The following collection of notes is based on a series of lectures that I attended by Eran Palti at SiftS 2019 at Universidad Autonoma de Madrid. 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 [hep-th]]. The reader is directed to this paper in addition to supplementary references that I also provide at the end of each set of notes.
In the following entry, the notes presented follow the second lecture of Palti’s series.
1. Introduction
In this collection of notes, we continue to build toward the view of why it is valid to think of the Weak Gravity Conjecture (WGC) and the Distance Conjecture (DC) as being almost axiomatic to the Swampland programme. In other words, we are working toward the understanding of how and why these two conjectures are two fundamental pillars of the Swampland programme, from which every other conjecture is related or connected in some way.
In the last post, one will recall that we began by considering a general introduction to the Swampland programme in the context of arguments about constraining effective field theories. We also considered a very basic introduction to the Magnetic Weak Gravity Conjecture (WGC) and reviewed how, if we have some U(1) gauge field with a gauge coupling , then as we should have an infinite tower of states [1]. This infinite tower of states was found to have a mass scale set by the product of the Planck scale and the gauge coupling.
In the present entry, we will continue our review of the Swampland, turning our attention to the WGC in the context of the 10-dimensional superstring. Following this, we will also outline a basic introduction to the DC. Then, to close, we will perform our first tests of the DC and discuss how this conjecture relates to the parameter space of M-theory.
2. An Infinite Tower of States: From the Weak Gravity Conjecture to the Distance Conjecture
In this section, we will work toward a gentle (and generally informal) introduction to the DC (much like we did for the WGC in the first entry) following Palti’s second lecture at SiftS 2019. To begin, let’s recall some basic facts about the mass scale in bosonic string theory. We start with the following spacetime action for the low-energy effective theory,
This is the low-energy effective action of the bosonic string. For review of the construction of this action, see Section 3.7 in [2]. What is important to note is that, neglecting the tachyon mode, this action contains the massless spectrum. But we are not particularly interested in the massless spectrum, as Palti emphasised in his talk. Instead, our present interest has to do with the massive modes. When we study the string massive modes, it can be reviewed in [2,3] and in other textbooks that we have for the nth harmonic of the string , where . In other words, if we increase the internal excitations along the free bosonic string, we increase the mass with the mass of the string set by the string scale .
What we learn in performing such a study of the bosonic string is that string theory has an infinite tower of massive states. One can observe this infinite tower quite quickly through light-cone gauge quantisation of the Polyakov action, providing direct access to the study of the string spectrum. The key message emphasised at this point in Palti’s lecture is how, in this theory of massive states, the mass scale of the tower is . Crucially, this is UV data.
Rather than spending more time reviewing the bosonic string, what we want to do is investigate how this mass scale behaves when we vary the parameters of the low-energy theory. In particular, we want to know how this infinite tower of massive states behaves when we vary the string coupling . To this end, let us for the moment introduce some more basic ideas by considering a generic 10-dimensional string from the action (1), which may be written as follows,
We are currently not worried about the extra terms `…’ in (2). As Palti explains, what we really want to focus on is the relation between the string scale and the Planck scale. Recall the d-dimensional Planck mass, . In the Swampland programme we often want to work in Planck units (a point which is emphasised in lecture 2), and it is useful to fix the Planck mass such that . Furthermore, we should also note that the d-dimensional Planck mass from the effective string action is defined as, . To convert the string scale to the Planck scale for the action (2), we look at the Ricci scalar pre-factor and consider dimensional reduction. In 10-dimensions, it is fairly trivial to see that we have (remembering that and so ),
Notice, upon rewritting the string scale in terms of the Planck scale, in the weak coupling limit where we find , which implies that we have an infinite tower of states that become light relative to the Planck mass. In another way, as . What is curious about this is that it is very reminiscent of the WGC that we discussed in the last entry, where we take a U(1) gauge field and make it weakly coupled to a find a light tower of states.
More pointedly, great emphasis is placed at this juncture about what is generally an interesting feature of string theory. If we make our theory weakly coupled, we obtain an infinite tower of light states relative to . That is to say, weakly coupled string theory doesn’t have light states close to the Planck scale as one may expect or anticipate. In fact, what we see is that in weakly coupled string theory we have states arbitrarily lower than . From a traditional effective theory point of view, this is quite striking behaviour. So let’s think about this more deeply.
