# Are We In the Swampland?

Way back in 2005, soon after the emergence of the “String Landscape” and the ensuing debate over whether this made string theory untestable pseudo-science, Cumrun Vafa in response started writing about the “Swampland”. In contrast to the “Landscape” of effective field theories that are low energy limits of string theory, the “Swampland” is the space of effective field theories that are not low energy limits of string theory. One motivation here is to be able to claim that string theory is predictive, since if you can show a theory is in the Swampland, then string theory predicts that theory doesn’t describe our world.

I wrote a couple blog postings about this back then, see here and here. The situation was rather comical, with Jacques Distler unintentionally making clear one problem with the whole idea. He was enthusiastic about it, and gave as one example the suggestion that one and two generation versions of the Standard Model were effective theories that could not be derived from string theory, but Volcker Braun immediately wrote in to tell him about such a derivation. At this time I started having my problems with the arXiv (and its moderator, Distler) about trackbacks, but that’s another story.

I haven’t paid much attention to the Swampland business since then, but noticed last night a new preprint with the title What if string theory has no de Sitter vacua?. The authors summarize their argument:

From this analysis we conclude that string theory has not made much progress on the problem of the cosmological constant during the last 15 years. There is a general agreement that the presence of dark energy should be an important clue to new physics. So far, string theory has not been up to the challenge. Or to be more precise, string theorists have not been up to the challenge.

The well-motivated introduction of the anthropic principle and the multiverse, was a big relief. The mathematical standards were lowered, and unconstrained model building could set in exploring a wild and free landscape of infinite possibilities. But beyond this suggestive connection between a possible multiverse and the rich mathematical structures of string theory not much solid results have been achieved. We reviewed some fraction of the mounting evidence that most, if not all of this landscape, is a swampland and we refer to [14,16,149] for similar lines of thought. We believe it makes more sense to listen to what string theory is trying to tell us, then to try to get out of the theory what one would like to have. In recent years, especially with the program of the Swampland [14, 150–152], there is luckily a growing community that embraces this idea. Perhaps this program really already made its first prediction: no measurable tensor modes in the CMB.

From what we have seen so far, we believe that the most sensible attitude is to accept there are no dS vacua at all because string theory conspires against dS vacua.

The suggestion here is basically that effective field theories on a deSitter background are in the Swampland, so can’t be derived from string theory. Since we seem to live in a deSitter space, the obvious conclusion to draw from this is that string theory is falsified: it can’t be the fundamental theory we are looking for. The authors discuss various unconvincing ways to try and avoid this conclusion.

By the way, the authors make the usual flawed argument that “the string theory landscape is just like the Standard Model”:

This kind of criticism is, however, misguided [10, 11]. One might compare with quantum field theory, where there is an infinity of fully consistent theories. Experiments are needed to pick the right one, and parameters must be fitted. When this is done the theory still has enormous predictive power, and no one would claim that the Standard Model is useless. One could argue in a similar way concerning the string landscape.

They also claim:

Paradoxically the critics of string theory and the proponents of the string landscape all agree on one thing: the landscape exists and we more or less know its properties.

At least this string theory critic has never agreed on this. I don’t believe “string theory” is a well-defined enough framework to answer the question of what all its ground states might be, or to properly characterize them. If you accept conjectures about the theory put forward by the landscapeologists, all evidence is now that the set of ground states they identify is so large as to make predictions impossible. The argument against string theory is that there are two possibilities here: either the theory is too poorly understood to tell us what its ground states are, or it does tell us something, and there are too many ground states to make useful predictions. Either possibility leads to the same conclusion that this is a failed idea.

It is rarely acknowledged just how serious the problem of a lack of a definition of “string theory” really is. To get some idea of how bad this problem is, one can consult one of the main references in this paper, a survey of the Landscape and the Swampland, based on Vafa’s 2017 TASI lectures (this paper also discusses the idea that deSitter is not in the Swampland). Claims are often made that AdS/CFT resolves the problem of defining a non-perturbative string theory of quantum gravity, but in the paper one finds:

