In this collection of notes, we look to review some more basic tests of the Weak Gravity Conjecture. In the last entry, recall that we reviewed a basic relation between the WGC and the Distance Conjecture. We then considered a first test of the Distance Conjecture having compactified our theory on a circle. Additionally, we reviewed evidence for the DC where we found that if we have large expectation values for the scalar fields in string theory, we obtain an infinite tower of exponentially light states. In this sense, we also reviewed the extreme parameter regime for weak and strong coupling. Finally, we reviewed a number of lessons about the DC and T-duality, concluding with a brief review of the parameter space of M-theory.
In the present entry – the third in this series of notes – we continue to expand on past discussions, turning particular attention to another basic test of the WGC. In further testing of the WGC we will also focus on a number of related topics ranging from gauge fields to Dp-branes and Type II strings, ending with a few brief comments on F-theory Heterotic duality. This will then lead us directly into the fourth and second-last entry of the series, where we will begin to review more advanced tests of the DC and WGC, using for instance arbitrary Calabi-Yau manifolds.
2. Weak Gravity Conjecture
In this section we return to the WGC, which we have already grown to understand as being closely related to the DC. Following Palti’s lecture series, although the WGC is studied quite extensively from the infrared point of view, we shall instead be studying it from the ultraviolet and maximally stringy perspective.
Proceeding directly from the last entry we return to the simple example of string compactification on a circle and consider some of the physics in  as discussed in . This time, in compactifying on , we are going to instead consider a more general solution for the metric. The reason for this is because we want to study in particular the case of compactification with gauge fields. The metric may be written as follows,
A few comments are necessary before proceeding. First, remember that we are working in perturbative superstring theory, so this metric is very similar to the one before, where the first term in the equality is a 9-dimensional object. Second, also remember from the last entry that our original metric encoded the parameter such that it became a dynamical field in the lower d-dimensional theory. But, as Palti notes, there is also an additional degree of freedom in the metric. What does this mean? This additional degree of freedom becomes a U(1) gauge field in the d-dimensional theory, as opposed to a scalar field, which will also have a coupling . Furthermore, in that we have added another component to the metric, namely the 9-dimensional term on the right-hand side, this is in fact the graviproton. Altogether, it follows that this is the most general solution for stringy compactification on a circle.
Now, what is of present interest is the Ricci scalar. So let’s look at what dimensional reduction now gives for the Ricci scalar,
Where is the gauge field kinetic term or, in other words, the field strength of the gauge field. Recall, also, from before that the in the exponential is related to the radius in the extra dimensions. So from (2) we can read off the gauge coupling for the U(1) gauge field as follows,
Which is telling us, similar to the last entry, that if we make the circle very large the theory becomes weakly coupled. But what is the symmetry of the U(1) gauge field? How do we know that symmetry of the gauge field? Consider a general U(1) gauge symmetry transformation of the form (i.e., the circle isometry),
Where is a local gauge parameter. Notice that the metric remains invariant, and from this we can indeed see how lower d-dimensional theory has a U(1) gauge field with the above gauge coupling.
Now, just like in the past entry, we want to look at the Kaluza-Klein expansion. Moreover, recalling the KK expansion for the higher D-dimensional field , notice that the gauge transformation (4) reveals that the KK modes obtain a charge under the U(1) gauge field. This charge is quantised, as anticipated, and for the nth KK mode it may be given as,
But what is the relation between the charge and the KK modes? Note, firstly, that the charge of are just the phases of these objects. Secondly, the emphasis at this point in Palti’s talk is to remember that the mass of the KK states calculated in a past entry in the Einstein frame, , is related to the charge. More pointedly, we are already familiar with how, for the KK modes, there is an infinite tower of states. We see that the mass increases along this tower, and so too does the charge. In other words, it is argued that we have a charge-mass relation for the infinite tower of states. Here it is for arbitrary ,
This relation between the charge, mass, and couping may have already been anticipated. Since all we’ve considered here is really just a reduction of Einstein gravity, let us consider the effective string action from a past set of notes, written below for convenience,
If we compactify this action on a circle, as we are so inclined, there is a gauge field obtained from the gravitational sector. This is similar to before, and is nothing new. What is new is that we also now obtain a second gauge field, , which comes from the Kalb-Ramond B-field with a single index in the direction. For this Kalb-Ramond field we may write,
Where we note that, generally, is an antisymmetric 2-form. If we also reduce , this also leads to a gauge field. Additionally, look at in (8). The kinetic terms for this additional gauge field are produced by the dimensional reduction of the kinetic terms from the Kalb-Ramond field. In other words, we can compute the kinetic term for the gauge field, , as it comes from the strength of the 2-form in 10-dimensions,
