# Literature: Duality Symmetric String and the Doubled Formalism

When it comes to a T-duality invariant formulation of string theory, there are two primary actions that are useful to study as a point of entry. The first is Tseytlin’s non-covariant action. It is found in his formulation of the duality symmetric string, which presents a stringy extension of the Floreanini-Jackiw Lagrangians for chiral fields. In fact, for the sigma model action in this formulation, one can directly reproduce the Floreanini-Jackiw Lagrangians for antichiral and chiral scalar fields. The caveat is that, although we have explicit $O(D,D)$ invariance, which is important because ultimately we want T-duality to be a manifest symmetry, we lose manifest Lorentz covariance on the string worldsheet. What one finds is that we must impose local Lorentz invariance on-shell, and from this there are some interesting things to observe about the constraints imposed at the operator level.

The main papers to study are Tseytlin’s 1990/91 works listed below. Unfortunately there is no pre-print available, so these now classic string papers remain buried behind a paywall:
1) Tseytlin, ‘Duality Symmetric Formulation of String World Sheet Dynamics
2) Tseytlin, ‘Duality Symmetric Closed String Theory and Interacting Chiral Scalars

For Hull’s doubled formalism, on the other hand, we have manifest 2-dimensional invariance. In both cases the worldsheet action is formulated such that both the string coordinates and their duals are on equal footing, hence one thinks of the coordinates being doubled. However, one advantage in Hull’s formulation is that there is a priori doubling of the string coordinates in the target space. Here, $O(D,D)$ invariance is effectively built in as a principle of construction. This is because for the covariant double sigma model action, the target space may be written as $R^{1, d-1} \otimes T^{2D}$, in which we have a non-compact spacetime and a doubled torus. From the torus identifications we have manifest $GL(2D; Z)$ symmetry. Then after imposing what we define as the self-duality constraint of the theory, which contains an $O(D,D)$ metric, invariance of the theory reduces directly to $O(D,D; \mathbb{Z})$.

1. Hull, ‘Doubled Geometry and T-Folds
2. Hull and Reid-Edwards, ‘Non-geometric backgrounds, doubled geometry and generalised T-duality

What is neat about the two formulations is that, turning off interactions, they are found to be equivalent on a classical and quantum level. It is quite fun to work through them both and prove their equivalence, as it comes down to the constraints we must impose in both formulations.

I think the doubled formalism (following Hull) for sigma models is most interesting on a general level. I’m still not comfortable with different subtleties in the construction, for example the doubled torus fibration background or choice of polarisation from T-duality. The latter is especially curious. But, in the course of the last two weeks, things are finally beginning to clarify and I look forward to writing more about it in time.

Related to the above, I thought I’d share three other supplementary papers that I’ve found to be generally helpful:

1) Berman, Blair, Malek, and Perry, ‘O(D,D) Geometry of String Theory
2) Berman and Thompson, ‘Duality Symmetric String and M-theory
3) Thompson, ‘T-duality Invariant Approaches to String Theory

There are of course many other papers, including stuff I’ve been studying on general double sigma models and relatedly the Pasti, Sorokin and Tonin method. But those listed above should be a good start for anyone with an itch of curiosity.

# Double Field Theory: The Courant Bracket

1. Introduction

In this post we are going to briefly and somewhat schematically discuss the appearance of the Courant bracket in Double Field Theory (DFT), following [1]. The point here is mainly to set the stage, so we jump straight into motivating the Courant bracket. In the next post, we will then study the B-transformations from the maths side and the C-bracket, following and expanding from [1] and others, with an emphasis in the end on how all of this relates to T-duality and strings.

What follows is primarily based on a larger collection of study notes, which I will upload in time.

2. Motivating the Courant Bracket

To understand the appearance of the Courant bracket in DFT, one way to start is by considering some general theory with a metric ${g_{ij}(X)}$ and a Kalb-Ramond field (i.e., an antisymmetric tensor field) ${b_{ij}(X)}$, where ${X \in M}$. The symmetries and diffeomorphisms of ${g_{ij}(X)}$ are generated by vector fields ${V^{i}(X)}$, where ${V \in T(M)}$ with ${T(M)}$ being the tangent bundle. As for ${b_{ij}(X)}$, the transformations are generated by one-forms ${\xi_{i}(X)}$, where ${\xi^{i}(X) \subset T^{\star}(M)}$ with ${T^{\star}(M)}$ being the cotangent bundle. We may combine ${V^{i}(X)}$ and ${\xi^{i}(X)}$ as a sum of bundles, such that (dropping indices) ${V + \xi \in T(M) \oplus T^{\star}(M)}$.

With these definitions, the opening question now is to ask, ‘what are the gauge transformations?’ To make sense of this, consider the following gauge parameters,

$\displaystyle \delta_{V + \xi} g = \mathcal{L}_{V} g$

$\displaystyle \delta_{V + \xi} b = \mathcal{L}_{V} b + d\xi \ \ \ (1)$

Here ${\mathcal{L}}$ is the Lie derivative. Furthermore, note given that ${V}$ generates diffeomorphisms, in (1) we get the Lie derivative in the direction of ${V}$. Also notice that ${\xi}$ does not enter the gauge transformation of ${g}$; however, for the gauge transformation of ${b}$, we do have a one-form ${\xi}$ and so we can take the exterior derivative. We should also note the following important properties of ${\mathcal{L}}$. For instance, when acting on forms the Lie derivative is,

$\displaystyle \mathcal{L}_{V} = i_{V}d + di_{V} \ \ \ (2)$

Where ${iV}$ is a contraction with ${V}$. We’re just following the principle of a contraction with a vector times the exterior derivative. It is also worth pointing out that ${\mathcal{L}}$ and the exterior derivative commutate such that,

$\displaystyle \mathcal{L}_{V}d = d\mathcal{L}_{V} \ \ \ (3)$

There are also some other useful identities that we are going to need. For instance, for the Lie algebra,

$\displaystyle [\mathcal{L}_{V_1}, \mathcal{L}_{V_2}] = \mathcal{L}_{[V_1,V_2]} \ \ (4)$

Where ${[V_1,V_2]}$ is just another vector such that ${[V_1,V_2]^{k} = V_{1}^{p}\partial_{p}V_{2}^{k} - (1,2)}$.