To begin with, it should be highlighted that the string coupling is a scalar field in the theory. More generally, we should remember that there are no coupling constants in string theory such that they are in fact expectation values of fields. As a quick review, remember that the expectation value of the massless scalar field, , which is the dilaton, controls the string coupling. This can be explained a bit more eloquently. Consider , where is some constant. The dilaton coupling reduces to , where is the Euler characteristic of the string worldsheet. The lesson we learn, often first in the study of the bosonic string, is how the constant dilaton mode taken to be the asymptotic value determines the string coupling constant, , such that we find , corresponding to the amplitude to emit a closed string [2].
All of that is to say that we should remember that the string coupling, , is a dynamical parameter – i.e., a field – that is determined by the dilaton. In this light, the previous statement about how we obtain a light tower of states relative to implies that (3) can be rewritten in the following way,
Where is some constant such that and . Following Palti, notice how if we send , we get a light tower of states. In studying this behaviour, it is nice to reflect back on the recent reference made to the WGC; however, this behaviour implies an encounter with another Swampland Conjecture, namely the Distance Conjecture.
The main segment of lecture 2 was structured on this idea: namely, one will find that the simplest example of the DC is by going to weak coupling in string theory, As we have hinted so far, this implies going to large distances in the dilaton field space in which one obtains, indeed, a light tower of states. To put it another way, the DC states that this kind of behaviour – where we obtain an infinite tower of light states – is universal. This means that whenever we give a value to a scalar field that is very large, we obtain a light tower of states that is, as Palti put it, exponential in the expectation value of the scalar field [1,5].
We have already sent . An interesting question now is to ask: what happens when we send ? This implies the strong coupling limit, which raises many curiosities. We will explore more deeply and define this limit more carefully in time. For now, as a point of introduction, we note that the DC states: given a scalar field , there is an infinite tower of states whose mass relative to may be written,
Let’s now test this conjecture with a simple example.
3. Testing the Distance Conjecture: Compactification of String Theory on a Circle
In the entry on Palti’s first lecture, it was mentioned that one approach to the study of the Swampland Conjectures is by way of a direct study of certain deep patterns to have emerged in string theory over time. What we will see, now that a gentle introduction to the WGC and the DC is out of the way, is that both of these conjectures are increasingly general. Indeed, we will see that they describe very general and deep patterns in string theory. But are they fundamentally true? The argument is that we may take the persistence of the sort of behaviour we find in string theory as evidence that these conjectures are true (or seem to be increasingly true). This is one of the main messages to come from Palti’s second lecture, wherein we consider a first basic test of the DC.
We construct this basic test by seeking to study the symmetry of the massive spectrum when string theory is compactified on a circle. We take the approach of first studying field theory compactified on a circle and then focus on the string case.
3.1. Field Theory Compactified on a Circle
Consider, for instance, spacetime. As before, we are working in Planck units where is the d-dimensional Planck mass. For the circle there is also of course a periodic identification of the form . We must also be mindful of notation when working in the higher dimensional and lower dimensional space. For the higher D-dimensional spacetime we have the following product metric,
This is the Einstein frame, where are D-dimensional coordinates such that while . If the D-dimensional metric is , the lower d-dimensional metric is . Notice also that we have the parameter , which is a d-dimensional scalar field. The and terms are constants. To aid in the production of a canonically normalised theory, Palti notes in his talk that we choose and . The reason for this choice will become clear in just a moment.
The circumference of the circle on which we will be compactifying our theory is given by,
Where we can see quite explicitly the relation between and the radius of the circle. Crucially, the radius of the circle becomes a dynamical field in d-dimensions. As it is a dynamical field, we will want to study the behaviour of the d-dimensional theory when we vary the expectation value of . Also important is that, when we reduce the higher D-dimensional Ricci scalar, , we obtain something in the Einstein frame in lower dimensions,
Moreover, to obtain (8) we have decomposed the Ricci scalar on the left-hand side of the equality for the metric (6). To do this, we take the metric ansatz and plug it into the higher dimensional Ricci scalar, which gives us a lower dimensional Ricci scalar from restricting the higher dimensional indices to lower dimensional indices. From one’s knowledge of scalar curviture, it can also be seen that the higher dimensional Ricci scalar is a two derivative object. This means that those derivatives act on the field ; however, the choice for and ensure no factorises in front of (hence the chosen definitions of and ). One can see that, after all this, we end up with a kinetic term that is canonically normalised.
Now that some notation has been established and we have dimensionally reduced to a circle, the idea is to consider a massless D-dimensional scalar field,
Where we have performed a Fourier expansion of the higher dimensional field in terms of the lower dimensional modes along the circle. Note, is made to be periodic because it depends here on the lower dth dimension, hence the decomposition already implicit in (9). Moreover, notice the coefficients depend on the external spacetime (lower dimensional coordinates). This means they are like lower dimensional fields. To word it another way, a higher dimensional field gives an infinite number of lower dimensional fields. The modes are d-dimensional scalar fields, where is the zero mode of and are the nth Kaluza-Klein (KK) modes of the higher dimensional field.
