Learning M-theory: Gauge theory of membranes, brane intersections, and the self-dual string

I’ve been learning a lot about M-theory. It’s such a broad topic that, when people ask me ‘what is M-theory?’, I continue to struggle to know where to start. Right now, much of my learning is textbook and I have more questions than answers. I naturally take the approach of first wanting as broad and general of a picture as possible. In some sense, it is like starting with the general and working toward the particular. Or, in another way, it’s like when being introduced to a new landscape and wanting, at the outset, a broad orientation to its general geographical features, except in this case we are speaking in conceptual and quantitative terms. I may not ever be smart enough to grasp M-theory in its entirety, but what is certain is that I am working my hardest.

In surveying its geographical features and charting my own map, if I may continue the analogy, obtaining a better sense of the fundamental objects of M-theory is a particular task; but my main research interest has increasingly narrowed to the study and application of gauge theory and higher gauge theory. This can be sliced down further in that I am very interested in the relationship between string and gauge theory, and furthermore in studying the higher dimensional generalisation of gauge theory. This interest naturally follows from the importance of gauge theory in contemporary physics, and then how we may understand it from the generalisation of point particle theory to string theory and then to other higher dimensional extended objects (i.e., branes). We’ve talked a bit in the past about how the dynamics on the D-brane worldvolume is described by a gauge theory. We’ve also touched on categorical descriptions, and how in p-brane language when we study the quantum theory the resemblance of the photon can be seen as a p-dimensional version of the electromagnetic field (by the way, we’re going to start talking about p-branes in my next string note). That is to say, we obtain a p-dimensional analogue of Maxwell’s equations. More advanced perspectives from the gauge theory view, or in this case higher gauge theory view in M-theory, illuminate the existence of new objects like self-dual strings.

There is so much here to write about and explore, I look forward to sharing more as I progress through my own studies and thinking. In this post, though, I want to share some notebook reflections on things I’ve been learning more generally in the context of M-theory: some stuff about membranes, 11-dimensional supergravity, and the self-dual string. This post is not very technical; it’s just me thinking out loud.

11-dimensional supergravity

The field content of 11-dimensional supergravity consists of the metric g_{\mu \nu}  , with 44 degrees of freedom; a rank 3 anti-symmetric tensor field C_{\mu \nu \rho}  , with 84 degrees of freedom; and these are paired off with a 32 component Majorana gravitino \Psi_{\alpha \mu}  , with 128 degrees of freedom. Although much has progressed since originally conceived, the Lagrangian for the bosonic sector is similar to as it was originally written [3]

S_{SUGRA} = \frac{1}{2k_{11}^2} \int_{M_{11}} \sqrt{g} \ (R - \frac{1}{48}F^{2}_{4}) - \frac{1}{6} F_{4} \wedge F_{4} \wedge C_3. \ \ (1)

The field strength is F_4 = dC_3  and k_{11}  is the 11-dimensional coupling constant. The field strength is defined conventionally,

\mid F_n \mid^2 = \frac{1}{n !} G^{M_1 N_1} G^{M_2 N_2} ... G^{M_n N_n}F_{M_{1}M_{2} ... M_{n}}F_{N_1 N_2 ... N_n}. \ \ (2)

The 11-dimensional frame field in the metric combination is G_{MN} = \eta_{AB}E^{A}_{M}E^{B}_{N}  , where we have the elfbeins E^{B}_{N}  , M,N  are indices for curved base-space vectors, and A,B  are indices for tangent space vectors. The last term in (2) is the Cherns-Simons structure. This is a topological dependent term independent of the metric. We see this structure in a lot of different contexts.