And, finally, we have,

$\displaystyle [\mathcal{L}_{X}, i_{Y}] = i_{[X,Y]} \ \ (5)$

Now follows the fun part. Given the transformation laws provided in (1), we want to determine the gauge algebra. To do this, we must compute in reverse order the gauge transformations on the metric ${g}$ and the ${b}$-field. For the metric we evaluate the bracket,

$\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]g = \delta_{V_2 + \xi_2} \mathcal{L}_{V_1}g - (1,2)$

$\displaystyle = \mathcal{L}_{V_1}\mathcal{L}_{V_2}g - (1,2)$

Using the identity (4) we find,

$\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]g = \mathcal{L}_{[V_1, V_2]} g \ \ \ (6)$

For the ${b}$-field we have,

$\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]b = \delta_{V_2 + \xi_2} (\mathcal{L}_{V_1}b + d\xi_{1}) - (1,2)$

$\displaystyle = \mathcal{L}_{V_1}(\mathcal{L}_{V_2}b + d\xi_2) - (1,2)$

$\displaystyle = \mathcal{L}_{[V_1, V_2]} + d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}V_{2}\xi_{1} \ \ \ (7)$

It turns out that this bracket satisfies the Jacobi identity, although it is not without its problems because, as we will see, there is a naive assumption present in the above calculations. In the meantime, putting this aside until later, the idea now is to compare the above with (1) and see what ‘pops out’. Notice that we find,

$\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}] = \delta_{[V_1,V_2]} + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} \ \ \ (8)$

In which we have discovered a bracket defined on ${T(M) \oplus T^{\star}(M)}$,

$\displaystyle [V_{1} + \xi_{1}, V_{2} + \xi_{2}] = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2} \xi_{1} \ \ \ (9)$

On the right-hand side of the equality we see a vector field in the first term and a one-form given by the final two terms. This Lie bracket is reasonable and, on inspection, we seem to have a definite gauge algebra. Here comes the problem allude a moment ago: there is a deep ambiguity in (9) in that we cannot, however much we try, determine unique parameters in our theory. Notice,

$\displaystyle \delta_{V + \xi}b = \mathcal{L}_{V}b + d\xi$

$\displaystyle = \mathcal{L}_{V + (\xi + d \sigma)} \ \ \ (10)$

The point being that the ambiguity of the one-form ${\xi}$ is so up to some exact ${d\sigma}$. To put it another way, if we change ${\xi}$ by ${d\sigma}$, we’re not actually changing anything at all. We would just get ${\mathcal{L}_{V}b + d(\xi + d\sigma)}$ where, when the exterior derivative hits ${d\sigma}$ we simply get nothing. So, given that ${\xi}$ is ambiguous up to some exact ${d\sigma}$, in a sense what we have is a symmetry for a symmetry. In other words, the present construction is not sufficient.

What we can do to correct the situation is analyse the mistake in (7). Let us, for instance, look at the right-hand side of the summation sign in this equation,

$\displaystyle d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}V_{2}\xi_{1}) = d(di_{V_1}\xi_{2} + i_{V_1}d\xi_{2}) - (1,2)$

The logic follows that the first term ${di_{V_1}\xi_{2}}$ is killed by ${d}$. It doesn’t make any contribution and its coefficient is just ${1}$. The trick then is to see, without loss of generality, that we may change the implicit coefficient ${1}$ in front of ${di_{V_1}\xi_{2}}$. It turns out, the coefficient that we can use is ${1 - \frac{\beta}{2}}$,

$\displaystyle = d((1-\frac{\beta}{2} di_{V_1}\xi_{2} + i_{V_1}d\xi_{2}) - (1,2)$

$\displaystyle = d(\mathcal{L}_{V_1}\xi_{2} - \frac{1}{2}\beta di_{V_1}\xi_{2}) - (1,2)$

$\displaystyle = d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2}\beta d [i_{V_1}\xi_{2} - i_{V_2}\xi_{1}]) \ \ \ (11)$

What we end up achieving is the construction of a much more general bracket,

$\displaystyle [V_{1} + \xi_{1}, V_{2} + \xi_{2}]_{\beta} = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2}\beta d(iV_{1}\xi_{2} - iV_{2}\xi_{2}) \ \ \ (12)$

What is so lovely about this result might at first seem counterintuitive. It turns out, as one can verify, for ${\beta \neq 0}$, we do not satisfy the Jacobi identity. So at first (12) may not seem lovely at all! But it makes perfect sense to consider cases of non-vanishing ${\beta}$. In mathematics, the case for ${\beta = 1}$ was introduced by Theodore James Courant in his 1990 doctoral dissertation [5], where he studied the bridge between Poisson geometry and pre-symplectic geometry. The idea here is to forget about the Jacobi identity – consider its loss an artefact of field theory with anti-symmetric tensors and gravity – and impose ${\beta = 1}$. When we do this what we obtain is indeed the famous Courant bracket. That is, given ${\beta = 1}$, the case of maximal symmetry is described by,

$\displaystyle [V_{1} + \xi_{1}, V_{2} + \xi_{2}]_{\beta=1} = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2} d(iV_{1}\xi_{2} - iV_{2}\xi_{2} \ \ \ (13)$

Although the Jacobi identity does not hold, one can show that for ${Z_{i} = V_{i} + \xi_{i}, i = 1,2,3}$, the Jacobiator assumes the form,

$\displaystyle [Z_1, [Z_2,Z_3]] + \text{cyclic} = dN(Z_1, Z_2, Z_3)$

Which is an exact one-form. This gives us a first hint that the unsatisfied Jacobi identity does not provide inconsistencies, because exact one-forms do not generate gauge transformations.

