I start with the Large $N$ QCD paper by 't Hooft. When 't Hooft published his paper on Large $N$ QCD it was clear why the string theory of hadrons due to Gabriele Veneziano could make sense. But at the same time, it was an end to strings as fundamental entities responsible for strong nuclear force. In fact, it turned out that the large $N$ QCD gives rise to Flux tubes and turned out as an effective field theory of QCD.

My question is, why don't we interpret AdS/CFT in the same manner, in the sense that the large $N$ limit of super Yang-Mills theory gives rise to a Stringy picture that is by definition only an effective field theory of the Super Yang-Mills in the large $N$ limit?

The fact that such duality is proven in that particular limit and not for arbitrary finite $N$ is the core of my question.


1 Answer 1


If string theory were really a UV completion of gravity then it would make a prediction for quantum gravity S-matrix elements when Newton's constant is large. But this cannot be done with the perturbative formulation based on the genus expansion. Rather, the main way people have of computing these large $G$ observables is to "believe the duality" and then do a small $N$ calculation in $\mathcal{N} = 4$ Super Yang-Mills using field theory methods (Feynman diagrams, integrability, bootstrap, localization and maybe lattice).

So for now, $\mathcal{N} = 4$ Super Yang-Mills with small $N$ defines what we mean by strongly coupled type IIB string theory around $AdS_5 \times S^5$. I.e. it admits a limit which agrees with weakly coupled string theory and stays well defined in principle for all other values of its parameters. In other words, we never understood strings well enough to even know what it would mean for the duality to "break down". It is conceivable that somebody might find another way of doing non-perturbative string theory which disagrees with Super Yang-Mills despite passing the same checks but so far that hasn't happened.

  • $\begingroup$ There are two problems with old school string theory. (1) no one can solve it and (2) no one can show that it even exists. $\mathcal{N} = 4$ SYM only has the first problem so that's why the "duality" is useful. But really, proving that something known to exist is dual to something not known to exist is impossible by definition. $\endgroup$ Jan 7, 2022 at 19:14
  • $\begingroup$ I'm not claiming that every possible check has been done since it's always possible to go to 8 loops or whatever the record is and check one more correction. If this were all the string theory community had done since 1997 it would indeed be a dead end game but the exploration of topics has been much broader than that. $\endgroup$ Jan 7, 2022 at 19:16
  • $\begingroup$ As a final comment: Can one falsify string theory as a quantum gravity theoretically if someone makes a nonperturbative computation in terms of the dual string theory (in particular in the strongly coupled regime) and find a result that agrees with Super Yang-Mills only in the large N limit? If so, has there been any attempt in this direction? $\endgroup$ Jan 7, 2022 at 21:28
  • $\begingroup$ Non-perturbative string theory is not a god given thing. There are attempts to find formulations of it other than Super Yang-Mills. If one of them succeeds and turns out to only agree with Super Yang-Mills at large $N$ then it will just mean type IIB string theory on $AdS_5 \times S^5$ is not unique and people will have to specify which one they're using. $\endgroup$ Jan 7, 2022 at 23:03

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