The factor in front of the kinetic terms is produced when we reduce . From (9) one can again read off the gauge coupling,
What is different here? Notice, if we now make the circle of radius very large, we obtain a strongly coupled theory. So, in taking from what we reviewed in the last entry, we know that charges under this gauge field are the winding modes of the string. That is, we have stringy or indeed quantum gravity states. Moreover, think about how if we take the basic Polyakov action for a string wrapping in the direction times in the Einstein frame, which means that we can set , then notice we have
Which is the worldline action for a charged particle,
Or we can think of this in another way by remembering that if we have some antisymmetric form of rank , there is going to be some object coupling to it. Hence, we may notice that, if we integrate some Kalb-Ramond 2-form on the string worldsheet, where the 2-form has one leg along the 9th direction and one leg along the extra dimension, and if we consider a string winding around the extra dimension, we find the string worldsheet is just a worldline in the 9th direction times a circle. If we then perform the integral along the extra direction, we obtain the coupling . And so, we may write,
Where a worldline coupled to a gauge field means that, as in (4.11), we have a particle in the lower d-dimensional theory. What this is telling us is that winding modes in the d-dimensional theory produce charged particles that are gauge fields under the Kalb-Ramond field. Consider again (4.12), we find once again a relation between the coupling, charge, and mass, except this time it is for the winding modes. These are interesting relations,
Which are strictly stringy – or quantum gravitational – in nature. Moreover, what we are discovering are what appear to be deeply general relations, where there is always some particle with a relation between its charge and its mass. And if these relations are, in fact, deeply general, then this means they are also intrinsic properties of quantum gravity. We will investigate this idea more deeply in the context of the Swampland in a moment.
In the meantime, also notice something else that is interesting. If we send the gauge coupling to zero (either by making the circle small or large), , we obtain an infinite tower of light states. But this is just a special case of the DC, emphasising again the relation between the DC and the WGC. Furthermore, notice that the gauge coupling depends on the scalar field. So should we want to go to weak coupling, we must give the scalar a large expectation value that directly implies an infinite tower of states.
Also notice that, in the context of our wider discussion in these notes, there is a noticeable symmetry in the theory, which until now has been left implicit; because we can exchange the two gauge fields and also the KK and winding modes. This is T-duality.
3. Quick Review: Type IIA String Theory
Let us quickly review another example and think about Type IIA string theory (from the last entry). Remember, Type IIA in the strongly coupled regime is just 11-dimensional supergravity reduced on a circle. Also remember, in thinking of the Type IIA string we have a massive Ramond-Ramond 1-form, , which is just a gauge coupling that is the graviphoton. The gauge group is U(1) and, it follows,
The states charged under this gauge field? A D0-brane, with a D6-brane representing the magnetic dual. Again, we find the following mass-charge relation,
So, as Palti summarises, we have another piece of evidence that the mass-charge-coupling relation is indeed general. And, in fact, the more we search the more we become convinced this relationship is a property of quantum gravity.
4. Weak Gravity Conjecture (d-dimensions)
These considerations bring us to a more formal definition of the WGC than what we have so far previously offered. Consider the following: take a theory coupled to gravity with a U(1) gauge coupling, ,
For the Electric WGC, there exists a particle with mass and charge satisfying,
And for the Magnetic WGC, the cutoff scale of the effective theory is bounded from above by the gauge coupling, such that we have the general statement,
Where the cutoff, as we understand, should correspond to the mass scale of an infinite tower of charged states. It is argued to be completely general.
5. Testing the WGC: The Heterotic String
Following Palti, let’s now consider testing the WGC even more than what we have done previously. For example, a leading question might be: Is the WGC true for the Heterotic string? The first formal test of the WGC was for the Heterotic string on a . Again, much of the following discussion also echoes , where a summary with additional pedagogical references can be found.
One of the first things we must consider is that we have the non-abelian gauge group . This is important to note because compactifying on a yields the following 4-dimensional gauge fields: . To understand why there are 28 U(1) gauge fields, simply remember that a may be thought of as a product of 6 circles. In 4-dimensions we obtain 12 gauge fields from the metric and the Kalb-Ramond field. We may break these up into 6 yielding 6 U(1)’s and 6 graviphotons. Additionally, particular to the Heterotic string is a 10-dimensional gauge group. This gauge group may be broken by Wilson lines on a circle to its Cartan subalgebra. That is to say, if we have a circle and take a gauge field on that circle, this will give us a Wilson line to which we can then give an expectation value. The Wilson line will break the non-abelian group to its Cartan subalgebra. For these reasons, one can see what the Cartan subalgebra gives .
