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I often hear people talk about finding a non-perturbative formulation of string theory.

What does this mean exactly? To my knowledge string theory is a perturbative method. Just like Feynman graphs are a perturbative method. But people don't talk about a non-peturbative formulation of Feynman graphs. (Or do they?)

If someone found a theory to which could be calculated with the perturbation method of strings. Should this theory really be called a "string theory"? Wouldn't it be a different separate theory? What properties would this theory have, and how do we know some theory like Supergravity isn't this theory?

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    $\begingroup$ Feynman diagrams are a perturbative expansion, but QFT is more than Feynman diagrams. $\endgroup$
    – Javier
    Sep 21, 2018 at 15:22
  • $\begingroup$ Feynman diagrams are a perturbative expansion in QFT. One can separately talk about nonperturbative QFT (often using path integrals). Analogously, the worldsheet formulation of string theory is perturbative. Similarly, one hopes to find a non-perturbative formulation of string theory. $\endgroup$
    – Prahar
    Sep 23, 2018 at 19:38

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"But people don't talk about a non-peturbative formulation of Feynman graphs. (Or do they?)"

Actually sometimes they do. It is common in quantum theory to apply variational techniques to approximate a ground state configuration and there are some QFT methods that make use of this approach. They are equivalent to summing a subset of the associated Feynman diagrams to infinite order and are thus refered to as nonperturbative methods. In many body physics these are sometimes called mean field methods.

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  • $\begingroup$ This is not an answer to the question, but a side-comment to the argumentation. $\endgroup$
    – DanielC
    Sep 21, 2018 at 18:26
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    $\begingroup$ @DanielC You are right, I was only addressing the "or do they?" part, not the question in the title. $\endgroup$ Sep 21, 2018 at 18:31
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In the case of Yang-Mills theory, QCD for example, we have a complete (non-perturbative) formulation of the theory in terms of the Lagrangian and the path integral. Feynman diagrams are then a perturbative method to calculate scattering amplitudes, but you can perform non-perturbative calculations by doing lattice QCD for example.

The formulation of string theory from the world sheet is inherently perturbatively stated in terms of an expansion in the string coupling. It gives the prescription for calculating the S-matrix in perturbation theory, but does in a sense not explicitly tells us fundamentally which theory we are dealing with. Likewise the 10D Lagrangians one works with in string theory are only low-energy effective Lagrangians, namely supergravity. The approaches to understanding the non-perturbative formulation of string theory would seem to go through M/F-theory and string field theory.

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  • $\begingroup$ So we would be looking for a theory like lattice QCD, so maybe some kind of lattice string theory? Are you saying that there may be more than one non-perturbative theory, which have the same S-matrix? I was lead to believe that the S-matrix uniquely determines the theory? $\endgroup$
    – user84158
    Sep 23, 2018 at 20:36
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Feynman graphs are perturbative. If you wish to discuss Non-Perturbative QFT, a Perturbative Expansion should not be brought up. An equivalent statement to Feynman graphs in QFT is a worldsheet approach to SST. Non-perturbative approaches to SST are: M-theory (or F-theory), matrix models, field theory, etc. In summary, "what is meant by non-perturbative SST" is equivalent to asking "what is meant by non-perturbative QFT" in a more specific context. If you are unfamiliar with the answer to the general question regarding the case of non-perturbative QFTs, I am unsure of why at all you are asking it in the case of SSTs.


For further readings on the topic of non-perturbative SST, a brief introduction is given in the text "An Introduction to Non-Perturbative String Theory" [Ashoke, S. 98]. For readings in the seminal papers introducing non-perturbative effects, "Combinatorics of coundaries in String Theory" [Polchinski, J. 94], "Five-Branes, Membranes and Non-Perturbative String Theory" [Strominger, A. et all. 95], "String Theory Dynamics in Various Dimensions" [Witten, E. 95], "Evidence for F-theory" [Vafa, C. 96], "Proposals on Non-Perturbative Superstring Interactions" [Motl, L. 97] and "Matrix String Theory" [Verlinde, E. 97], "Noncommutative Geometry and String Field Theory" [Witten, E. 86] and "Introduction to String Field Theory" [Siegel, W. 01], etc. respectively. In some cases I could not recall the original text and instead listed introductory resources on the subject.

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  • $\begingroup$ Not really because string theory was define perurbatively. So you can't avoid talking about the perturbative theory when you ask what is the non-perurbative sector? The difference is in QFT the non-perturbative formulation was already known. $\endgroup$
    – user84158
    Jan 30, 2019 at 15:06
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    $\begingroup$ Sure, I assume you’re reacting to the blanket statement “what is meant by non-perturbative methods in SST is equivalent to asking what is meant by non-perturbative methods in QFT”. If one studies non-perturbative sectors in SST, perturbative expansion is rarely brought up, so I disagree with the statement “you cannot avoid talking about perturbative theory when asked about non-perturbative methods” — especially because the argument you’ve made for that appears to be historical? $\endgroup$ Jan 30, 2019 at 17:39
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    $\begingroup$ Again, if one is interested in the non-perturbative formulations of SST I have provided them in the answer above. Your question appeared to be very broad, and of course my answer was therefore slightly less technical than it could have been because of this. Of course asking of non-perturbative methods in SST is not EXACTLY equivalent to asking of non-perturbative methods in QFT, this should be obvious, but I drew the comparison assuming you’re familiar w/ non-perturbative QFT. $\endgroup$ Jan 30, 2019 at 17:51

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