Another point worth highlighting as a natural consequence of compactification concerns how we also see that the in the exponential is quantised. This means it should be an integer, since we should have periodicity. This indicates that the momentum of the lower dimensional fields is quantised in the compact direction allowing us to write,
For simplicity, we shall restrict to flat space in lower dimensions. This means . And from this, we look at the equations of motion for in (9). We find,
Where is just the Klein-Gordon equation. When we expand this equation we obtain (by restricting the indices to be external indices plus the inverse metric) what is written to the right of the first equality. From this, we can look at the equations of motion for each of the modes,
The question that is raised: what is this lower dimensional equation for each of the KK modes? It is a Klein-Gordon equation for a massive field. But what is the mass of this field? Quite simply, it is set by the radius of the circle, , and the KK number. So the mass of the nth KK mode is given by,
What is this telling us? It says that when we dimensionally reduce on a circle, as we have done, we obtain something similar to the string (which we will look at in a moment). Notice, moreover, that we have a lower dimensional theory and that theory has an infinite number of massive states. What we have found, as Palti emphasises in his lecture, is that in the lower d-dimensional theory the KK modes are a massive tower of states. The masses here are increasing. Why is this so? Recall that the radius of the circle, , is a dynamical field in the lower dimensional theory. As such, the mass of the infinite tower of states that we observe depends on the expectation value of the field in the lower dimensional theory.
However, this isn’t quite the spectrum of string theory on a circle. We have so far only been considering field theory compactified on a circle. What we observe is thus the massive spectrum of Einstein gravity. For the complete string spectrum on a circle we need to add another important piece to the picture. So let us go to the string theory picture, and then connect the results.
3.2. Compactification of String Theory on a Circle
In this section we consider generally the propagation of a string in spacetime in which one spatial dimension is curled up into a circle. One can review the full procedure in section 2.2.2 in [1]. For further review on compactifying on a circle, see [2,4]. To save space, and in following Palti’s lecture, we move directly toward the main point of focus: namely, when we compactify a dimension we modify the string mass spectrum. And, indeed, much like before it is the massive spectrum that we are interested in studying.
Working in 10-dimensions, as we have been, one will find that when compactifying the 10th dimension we obtain for the compactified direction,
Where one will notice that we now have winding states. In (14), is the winding number such that . This comes from the fact that the string can wind around the circle times. We can also define the winding as . As we will discuss in a moment, the winding is actually a type of momenta. In review of the mode expansions, one will find both left and right-moving modes, which, together, for the compact direction may be written as,
The total center of mass momentum is therefore . Importantly, when we compactify a dimension, the center of mass momentum is quantised along that direction. Moreover, it turns out that along the circle the string acts like a D0-brane, i.e. a particle with quantised momentum . This term is, in fact, the Kaluza-Klein excitation number. And what we observe is how, in (15), we have the momentum mode in the form of and another form of momentum in the form of , which is the winding mode of the string satisfying,
To realign with Palti’s talk, notice that we now have additional states that we must consider: i.e., when we compactify on a circle there are also winding modes. We will talk more about these in a moment. For now, we should remember that the entire point of the exercise is to look at the massive spectrum. If we go to the target space light-cone gauge, the mass spectrum of the string reads as,
If one were to look deeper it is not too difficult to prove (17) and see why the level matching condition no longer holds. Indeed, we find . And, if we drop the excited oscillators, for the mass formula we have,
Following Palti,the task in these notes is to now think of this result (which is standard and can be reviewed in any string textbook) in relation to what we found in (13). This involves changing to the Einstein frame (6). In changing from the string frame to the Einstein frame, Palti explains how the massive spectrum which now includes both the KK number and winding number matches the field theory result for the KK masses (13),
3.3. Testing the Distance Conjecture
What we now want to do is test the DC by studying the d-dimensional effective theory, with the action (8) and the mass spectrum (19). One can, and perhaps should, anticipate a discussion on T-duality. Although it has not yet been introduced, its presence is ubiquitous.
Looking at (8) and (19) recall the fact that we have a scalar field in our theory. As has so far been described, this scalar field gives the radius of the circle. So a natural question to ask is, what happens when we change the expectation value of ? Do we obtain exponentially light states?