Let us focus on these last 16 U(1) gauge fields that come from breaking the gauge group. The states charged under these are string oscillators from which we once again obtain an infinite tower of states. The first massive excitation is the spinor with mass,
When we compactify on a we obtain charged states that correspond to the 16-dimensional charge vectors,
The idea now is to consider how, in the Einstein frame, and working in Planck units, we have the following gauge coupling for any of the U(1) gauge fields,
In which the gauge coupling is equal to the string coupling, and where depends on the expectation value of the dilaton. To put it explicitly, we have a dilatonic coupling. And, so, in terms of the bound set by the WGC for the mass the following inequality is satisfied,
Which is the limit of the expectation values of the small Wilson lines. As Palti notes, an interesting further test would be for arbitrary Wilson lines, but what he focuses on in his presentation is the way in which the entire analysis may be generalised for the complete gauge fields in which the U(1)’s from the are included. So now we consider the mass of the higher oscillator modes,
For which, in his talk, Palti gives the possible charges,
Where and . In that the charges should be integer, they must satisfy the lattice condition .
Now, the whole point of the analysis up to the present is to consider the mass-charge relation. And, in fact, what we find is the following mass-to-charge ratio,
Or, to put the matter differently, notice in (24) the factor is just , and . And so,
Where we find quite explicitly that the mass is bounded by the charge for all of the states. This again satisfies the WGC, where, for all the U(1)’s, the mass is less than the charge. We also find that there is an infinite tower of states charging at , and as we go further up the tower (so to speak) the bound in (27) becomes saturated but never violated. So all of our results so far are consistent, and the WGC indeed proves true for the Heterotic string.
6. What About Other Gauge Fields?
The following question we might now ask, as Palti motivates it: what other gauge fields might we consider? So far we have consider some fairly straightforward or simple examples. Can we continue to generalise?
6.1. Testing the Electric WGC: Open String U(1)’s
Another U(1) we get in string theory is an open string U(1), which, considering again Dp-branes, it is a U(1) gauge field on the world-volume. D-branes of course live in Type II string theory, so we could in general consider Type IIA/IIB on , where there is equal radius for the torus. The D-brane can be thought of as filling the non-compact spacetime. In considering string theory on this background, take in particular a Type IIB on a with 6 circles of radius R as an example. We therefore have some 4-dimensional , and what we want to do is specifically put a D3-brane with its 4-dimensional world-volume completely in the external spacetime. The D3-brane of course carries U(1), so we therefore now have a U(1) gauge symmetry in our 4-dimensional theory.
Now, with the scenario partly constructed, notice we only have one spacetime filling D-brane, which, impliedly, means that we have some fundamental open string with its endpoints ending on this brane. But this is not consistent. Why? The gauge symmetry we have included is an open string gauge symmetry, and so it is a gauge symmetry being carried by the non-perturbative D3-brane. But if we have just the single D3-brane, it will source the charge inside the 6-dimensional torus, and, one way to put it is that this scenario is akin to inserting a charged particle in a confined space in which there is nowhere for the field lines to propagate. In other words, we have a U(1) neutral state; but D-branes also source R-R fields. This is one of the great facts about D-branes, because insofar that they carry R-R charges, this gives string theory its power of being able to have a source for every gauge field . In our current construction, however, the presence of the D-brane means that it will provide a source in the compact whilst we lack an appropriate sink for the R-R field lines. This is obviously a problem because the field lines must end somewhere. This is why Palti points in another direction in his talk.
One option is that we could add an anti-brane; but means that the branes will then annihilate one another and, as this is an unstable configuration, it doesn’t really remedy the situation. Instead, the solution is based on a well known fact that orientifold planes are sinks for R-R charge. We might therefore instead introduce the needed negative charges by way of invoking orientifold planes. In doing so, this implies that the spectrum now also contains unoriented strings. These unoriented strings have charge 2 under the U(1), as, under orientifold involution, they stretch between the D-brane and its image. With this configuration, we have a consistent construction, which, with the presence of the orientifold, then means we have a second D3-brane as illustrated below.
In considering the scenario we have constructed, the actual states being charged under the U(1) are open strings whose endpoints end on the D3-branes with a charge .
Now let us think more deeply about the scenario in relation to the WGC. Is it not possible to violate the WGC? For instance, if the state has charge , what if we pull the D3-branes apart (i.e., moving away from the orientfold)? The string that is already stretched between the D3-branes would stretch even more over some spatial distance. This would make it massive. But what of the charge? Well, the charge would remain constant. On first inspection, this would seem to violate the WGC. Let us quantify these ideas as follows.
In , the relation between the string scale and the Planck scale can be found as (from dimensional reduction and re-writing everything in Planck units),
And the gauge coupling on the D3-brane is simply,
Now, for the stretched string, the mass is given as
Rearranging (30) it can be found that,
If the main task was to try and violate the WGC by stretching the string to great length, as we pull the D3-branes away from the orientfold, the question is: have we succeeded? More precisely, to violate the WGC (31) would have to be greater than 1. Is this the case? No, it is not! The reason is because, if we’re working in the perturbative string description – i.e., the controlled weak-coupling regime – than the coupling . So, in fact, the WGC is satisfied. That is,
As we stretch the string and make it massive, with the orientfold growing very large, the gauge coupling does not change. What we are doing, in effect, is diluting gravity. What’s more, we are diluting gravity faster than the mass can increase. And, it turns out, when we obtain a weakly coupled theory.