Although, from what I presently understand, the total degrees of freedom of M-theory are not yet completely nailed down, we can of course begin to trace a picture in parameter space. As we’ve discussed before on this blog, it can be seen how 10-dimensional type IIA theory in the strong coupling regime behaves as an 11-dimensional theory whose low-energy limit is captured by 11-dimensional supergravity. Reversely, compactify 11-dimensional supergravity on a circle of fixed radius in the x^{10} = z  direction, from the 11-dimensional metric we then obtain the 10-dimensional metric, a vector field and the dilaton. The 3-form potential leads to both a 3-form and a 2-form in 10-dimensions. The mysterious 11-dimensional theory can also be seen to give further clue at its parental status given how supergravity compactified on unit interval {\mathbb{I} = [0,1]}  , for example, leads to the low-energy limit of E8 \times E8  heterotic theory.

Non-renomoralisability of 11-dimensional SUGRA

One thing that I’ve known about for sometime but I have not yet studied in significant detail concerns precisely how 11-dimensional supergravity is non-renormalisable [4,5,6]. Looking at the maths, what I understand is that above two-loops the graviton-graviton scattering is divergent. Moreover, as I still have some questions about this, what I find curious is that in the derivative expansion in 11-dimensional flat spacetime (using a 1PI/quantum effective Lagrangian approach) the generating functional for the graviton S-matrix is non-local. But due to supersymmetry, low order terms in the derivative expansion can be separated into local terms, such as t_8 t_8 R^4  , and non-local (or global) terms that correspond to loop amplitudes. But what happens is that, at 2-loops, a logarithmic divergence that is cut off at the Planck scale mixes with a local term of the schematic form D^{12}R^4  , where R^4  is the supersymmetrised vertex. In the literature, one will find a lot of discussion about this R^4  vertex. But like I said, I really need more time looking at this.

In short, the important mechanism in string theory that allows us to avoid UV divergences is absent, or appears absent, in maximal supergravity. What could the UV regulator be? As in any supergravity, from what I understand, it is not clear that a Lagrangian description is sufficient at the Planck scale.

Membranes, D-branes, and AdS/CFT

The facts of 11-dimensional supergravity and how it relates to 10-dimensional string theory are textbook and well-known. Going beyond dualities relating different string theories, an obvious question concerns what M-theory actually constitutes. One thing that is known is that M-theory reduces to 11-dimensional SUGRA at low-energies, as we touched on, and it is known that fundamental degrees of freedom are 2-dimensional and 5-dimensional objects, known as M2-branes and M5-branes. Study of these non-perturbative states offer several intriguing hints. There are also solutions to classical supergravity known as F1 – the fundamental string – and its magnetic dual, the NS5-brane. As it relates to the story of the five string theories, the M-branes realize all D-branes, and this is why D-branes are considered consistent objects in quantum gravity.

The way that M-theory sees D-branes is via the net of dualities. All of the D-branes and the NS5 brane are solutions to type II theories, both A and B. So, when you reduce M-theory on a circle, in that you get back to Type IIA, the M2-branes and M5-branes reduce to the various D-branes such that under S-duality from the D5-brane you get the NS5.

The worldvolume theory of the M5-brane is always strongly coupled, which can be seen in moduli space (its parameters are simply a point). So there is no Lagrangian for this theory, and it suggests something deep is needed or is missing. It is expected that its worldvolume theory will be a 6-dimensional superconformal field theory, typically known as the 6d(2,0) theory. The worldvolume theory for M2-branes (on an orbifold) has been found to be a 3-dimensional superconformal Chern-Simons theory with classical \mathcal{N} = 6 supersymmetry.

If one considers a single M5-brane, a theory can be formulated in terms of an Abelian (2,0)-tensor multiplet, consisting of a self-dual 2-form gauge field, 5 scalars, and 8 fermions, but it is not known how to generalise the construction to describe multiple M5-branes. To give an example, using AdS/CFT [7] it is described how the worldvolume theory for a stack of N  M5-branes is dual to M-theory on AdS7 \times S4  with N  units of flux through the 4-sphere, which reduces to 11-dimensional SUGRA on this background in the limit large N  limit.

Brane intersections and stacks

The existence of branes is one of the most fascinating things about quantum gravity. There is a lot to unpack when learning about D1-branes, D3-branes, D5-branes, M2-branes, and M5-branes, as well as how they may intersect and what sort of consistent solutions have already been found [8,9, 10, 11, 12].

