Stringy Things

Notes on the Swampland (1): Constraining Effective Field Theories

1. Introduction

This is the first of a collection of several notes based on a series of lectures that I attended by Eran Palti at SiftS 2019. The theme of the lecture series was ‘String Theory and the Swampland‘. Palti’s five lectures were supported by his most recent and impressive 200 page review paper on the Swampland, which includes over 600 references [arXiv: 1903.06239]. The reader is directed to this paper and also to its primary references for more detailed information.

2. Review – Effective Field Theory

2.1. Schematic Overview

There are a few different ways in which one can approach the concept of the Swampland. One approach is through a direct study of certain deep patterns that have emerged in string theory (ST) over time [1], but were generally not appreciated until particularly important papers were developed conjecturing gravity as the weakest force [2] and conjecturing how there is a general geometry of the string landscape [3]. These are known as the Weak Gravity Conjecture and the Distance Conjecture, respectively.

It can be argued that these two conjectures are the two pillars of the Swampland programme. Their logic and rationale is deeply stringy, and potentially very general. It is, in a sense, an injustice to discuss the Swampland without first studying within a purely stringy context the general features that are observed to emerge in all string theory vacuum constructions and what we might consider as the two primary conjectures. On the other hand, there is a way to build toward this aim by way of a gentler introduction, which begins with a discussion of effective field theory (EFTs). We may take a few moments to consider a brief and schematic review of EFTs, beginning with motivation.

Effective field theory is a standard tool today in theoretical physics. Anyone who is familiar with EFTs will know that the story in many ways begins in parameter space. The nature of reality is such that there appears interesting physics at all scales. In almost every regime of energy, time or distance there exists physical phenomena that present themselves to be studied. Howard Georgi describes this incredible fact about the nature of reality in terms of a striking if not miraculous richness of phenomena [4]. In the context of this remarkable richness, a commonly cited motivation for the use of effective theory is that of convenience. We learn, much like in the example of Feynman’s glass of wine, that it is perfectly valid to partition parameter space, isolate a particular set of phenomena from the rest, and then proceed to describe that set of phenomena without requiring to understand the complete or total theory.

The intuition behind the use of EFTs is rather practical. An engineer building a bridge isn’t required to account for quantum gravity. The same idea applies to the example in the last paragraph. When considering different energy scales, should we choose to describe physics at a particular scale, it is perfectly valid within the philosophy of effective theory to isolate a set of phenomena at that scale from the complete theory, so that we may then study and describe its particulars without requiring to know the detailed dynamics at the other scales. So, if for instance we are interested in the physics at some scale {m^{2}}, it is not required that we know the dynamics at {\Lambda >> m^{2}}.

Much of the Swampland is based on a critique of how EFTs are constructed. As a matter of review consider, for example, a path integral {S} for some fields {\phi^{\prime}},

\displaystyle  \int D \phi^{\prime}e^{iS[\phi^{\prime}]} \ \ (1)

In principle, we can compute some of this integral but not all of it. So what we do is perform integration by splitting the fields into the momentum modes of each of the fields. This means may we perform integration over the {k} momentum modes. We also set {k > \lambda}, where {\lambda} is some energy scale. Hence, this energy scale {\lambda} is now the cutoff of the theory. And so, in integrating over the momenta, we are left with a path integral for these modes less than the cutoff of the theory,

\displaystyle  \int D \phi^{\prime}_{k < \lambda} e^{i S_{eff}[\phi^{\prime}]} \ \ (2)

What is left after integrating over all of the high energy modes is the effective action. The effective action is a function of some fields with modes less than the energy scale or cutoff, {\lambda}. It is also a function of the cutoff, such that {S_{eff} [\phi, \lambda]}. This effective action is valid below the cutoff scale, and in principle you don’t lose information.

However, this approach is problematic because one is required to know the ultraviolet (UV) theory. It can quite simply be said that we often don’t know the UV theory. Another issue is that integration can be very difficult. What to do?

2.2. Alternative Approach to EFTs

There is an alternative approach to constructing EFTs that we might pursue, which simplifies the computation and avoids some of the other issues stated above. This approach requires some guesswork and approximation based on what we believe the EFT should look like, filling in some of the gaps when it comes to our lack of knowledge of (in this case) the UV theory.

One may anticipate a problem with this. Generally it is the case that the following alternative approach is what allows for ambiguities in our theoretical picture, considering that a number of guesses are often made. The trade-off, though, is that it is easier to manage than the original approach described above.

