For example the eq 2.1 here with regards to Type IIB.

Unless I am terribly missing/misreading something Polchinski doesn't ever seem to derive these low energy supergravity actions.

I would like to see a beginner's explanation (maybe together with review paper for further information) to getting these actions from string theory (and hopefully also something about deriving the black-hole and the brane metrics from them)

  • 2
    $\begingroup$ You just have to compute a lot of string amplitudes, which Polchinski doesn't want to waste time doing. Some of these are left as exercises. Once you've done this you can look for an effective field theory that reproduces the amplitudes, and you'll find supergravity. Alternatively, once you've convinced yourself that string theory preserves spacetime supersymmetry, you know from the fact that there aren't very many consistent supergravity theories that you're going to get the right answer. $\endgroup$
    – Matthew
    Jul 31, 2013 at 16:27
  • $\begingroup$ This question (v1) appears to be off-topic because it is a book request, cf. discussion on meta. Phys.SE strives to be more than just a link farm. $\endgroup$
    – Qmechanic
    Jul 31, 2013 at 16:31
  • $\begingroup$ @Matthew You say that "you can look for an effective field theory" - can you give a reference as to how this is done? It otherwise looks like a wild hunt question - one can get the field content but does that somehow uniquely fix these complicated structures of the supergravity actions? Also is there a reference which explains how these amplitudes are to be calculated in supergravity? (..its not obvious to a beginner!..) $\endgroup$
    – user6818
    Jul 31, 2013 at 16:58
  • 1
    $\begingroup$ @Qmechanic, asking how effective actions are derived in string theory is in my opinion a very legitimate technical physics question. I'd like to see an answer here too and have reformulated the questions, such that it is no longer a reference request. Can you reopen the question? $\endgroup$
    – Dilaton
    Jul 31, 2013 at 16:59
  • 1
    $\begingroup$ @Qmechanic : This is an interesting question, the problem of book is secondary. $\endgroup$
    – Trimok
    Jul 31, 2013 at 18:13

1 Answer 1


I think that the quickest route to the effective action actually isn't through string amplitudes, but through the beta functions. The conditions for worldsheet conformal invariance are equivalent to the spacetime equations of motion, and from these you can infer an on-shell effective action. This is all you can hope for in string theory (or any theory of quantum gravity), which isn't well-defined off-shell.

As for references, Polchinski does the bosonic case. These computations are easiest in dim reg, but his scheme is fine also. I think the original analysis was by Callan et. al, http://adsabs.harvard.edu/abs/1985NuPhB.262..593C.

There are also really fun technicalities that come up in higher genus corrections. These were pointed out by Fischler and Susskind http://inspirehep.net/record/17879, and expanded on in a series of papers by Fradkin and Tseytlin, and Callan et. al.

  • $\begingroup$ So you mean that the equations of motion of the LEEA somehow arise from the demand of worldsheet conformal invariance? Thanks for the references! I guess I will try to work through them. So you suggest that these references will help me understand where the equation 2.1 of my linked paper comes from? [...just a curiousity - are you Prof.Matthew Schwartz from Harvard?...] $\endgroup$
    – user6818
    Aug 1, 2013 at 20:16
  • $\begingroup$ That's right, the two are equivalent. I think you should be able to understand the effective action by reading Polchinski, but that paper by Callan etc. might help also. $\endgroup$
    – Matthew
    Aug 2, 2013 at 23:03
  • $\begingroup$ Thanks for the general directions - also in the same context I would like to mention this other thing that I often run into - the emergence of supergravity that we have been talking of till now did not need D-branes - right? - but isn't there a statement that if the supergravity is emerging as a LEEA of a collection of N D-branes then apparently in the limit of N going to infinity this LEEA is exact and there are really no string effects left! $\endgroup$
    – user6818
    Aug 7, 2013 at 17:47
  • $\begingroup$ Can you kindly help make this idea exact and say as to how it is related to whatever we have said till now and if you can give some pegdagogic reference towards that. $\endgroup$
    – user6818
    Aug 7, 2013 at 17:50
  • $\begingroup$ Actually there is a method that is more direct than the (super-)conformal invariance at quantum level. If you fix the BRST charge for the string in a general curved background, the nilpotence and holomorphicity of the BRST current gives the equation of motion for the background. A tree level computation in the BRST method already gives the leading order in alpha prime for the equations of motion. In the (super-)conformal invariance method the leading order is only obtained at one-loop computation. $\endgroup$
    – Nogueira
    Aug 24, 2018 at 0:58

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