But why ${\beta = 1}$? Courant argued that the correct value of ${\beta}$ is in fact ${1}$ because, as he discovered, there is an automorphism of the bracket. This means that if do an operation on the elements, it respects the bracket. This automorphism is, moreover, an extra symmetry known in mathematics as a B-transformation. What follows from this is, I think, actually quite special. Given the Courant bracket is a generalisation of the Lie bracket, particularly in terms of an operation on the tangent bundle ${T(M)}$ to an operation on the direct sum of ${T(M)}$ and the p-forms of the vector bundle, what we will discuss is how the B-transformation in mathematics relates in a deep way to what in physics, especially string theory, we call T-duality (target- space duality). This is actually one of the finer points where mathematics and physics intersect so wonderfully in DFT.

In the next post we’ll carry on with a discussion of the B-transformation and then also the C-bracket, finally showing how everything relates.

References

[1] Zwiebach, B. (2010). ‘Double Field Theory, T-Duality, and Courant Brackets’ [lecture notes]. Available from [arXiv:1109.1782v1 [hep-th]].

[2] Hohm, O., Hull, C., and Zwiebach, B. (2010). ‘Generalized metric formulation of double field theory’. Available from [arXiv:1006.4823v2 [hep-th]].

[3] Hull, C. and Zwiebach, B. (2009). ‘Double Field Theory’. Available from [arXiv:0904.4664v2 [hep-th]].

[4] Hull, C. and Zwiebach, B. (2009). ‘The Gauge Algebra of Double Field Theory and Courant Brackets’. Available from [arXiv:0908.1792v1 [hep-th]].

[5] Courant, T. (1990). ‘Dirac manifolds’. Trans. Amer. Math. Soc. 319: 631–661. Available from [https://www.ams.org/journals/tran/1990-319-02/S0002-9947-1990-0998124-1/].

# Generalised Geometry, Non-commutativity, and Emergence

The project I am working on for my dissertation has to do with the notion of emergent de Sitter space. Of course an ongoing problem in string theory concerns whether asymptotic de Sitter spacetime can exist as a solution, and needless to say this question serves as one motivation for the research. With what appears to be the collapse of KKLT (this is something I will write about, as from my current perspective, the list of complaints against KKLT have not yet seemed to be satisfactorily answered), this academic year I wanted to start picking at the question of perturbative string de Sitter vacua from a different line of attack (or at least explore the possibility). Often, for instance, we approach de Sitter constructions by way of a classical supergravity approach with fluxes, non-geometry, or KKLT-like constructions which add quantum effects to stablise the moduli. One could also look at an alternative to compactification altogether and invoke the braneworld formalism. But, as it is, I’ve not been entirely satisfied with existing programmes and attempts. So the question over the autumn months, as we approached the winter break, concerned whether there was anything else clever that we can think of or take inspiration from. I’m not comfortable in divulging too much at this time, not least until we have something solid. Having said that, in this post let’s talk about some of the cool and fun frontier mathematical tools relevant to my current research.

For my project the focus is on a number of important concepts, including generalised and non-commutative geometry. Within this, we may also ask questions like whether spacetime – and therefore geometry – is emergent. Sometimes in popular talks, one will hear the question framed another way: ‘is spacetime dead?’ But before getting ahead of ourselves, we may start with a very well known and familiar concept in string theory, namely T-duality. Indeed, one motivation to study generalised geometry relates to T-duality, particularly as T-duality expresses how a string experiences geometry. For example, one will likely be familiar with how, in string theory, if we consider the propagation of a string in spacetime in which one spatial dimension is curled up into a circle, the idea is that when we compactify a dimension (in this case on a circle) we modify the string mass spectrum. Less abstractly, take some 10-dimensional string theory and then compactify on a circle $S^{1}$ of radius $R$. The string moves along the circle with the momenta quantised such that $p = n / R (n \in \mathbb{Z})$. When compactifying the 10th dimension we obtain for the compactified direction, $\displaystyle X_{(s)}^{d} (\tau, \sigma + 2\pi) = X_{(s)}^{d}(\tau, \sigma) + 2 \pi \omega R$, where we now have winding modes. This is because, as one will learn from any string theory textbook, the string winds around the circle with coordinate X. We can thus write the statement $\delta X = 2\pi R m (m \in \mathbb{Z})$. In this basic example T-duality is the statement $R \rightarrow \frac{\alpha^{\prime}}{R}$ with $n \longleftrightarrow m$. The winding modes that appear are of course a deeply stringy phenomenon. And what is interesting is the question of the generalisation of T-duality. Moreover, how might we think of string geometry in such a way that T-duality is a natural symmetry? Generalised geometry was largely motivated by this duality property, such as in the work by Nigel Hitchin. The basic mathematical statement is that the tangent bundle $TM$ of a manifold $M$ is doubled in the sum of the tangent and co-tangent bundle $TM \oplus T \star M$. In this formalism we also replace the Lie bracket with a Courant bracket, which we may write as something of the form $[X + \xi, Y + \eta]_{C} = [X, Y] + L_{X} \eta - L_{Y}\xi - \frac{1}{2} d(i_{X} \eta - i_{Y}\xi)$ such that $X \xi, Y + \eta \in \Gamma (TM \oplus T \star M)$. In physics, there is also motivation to ask about the geometry of spacetime in which strings propagate. For instance, the existence of winding modes and the nature in which T-duality connects these winding modes to momentum hints that perhaps the fundamental geometry of spacetime should be doubled. This idea serves as one motivation for the study and development of Double Field Theory, which is something the great Barton Zwiebach has been working on in recent years and which uses the SO(d,d) invariant formalism (see his lecture notes).

Additionally, in these areas of thinking, one will often also come across notions like non-geometry or fuzzy geometry. Sometimes these words seem used interchangably, but we should be careful about their meaning. For instance, non-geometry possess a number of characteristics that contribute to its formal definition, one being that it refers strictly to non-Riemannian geometry. Furthermore, we are also speaking of non-geometry as non-commutative geometry $[X_{i}, X_{J}] \approx \mathcal{O}(l_{s})$ as well as non-associative geometry $[X_{i}, X_{J}] X_{k} \approx (l_{s})$. One of many possible ways to approach the concept in this regard is to think quantum mechanically. If General Relativity is a very good approximation at long distances, in which we may think of smooth and continuous manifolds; at the smallest scale – such as the string scale – there are important hints that our typical understanding of geometry breaks down.

