nLab membrane instanton

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.

Related concepts

• hierarchy problem

• non-perturbative effect

• complex volume

• associative submanifold

• D3-brane instanton

• M5-brane instanton

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

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super 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-brane	type I, II, heterotic	circle n-connection	$\,$
string	$\,$	B2-field	2d SCFT
NS5-brane	$\,$	B6-field	little string theory
D-brane for topological string			$\,$
A-brane			$\,$
B-brane			$\,$
M-brane	11D SuGra/M-theory	circle n-connection	$\,$
M2-brane	$\,$	C3-field	ABJM theory, BLG model
M5-brane	$\,$	C6-field	6d (2,0)-superconformal QFT
M9-brane/O9-plane			heterotic string theory
M-wave
topological M2-brane	topological M-theory	C3-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-brane	6d (2,0)-superconformal QFT
self-dual string		self-dual B-field
3-brane in 6d

References

• Katrin Becker, Melanie Becker, Andrew Strominger, section 2.1 of Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456, 130 (1995) (hep-th/9507158)

• Jeff Harvey, Greg Moore, Superpotentials and Membrane Instantons (arXiv:hep-th/9907026, spire:503172)

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 $G_2$ 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.

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1. This relation was pointed out by Hisham Sati. ↩