duality between M-theory and type IIA string theory in nLab

Contents

Contents

Idea

The conjectured duality between type IIA string theory and the theory with working title M-theory is essentially one half of what defines M-theory in the first place, namely the statement that M-theory is the non-perturbative theory of which the string perturbation series of type IIA string theory is the perturbation series.

The second defining characteristic of M-theory is that at low energy, hence at large wavelength, it is approximated by 11-dimensional supergravity. On the other hand, the low energy limit of type IIA string theory is 10d type IIA supergravity.

In view of this, the conjectured relation between M-theory and type IIA string theory says, more in detail, that in the low energy limit the relation becomes that exhibited exhibited by the Kaluza-Klein compactification of 11-dimensional supergravity to 10d type IIA supergravity on a circle fiber, such that the radius of the circle fiber is proportional to the coupling constant of type II A string theory.

M−Theory(?) →lowenergylimit 11d Supergravity small coupling limit↓ ↓ KKreduction onS 1 type IIA string theory →lowenergyeffectiveQFT 10d Supergravity \array{ M-Theory(?) &\stackrel{low\;energy\;limit}{\to}& \text{11d Supergravity} \\ {}^{\mathllap{ \array{ small \\ coupling \\ limit}}}\bigg\downarrow && \bigg\downarrow{}^{\mathrlap{ \array{KK\;reduction \\ on S^1 }}} \\ \text{type IIA string theory} &\stackrel{low\;energy\;effective\;QFT}{\to}& \text{10d Supergravity} }

(e.g. Obers-Pioline 98, p. 12)

One indication for this conjecture is that the double dimensional reduction of the Green-Schwarz sigma-model that describes the super membrane (later called the M2-brane) propagating on an 11-dimensional supergravity background spacetime yields the Green-Schwarz sigma-model for the type IIA superstring propagating on a 10d type IIA supergravity background (Duff-Howe-Inami-Stelle 87).

This observation is the origin of the name “M-theory”, in the first place: As a non-commital shorthand for “membrane theory”, where the caveat is that, there is at least no direct analog of the string perturbation series with strings replaced by membranes.

What convinced the community that 11-dimensional supergravity really is the low energy limit of some non-perturbative version of perturbative string theory was the arguments given in (Witten 95).

• graviphoton

• duality in string theory

• duality between F-theory and heterotic string theory

• duality between type I and heterotic string theory

KK-compactifications of M-theory – table

References

The original insight is due to

• Michael Duff, Paul Howe, T. Inami, Kellogg Stelle, Superstrings in D=10D=10 from Supermembranes in D=11D=11, Phys. Lett. B 191 (1987) 70 [doi:10.1016/0370-2693(87)91323-2] and in: Michael Duff (ed.) The World in Eleven Dimensions 205-206 (1987) [spire]

• Paul K. Townsend, The eleven-dimensional supermembrane revisited, Phys. Lett. B 350 (1995) 184-187 [arXiv:hep-th/9501068, doi:10.1016/0370-2693(95)00397-4]

but it gained popularity only with:

• Edward Witten, §2.2, §2.3 in: String Theory Dynamics In Various Dimensions, Nucl. Phys. B 443 (1995) 85-126 [arXiv:hep-th/9503124, doi:10.1016/0550-3213(95)00158-O]

Review:

• Michael Duff, §2(ii) in: M-Theory (the Theory Formerly Known as Strings), Int. J. Mod. Phys. A 11 (1996) 5623-5642 [arXiv:hep-th/9608117, doi:10.1142/S0217751X96002583]

• Mike Duff, The World in Eleven Dimensions: Supergravity, Supermembranes and M-theory, IoP 1999 (publisher)

• Niels A. Obers, Boris Pioline, §2.1 in: U-duality and M-Theory, Phys. Rept. 318 (1999) 113-225 [arXiv:hep-th/9809039, doi:10.1016/S0370-1573(99)00004-6]

The actual double dimensional reduction over a non-trivial circle-principal bundle of the supergravity C-field flux densities G 4G_4 and G 7G_7 to the B-field and (most of) the RR-field flux densities ois worked out (only?) in:

• Varghese Mathai, Hisham Sati, §4 in: Some Relations between Twisted K-theory and E 8E_8 Gauge Theory, J. High Energ. Phys. 2004 03 (2004) 016 [arXiv:hep-th/0312033, doi:10.1088/1126-6708/2004/03/016]