We will spend a lot of time on this blog discussing technicalities. For now, I just want to highlight some of the different formalisms and tools. In taking a larger view, one thing that is interesting is how there are many similarities between non-commutative and non-associative algebra and generalised geometry, fuzzy geometry, and finally ideas of emergence and a generalised quantum mechanics, although a precise formulation of their relation remains lacking. But this is the arena, if you will, which I think we might be able to make some progress.

As for my research, the main point of this post is to note that these are the sorts of formalisms and tools that I am currently learning. The thing about string theory is that it allows for is no sharp distinction between matter and geometry. Then to think about emergent space – that spacetime is an emergent phenomena – this infers the idea of emergent geometry, and so now we are also starting to slowly challenge present comforts about such established concepts as locality. When we think about emergent geometry we might also think of the structure of perturbative string vacua and ultimately about de Sitter space as a solution that escapes the Swampland. There is a long way to go, but right now I think in general there is an interesting line of attack.

For the engaged reader, although dated the opening article by Michael Douglas in this set of notes from the 2002 summer school at the Clay Mathematics Institute may serve an engaging introduction or overview. A basic introduction to some of the topics described in this post can also be found for instance in this set of notes by Erik Plauschinn on non-geometric backgrounds. Non-commutative (non-associative) geometry is covered as well as things like doubled geometry / field theory. Likewise, I think this paper on non-associative gravity in string theory by Plauschinn and Ralph Blumenhagen offers a fairly good entry to some key ideas. Dieter Lüst also has some fairly accessible lecture notes that offer a glance at strings and (non)-geometry, while Mariana Graña’s lecture notes on generalised geometry are a bit more detailed but serve as a basic entry. Then there are Harold Steinacker’s notes on emergent geometry from matrix models and on non-commutative geometry in relation to matrix models. Finally, there are these lecture notes by Maxim Zabzine on generalised complex geometry and supersymmetry. This is by no means comprehensive, but these links should at least help one get their feet wet.

Maybe in one of the next posts I will spend some time with a thorough discussion on non-commutativity or why it is a motivation of Double Field Theory to make T-duality manifest (and its importance).

**Cover Image: Study of Curve Folding [http://pr2014.aaschool.ac.uk/EMERGENT-TECHNOLOGIES/Curved-Folding-Workshop]

# Notes on the Swampland (3): Testing the Weak Gravity Conjecture – Gauge Fields, Dp-branes, Type II Strings, and F-Theory-Heterotic Duality

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 third lecture of Palti’s series.

1. Introduction

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 ${\longleftrightarrow}$ 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 [3] as discussed in [1]. This time, in compactifying on ${S^{1}}$, 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,

$\displaystyle ds^{2} = e^{2 \alpha \phi}g_{\mu \nu}dX^{\mu}dX^{\nu} + e^{2 \beta \phi}(dX^{d} + A_{\mu}dX^{\mu})^{2} \ \ (1)$

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 ${\phi}$ 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 ${A_{\mu}}$ in the d-dimensional theory, as opposed to a scalar field, which will also have a coupling ${g}$. Furthermore, in that we have added another component to the metric, namely the 9-dimensional ${A_{\mu}}$ 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,

$\displaystyle \int d^{D}X \sqrt{-G}R^{D} = \int d^{d}X\sqrt{-g} [R^{d} - \frac{1}{2}(\partial \phi)^{2} - \frac{1}{4}e^{-2(d - 1)\alpha \phi}F_{(A), \mu \nu} F^{\\mu \nu}_{(A)}] \ \ (2)$

Where ${F_{(A), \mu \nu} =\frac{1}{2} \partial_{[\mu}A_{\nu]}}$ is the gauge field kinetic term or, in other words, the field strength of the gauge field. Recall, also, from before that the ${\phi}$ 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,

$\displaystyle g_{(A)} = e^{d - 1}\alpha \phi = \frac{1}{2 \pi R} (\frac{1}{2 \pi R})^{\frac{1}{d - 2}} \ \ (3)$

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),

$\displaystyle A_{\mu} \rightarrow A_{\mu} - \partial_{\mu} \lambda (X^{\nu}), \ \ X^{d} \rightarrow X^{d} + \lambda (X^{\nu}) \ \ (4)$

Where ${\lambda (X^{\nu})}$ 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 ${\Psi (X^{\mu}) = \sum_{n = -\infty}^{\infty} \psi_{n} (X^{\mu})e^{2\pi i n X^{d}}}$, notice that the gauge transformation (4) reveals that the KK modes ${\psi_{n}}$ 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,

$\displaystyle q_{n}^{A} = 2\pi n \ \ (5)$

But what is the relation between the charge and the KK modes? Note, firstly, that the charge of ${\psi_{n}}$ 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, ${M^{2}_{\text{n kk mode}} = (\frac{n}{R})^{2} (\frac{1}{2 \pi R})^{2 \ d - 2}}$, 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 ${n}$,

$\displaystyle g_{(A)} q_{(n)}^{(A)} = M_{n, 0} \ \ (6)$

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,

$\displaystyle S_{D} = 2\pi M_{s}^{D - 2} \int d^{D} X \sqrt{-G}e^{-2 \phi} (R - \frac{1}{12} H_{\mu \nu \rho} H^{\mu \nu \rho} + 4\partial_{\mu} \Phi \partial^{\mu} \Phi) \ \ (7)$

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, ${V_{\mu}}$, which comes from the Kalb-Ramond B-field with a single index in the ${X^{d}}$ direction. For this Kalb-Ramond field we may write,

$\displaystyle V_{\mu} \equiv B_{[\mu d]} \ \ (8)$

Where we note that, generally, ${B_{[mn]}}$ is an antisymmetric 2-form. If we also reduce ${B_{[mn]}}$, this also leads to a gauge field. Additionally, look at ${V_{\mu}}$ 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, ${V_{\mu}}$, as it comes from the strength of the 2-form in 10-dimensions,