For example, an M2-brane, or a stack of coincident M2-branes, can end on a D5-brane. This is similar to the more simplified story of how D-branes, coincident D-branes, can intersect in string theory. Typically, D1-D3 systems in Type IIB string theory are studied because this system relates to the M2-M5 system by dimensional reduction and T-duality.

Self-dual string

For a membrane to end on a D5-brane, the membrane boundary must carry the charge of the self-dual field B on the five-brane worldvolume. There are different solutions to the field equations of B. For instance, a BPS solution was found [10] by looking at the supersymmetry transformation.

The linearised supersymmetry equation is

\delta_{\epsilon} \Omega^{j}_{\beta} = \epsilon^{\alpha i}(\frac{1}{2} (\gamma^{a})_{\alpha \beta}(\gamma_{b^{\prime}})^{j}_{i}\partial_a X^{b^{\prime}} - \frac{1}{6}(\gamma^{abc})_{\alpha \beta}\delta^{j}_i h_{abc}) = 0. \ \ (3)

Here b^{\prime}  labels transverse scalars, a indices label worldvolume directions, \alpha, \beta  denote spinor indices of spin(1,5), and i,j are spinor indices of USp(4)  . The solution balances the contribution of the 3-form field strength h with a contribution from the scalars. Additionally, the worldvolume of the string soliton can be taken to be in the 0,1 directions with all fields independent of x^0  and x^1  . An illustration of the solution is given below, showing an M2-brane ending on an M5-brane with a cross section S^3 \times \mathbb{R}  .

M2-branes ending on a M5-brane. The endpoint is a string. Courtesy of N. Copland, Aspects of M-Theory Brane Interactions and String Theory Symmetries [https://arxiv.org/abs/0707.1317].

As I am still trying to understand the calculation, I am currently looking at the following string solution

H_{01m} = \pm \frac{1}{4} \partial_m \phi,

H_{mnp} = \pm \frac{1}{4} \epsilon_{emnpq}\delta^{qr}\partial_r \phi,

\phi = \phi_0 + \frac{2Q}{\mid x - x_0 \mid^2}, \ \ (4)

where \phi  may be replaced by a more general superposition of solutions. We denote \pm Q as the magnetic and electric charge. There is a conformal factor in the full equations of motion which guarantees that they are satisfied even at x = x_0  , which means the solution is solitonic. This string soliton is said to possess its own anomalies that require cancellation (I assume Weyl, Lorentz). What is neat is that this string can be dimensionally reduced to get various T-duality configurations, which is something that would be fun to look into at some point down the road.

References

[1] D. Fiorenza, H. Sati, and U. Schreiber, The rational higher structure of m-theory. Fortschritte der Physik, 67(8-9):1910017, May 2019. [arXiv:1903.02834 [hep-th]].

[2] E. Witten, String theory dynamics in various dimensions. Nuclear PhysicsB, 443(1):85 – 126, 1995.

[3] E. Cremmer, B. Julia, and J. Scherk, Supergravity Theory in 11-dimensions. Phys. Lett. B76, No. 4, (409-412) 19 June 1978.

[4] S. Chester, S. Pufu, and X Yin, The M-Theory S-Matrix from ABJM: Beyond 11D supergravity. (2019). [arXiv:1804.00949v3 [hep-th]].

[5] A. Tseytlin, R4 terms in 11 dimensions and conformal anomaly of (2,0) theory. (2005). [arXiv:hep-th/0005072v4 [hep-th]].

[6] G. Russo, and A. Tseytlin, One-loop four-graviton amplitude in eleven-dimensional supergravity. (1997). [arXiv:hep-th/9707134v3 [hep-th]].

[7] P. Heslop, and A. Lipstein, M-theory Beyond The Supergravity Approximation. (2017). [arXiv:1712.08570 [hep-th]].

[8] P.K. Townsend, D-branes from M-branes. (1995). [arXiv:hep-th/9512062 [hep-th]].

