# Steps for going from Polyakov action to low-energy effective action (SUGRA?) in String Theory

A lot of string-theory questions have been asked here. This one is, hopefully, different in that this inquiry is specific and the expected answer would be more mathematical than philosophical in character.

To give some context, I'm familiar with LQG, QFT etc. I'm a beginner in CFT and string-theory. On reading papers on AdS/CFT and String Theory the following sentence often appears: "The low-energy effective action for $x$ dimensional ST of type $y$ reduces to that of supergravity (SUGRA) in $z$ dimensions with matter fields of types $a$, $b$ and $c$".

The \$64,000 dollar question any noob, such as me, wants answered is what are the mathematical/technical steps or arguments which lead us from the fundamental action of string-theory (the Polyakov/Nambu-Goto action) to a low-energy effective description. Of course, I might be asking a question for which the only concise answer is: "go read GSW". Here's hoping otherwise ;)

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Nice question, space_cadet. Furthermore, I agree with your point concerning the kind of language on a site about physics - mathematics :) I might use "You don't like it? Go somewhere else!" ;) –  Robert Filter Jan 7 '11 at 9:56
If you can get your hand on them, there are lectures by Callan and Thorlacius called sigma models and string theory, which spell out a lot of the calculations in detail. –  user566 Jan 7 '11 at 16:26

The basic answer is indeed "go read GSW or Polchinski" but if your goal was to get from the Polyakov action to a low-energy effective description as quickly as possible there are parts you could skip. In brief you want to

1. Work out the spectrum of the first quantized string. The "best" method is modern, covariant quantization, but the quickest method is light-cone quantization. If you want to know how to do this for say type IIA or IIB superstring you need to read GSW Chap. 2 and 4.3.

2. Deduce the existence of spacetime supersymmetry in the theory. This is the content of GSW Chap. 5.

3. There is a unique low-energy action with the massless field content and spacetime supersymmetry of type IIA or IIB string theory, namely the actions for IIA and IIB Sugra, so you know on general grounds that the action for IIA or IIB supergravity must be the low-energy effective action for IIA or IIB string theory.

4. If the general argument of 3. leaves you feeling uneasy you can remove this unease by computing a variety of 3 and 4 point tree amplitudes directly in string theory (now you need to read Chap. 7 of GSW), taking their low-energy limit, and comparing to the 3 and 4 point amplitudes you compute using IIA and IIB Sugra. You will find perfect agreement.

5. If you want to generalize this to include additional matter fields you will need to extend your analysis to the heterotic string and/or learn how to include D-branes into your type II analysis. Now you need to start reading Polchinski or various reviews, but you'll have a good place to start if you have done 1,2 and part of 4.

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Thank you @Jeff. And yes, that is my goal. I'm at a stage in my career when I don't have the time to sit around reading through various tomes to reach the meat. Of course, when I do have time I will read GSW cover to cover. In the meantime your list will help me get a quick start. Thanks again. –  user346 Jan 7 '11 at 15:40

To answer your question including the full supergravity, explicitly enough, one needs an amount of space not much smaller than the usual textbooks. And supergravity usually appears in their second volumes only. ;-)

However, one may divide the question to the proof of supersymmetry, and the proof of gravity in string theory. The proof that superstring theory obeys spacetime supersymmetry is a separate thing. A combination of gravity and supersymmetry immediately implies that the spacetime action has to be one of supergravity - this is pretty much by definition of supergravity, even though this verbal exercise doesn't really construct the action in components.

So the remaining question is why string theory predicts gravity.

It's because the world sheet action describes the proper area of a world sheet that propagates in a background spacetime. This spacetime may be infinitesimally changed by changing the background metric. Because the world sheet theory is scale-invariant, every infinitesimal deformation of the world sheet Lagrangian has to correspond to a marginal deformation, i.e. a dimension (1,1) tensor field.

By the state-operator correspondent, a local field of this dimension translates to an allowed, physical massless field in spacetime. Its spacetime spin is 2 because it carries the same indices as the background graviton. By adding a condensate of such closed strings in the gravitational vibrational pattern, one impacts the behavior of all other strings in the same way as if he deformed the background. This follows from the state-operator correspondence.

So the principle of equivalence is automatically satisfied: the graviton closed strings behave just like perturbations of the background, and they act on the motion of all objects and fields in the same way. The spacetime geometry itself follows Einstein's equations because they're the only diff-invariant locally Lorentz-invariant equations that they could satisfy - whose linear part is a wave equation.

More explicitly, one may show that one gets the Ricci tensor in spacetime because it's a consistency condition for the allowed backgrounds where strings may propagate. The string dynamics may only be defined by scale-invariant world sheet theories, and the change of the world sheet action with respect to the scale - also known as the beta-function (for each coupling constant) - has to vanish as a consequence.

For the worldsheet, the metric tensor at each spacetime point is a coupling constant and the beta-function may be calculated from one-loop graphs in the world sheet theory, and it ends up being the Ricci tensor, roughly speaking. It follows that the background has to satisfy the right equations of general relativity at long distances.

The state-operator correspondence guarantees that the closed-string excitations also interact according to the same equations because the closed strings are equivalent to infinitesimal deformations of the background.

Supersymmetry is added as a lot of extra fermionic stuff and leads to supergravity in spacetime - which is still just gravity coupled to other matter fields in a way that preserves spacetime supersymmetry. See also

http://motls.blogspot.com/2007/05/why-are-there-gravitons-in-string.html

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@Lubos it is a little surreal to actually be communicating with you. I am a big fan of your blog, even though I don't necessarily agree with everything you say. I'm not used to the level that you bring to the answers on this site, so I'll get back with a more useful comment when I have managed to fully assimilate your response. –  user346 Jan 14 '11 at 4:18
Ok. Based on what I've gathered so far ... your explanation, starting in the 4th para, for why the world-sheet action encodes the dynamics of the bulk geometry is very nice, modulo my understanding of some terms such as "marginal deformation" and "state-operator correspondence". For the second term I understand the general notion but not in its manifestation in terms of the properties of a string. I also need to go deeper into the other calculations which you briefly touch on. But overall A+ :) –  user346 Jan 14 '11 at 5:23