So what is the alternative approach? In short, it can be defined according to a set of rules. To name a few such rules, consider:

1) There should be no processes with energy scales greater than the cutoff. So the theory should not be able to access energy scales greater than {\lambda}. Another way to put it is how one should not have processes with momenta greater than {\lambda}. Consider, for instance, some kinetic term for a scalar field,

\displaystyle  (\partial \phi)^{2} \ \ (3)

We can write an EFT like this, but since we don’t know the UV theory it could be that there other other terms in the EFT that we have neglected, which are a sum of higher derivative terms. By way of dimensional analysis, we can see there should be suppression of these higher derivative terms. For instance,

\displaystyle  (\partial \phi)^{2} + \sum_{k} \frac{1}{\lambda^{2k}(\partial \phi)^{2 + 2k}} \ \ (4)

We can see in (4) that there is suppression provided by the cutoff term. But we don’t know if this is actually correct. It could very well be that if we did the complete integration, a different cutoff would appear. In short, we are performing guesswork.

2) We should include all operators allowed by the symmetries of the theory. That is to say, we should include in the Lagrangian some objects that look like,

\displaystyle  \mathcal{L} > \frac{1}{\lambda^{k}}O^{d+k} \ \ (5)

Where we have some operators suppressed by the energy scale.

3) Often we will work in a perturbative expansion, in which case {g << 1} in order to have trust in the theory.

4) There should be no anomalies in the theory, especially for massless gauge fields.

The main idea, in summary, is that from the particular rules stated above one can essentially construct whatever EFT they might choose. Now, a natural pedagogical question may be as follows: why is a lack of knowledge about the UV theory a problem, considering one may still simply construct an EFT as described above?

Given that more often than not the UV theory is not known, as already stated, the main problem should be fairly obvious: EFTs rely on guesswork. In our previous example, one may rightly raise the concern that a very important guess and therefore working assumption was made about the value of the cutoff scale. Another person might then reply, ‘what is the problem? We make educated guesses all the time in physics!’ The answer to this question is something in which we will more thoroughly elaborate in just a moment. For now, in the context of EFTs, it can quite simply be stated that when it comes to an EFT coupled to gravity, there is a sort of induced universal expectation about the nature of the cutoff scale. And so there is some tension, and this brings us to the next rule.

2.3. EFT Coupled to Gravity

(5) With gravity, the cutoff scale is universally accepted to be less than the Planck scale. This means {\lambda < M_{P}}. In 4-dimensions, for example, the value of {M_{P}} is approximated as,

\displaystyle  M_{P} \sim 10^{18} GeV \ \ (6)

The reason for rule (5) generally is because if one reaches the Planck scale, the theory will be strongly coupled. It is unlikely the EFT will be valid at this scale, considering also the inclusion of both quantum mechanics and gravity. Moreover, although this is where string theory (ST) may enter into the picture, as it is valid at such energy scales, there are nuances that must be considered and appreciated.

To offer one example, in perturbative ST where the string coupling is sent to zero, {g_{s} \rightarrow 0}, this is valid at arbitrary UV physics. But perturbative ST is a small piece of a much richer theory, and it is generally true that deep physical insight may be drawn from non-perturbative methods. We may further emphasise this last point by noting that, when some finite value is attributed to {g_{s}}, non-perturbative effects appear prior to the Planck scale that suggest one’s theory is incomplete.

Putting such issues to one side for a moment, we may focus and concentrate the discussion according to this important summary message: some of the EFT rules discussed are stronger than others. Rule (1), for instance, is much stronger than rule (2). This last rule (5) is argued to be necessary; but we may still question whether it is sufficient. And it is is in the context of this question that we may also introduce the concept of the Swampland.

3. EFTs and the Swampland

Traditionally, when working in effective theory it is fairly simple to state or assert some cutoff below the Planck scale. Consequently, one may suppress their worries about quantum gravity. In fact, this is quite a common approach.

On the other hand, the Swampland programme is about how this assumption is wrong. Why is it considered wrong?

The Swampland is at least partly about how it is wrong to assume that, if one is working at scales much less than {M_{p}}, one need not worry about quantum gravity [5]. Instead, and for reasons that will become clear, the Swampland represents EFTs that are self-consistent but which are not or cannot be completed with the addition of quantum gravity in the UV.

But let us pause for a moment and reflect on this statement. The reason we have opened with a discussion of EFTs is to, at least in part, emphasise the manner in which self-consistency is an important tool at high-energies. Self-consistency allows us to assess the structure of physical theories at high-energy scales, especially with the absence of empirical constraints [5]. But at low-energies, the concept of self-consistency becomes much less sharp or effective as a tool for assessing physical theories.

In ST, the reason for this relates to the lack of unique predictions for low-energy physics. The picture we are about to describe is one already widely known and publicised. In bosonic string theory, spacetime is 26-dimensional. In superstring theory, it is 10-dimensional. Finally, in M-theory, it is 11-dimensional. That string theory implies extra dimensions is not a problem; it just means that in order to give description to nature – physical phenomena – we are required to compactify these extra dimensions to six-dimensional spaces. However, from our current perspective and understanding within ST, this situation gives rise to an order {10^500} four-dimensional vacua. This means that ST allows for many different low-energy effective theories, which may also be self-consistent.