[9] A. Strominger, \textit{Open p-branes}. Phys. Lett. B 383 (1996) 44. [arXiv:hep-th/9512059 [hep-th]].

[10] P.S. Howe, N.D. Lambert, and P.C. West, The self-dual string soliton. Nucl. Phys. B 515 (1998) 203. [arXiv:hep-th/9709014 [hep-th]].

[11] M. Perry and J.H. Schwarz, Interacting chiral gauge fields in six dimensions and Born-Infeld theory. Nucl. Phys. B 489 (1997) 47. [arXiv:hep-th/9611065 [hep-th]].

[12] D.S. Berman, Aspects of M-5 brane world volume dynamics. Phys. Lett. B 572 (2003) 101. [arXiv:hep-th/0307040 [hep-th]].

[13] J. Huerta, H. Sati, and U. Schreiber, Real ADE-equivariant (co)homotopy and Super M-branes. (2018). [arXiv:1805.05987 [hep-th]].

[14] N. Copland, Aspects of M-Theory Brane Interactions and String Theory Symmetries. [https://arxiv.org/abs/0707.1317].

[15] S. Palmer, Higher gauge theory and M-theory. [https://arxiv.org/abs/1407.0298].

Notes on the Swampland (4): The Distance Conjecture for Arbitrary Calabi-Yau Manifolds, the Emergence Proposal, and the de Sitter Conjecture

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 fourth and fifth lectures of Palti’s series.

1. Introduction

In this final entry, we approach the conclusion of this collection of notes by focusing on the fourth and fifth of Palti’s lectures. Due to lack of space, we will not cover every topic in lectures four and five. Instead, we shall focus our energy on paying particular attention and detail to one of the most important and interesting subjects of study presented by Palti (from lecture four): namely, the study of the Distance Conjecture in the context of arbitrary Calabi-Yau (CY) manifolds. Then we will conclude these notes by briefly thinking about a few choice cosmological implications of the Swampland (the topic of Palti’s fifth lecture), particularly the de Sitter conjecture in the context of Type IIA string theory on CY with flux and in the context of 11-dimensional supergravity.

But before all this, we spend a short amount of time reflecting on the ‘Emergence Proposal’ (a concept introduced at the end of lecture four) and on some timely issues facing the Swampland programme.

2. A House of Cards? Emergence and the Swampland

As a summary review, let us quickly recall what we have so far emphasised in this series of notes. One of the featured viewpoints to be highlighted in Palti’s lectures is the observation that, among the growing list of Swampland Conjectures, there is ample reason to suggest that the Distance Conjecture and the Weak Gravity Conjecture are among two of the most established in terms of evidence. That is to say, short of complete and formal proof, the amount of evidence supporting these two conjectures in particular is very solid. We have already spent quite a bit of time exploring a number of tests and we have already begun to develop a deeper understanding for why the DC and the WGC are supported by significant evidence.

This emphasis on mathematical testability is important. In general, it does not seem too egregious to admit that the Swampland programme as a while has being experiencing both internal and external controversy. This controversy would seem as much scientific as philosophical. For example, consider the most recent Swampland conference (also held at IFT) that followed a couple months after the summer school from which these notes were originally written. One description of the situation at the September conference is this: the list of conjectures has experienced unrelenting growth but at the result of questionable rigour. In a moment of hyperbole issued for exaggerating effect, we might say that the programme itself is developing analogously as an infinite tower of conjectures. It is a well-established concern among portions of the Swampland research community that we are not proving/disproving conjectures faster than the rate in which new conjectures are being introduced. And from the perspective of this humble student, the situation has reached a point where proof and disproof are imperative.