$\displaystyle \int d^{d}X \sqrt{-g} [R^{d} - \frac{1}{4}e^{-2(\alpha + \beta)\phi}F_{(V), \mu \nu}F^{\mu \nu}_{(V)}] \ \ (9)$

The factor in front of the kinetic terms is produced when we reduce ${\sqrt{-G}H_{\mu \nu \rho}H^{\mu \nu \rho}}$. From (9) one can again read off the gauge coupling,

$\displaystyle g (v) = e^{(\alpha + \beta)\phi} = 2\pi R (\frac{1}{2 \pi R})^{\frac{1}{d - 2}} \ \ (10)$

What is different here? Notice, if we now make the circle of radius ${R}$ 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 ${X^{d}}$ direction ${w}$ times in the Einstein frame, which means that we can set ${\sigma = \frac{2\pi}{w}X^{d}}$, then notice we have

$\displaystyle S_{P} = -\frac{T}{2} \int_{\sum} d\tau d\sigma [2i V_{\mu} \partial_{\tau} X^{\mu} \partial_{\sigma} (\frac{w\sigma}{2 \pi})]$

$\displaystyle = -i\frac{w}{2 \pi \alpha^{\prime}} \int_{\gamma} d\tau (\partial_{\tau} X^{\mu})V_{\mu} \ \ (11)$

Which is the worldline action for a charged particle,

$\displaystyle q_{w}^{(V)} = \frac{w}{2 \pi \alpha^{\prime}} (2\pi R)^{\frac{2}{d - 2}} \ \ (12)$

Or we can think of this in another way by remembering that if we have some antisymmetric form of rank ${n}$, 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 ${V_{\mu}}$. And so, we may write,

$\displaystyle \int_{\sum = C \times S^{1}} B_{[\mu d]} dX^{\mu} \wedge dX^{d} \sim \int_{C} V_{\mu} \ \ (13)$

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,

$\displaystyle g_{(V)}q_{w}^{(V)} = M_{0, w} \ \ (14)$

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), ${g \rightarrow 0}$, 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, ${C^{(1)}}$, which is just a gauge coupling that is the graviphoton. The gauge group is U(1) and, it follows,

$\displaystyle g_{C^{(1)}} \sim \frac{1}{g_{s}^{3/4}} \ \ (15)$

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,

$\displaystyle M_{D0} = g_{c^{(1)}} q_{D0} \ \ (16)$

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, ${g}$,

$\displaystyle S = \int d^{d}X \sqrt{-g} [] (\frac{M_{p}^{d}}{2})^{d-2}R^{d} - \frac{1}{4g_{s}^{2}} F^{2} + ... ] \ \ (17)$

For the Electric WGC, there exists a particle with mass ${m}$ and charge ${q}$ satisfying,

$\displaystyle M \leq \sqrt{\frac{d - 2}{d - 3}} gq (M_{p}^{d})^{\frac{d - 2}{2}} \ \ (18)$

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,

$\displaystyle \Lambda \lesssim g(M_{p}^{d})^{\frac{d - 2}{2}} \ \ (19)$

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 ${T^{6}}$ [3]. Again, much of the following discussion also echoes [1], 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 ${SO(32)}$. This is important to note because compactifying on a ${T^{6}}$ yields the following 4-dimensional gauge fields: ${U(1)^{28}}$. To understand why there are 28 U(1) gauge fields, simply remember that a ${T^{6}}$ 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 ${B_{[mn]}}$ 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 ${U(1)^{16}}$.

Let us focus on these last 16 U(1) gauge fields that come from breaking the ${SO(32)}$ gauge group. The states charged under these are string oscillators ${\underbar{q} = (q_{1}, ..., q_{16})}$ from which we once again obtain an infinite tower of states. The first massive excitation is the ${SO(32)}$ spinor with mass,

$\displaystyle m^{2} = \frac{4}{\alpha^{\prime}} \ \ (20)$

When we compactify on a ${T^{6}}$ we obtain charged states that correspond to the 16-dimensional charge vectors,

$\displaystyle \textbf{q} = (\pm \frac{1}{2}, ..., \pm \frac{1}{2}) \ \ (21)$

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,

$\displaystyle g^{2} = g_{s}^{2} = \frac{2}{\alpha^{\prime}} \ \ (22)$

In which the gauge coupling is equal to the string coupling, and where ${\alpha^{\prime}}$ 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,

$\displaystyle m^{2} \leq g^{2} \mid \textbf{q} \mid^{2} = \frac{8}{\alpha^{\prime}} \ \ (23)$

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 ${U(1)^{28}}$ gauge fields in which the U(1)’s from the ${T^{6}}$ are included. So now we consider the mass of the higher oscillator modes,

$\displaystyle m^{2} = \frac{2}{\alpha^{\prime}} (\mid \underbar{q} \mid^{2} - 2) \ \ (24)$

For which, in his talk, Palti gives the possible charges,

$\displaystyle \textbf{q} = (q_{1} + \frac{c}{2}, ..., q_{16} + \frac{c}{2}) \ \ (25)$

Where ${q_{i} \in \mathbb{Z}}$ and ${c = 0,1}$. In that the charges should be integer, they must satisfy the lattice condition ${\mid \underbar{q} \mid^{2} \in 2N}$.

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,

$\displaystyle \mid \textbf{z} \mid^{2} = \frac{\mid \textbf{q} \mid^{2}}{\mid \textbf{q} \mid^{2} - 2} \ \ (26)$

Or, to put the matter differently, notice in (24) the ${\frac{2}{\alpha^{\prime}}}$ factor is just ${g_{s}^{2}}$, and ${g_{s}^{2} = \frac{m^{2}}{M_{P}^{2}}}$. And so,

$\displaystyle \frac{m^{2}}{g^{2} \mid \textbf{q} \mid^{2}} = \frac{\mid \underbar{q} \mid^{2} - 2}{\mid \underbar{q} \mid^{2}} < 1 \ \ (27)$

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 ${g}$, 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 ${\mathbb{R}^{1, (q - n)} \times T^{6}}$, 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 ${T^{6}}$ with 6 circles of radius R as an example. We therefore have some 4-dimensional ${M_{1,3} \times T^{6}}$, and what we want to do is specifically put a D3-brane with its 4-dimensional world-volume completely in the ${M_{1,3}}$ 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 [8]. In our current construction, however, the presence of the D-brane means that it will provide a source in the compact ${T^{6}}$ 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 ${+1}$.

