Supergravity as the low energy effective action of Superstring- / M-theory

[jcdenton]

Hello.
I have a question for the specialists on Superstring Theory:

I am looking for a reference to original research, where the following is proven / shown / evidence is given / made plausible / etc.

- 10D supergravities in their various forms are the low energy limits of superstring theories

- 11D supergravity is the low energy limit of M-theory


I am aware of the standard textbooks on the subject, however the best I got out of them is that

Green/Schwarz/Witten: Superstring theory Volume 2 Chapter 13:
- The Field content of IIB Sugra matches the massless fields in IIB Superstring theory

Ortin: Gravity and Strings, Chapter 15
- (page 432f) When studying Superstring theory on curved backgrounds, one wants Weyl invariance to be preserved on the quantum level. Vanishing of the beta-functions leads to the same equations of motion as the bosonic part of 10D Sugra calculated in 26 dimensions, so that a certain term drops. There is a comment, that for superstrings, you should get D=10, but without reference.

Becker/Becker/Schwarz: Superstring theory and M-theory
The text says nothing about arriving at Sugra as a low energy effective action, altough broadly elaborating about the field content of supergravity. In the reference section, there is a paper by Witten, hep-th/9503124, where he argues that "that eleven-dimensional supergravity arises as a low energy limit of the ten-dimensional Type IIA superstring" in the strong coupling limit. However in his paper, he uses that the low energy effective action of IIA Superstring theory is given by 10D Supergravity on page 5.

Michael Dine: Supersymmetry and String theory
The book contains a bit of supergravity in the low energy effetive action section, besides the claim, that Sugra is the low energy limit of Superstring theory. The suggested reading refers to Green/Schwary/Witten, see above.

I am especially disappointed with the references in Ortins book concerning this topic, because he has like 900 references, but there seems to be no need to reference a sentence (page 430) like "Actually, for some superstring theories, it is possible to arrive at the effective theory using (supersymmetry) arguments." He then gives 10 references to work on 10D supergravity, but none of them concerns the relation to string theory.

In Green/Schwarz/Witten, we find (page 314) "The ten-dimensional supergravity theories are, of course, the low-energy lmits of certain string theories."
The reference section for chapter 13 contains many references on supergravities, their anomalies etc., but there is nothing which actually explicitly related string theory and supergravity.


Am I missing something?

Thanks for your answers!

 

[fzero]

The most important fact, apparently alluded to in Ortin, is that the 11d and 10d type IIA/B supergravity theories have a field content and tree-level Lagrangian that is completely constrained by supersymmetry. These theories were classified algebraically by W. Nahm in Supersymmetries and their representations, Nucl. Phys. B135 (1978), 149-166 (http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=197709213&OF=4. ).

For 11D sugra, it's considered as the low-energy limit of M-Theory as a definition, but you can connect to the terms in the 11D action either explicitly using Matrix Theory or by using dualities to relate to IIA or some other string theory that allows a relevant calculation. One of the basic references would be Townsend, "The eleven-dimensional supermembrane revisited," hep-th/9501068.

 

[lumidek]

Dear J.C. Denton,

the original papers you're looking for may exist but they're surely not important if they do exist because the point you want to prove is trivial and doesn't have to be checked explicitly (even though the textbooks you mention do show many explicit things about the interactions).

The 2-derivative terms for the massless fields in supergravities with 32 supercharges - and essentially also 16 supercharges - are totally determined by the supersymmetry algebra. That's how the actions for SUGRA were derived in the first place, in the field-theoretical context.

The same spacetime supersymmetry algebra may be easily proved in the relevant superstring theories. They also have the same spectrum, so it's guaranteed that at the level of the 2-derivative interactions, superstring/M-theory inevitably produces the same interactions, too.

You will find lots of papers from the early 1980s such as

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVN-470VSH9-1V9&_user=10&_coverDate=03/03/1983&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1558528350&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=128043e45df971b0fd7c8dee3db1aa86&searchtype=a

that actually study both superstring theory and supergravity - in various formalisms such as the light-cone gauge above (favorite approach of Green and Schwarz). The transition from string theory to SUGRA is a simple process in which one neglects all modes on the string except for the zero modes: that's the way to determine the spectrum. The interactions are guaranteed to coincide even though only the stringy approach makes their calculation "natural". Of course, it has been checked at least partially in many papers - which usually had another, more ambitious, goal as well.

When you go beyond the two-derivative (or quartic-fermionic) terms, it's questionable whether supersymmetry fully determines the action: probably not. But if you try to go in this way, you're also going to encounter the non-renormalizability of SUGRA etc. and the full string/M-theory is the only way how to "define the SUGRA" at this more accurate level.

Best wishes
Lubos

 

[jcdenton]

Thanks for both your answeres! They help a lot to understand the underlying logic.

However, although the derivation of the low energy limit might seem trivial considering Nahm's paper, it seems funny that this point is not spelled out explicitly in the textbooks on the subject (or I just overlooked it).

Now given your above arguments, let me rephrase my question:

Is there a reference where it is shown that taking the low energy limit of String theory can be done by neglecting string states with non-vanishing masses? I mean, is this limit well defined (consistent etc.) in some way?

Looking at the textbooks and your answers, it also seems trivial to take the low energy limit like this. However it sounds rather heuristic, and I would be interested in papers where they discuss possible caveats in this derivation, or better, show that the low energy limit has to be taken exactly like this.