In listening to and monitoring debates about the programme, I have come to be of the mind that we should proceed with particular caution. The caution is this: there is a genuine concern growing about a lack of mathematical rigour, which would seem verified by the observation that the list of conjectures is growing at a much faster rate than formal proof/disproof. This concern gains further urgency when considering the surrounding sociology, where calls for systematic evidence have been matched with what seems a more generally developing narrative against String Theory / M-theory writ large. It is understandable that the Swampland programme is compelled to react with a mission to make predictions and provide such evidence. But these sorts of commitments may still be premature. Predictions are key, but if we do not even know the theory to be accurate, any predictive claims or evidence would seem to put not just the Swampland but the entire reputation of String Theory at risk. In other words, it would seem reckless to begin contemplating demands for evidence without exhaustive rigour and confirmation of the theory at hand. In the very least, and in the best possible scenario, we have a theory not completely understood. But in either case, to then make predictions on these grounds – on a tower of conjectures, which, at the end of the day could very well be a house of cards – is risky.

But perhaps it is in this context that the moral tone of Palti’s lecture series earlier in the summer (prior to the Swampland conference in September) might be seen to be profoundly insightful and of timely inclination. In Palti’s fourth lecture, for example, the message becomes much more pronounced – that we may take the view that the DC and the WGC are in fact two fundamental pillars of the Swampland. Let me state this slightly differently. In reflecting on Palti’s lectures, to take the view that the entire programme depends on the DC and the WGC, and to map the relation of all the other conjectures from this foundation, it provides clarified view on a programme of proof/disproof.

Furthermore, if what we have done so far is focused on studying and reviewing examples of why the DC and the WGC can be trusted – why, short of complete formal proof – the evidence for these two conjectures is both substantial and inescapable, what we are coming to learn is precisely why both the DC and the WGC are an example of two first-class constraints. Taking this view has consequences. If the DC and the WGC are first-class constraints, it follows that if one understands these two conjectures they may then go on to understand all of the other conjectures. If we can disprove any number of the second-class conjectures, the Swampland programme would not collapse. If, on other hand, we should disprove the DC or the WGC, it is likely the entire programme collapses in on itself. The picture is illustrated quite explicitly in the above image.

The above picture describes what is called the ‘Emergence Proposal’ [1], based, in a sense, on the idea that Swampland Conjectures are consequences of the emergent nature of dynamic fields in quantum gravity. In lecture 4, we learned that if a coherent picture is emerging that outlines the relations between the growing conjectural assertions of the Swampland, a related internal programme of proof/disproof may also most effectively work from the bottom-up. But with the Emergence Proposal (as I currently understand it), not only is there the idea of first and second-class constraints – an idea for how we may perhaps pursue a foundational line of enquiry – another deep idea also comes to the fore: namely, the Swampland constraints are rooted in some underlying microscopic physics to be discovered. We don’t know what defines this microscopic physics, if it exists at all. That is a subject for another time. But we know, currently placed just above it in an overall web of constraints, the DC and the WGC may still offer some direct insight.

On that end, we now turn our attention to one of the deepest tests yet of the DC, beginning with a brief discussion of the refined version of the conjecture.

3. The Distance Conjecture (Refined)

We begin with the following message in mind: already we have seen several tests of the WGC and the DC. Each time, we have focused on increasing the complexity of the test and each time we have found strong evidence that both the WGC and the DC are deeply general. What we want to do now is proceed to review more tests of the DC, this time for even more complex geometry: namely, arbitrary Calabi-Yau manifolds.

Formally, the DC can be understood as follows [2]. As Palti put it in lecture four, consider how if we have a scalar field that is canonically normalised then we have already come to expect that there should be an infinite tower of states that goes something like,

\displaystyle  (\partial \phi)^{2}, \ m \sim e^{- \alpha \phi} \ \ (1)

Indeed, we are starting to understand that the behaviour in (1) would seem a general property of string theory. But we might ask, following Palti, what if the scalar is not canonically normalised? Consider, for instance, the scenario where we have some complicated function {f (\phi)} in front of the kinetic term,