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 ${+1}$, 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 ${D=10}$, the relation between the string scale and the Planck scale can be found as (from dimensional reduction and re-writing everything in Planck units),

$\displaystyle M_{s}^{2}g_{s}^{-2} (RM_{s})^{6} \sim M_{P}^{2} \ \ (28)$

And the gauge coupling on the D3-brane is simply,

$\displaystyle g \equiv \sqrt{g_{s}} \ \ (29)$

Now, for the stretched string, the mass is given as

$\displaystyle m^{2} \sim (RM_{s})^{2}M_{s}^{2} \sim \frac{g_{s}^{2}M_{P}^{2}}{(RM_{s})^{4}} \ \ (30)$

Rearranging (30) it can be found that,

$\displaystyle \frac{m^{2}}{g^{2}_{s} M_{P}^{2}} \sim \frac{g_{s}}{(RM_{s})^{4}} \ \ (31)$

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 ${g_{s} 1}$. So, in fact, the WGC is satisfied. That is,

$\displaystyle \frac{m^{2}}{g^{2}_{s} M_{P}^{2}} \sim \frac{g_{s}}{(RM_{s})^{4}} < 1 \ \ (32)$

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 ${M_{P} \rightarrow \infty}$ we obtain a weakly coupled theory.

6.2. In General for different cases of ${n}$

Notice that, in general, the scenario constructed above may be considered in terms of compactification of Type IIA/B string theory on ${4\mathcal{R}^{1, 9-1} \times T^{n}}$. We considered the case for ${n >2}$ when we compactified on a ${T^{6}}$. But other subtleties arise when considering the case of ${n = 2}$ and especially ${n < 2}$, 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 [1].

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 [5] in particular, 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 ${B_{2}}$ 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 ${C_{0}}$ contained in ${B_{2}}$ 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 ${g_{D7} \rightarrow 0}$, 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 ${M_{P}}$ 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!

$\displaystyle volume \ fixed \rightarrow small \ 2-cycle$

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 propagating in the 6-dimensions is tensionless as the volume of the curve ${C_{0}}$ contained in $B_{2}$ goes to zero, $\text{vol}_{j}(C_{0}) \rightarrow 0$. 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,

$\displaystyle F-theory \longleftrightarrow Heterotic \ duality$

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.

Reference

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

[2] B. Heidenreich, M. Reece, and T. Rudelius, Sharpening the Weak Gravity Conjecture with Dimensional Reduction, JHEP 02 (2016) 140, [arXiv:1509.06374 [hep-th]].

[3] 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].

[4] B. Heidenreich, M. Reece, and T. Rudelius, Evidence for a sublattice weak gravity conjecture, JHEP 08 (2017) 025, [arXiv:1606.08437].

[5] S.-J. Lee, W. Lerche, and T. Weigand, Tensionless Strings and the Weak Gravity Conjecture, JHEP 10 (2018) 164, [arXiv:1808.05958].

[6] S.-J. Lee, W. Lerche, and T. Weigand, Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions, [arXiv:1901.08065].

[7] 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].

[8] J. Polchinski, String theory. Vol. 2: Superstring theory and beyond. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2007.

# Notes on the Swampland (2): Weak Gravity Conjecture, Distance Conjecture, and the Parameter Space of M-theory

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 ${g}$, then as ${\Lambda \sim M \sim gM_{p}}$ we should have an infinite tower of states [1]. This infinite tower of states was found to have a mass scale ${M}$ 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,

$\displaystyle S_{D} = 2\pi M_{s}^{D-2} \int d^{D}X \sqrt{-G} e^{-2 \Phi} (R - \frac{1}{12} H_{\mu \nu \rho} H^{\mu \nu \rho} + 4 \partial_{\mu} \phi \partial^{\mu} \phi) \ \ (1)$

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 ${M^{2} = \frac{4}{\alpha^{\prime}} (N_{\perp} - 1)}$, where ${N_{\perp} = \sum_{n =1}^{\infty} : \alpha^{\dagger}_{-n} \alpha_{n} :}$. In other words, if we increase the internal excitations ${N_{\perp}}$ along the free bosonic string, we increase the mass with the mass of the string set by the string scale ${M_{s} \sim \frac{1}{\sqrt{\alpha^{\prime}}}}$.

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 ${M \sim M_{s}}$. 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 ${g_{s}}$. 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,

$\displaystyle S = \frac{2\pi M_{s}^{8}}{g_{s}^{2}} \int d^{10}X \sqrt{-G} [R + ...] \ \ (2)$

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, ${M_{p}^{d}}$. 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 ${M_{p}^{d} = 1}$. Furthermore, we should also note that the d-dimensional Planck mass from the effective string action is defined as, ${\frac{M_{P}^{d - 2}}{2}R \equiv 2\pi M_{s}^{D-2}}$. 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 ${g_{s} \sim e^{\Phi}}$ and so ${e^{-2\Phi} \sim 1 / g_{s}^{2}}$),

$\displaystyle \frac{M_{P}^{8}}{2} = \frac{2\pi M_{s}^{8}}{g_{s}^{2}} \implies M_{s} \sim g_{s}^{1/4}M_{P} \ \ (3)$

Notice, upon rewritting the string scale in terms of the Planck scale, in the weak coupling limit where ${g_{s} \rightarrow 0}$ we find ${\frac{M_{s}}{M_{P}} \rightarrow 0}$, which implies that we have an infinite tower of states that become light relative to the Planck mass. In another way, ${\frac{M_{oscillator}}{M_{P}} \rightarrow 0}$ as ${g_{s} \rightarrow 0}$. 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 ${M_{P}}$. 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 ${M_{P}}$. 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 ${g_{s}}$ 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, ${\Phi}$, which is the dilaton, controls the string coupling. This can be explained a bit more eloquently. Consider ${\Phi (X) = \lambda}$, where ${\lambda}$ is some constant. The dilaton coupling reduces to ${\lambda \chi}$, where ${\chi}$ 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 ${\Phi = \lim_{X \rightarrow \infty} \Phi (X)}$ determines the string coupling constant, ${g_{s}}$, such that we find ${g_{s} \sim e^{\Phi}}$, corresponding to the amplitude to emit a closed string [2].