Now, there is a lot that we still do not know about ST. Indeed, at the present time it is far from a complete theory and thus our knowledge and understanding is still quite limited. This incompleteness includes both the mathematical structure of the theory and how we understand it in terms of how ST relates to physical phenomena. I think it is always important to emphasise our present historical perspective when considering the ongoing development of a theory. That said, from where we sit, there is undoubtedly a vast landscape of possibilities, and this vast landscape of vacua suggests that an overwhelming number of different universes can exist, each with physical laws and constants.

The issues we face today are highly technical. As has so far been left implied, one problem has to do with how we construct EFTs. Another related issue has to do with the fact that it is a significant drop from the Planck scale to currently accessible energy scales. Regarding the latter, sometimes theories can be too general for a particular problem. For example, consider computing the energy spectrum of hydrogen within quantum field theory (QFT). It turns out to be much harder to do than in plain old quantum mechanics. This is because QFT is too general for the problem. The same logic and understanding can be applied to quantum gravity. To borrow the words of David Tong [6], to employ a quantum theory of gravity to formulate predictions for particle physics, this is in many ways like invoking QCD to formulate predictions on how coffee makers or kettles work.(From my own vantage, this gap is quite interesting to think about in the broader context of theory construction).

In addition to the above, the other more pressing issue is that, while there is an incredibly rich landscape of vacua – the String Theory Landscape – which corresponds to an incredibly large spectrum of EFTs, this fact often seems misconstrued as implying a complete or total absence of constraints [5]. But it is not so, and this is what defines the historical urgency of the Swampland programme: to establish, define, and prove necessary constraints on low-energy EFTs. At least in part, this is what might be taken to define the Swampland: even for effective theories that include gravity, there is a large set of apparently self-consistent low energy EFTs that ultimately produce an inconsistency in the UV [5].

Swampland theoryspace
[Image: Figure 1 from A. Palti, ‘The Swampland: Introduction and Review’, depicting theoryspace and the subset of EFTs which could arise from string theory.]

In the Swampland programme, one motivation is to uncover new rules for the construction of effective quantum field theories. Moreover, one can take it as a principle aim of the Swampland programme to quantify a set of low-energy constraints that enable us to delineate between EFTs that are in the string Landscape and those that are not. The constraints or criteria for such a delineation of theories must be formulated purely in terms of the low-energy effective theory.

4. From EFTs to the Rules of the Swampland

The question now is, how do we go about obtaining such new rules? To develop and study potential new rules, we focus on infrared (IR) aspects of quantum gravity. For instance, we study black holes / holography to probe the IR. We also study within the formalisms of ST.

Prior to 2014 (i.e, pre-primordial gravitational waves), the approach was to study specific constructions (compactifications) and from there extract phenomenologies. This proved difficult because, again, we don’t know the UV starting point. So, as described, the procedure was to make assumptions and attempt to construct something like our universe. Post ~2014, on the other hand, the approach is different in that it is now more or less conventional to use known ST constructions to determine general rules. Then, from there, one studies the phenomenology. As it presently stands, ST has an excellent track record of developing or discovering general rules (for example, think of black hole microstates or extra dimensions). This history of ST is one of its current strengths and something we can rely on – that is, we can be confident that it is likely the Swampland rules are not misleading us. To see this, a number of examples will be considered in this small collection of notes.

5. Weak Gravity Conjecture (Magnetic)

Let us consider, for example, a first encounter with the Weak Gravity Conjecture (WGC), one the new conjectured rules of the Swampland. There are two versions to this conjecture, the Electric WGC and the Magnetic WGC. For the moment, we shall consider a basic introduction to the Magnetic WGC. Arguments for why this may be general will be offered in following notes.

To start, we consider the following effective theory coupled to gravity, with a U(1) gauge symmetry and with a gauge coupling {g}. The action is of the form,

\displaystyle  S = \int d^{d}X \sqrt{g} [(M^{d}_{p})^{d-2} \frac{R^{d}}{2} - \frac{1}{4g^{2}} F^{2} + \ ... \ ] \ \ (7)

Now, the WGC tells us that there is a rule for any such low-energy EFT. The rule is that the cutoff scale of this theory is set by the gauge coupling times the Planck scale. In recent years, research has offered insights into what this cutoff means. We learn that for {\Lambda \sim M \sim g(M_{p}^{d})^{d-2 / 2}}, {\Lambda} is the mass scale in the theory and this mass scale is the mass of an infinite tower of charged states. Moreover, if an example of an effective theory is to be valid in ST, then we are lead to conclude that there must be a tower of states of increasing mass and charge. This tells us that \Lambda \sim g(M_{p}^{d})^{d-2 / 2} is the cutoff scale of the theory precisely because the EFT will breakdown under the infinite mass scale.