As Palti highlights in his lecture, we see that this is precisely the case. Recall how the exponential of goes like in (7). The mass of the state parallel in in (14) will grow exponential in . So, when considering the DC, we see that for (size of the circle) there is an infinite tower of KK modes that go something like the inverse power of in (14),
And we also have the winding mode tower,
Where, . What we see is that, if we make the circle very big we obtain an infinite tower of states that becomes very light. These are the KK modes. Reversely, if we make the circle very small we obtain an infinite tower of states that becomes very light. These are the winding modes. What is going on here? A discussion on T-duality is well anticipated. But another way to visualise this behaviour is first by reviewing the following log plot, where the mass scale for the KK and winding modes are plotted as a function of the expectation value of the scalar field .
The slope is , while the symmetry is indeed an expression of T-duality.
4. Lessons about the Distance Conjecture and T-duality
What did we learn in our first test of the DC? Several lessons can be gleaned, which then set-up for more advanced discussion:
1) We learn, for example, that the conjecture is deeply string theoretic. The presence of winding modes means we are learning about very stringy behaviour.
2) This term that we’ve just considered, which acts as the exponent for the exponential behaviour, it is roughly order one: . So our tower of states truly are exponentially light.
3) Think for instance of the case when . In this limit the effective theory breaks down. Why? Notice that when we send to negative infinity, this pulls down an infinite number of modes below the cutoff scale (i.e., an infinite tower of light KK modes). The implication is that we have new modes now appearing in the theory. Moreover, as discussed in the previous section, whenever we set the scalar field to have a very large expectation value, what we obtain is an infinite tower of light states and new description of the physics, which in this case is the higher dimensional theory.
4) What about the limit ? The effective theory still breaks down. Just as in 3), we obtain an infinite tower of lights states (winding modes). What about the description of the physics? Is there a new description in this limit? The answer is that it is, again, a D-dimensional theory because of T-duality.
To offer an example, consider the mass of the spectrum in the string frame,
The spectrum is invariant under the symmetry,
Which is T-duality. All that we are doing, as Palti puts it, is rearranging our degrees of freedom. To word this differently, T-duality is simply a special type of symmetry that allows us to relate our theory at a short distance with our theory at a long distance. They are the same theory, except from the vantage that we are viewing that theory from different perspectives: i,e., T-duality allows us to transform between small and large distance scales. In the case of compactification of some spatial dimension to a circle of radius , as we have been considering throughout these notes, the simple idea to begin with is that we may transform the original radius to a larger (or smaller) radius , such that . One can then see that with such a transformation we must also transform the winding states, such that . The main premise is that high-momentum states in the one theory is exchanged for the winding number in the other (and vice versa). Under this transformation the whole theory stays the same (T-duality invariance), including the spectrum, it is just that we are transforming from the KK modes to the winding modes (and vice versa).
This is why, for instance, in the limit the circle becomes very small and the winding modes become very light; but the physics in this limit is the same as in the case when the circle is very big.
5. The Dilaton Revisited in Type IIA String Theory
From the test of the DC by studying field theory compactified on a circle, we have already gained some interesting insights. We have observed that, in the case of a scalar field, we may go to opposite limits, or . In both cases we obtain an infinite tower of light states.
Now, recall that in the much earlier example of the dilaton, where we considered the string coupling, we only studied one limit: (see section 2). That is, we only considered what happens in the weakly coupled theory. Let’s now revisit this example, and consider what happens in the strong coupling limit where .
In going back to ask this question about the dilaton, remember that in the D-dimensional theory . We also know that and that when , which is similar to lesson 3) above where we obtained light KK modes.
In short, if we send , we are lead to believe by the logic of the DC that in this strong coupling limit we should obtain a light tower of states. Is this true?
To think of the strongly coupled theory, let’s go to the superstring theory. Consider, for instance, the Type IIA string. This also has a massive spectrum, which we may consider. There is the universal Neveu-Schwarz sector in brackets and then also the Ramond-Ramond sector, which contains all odd-dimensional anti-symmetric forms,
With the presence of these anti-symmetric forms, we can study what sort of objects are in our theory. Moreover, recall that if we have anti-symmetric forms, this means we have some object that couples to it. We may restate this fact as follows,
* is a 1-form, which couples to a particle (i.e., D0-brane).
* is a 2-form and couples to a string (i.e., the fundamental string).
* is a 3-form and it couples to a membrane (D2-brane).
Notice the pattern that, as we have only odd-ranked anti-symmetric forms in our theory, this means we have only even ranked Dp-branes. Just as the fundamental string is an object that exists in our theory, branes are legitimate objects in our theory.