6.2. In General for different cases of
Notice that, in general, the scenario constructed above may be considered in terms of compactification of Type IIA/B string theory on . We considered the case for when we compactified on a . But other subtleties arise when considering the case of and especially , particularly due to backreaction on the space. In all cases, it can be seen that the Electric WGC holds for open string U(1)s .
7. Testing the Magnetic Weak Gravity Conjecture: Type IIB String Theory in 6d F-theory
In the last example we considered a test of the Electric WGC for open string U(1)s. What about the Magnetic WGC? Does the MWGC likewise hold for open string U(1)s? Recall from earlier in our discussion the MWGC is not making a statement about a single charged state but about an infinite tower of charged states. Where is the infinite tower of charged states in our scenario? The answer is rather non-trivial and can be reviewed in a series of incredibly interesting and mathematically rich papers [5, 6, 7], which display some lovely stringy physics.
We will save a detailed review of these papers for a separate entry (following the formal conclusion of this series of notes on Palti’s lectures). In the meantime, looking at  in particle, a brief if not altogether terse description may be considered. What the authors find is that, for the infinite tower of states, they turn out to be non-perturbative states of the theory.
To see these non-perturbative states is difficult. The set-up is this: consider Type IIB string theory on a 4-dimensional manifold, meaning compactification down to 6-dimensions. A powerful method to study non-perturbative type IIB string theory is by way of uplifting to F-theory (or, for Type IIA, uplifting to M-theory. So the framework is 6-dimensional F-theory. The 6-dimensional Planck mass is defined by the volume of the F-theory compactification space, which is a complex Kähler surface at the base of a Calabi-Yau 3-fold. In these notes, we have not yet considered such complex extended objects. But the idea is that we then consider a D7-brane filling the 6 external dimensions and wrapping a holomorphic curves on the Kähler surface in the 4-dimensional space. In the uncompactified 6 dimensions, the D3-brane wrapping the 2-cycle produces a solitonic ring. Associated strings on the curve contained in are sourced under the D7-brane gauge group.
From this construction, however roughly described, the idea is to uplift to a strong coupling (using F-theory). From this, if the goal is , where the tower of states become light according the WGC, then the 2-cycle must become very large. But, if the 2-cycle becomes big, the volume of the 4-dimensional manifold changes and, impliedly, the values of and the string scale also change. So one approach is to keep the volume fixed. However, fixing the volume while making the 2-cycle big means that another 2-cycle needs to be small!
Now consider the following. If a D3-brane wrapped in internal dimensions gives a string in external dimensions, impliedly, in the above construction, it seems a D3-brane wrapped on the small 2-cycle is found to produce a string in the 6 external dimensions. But this string but this string propagating in the 6-dimensions is tensionless as the volume of the curve contained in goes to zero, . Moreover, as the tension of the string is actually the size of the cycle, the string itself asymptotically describes an open Heterotic string. And so we observe,
And, as it is found that the string is charged under U(1), to finalise what is an incredible piece of evidence, the oscillator modes become massless and again what is found is an infinite tower of light states.
This concludes the summary. In a separate future entry we will study the technicalities in detail.
In the next collection of notes from Palti’s lecture series, we will continue our study by considering more complex manifolds – that is, arbitrary Calabi-Yau manifolds – to see if the WGC still holds! We will also looks to some more advanced tests of the DC, particularly in the context of Type IIB string theory.
 E. Palti, `The Swampland: Introduction and Review’, [arXiv:1903.06239v3[hep-th]]
 B. Heidenreich, M. Reece, and T. Rudelius, Sharpening the Weak Gravity Conjecture with Dimensional Reduction, JHEP 02 (2016) 140, [arXiv:1509.06374 [hep-th]].
 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, [hep-th/0601001].
 B. Heidenreich, M. Reece, and T. Rudelius, Evidence for a sublattice weak gravity conjecture, JHEP 08 (2017) 025, [arXiv:1606.08437].
 S.-J. Lee, W. Lerche, and T. Weigand, Tensionless Strings and the Weak Gravity Conjecture, JHEP 10 (2018) 164, [arXiv:1808.05958].
 S.-J. Lee, W. Lerche, and T. Weigand, Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions, [arXiv:1901.08065].
 S.-J. Lee, W. Lerche, and T. Weigand, A Stringy Test of the Scalar Weak Gravity Conjecture, Nucl. Phys. B938 (2019) 321–350, [arXiv:1810.05169].
 J. Polchinski, String theory. Vol. 2: Superstring theory and beyond. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2007.