; Orgera, J.; Polchinski, J. Nonperturbative Instability of AdS(5) × S**5/Z(k). Phys. Rev. D
**2008**, 77, 024004. [Google Scholar] [CrossRef] [Green Version] - Guarino, A.; Varela, O. Dyonic ISO(7) supergravity and the duality hierarchy. J. High Energy Phys.
**2016**, 2, 079. [Google Scholar] [CrossRef] [Green Version] - Guarino, A.; Malek, E.; Samtleben, H. Stable Nonsupersymmetric Anti–de Sitter Vacua of Massive IIA Supergravity. Phys. Rev. Lett.
**2021**, 126, 061601. [Google Scholar] [CrossRef] - Guarino, A.; Tarrio, J.; Varela, O. Brane-jet stability of non-supersymmetric AdS vacua. J. High Energy Phys.
**2020**, 9, 110. [Google Scholar] [CrossRef] - Witten, E. Constraints on Supersymmetry Breaking. Nucl. Phys. B
**1982**, 202, 253. [Google Scholar] [CrossRef] - Bomans, P.; Cassani, D.; Dibitetto, G.; Petri, N. Bubble instability of mIIA on AdS
_{4}×S^{6}. arXiv**2021**, arXiv:2110.08276. [Google Scholar] - Hull, C.M. A New Gauging of N=8 Supergravity. Phys. Rev. D
**1984**, 30, 760. [Google Scholar] [CrossRef] - Cremmer, E.; Julia, B. The SO(8) Supergravity. Nucl. Phys. B
**1979**, 159, 141–212. [Google Scholar] [CrossRef] - DeWit, B.; Samtleben, H.; Trigiante, M. The Maximal D = 4 supergravities. J. High Energy Phys.
**2007**, 6, 049. [Google Scholar] - Hull, C.M.; Warner, N.P. Noncompact Gaugings From Higher Dimensions. Class. Quant. Grav.
**1988**, 5, 1517. [Google Scholar] [CrossRef] [Green Version] - Dall’Agata, G.; Inverso, G.; Marrani, A. Symplectic Deformations of Gauged Maximal Supergravity. J. High Energy Phys.
**2014**, 7, 133. [Google Scholar] [CrossRef] [Green Version] - Guarino, A.; Varela, O. Consistent N=8 truncation of massive IIA on S
^{6}. J. High Energy Phys.**2015**, 12, 020. [Google Scholar] - Breitenlohner, P.; Freedman, D.Z. Stability in Gauged Extended Supergravity. Ann. Phys.
**1982**, 144, 249. [Google Scholar] [CrossRef] - Danielsson, U.; Dibitetto, G. Fate of stringy AdS vacua and the weak gravity conjecture. Phys. Rev. D
**2017**, 96, 026020. [Google Scholar] [CrossRef] [Green Version] - Danielsson, U.H.; Dibitetto, G.; Vargas, S.C. A swamp of non-SUSY vacua. J. High Energy Phys.
**2017**, 11, 152. [Google Scholar] [CrossRef] [Green Version] - Danielsson, U.H.; Dibitetto, G.; Vargas, S.C. Universal isolation in the AdS landscape. Phys. Rev. D
**2016**, 94, 126002. [Google Scholar] [CrossRef] [Green Version] - Coleman, S.R.; De Luccia, F. Gravitational Effects on and of Vacuum Decay. Phys. Rev. D
**1980**, 21, 3305. [Google Scholar] [CrossRef] [Green Version] - Witten, E. Instability of the Kaluza-Klein Vacuum. Nucl. Phys. B
**1982**, 195, 481–492. [Google Scholar] [CrossRef] - Brown, J.D.; Teitelboim, C. Neutralization of the Cosmological Constant by Membrane Creation. Nucl. Phys. B
**1988**, 297, 787–836. [Google Scholar] [CrossRef] - Cvetic, M.; Soleng, H.H. Supergravity domain walls. Phys. Rep.
**1997**, 282, 159–223. [Google Scholar] [CrossRef] [Green Version] - Cvetic, M.; Griffies, S.; Rey, S.-J. Nonperturbative stability of supergravity and superstring vacua. Nucl. Phys. B
**1993**, 389, 3–24. [Google Scholar] [CrossRef] - Freedman, D.Z.; Nunez, C.; Schnabl, M.; Skenderis, K. Fake supergravity and domain wall stability. Phys. Rev. D
**2004**, 69, 104027. [Google Scholar] [CrossRef] [Green Version] - Nester, J.M.; Tung, R.-S.; Zhang, Y.Z. Ashtekar’s new variables and positive energy. Class. Quant. Grav.
**1994**, 11, 757–766. [Google Scholar] [CrossRef] [Green Version] - Townsend, P.K. Positive Energy and the Scalar Potential in Higher Dimensional (Super)gravity Theories. Phys. Lett. B
**1984**, 148, 55–59. [Google Scholar] [CrossRef] - Boucher, W. Positive Energy without Supersymmetry. Nucl. Phys. B
**1984**, 242, 282–296. [Google Scholar] [CrossRef] - Breitenlohner, P.; Freedman, D.Z. Positive Energy in anti-De Sitter Backgrounds and Gauged Extended Supergravity. Phys. Lett. B
**1982**, 115, 197–201. [Google Scholar] [CrossRef]

