# 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:
- Accepted Manuscript

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

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); 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)},

number = 5,

volume = 2018,

place = {United States},

year = {2018},

month = {5}

}

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Localised anti-branes in non-compact throats at zero and finite T

journal, February 2015

- Blåbäck, J.; Danielsson, U. H.; Junghans, D.
- Journal of High Energy Physics, Vol. 2015, Issue 2

##
New families of interpolating type IIB backgrounds

journal, April 2010

- Minasian, Ruben; Petrini, Michela; Zaffaroni, Alberto
- Journal of High Energy Physics, Vol. 2010, Issue 4

##
Smooth Horizonless Geometries Deep Inside the Black-Hole Regime

journal, November 2016

- Bena, Iosif; Giusto, Stefano; Martinec, Emil J.
- Physical Review Letters, Vol. 117, Issue 20

##
String cosmology and the landscape

journal, March 2017

- Bena, Iosif; Graña, Mariana
- Comptes Rendus Physique, Vol. 18, Issue 3-4

##
Microscopic origin of the Bekenstein-Hawking entropy

journal, June 1996

- Strominger, Andrew; Vafa, Cumrun
- Physics Letters B, Vol. 379, Issue 1-4

##
Momentum fractionation on superstrata

journal, May 2016

- Bena, Iosif; Martinec, Emil; Turton, David
- Journal of High Energy Physics, Vol. 2016, Issue 5

##
M-theory superstrata and the MSW string

journal, June 2017

- Bena, Iosif; Martinec, Emil; Turton, David
- Journal of High Energy Physics, Vol. 2017, Issue 6

##
One ring to rule them all and in the darkness bind them?

journal, January 2005

- Bena, Iosif; Warner, Nicholas P.
- Advances in Theoretical and Mathematical Physics, Vol. 9, Issue 5

##
Supergravity microstates for BPS black holes and black rings

journal, June 2006

- Berglund, Per; Gimon, Eric G.; Levi, Thomas S.
- Journal of High Energy Physics, Vol. 2006, Issue 06

##
The geometry of *D* = 11 Killing spinors

journal, April 2003

- Gauntlett, Jerome P.; Pakis, Stathis
- Journal of High Energy Physics, Vol. 2003, Issue 04

##
Kinetic terms in warped compactifications

journal, May 2009

- Douglas, Michael R.; Torroba, Gonzalo
- Journal of High Energy Physics, Vol. 2009, Issue 05

##
6D microstate geometries from 10D structures

journal, November 2013

- Giusto, Stefano; Martucci, Luca; Petrini, Michela
- Nuclear Physics B, Vol. 876, Issue 2

##
Massless black holes and conifolds in string theory

journal, September 1995

- Strominger, Andrew
- Nuclear Physics B, Vol. 451, Issue 1-2

##
Supersymmetric solutions in six dimensions: a linear structure

journal, March 2012

- Bena, Iosif; Giusto, Stefano; Shigemori, Masaki
- Journal of High Energy Physics, Vol. 2012, Issue 3

##
Black hole entropy in M-Theory

journal, December 1997

- Maldacena, Juan; Strominger, Andrew; Witten, Edward
- Journal of High Energy Physics, Vol. 1997, Issue 12

##
de Sitter vacua in string theory

journal, August 2003

- Kachru, Shamit; Kallosh, Renata; Linde, Andrei
- Physical Review D, Vol. 68, Issue 4

##
Towards inflation in string theory

journal, October 2003

- Kachru, Shamit; Kallosh, Renata; Linde, Andrei
- Journal of Cosmology and Astroparticle Physics, Vol. 2003, Issue 10

##
Supergravity solutions from floating branes

journal, March 2010

- Bena, Iosif; Giusto, Stefano; Ruef, Clément
- Journal of High Energy Physics, Vol. 2010, Issue 3

##
Supergravity and a confining gauge theory: duality cascades and χSB-resolution of naked singularities

journal, August 2000

- Klebanov, Igor R.; Strassler, Matthew J.
- Journal of High Energy Physics, Vol. 2000, Issue 08

##
Habemus superstratum! A constructive proof of the existence of superstrata

journal, May 2015

- Bena, Iosif; Giusto, Stefano; Russo, Rodolfo
- Journal of High Energy Physics, Vol. 2015, Issue 5

##
Hair-brane ideas on the horizon

journal, November 2015

- Martinec, Emil J.; Niehoff, Ben E.
- Journal of High Energy Physics, Vol. 2015, Issue 11