Interestingly, notice also that this tells us that the cutoff goes to zero when {g \rightarrow 0}, which is quite different from traditional pre-Swampland rules about how to construct EFTs. Consider it this way, when {g \rightarrow 0} the cutoff is low, and in this limit the theory is weakly coupled. According to what we may now consider as the traditional rules of EFTs, a weakly coupled theory is undoubtedly better from an EFT perspective, and generally the theory is considered more trustworthy in such a limit. So already there is a noticeable contrast, because the MWGC is saying something quite different: when the theory is weakly coupled, the cutoff is extremely low; thus instead of the cutoff scale for quantum gravity being at {M_{p}}, the conjecture is saying that the cutoff could actually be far lower than {M_{p}}.

From a traditional effective theory perspective, this may be perceived as somewhat shocking; there is no energy scale in this theory associated to the gauge coupling {g}. At weak coupling, there is also less control over the theory (instead of the traditional benefit of having more control).

Notice some other interesting characteristics for the conjecture. Firstly, it is gravitational – it is completely tied to coupling the theory to gravity. Consider, for example, the case where {M_{p} \rightarrow \infty}. In this case, the theory becomes decoupled from gravity such that {M_{p}} is like the coupling strength of gravity. What does this tell us? Quite simply, the theory becomes trivial when {M_{p} \rightarrow \infty} (a statement true for almost all Swampland conjectures).

Notice also that we have a statement about some energy scale. The statement is such that at some point, the effective theory must be modified. More pointedly, at higher energies the theory necessarily becomes increasingly constrained. This point about modification is particularly interesting. The implication is as a follows.

Consider again the image of theoryspace. Consider, also, starting with some theory at very low energy that gives the Einstein-Maxwell equations. Now, remaining at the same point in theoryspace, we begin increasing the energy scales of the theory as illustrated. We can do this for some amount of time leaving the theory unmodified. But, as pictured, the idea is that eventually we will reach a point, at the cone, where must modify the effective theory to focus on the constrained theory in the UV.

This is one way to visualise the statement that even for effective theories that include gravity, if we don’t modified our apparently self-consistent low energy EFT, we will ultimately produce a theory that is not consistent in the UV. In other words, what this is saying is that we must modify the EFT such that it conforms to the increasingly constrained theory in UV along the upward slope of the cone. That is, the theory must be modified so that it flows in energy toward the constrained theory of quantum gravity. And in the broader context of the Swampland programme, particularly in terms of defining criteria to distinguish the Landscape of vacua from the Swampland, it should be noted that interesting consistency requirements tested against WGC are currently being formulated, including studies on the behaviour of quantum gravity under compactification [7]. These ideas will be subject to further discussion in following entries.

Of course, the WGC is still a conjecture. That is to say, there is still no formal proof. But in this series of notes, several examples will be explored that offer very strong evidence that the WGC should be true.

6. Summary

To conclude this note, the statement that we must modify the EFT such that it conforms to the increasingly constrained theory in the UV – this very much captures all of the Swampland conjectures. The emphasis is that the implications of the WGC are in stark contrast to the approach for the traditional construction of EFTs, wherein for the latter the attitude is that at very high energies one may leave the theory unmodified until approaching somewhere near the Planck scale in which lots of new degrees of freedom appear in the theory, thus magically completing it as a quantum theory of gravity. The Swampland is saying, directly and explicitly, this is not a valid approach to effective theory construction and that modification of the theory can and likely will occur at energy levels far below the Planck scale.

In the next post, we will look at bit more at the WGC in the context of the 10D superstring. We will also begin to study the Distance Conjecture and, finally, look a bit at M-theory.

References

[1] C. Vafa, ‘The String Landscape and the Swampland’, [arXiv:hep-th/0509212 [hep-th]].

[2] N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, ‘The String Landscape, Black Holes and Gravity as the Weakest Force’, JHEP 06 (2007) 060, [arXiv:hep-th/0601001 [hep-th]].

[3] H. Ooguri and C. Vafa, ‘On the Geometry of the String Landscape and the Swampland’, Nucl.Phys.B766: 21-33, 2007, [arXiv:hep-th/0605264 [hep-th]].

[4] H. Georgi, ‘Effective Field Theory’, Ann.Rev.Nucl.Part.Sci. 43 (1994) 209-252.

[5] E. Palti, ‘The Swampland: Introduction and Review’, [arXiv:1903.06239v3[hep-th]].

[6] D. Tong, ‘String Theory’ [lecture notes], [arXiv:0908.0333 [hep-th]].

[7] Y. Hamada and G. Shiu, ‘Weak Gravity Conjecture, Multiple Point Principle and the Standard Model Landscape’, JHEP 11 (2017) 043, [arXiv:1707.06326 [hep-th]].

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