All of that is to say that we should remember that the string coupling, ${g_{s}}$, 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 ${M_{P}}$ implies that (3) can be rewritten in the following way,

$\displaystyle \frac{M_{oscillator}}{M_{P}} \sim e^{\alpha \phi} \ \ (4)$

Where ${\alpha}$ is some constant such that ${\alpha > 0}$ and ${\alpha \sim \mathcal{O}(1)}$. Following Palti, notice how if we send ${\phi \rightarrow -\infty}$, 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 ${\Phi \rightarrow -\infty}$. An interesting question now is to ask: what happens when we send ${\Phi \rightarrow + \infty}$? 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 ${\phi}$, there is an infinite tower of states whose mass relative to ${M_{P}}$ may be written,

$\displaystyle \frac{M}{M_{P}} \sim e^{-\alpha \phi} \ \ (5)$

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, ${D = d + 1}$ spacetime. As before, we are working in Planck units where ${M_{P}^{d}}$ is the d-dimensional Planck mass. For the circle there is also of course a periodic identification of the form ${X^{d} = X^{d} + 1}$. 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,

$\displaystyle ds^{2} = G_{MN} dX^{M}dX^{N} = e^{2 \alpha \phi}g_{\mu \nu} dX^{\mu}dX^{\nu} + e^{2 \beta \phi}(dX^{x})^{2} \ \ (6)$

This is the Einstein frame, where ${X^{M}}$ are D-dimensional coordinates such that ${M = 0,...,d}$ while ${\mu = 0,...,d-1}$. If the D-dimensional metric is ${G_{MN}}$, the lower d-dimensional metric is ${g_{\mu \nu}}$. Notice also that we have the parameter ${\phi}$, which is a d-dimensional scalar field. The ${\alpha}$ and ${\beta}$ terms are constants. To aid in the production of a canonically normalised theory, Palti notes in his talk that we choose ${\alpha = \frac{1}{2 (d-1)(d-2)}}$ and ${\beta = -(d-2)\alpha}$. 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,

$\displaystyle 2 \pi R = \int_{0}^{1} \sqrt{G_{dd}} dX^{d} = e^{\beta \phi} \ \ (7)$

Where we can see quite explicitly the relation between ${\phi}$ 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 ${\phi}$. Also important is that, when we reduce the higher D-dimensional Ricci scalar, ${R}$, we obtain something in the Einstein frame in lower dimensions,

$\displaystyle \int d^{D}X \sqrt{-G} R^{D} = \int d^{d}X \sqrt{-g} [R^{d} - \frac{1}{2} (\partial \phi)^{2}] \ \ (8)$

Moreover, to obtain (8) we have decomposed the Ricci scalar ${R^{D}}$ 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 ${R^{d}}$ 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 ${\phi}$; however, the choice for ${\alpha}$ and ${\beta}$ ensure no ${\phi}$ factorises in front of ${R^{d}}$ (hence the chosen definitions of ${\alpha}$ and ${\beta}$). 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,

$\displaystyle \Psi (X^{M}) = \sum_{n = -\infty}^{\infty} \psi_{n} (X^{\mu})e^{2\pi i n X^{d}} \ \ (9)$

Where we have performed a Fourier expansion of the higher dimensional field in terms of the lower dimensional modes along the circle. Note, ${\Psi}$ 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 ${\psi_{n}}$ modes are d-dimensional scalar fields, where ${\psi_{0}}$ is the zero mode of ${\Psi}$ and ${\psi_{n}}$ 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 ${n}$ 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,

$\displaystyle -i \frac{\partial}{\partial X^{d}} \Psi = 2\pi n \Psi \ \ (10)$

For simplicity, we shall restrict to flat space in lower dimensions. This means ${g_{\mu \nu} = \eta_{\mu \nu}}$. And from this, we look at the equations of motion for ${\Psi}$ in (9). We find,

$\displaystyle \partial^{M}\partial_{M} \Psi = (e^{-2 \alpha \phi}\partial^{\mu}\partial_{\mu} + e^{-2 \beta \phi}\partial^{2}_{X^{d}}) \Psi = 0 \ \ (11)$

Where ${\partial^{M}\partial_{M} \Psi = 0}$ is just the Klein-Gordon equation. When we expand this equation we obtain (by restricting the ${M}$ 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 ${\psi_{n}}$ modes,

$\displaystyle [\partial^{\mu}\partial_{\mu} - (\frac{1}{2\pi R})^{2} (\frac{1}{2\pi R})^{2 / d - 2} (2\pi n)^{2}] \psi_{n} = 0 \ \ (12)$

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, ${R}$, and the KK number. So the mass of the nth KK mode is given by,

$\displaystyle M^{2}_{\text{n kk mode}} = (\frac{n}{R})^{2} (\frac{1}{2 \pi R})^{2 \ d - 2} \ \ (13)$

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, ${R}$, 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,

$\displaystyle X_{(s)}^{d} (\tau, \sigma + 2\pi) = X_{(s)}^{d}(\tau, \sigma) + 2 \pi \omega R \ \ (14)$

Where one will notice that we now have winding states. In (14), ${\omega}$ is the winding number such that ${\omega \in \mathbb{Z}}$. This comes from the fact that the string can wind around the circle ${\omega}$ times. We can also define the winding as ${n = \frac{\omega R}{\alpha^{\prime}}}$. As we will discuss in a moment, the winding ${n}$ 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,

$\displaystyle X^{d}(\tau, \sigma) = x^{d}_{0} + \frac{\alpha^{\prime}}{2}(p_{L}^{d} + p_{R}^{d})\tau + \frac{\alpha^{\prime}}{2} (p_{L}^{d} - p_{R}^{d})\sigma + \ \text{oscillator modes} \ \ (15)$