We can now ask the question if using this AdS/CFT correspondence gives a non-perturbative definition of string theory. The motivation for this is that we can give a non-perturbative definition of SYM theory, for example by lattice regularization, whereas the holographic quantum gravity dual theory in AdS has no complete definition. The fact that the CFT side, i.e. the non-perturbative definition of SYM, gives in principle, a non-perturbative definition of the AdS side, is of course true. But this may be not very useful for deeper questions of quantum gravity. In fact the regime that the gravity side is weakly coupled is big corresponds to when the SYM is strongly coupled. In fact ‘t Hooft was trying to use string theory as a solution to the gauge theory question at strong coupling and not the other way around!…

we find ourself back at the beginning: we want to know fundamentally, what is quantum gravity? It should describe the quantum fluctuations of the metric. From a brief analysis of the standard Einstein-Hilbert action, we see that fluctuations of the metric at the Planck scale should become very violent, leading to potential changes in the topology of the spacetime [103, 104]. This leads naturally to the idea that quantum gravity should be equivalent to summing over all spacetime topologies and geometries:
$$Z_{QG} \sim \sum_{\text{top. and geo.}} e^{-S}$$
In general we have no idea about what description will lead to the correct sum over geometries and topologies. We only do know that there should be some mechanism that washes out the Planck scale fluctuations to produce a smooth space at lower energies. It seems that this description must come from some new fundamental principle, rather than from some duality such as mirror symmetry or AdS/CFT. This lack of knowledge of describing the gravity side quantum mechanically is “the missing corner” in our understanding of string theory.

In both this survey and the new paper, the tactic of trying to remove the Landscape to restore the predictivity of string theory hits up against the obvious problem: you’re left with no theory at all (the equation above, with an undefined sum and an undefined action, is the essence of no theory).

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### 21 Responses to Are We In the Swampland?

1. Jack Morava says:

Perhaps it’s relevant that it’s not known whether or not there is a countable number of smooth structures on the four-sphere?

2. Peter Woit says:

Jack,
I think the notation “sum over all topological and geometric structures” speaks for itself. As a good rule of thumb, whenever people in this field talk about having a “measure problem”, if you look into it you’ll find that they don’t even know what the space is on which they are looking for a measure. The space of “all topologies and geometries” is a good example.

3. Marc Nardmann says:

@Jack Morava: If by “countable” you mean “having cardinality at most aleph_0”, then it *is* known that the set of diffeomorphism classes of smooth structures on the 4-sphere is countable. Even the union, over all compact 4-manifolds M, of the sets Sm(M) of diffeomorphism classes of smooth structures on M is countable.

In the “sum over all topological and geometric structures” context, one should also consider *noncompact* 4-manifolds M, and on those Sm(M) has usually (conjecturally: always) the cardinality of the continuum. Therefore compact 4-manifolds would probably be irrelevant, because the set of geometric structures (whatever that means precisely) on them should be expected to have measure 0 within the space of all topological and geometric structures. Of course one needs a precise definition of the space and of the measure to verify this expectation.

The set Sm(M) can be uncountable only for (noncompact) manifolds of dimension 4. Some people have suggested that this explains why our universe is 4-dimensional: the set of universes of other dimensions should be expected to have measure 0 within the space of all universes. Again, it all depends on the chosen space and measure. (Probably M-theorists would prefer anyway to explain why our universe is 11-dimensional?)

4. Peter Woit says:

Sorry topologists, but that’s enough about this issue, which has nothing to do with the topic of the posting.

5. Søren Bro Thygesen says:

So this is the basis for your problems with trackbacks? I find that incredibly childish on Distler’s part. I’m aware that smart people often have eccentric behaviour but man..

6. Peter Woit says:

Søren Bro Thygesen,
To be accurate, I have no idea at all (despite signficant effort to find out) why trackbacks to my blog are now banned, or exactly what Distler’s role in this is.

7. Shantanu says:

8. Peter Woit says:

Shantanu,

Not that I know of. Seems unlikely that any would. Claims about string theory and the landscape/multiverse make a very appealing story (physicists have this far-out cool idea that explains everything!). The new claims aren’t such an appealing story, and the “swampland” designation isn’t very good marketing. No one wants to write or read stories like “you know that cool stuff you read about last year? Best to just forget about it, probably doesn’t work”. In addition, when asked “what replaces the landscape/multiverse”, I think all those who want to get rid of the landscape have is the old xkcd cartoon

https://xkcd.com/171/

9. new says:

Peter can you comment on this

” Perhaps this program really already made its first prediction: no measurable tensor modes in the CMB.”

if the above is a genuine prediction of string theory, what are the implications to string theory if they find measurable tensor modes in the CMB?