Let’s focus for a moment on the D0-branes and think about computing the mass of these particles. Given that Dp-branes have a mass/tension, we can write this in general in the following way
Which tells us the mass, because, for a D0-brane, we simply have . Now, at this point in Palti’s talk, we should take notice of something interesting about this object. If we send , Dp-branes become very heavy. When this happens, the Dp-branes decouple from our theory. So in the weakly coupling limit, we see that relative to the strings (), these extended objects are actually unseen. This is precisely why in perturbative string theory, one cannot see Dp-branes. It also implies that to see these objects, what is required is strong coupling and non-perturbative limits.
So, in considering D0-branes, whenever we consider the masses of objects in the Swampland programme, Palti makes the point to emphasise that for these reasons we always go to the Einstein frame (remember, in the Swampland programme, we’re always thinking in terms of the Planck scale),
Now we see something interesting. Up to this point, the leading question concerns what happens when we send . We know that we have a strongly coupled regime. As it follows that and . So the mass of the D0-brane goes to zero.
But, one might ask, can we trust this regime (namely, strongly coupled string theory)? In general, the strongly coupled limit sets off various alarms of concern. But, by extending much of the same logic displayed throughout this entire discussion, the answer is that we can trust it. Why? Notice that the present example is very similar to the previous one, where we made the circle very small and the description of the physics was of the higher dimensional theory. We know that string theory can handle such limits because of T-duality. And, moreover, in the above limit, we know that we can trust the regime because we can see that as we obtain an infinite tower of light modes that are bound states of branes, strongly coupled Type IIA at low-energies is nothing but 11-dimensional supergravity (SUGRA).
6. Parameter Space of M-theory
Just as in the case when we obtained a description of the physics of the higher dimensional theory, so, too, in our present example, have we obtained a higher dimensional description. The point of emphasis is how this is T-duality in practice, and it leads us directly to a picture of M-theory.
To summarise, in the figure above we begin with a point in parameter space. As an example, we begin with Type IIA string theory that we just considered in Section 5. And then we consider another point, which is 11-dimensional supergravity. What we have found, or at least reviewed, is how we can move between these two theories depending on the string coupling limit. If we go to the weak coupling limit (or when the dilaton has a large negative expectation value), then we go to a perturbative Type IIA string theory and we obtain light states (from the light oscillator modes). On the other hand, when we go to the strong coupling limit , we have strongly coupled Type IIA string theory and, in this case, we should transform to a description of SUGRA, in which, again, there is an infinite tower of light states.
There are also other ways we can transform in parameter space. In another example we consider Type IIA string theory on a circle. So consider another direction in parameter space, governed by the size of the circle. In the limit of Type IIA / when the circle is very big, such that , we obtain a 10-dimensional Type IIA stirng theory (where from the 9-dimensional perspective we have a tower of states that are the KK modes). There is also T-duality, where . That means, we can also go the other direction in parameter space and send . We can see that this is tantamount to sending in Type IIB string theory. So in Type IIA from the 9-dimensional perspective, this corresponds to the circle becoming very small and gives Type IIB on a circle that is very big, which is IIB string theory in 10-dimensions.
7. Summary
To conclude, what we see in these results is that the DC is an incredibly strong and powerful, if not a deeply insightful conjecture, that describes a provocative picture of the parameter space of M-theory. What we see moreover is how, when we look at the parameter space in string theory, those parameters are scalar fields. As we have been experimenting, we can give these scalar fields large expectation values, which then moves us to the limits of the parameters where we obtain an infinite tower of light states. These towers of states can offer us a different description of the physics in a new regime. To put it more concisely, the different limits correspond to the 5 string theories and 11-dimensional supergravity. All of the string theories are linked by dualities describing different parameterisations of the same theory, M-theory. Each of these string theories have their own unique characteristics, offering descriptions in their respective corners of parameter space.
In the next collection of notes, we will review the third lecture in Palti’s series and consider a more formal definition of the WGC. We will then look to perform a deeper test of the WGC than in previous discussions, focusing particularly in the context of the heterotic string.
References
[1] E. Palti, `The Swampland: Introduction and Review’, [arXiv:1903.06239v3[hep-th]]
[2] J. Polchinski, `String theory. Vol. 1: An introduction to the bosonic string’. Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2007.
[3] J. Polchinski, `String theory. Vol. 2: Superstring theory and beyond’. Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2007.
[4] K. Becker, M. Becker, J.H. Schwarz, `String Theory and M-Theory: A Modern Introduction’, 2006.
[5] 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]].