# BPS equations and non-trivial compactifications

## Abstract

We consider the problem of finding exact, eleven-dimensional, BPS supergravity solutions in which the compactification involves a non-trivial Calabi-Yau manifold, Y, as opposed to simply a T ^{6}. Since there are no explicitly-known metrics on non-trivial, compact Calabi-Yau manifolds, we use a non-compact “local model” and take the compactification manifold to be Y = M _{GH} × T ^{2}, where MGH is a hyper-Kähler, Gibbons-Hawking ALE space. We focus on backgrounds with three electric charges in five dimensions and find exact families of solutions to the BPS equations that have the same four supersymmetries as the three-charge black hole. Our exact solution to the BPS system requires that the Calabi-Yau manifold be fibered over the space-time using compensators on Y. The role of the compensators is to ensure smoothness of the eleven-dimensional metric when the moduli of Y depend on the space-time. The Maxwell field Ansatz also implicitly involves the compensators through the frames of the fibration. In conclusion, we examine the equations of motion and discuss the brane distributions on generic internal manifolds that do not have enough symmetry to allow smearing.

- Authors:

- Univ. of Southern California, Los Angeles, CA (United States). Dept. of Physics and Astronomy
- Univ. of Southern California, Los Angeles, CA (United States). Dept. of Physics and Astronomy and Dept. of Mathematics

- Publication Date:

- Research Org.:
- Univ. of Southern California, Los Angeles (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1499194

- Grant/Contract Number:
- SC0011687

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Volume: 2018; Journal Issue: 5; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 79 ASTRONOMY AND ASTROPHYSICS; Flux compactifications; M-Theory; Supergravity Models; Black Holes in String Theory

### Citation Formats

```
Tyukov, Alexander, and Warner, Nicholas P.
```*BPS equations and non-trivial compactifications*. United States: N. p., 2018.
Web. doi:10.1007/jhep05(2018)022.

```
Tyukov, Alexander, & Warner, Nicholas P.
```*BPS equations and non-trivial compactifications*. United States. doi:10.1007/jhep05(2018)022.

```
Tyukov, Alexander, and Warner, Nicholas P. Fri .
"BPS equations and non-trivial compactifications". United States. doi:10.1007/jhep05(2018)022. https://www.osti.gov/servlets/purl/1499194.
```

```
@article{osti_1499194,
```

title = {BPS equations and non-trivial compactifications},

author = {Tyukov, Alexander and Warner, Nicholas P.},

abstractNote = {We consider the problem of finding exact, eleven-dimensional, BPS supergravity solutions in which the compactification involves a non-trivial Calabi-Yau manifold, Y, as opposed to simply a T6. Since there are no explicitly-known metrics on non-trivial, compact Calabi-Yau manifolds, we use a non-compact “local model” and take the compactification manifold to be Y = MGH × T2, where MGH is a hyper-Kähler, Gibbons-Hawking ALE space. We focus on backgrounds with three electric charges in five dimensions and find exact families of solutions to the BPS equations that have the same four supersymmetries as the three-charge black hole. Our exact solution to the BPS system requires that the Calabi-Yau manifold be fibered over the space-time using compensators on Y. The role of the compensators is to ensure smoothness of the eleven-dimensional metric when the moduli of Y depend on the space-time. The Maxwell field Ansatz also implicitly involves the compensators through the frames of the fibration. In conclusion, we examine the equations of motion and discuss the brane distributions on generic internal manifolds that do not have enough symmetry to allow smearing.},

doi = {10.1007/jhep05(2018)022},

journal = {Journal of High Energy Physics (Online)},

issn = {1029-8479},

number = 5,

volume = 2018,

place = {United States},

year = {2018},

month = {5}

}