Contents

# Contents

## Idea

Membrane instantons are constituted by supermembranes (M2-branes) wrapped on spacelike (and supersymmetric) cycles. Also known as SM2-branes.

These appear as non-perturvative effects in M-theory model building. For instance they generate the gauge hierarchy in constructions of M-theory on G2-manifolds such as the G2-MSSM.

Basically the effect is that the action functional of the given 11-dimensional supergravity is multiplied by the exponentiated Nambu-Goto action of the wrapping membrane configurations $\phi \colon \Sigma \to X$. These are of the form (e.g. Harvey-Moore 99, (4.13), Atiyah-Witten 01 (4.17), Curio 02, (2.15))

$\exp\left( \int_{\Sigma} \left( \ell_P^{-3} vol(\phi^\ast g) + i \phi^\ast C \right) \right) \,,$

where $\ell_P$ is the Planck length of 11-dimensional supergravity, $T = 1/\ell^3$ is the membrane tension, $C$ is the background supergravity C-field, $g$ the background metric and $vol$ its volume form, .

## Properties

### Relation to worldsheet and D-brane instantons

Under the duality between M-theory and type IIA string theory, membrane instantons become string worldsheet instantons or D-brane instantons, depending on whether they wrapped brane wrape the M-theory circle fiber or not. See at non-perturbative effect the section Worldsheet and brane instantons for more.

### Relation to complex volume

When here $(\Sigma, \phi^\ast g)$ is a hyperbolic 3-manifold and $\phi^\ast C = CS(\omega)$ is the Chern-Simons invariant then this is the exponentiated “complex volume” of $\Sigma$1.

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
$(D = 2n)$type IIA$\,$$\,$
D(-2)-brane$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$D=7 super Yang-Mills theory
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D(-1)-brane$\,$$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(p,q)-string$\,$$\,$$\,$
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection$\,$
string$\,$B2-field2d SCFT
NS5-brane$\,$B6-fieldlittle string theory
D-brane for topological string$\,$
A-brane$\,$
B-brane$\,$
M-brane11D SuGra/M-theorycircle n-connection$\,$
M2-brane$\,$C3-fieldABJM theory, BLG model
M5-brane$\,$C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
S-brane
SM2-brane,
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

## References

Discussion specifically for M-theory on G2-manifolds includes

• Michael Atiyah, Edward Witten, around (4.17) of $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)

• Gottfried Curio, Superpotentials for M-theory on a $G_2$ holonomy manifold and Triality symmetry, JHEP 0303:024,2003 (arXiv:hep-th/0212211)

Discussion related to the cosmological constant in models of M-theory on G2-manifolds includes

• Beatriz de Carlos, Andre Lukas, Stephen Morris, Non-perturbative vacua for M-theory on G2 manifolds, JHEP 0412:018, 2004 (arxiv:hep-th/0409255)

which concludes that with taking non-perturbative effects from membrane instantons into account one gets 4d vacua with vanishing and negative cosmological constant (Minkowski spacetime and anti-de Sitter spacetime) but not with positive cosmological constant (de Sitter spacetime). They close by speculating that M5-brane instantons might yield de Sitter spacetime.

1. This relation was pointed out by Hisham Sati.

Last revised on September 23, 2019 at 00:22:07. See the history of this page for a list of all contributions to it.