10. Peter Woit says:

new,
There’s no such thing as a “genuine prediction of string theory”, for tensor modes or for anything else.

11. Cobi says:

One should mention that after writing down the qualitative “sum over spacetimes” formula, Vafa et al. review an explicit realization in the context of topological string theory.

12. clayton says:

Sav Sethi at Chicago has been making similar claims in seminars for quite a while now; this has been on the arXiv since last fall in 1709.03554, which Danielsson and Van Riet cite. I think that these issues are well known in the string theory community, but it doesn’t affect the day-to-day activity, which is no longer driven primarily by reproducing the Standard Model at low energy. Whether that is evidence of failure or (a good kind of) resilience seems to be a question of taste.

13. Peter Woit says:

Cobi,
Yes, a sum over some Calabi-Yau three-folds is discussed, but this isn’t a theory of quantum gravity. The xkcd cartoon applies.

Clayton,
“no longer driven primarily by reproducing the Standard Model at low energy” is a pretty weasel-worded way of saying “not driven at all by getting particle physics at any energy”. Also seems to be synonymous with “have given up on getting a theory that can be connected to the real world”.

If you read all the way through Sethi, at some point you finally come to a mention of what the real problem is: there is no well-defined theory capable of giving unambiguous answers to the questions being asked. More specifically, Sethi writes:

“To answer this question, we require a framework for computing quantum corrections. String theory should be that framework but we immediately face an obstacle. String theory requires an on-shell solution. There is no currently understood method of computing quantum corrections off-shell, and this is closely tied to the fact that observables in string theory are always correlation functions of vertex operators, which are only defined in an on-shell background.”

14. Petite Kabylie says:

Peter,

I do not quite agree with your argument this time:
“…either the theory is too poorly understood to tell us what its ground states are, or it does tell us something, and there are too many ground states to make useful predictions. Either possibility leads to the same conclusion that this is a failed idea.”

I am ok with the second possibility, but not with the first. If a theory is poorly understood it just means more work (and more time) is needed to get the idea off the ground rather than a right away dismissal.

15. DrDave says:

PW (2005): “I’ve heard from someone associated with the arXiv that it’s not their intention to allow trackbacks to my postings to be censored and that part of the problem has been both difficulties they’ve been having with new software….”
Curious what this person would say now.

16. Peter Woit says:

Petite Kabylie,
I was using “poorly understood” as a polite way of saying there is no theory (i.e. no non-perturbative version of M-theory for which the ground states are well-defined, and not in conflict with experiment).

It is fine to take the attitude that if one doesn’t have a viable theory, that just means one needs to put more time and work into finding one. In evaluating this question, in this case one should keep in mind that people have been trying unsuccessfully to solve this problem for over thirty years, and that most of the experts in the field who have worked on this have given up long ago and are doing other things (I know of no promising ideas for a viable definition of M-theory, and thus virtually no one working on this).

17. Curious Mayhem says:

The emptiness of that Z ~ SUM(topo,geo) exp(-S) — where we have no idea what the sum, its measure, or the action is — reminds me of Pauli’s reaction to Heisenberg’s proposal for a unified theory, with “just some details to fill in”:

“This is to show the world that I can paint like Titian. [A big drawing of an empty rectangle] Only technical details are missing.”

(Apparently from a 1958 letter to George Gamow, in reaction to Werner Heisenberg’s claim to a journalist that Pauli and Heisenberg had found a unified field theory. Quoted in Michio Kaku’s Hyperspace, p. 137 (1995).)

18. Peter Woit says:

Curious Mayhem,

Story also told in my book, didn’t realize Kaku also had it.

19. Moyses says:
20. Moyses, that article by Moritz, Retolaza, and Westphal appeared as preprint already in 07/17, and its claim is discussed in the above article by Danielsson-vanRiet:

We now highlight 3 problems with the IIB constructions that have been pointed out in the recent literature: […] 2) issues with anti-brane backreaction on the 4D moduli, which was found recently by Moritz, Retolaza, and Westphal [131]

21. Moyses says:

Urs Schreiber, I understand that such problems are part of the perturbative approach to IIB strings. Somehow, they are analogous to perturbative IR divergencies in QCD that can be cured by the lattice. For instance, a nonperturbative approach as IKKT shows different behavior at latter times: doi:10.1007/JHEP10(2012)147