**Figure 1.**The level curves of the scalar potential for the ${\mathrm{G}}_{2}$-invariant sector of $\mathrm{ISO}\left(7\right)$ gauged supergravity. The horizontal axis represents the $\phi $ direction, while the vertical axis represents $\chi $. Both the SUSY and non-SUSY ${\mathrm{G}}_{2}$ extrema are local minima, while the $\mathrm{SO}\left(7\right)$ critical point is actually a saddle.

**Figure 2.**The profile of the scalar potential of ${\mathrm{G}}_{2}$-invariant $\mathrm{ISO}\left(7\right)$ gauged supergravity (orange surface), against the three globally bounding functions $-3{f}^{2}$, obtained by solving the PDE (20) through a perturbative expansion around each critical point. The plot is drawn in $\ell =1$ units.

**Figure 3.**The profile of the scalar potential restricted to the straight line connecting vacua “1” and “2” (blue curve). Besides the previously determined globally bounding functions (in yellow and green), we also plot $-3{f}_{D{W}_{12}}^{2}$ (orange curve), having both critical points as local extrema, and thus defining the interpolating flow (zoomed on the right). The plots are both drawn in $\ell =1$ units.

**Figure 4.**The profile of the scalar potential restricted to the straight line connecting vacua “1” and “3” (blue curve). Besides the previously determined globally bounding functions (in orange and green), we also plot $-3{f}_{D{W}_{13}}^{2}$ (yellow curve), having both critical points as local extrema, and thus defining the interpolating flow (zoomed on the right). The plots are both drawn in $\ell =1$ units.

**Figure 5.**The profiles of $\phi \left(r\right)$ (

**left**) and $\chi \left(r\right)$ (

**right**), plotted against $\frac{r}{\ell}$. The scalar vevs asymptote to the ones in the SUSY-${\mathrm{G}}_{2}$ vacuum as $r\to -\infty $, while they approach those in the non-SUSY $\mathrm{SO}\left(7\right)$ as $r\to +\infty $.

**Figure 6.**The profiles of $\phi \left(r\right)$ (

**left**) and $\chi \left(r\right)$ (

**right**), plotted against $\frac{r}{\ell}$. The scalar vevs asymptote to the ones in the non-SUSY-${\mathrm{G}}_{2}$ vacuum as $r\to -\infty $, while they approach those in the non-SUSY $\mathrm{SO}\left(7\right)$ as $r\to +\infty $.

**Table 1.**The three different AdS critical points of $\mathrm{ISO}\left(7\right)$ gauged supergravity preserving at least ${\mathrm{G}}_{2}$ as residual symmetry. The squared masses of the ${\mathrm{G}}_{2}$ singlet modes are normalized to the absolute value of the cosmological constant. In these units, the BF bound [20] lies at ${m}_{\mathrm{BF}}^{2}=-\frac{3}{4}$.

ID | SUSY | ${\mathit{G}}_{\mathbf{res}\mathbf{.}}$ | $\mathit{\chi}$ | ${\mathit{e}}^{\mathbf{-}\mathit{\phi}}$ | ${\mathit{m}}_{{\mathbf{G}}_{\mathbf{2}}}^{\mathbf{2}}$ |
---|---|---|---|---|---|

1 | $\mathcal{N}=0$ | $\mathrm{SO}\left(7\right)$ | 0 | ${5}^{-1/6}$ | 2; $-\frac{2}{5}$ |

2 | $\mathcal{N}=1$ | ${\mathrm{G}}_{2}$ | ${2}^{-7/3}$ | ${3}^{1/2}{5}^{1/2}{2}^{-7/3}$ | $\frac{1}{3}\left(4\pm \sqrt{6}\right)$ |

3 | $\mathcal{N}=0$ | ${\mathrm{G}}_{2}$ | $-{2}^{-4/3}$ | ${3}^{1/2}{2}^{-4/3}$ | 2; 2 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dibitetto, G.
Positive Energy and Non-SUSY Flows in ISO(7) Gauged Supergravity. *Universe* **2022**, *8*, 293.
https://doi.org/10.3390/universe8050293

**AMA Style**

Dibitetto G.
Positive Energy and Non-SUSY Flows in ISO(7) Gauged Supergravity. *Universe*. 2022; 8(5):293.
https://doi.org/10.3390/universe8050293

**Chicago/Turabian Style**

Dibitetto, Giuseppe.
2022. "Positive Energy and Non-SUSY Flows in ISO(7) Gauged Supergravity" *Universe* 8, no. 5: 293.
https://doi.org/10.3390/universe8050293