The total center of mass momentum is therefore ${p^{d} = p_{L}^{d} + p_{R}^{d}}$. 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 ${p^{d} = \frac{n}{R}}$. This ${n}$ 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 ${(p_{L}^{d} + p_{R}^{d})}$ and another form of momentum in the form of ${(p_{L}^{d} - p_{R}^{d})}$, which is the winding mode of the string satisfying,

$\displaystyle \frac{\alpha^{\prime}}{2}(p_{L}^{d} + p_{R}^{d}) = \omega R \ \ (16)$

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,

$\displaystyle H = \frac{\alpha^{\prime}}{2} [\frac{1}{2}(p_{L}^{d} + p_{R}^{d})^{2} + p^{\alpha}p_{\alpha} + (p^{d})^{2}] + (N_{\perp}^{L} + \tilde{N}_{\perp}^{R} - 2) \ \ (17)$

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 ${N_{\perp} - \tilde{N}_{\perp} = n \omega}$. And, if we drop the excited oscillators, for the mass formula we have,

$\displaystyle M^{2} = (\frac{n}{R})^{2} + (\frac{\omega R}{\alpha^{\prime}})^{2} \ \ (18)$

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),

$\displaystyle (M_{n, w})^{2} = (\frac{1}{2\pi R})^{2 / D - 2} (\frac{n}{R})^{2} + (2\pi R)^{2 \ D - 2}(\frac{wR}{\alpha^{\prime}})^{2} \ \ (19)$

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 ${\phi}$ 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 ${\phi}$? 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 $\phi$ goes like $R$ in (7). The mass of the state parallel in $R$ in (14) will grow exponential in $\phi$. So, when considering the DC, we see that for $\phi$ (size of the circle) there is an infinite tower of KK modes that go something like the inverse power of $R$ in (14),

$\displaystyle M_{kk} \sim e^{\gamma \phi}, \ \ \text{as} \ \phi \rightarrow -\infty \ \ (20)$

And we also have the winding mode tower,

$\displaystyle M_{w} \sim e^{- \gamma \phi}, \ \ \text{as} \ \phi \rightarrow \infty \ \ (21)$

Where, ${\gamma = \sqrt{2}(\frac{d - 1}{d - 2})^{1/2}}$. 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 ${\phi}$.

The slope is $\gamma$, while the ${\mathcal{Z}_{2}}$ 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 ${\gamma}$ term that we’ve just considered, which acts as the exponent for the exponential behaviour, it is roughly order one: ${\gamma \sim \mathcal{O}(1)}$. So our tower of states truly are exponentially light.

3) Think for instance of the case when ${\phi \rightarrow - \infty}$. In this limit the effective theory breaks down. Why? Notice that when we send ${\phi}$ 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 ${\phi}$ 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 ${\phi \rightarrow \infty}$? 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,

$\displaystyle (M^{s}_{nw})^{2} = (\frac{n}{R})^{2} + (wR)^{2} \ \ (22)$

The spectrum is invariant under the symmetry,

$\displaystyle R \longleftrightarrow \frac{1}{R}, \ n \longleftrightarrow w \ \ (23)$

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 ${R}$, as we have been considering throughout these notes, the simple idea to begin with is that we may transform the original radius ${R}$ to a larger (or smaller) radius ${R^{\prime}}$, such that ${R^{\prime} \leftrightarrow \frac{\alpha^{\prime}}{R}}$. One can then see that with such a transformation we must also transform the winding states, such that ${n \leftrightarrow w}$. 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 ${\phi \rightarrow \infty}$ 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, ${\phi \rightarrow \infty}$ or ${\phi \rightarrow - \infty}$. 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: ${g_{s} \rightarrow 0}$ (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 ${g_{s} \rightarrow \infty}$.

In going back to ask this question about the dilaton, remember that in the D-dimensional theory ${g_{s} = e^{\phi}}$. We also know that ${M_{s} \sim g_{s}^{1/4}M_{p}}$ and that ${\phi \rightarrow - \infty}$ when ${g_{s} \rightarrow 0}$, which is similar to lesson 3) above where we obtained light KK modes.

In short, if we send ${\phi \rightarrow \infty}$, 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,

$\displaystyle \text{N-S}: \ \{G_{MN}, B_{[MN]}, \phi \}$

$\displaystyle \text{R-R}: \ C_{M}^{(1)}, C_{MN \rho}^{(3)}$

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,

*${C_{M}^{(1)}}$ is a 1-form, which couples to a particle (i.e., D0-brane).

*${B_{[MN]}}$ is a 2-form and couples to a string (i.e., the fundamental string).

*${C_{MN \rho}}$ 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

$\displaystyle T_{p} \sim \frac{M_{s}^{p + 1}}{g_{s}} \ \ (24)$

Which tells us the mass, because, for a D0-brane, we simply have ${M_{D0} \sim \frac{M_{s}}{g_{s}}}$. Now, at this point in Palti’s talk, we should take notice of something interesting about this object. If we send ${g_{s} \rightarrow 0}$, 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 (${M_{s}}$), 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),

$\displaystyle M_{D0} \sim \frac{M_{s}}{g_{s}} \sim \frac{M_{P}}{g_{s}^{3/4}} \ \ (25)$

Now we see something interesting. Up to this point, the leading question concerns what happens when we send ${\phi \rightarrow \infty}$. We know that we have a strongly coupled regime. As ${g_{s} \rightarrow \infty}$ it follows that ${\phi \rightarrow \infty}$ and ${M_{D0} \rightarrow 0}$. 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 ${g_{s} \rightarrow 0}$ (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 ${g_{s} \rightarrow \infty}$, 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 / ${S^{1}}$ when the circle is very big, such that ${R \rightarrow \infty}$, 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 ${\frac{IIA}{R} \longleftrightarrow \frac{IIB}{1/R}}$. That means, we can also go the other direction in parameter space and send ${R \rightarrow 0}$. We can see that this is tantamount to sending ${R \rightarrow \infty}$ 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]].