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

Moreover, let us consider for a moment a theory with a moduli space, {\mathcal{M}} (remember: a moduli space is a space parameterised by the value of some scalar fields). We will make it so {\mathcal{M}} is parameterised by {\phi^{i}}, and we should note that {\phi^{i}} has no potential (typically, this implies that there should be some supersymmetry in the theory). Now, take any point {P \in \mathcal{M}}, where a point in Moduli space is given by the expectation value for the scalar fields {\phi^{i}}. We define another point {Q \in \mathcal{M}} such that, in this set-up, the geodesic proper distance (i.e., the distance is equivalent to the vacuum expectation value in field space) between {P} and {Q} may be denoted as {d(P, D)} (note: we measure the distance using the field space metric in front of the kinetic terms). Crucially, the first statement of the DC says this geodesic distance is infinite, which is to say the scalar field obtains an infinite vacuum expectation value. The second statement describes the behaviour at this infinity. That is to say, the second state describes that there exists an infinite tower of states with mass scale m , such that m(Q) \sim m(P)e^{-\alpha d(P,Q) / M_{P}} as d(P,Q) \rightarrow \infty and where \alpha \sim \mathcal{O}(1) .

This is the key idea. Given two points of great distance in field space – at least greater than the Planck scale – we obtain an infinite tower of exponentially light states.

We have of course already started to become familiar with this statement. The point that ought to be highlighted here, however, is that if we have some basic canonically normalised scalar field {(\partial \phi)^{2}}, then all that we get is the familiar {m \sim e^{-\alpha \phi}}. In more complicated situations, such as when the scalar field is not canonically normalised, the refined DC tells us that we can apply it also to such completely general situations.

In these notes, we will not explore any further an example of the trivial canonical case. Instead, having discussed is the refined distance conjecture [1], what we want do is review whether it holds in the case of arbitrary complex extended structures.

4. Type IIB on Calabi-Yau C3-fold

4.1. Supergravity Set-up

In this example, we invoke Type IIB string theory on a Calabi-Yau C3-fold (i.e., we have a 6-dimensional CY space). In the construction we are about to study, the geometry we will be working with is about as complicated as it gets, so we start with some basics.

We should first note that Type IIB string theory on CY gives {\mathcal{N} = 2} supergravity (SUGRA) in 4-dimensions. Due to limited space, we are not going to establish the supergravity formalism in these notes. The reader is instead directed to ref. [1, 3-5] for an introduction, where, for these notes, we are of course following Palti in ref. [1] quite strictly. Another very important paper, which we will cover in some depth is ref. [6] on infinite distances in field space. In fact, majority of what follows is based on this paper.

Meanwhile, to continue establishing the basics, the general supergravity set-up is this: we have {n_{V}} vector multiplets with bosonic content of a complex scalar field. Similar in a sense to past discussion about the presence of scalar fields with regards to the radius of the circle, in the present case the scalar fields we are interested in studying are complex structure moduli, {t^{i}}, where {i = 1, ..., n_{V}}. These complex structure moduli parameterise the geometry of the CY.

We also have gauge fields, {A^{i}}. These gauge fields are quite interesting, as we will elaborate. For now, note that there is a gravity multiplet which contains a (bosonic fields) graviton and graviphoton, {A^{0}}. All of the gauge fields can be combined such that {I = \{0, i \}} for {A^{I}}.

The number of fields, {t^{i}} and {A^{I}}, is counted by the number of 3-cycles in the CY, which, for a typical CY, is {\sim \mathcal{O}(100)}. This means that for the field space in the effective field theory we find a space with {\sim 100} complex dimensions (and so we have a 200 dimensional field space in total).

Based on previous discussions, one might wonder whether there are charged states under the gauge field {A^{I}}. The answer is that there are charged states, they are BPS states which are {D3-branes} wrapping 3-cycles in the CY. Schematically, the moduli describe the size of the 3-cycle and then they describe the mass of the D3-branes that are wrapping the 3-cycles, behaving like particles in the external dimensions.

Generally, in this set-up, we find an action of the form,

\displaystyle  S_{\mathcal{N} = 2} = \int d^{4}x \sqrt{-g} [\frac{R}{2} - g_{ij} \partial_{\mu} t^{i} \partial^{\mu} \bar{t}^{j} -h_{\sigma \lambda} \partial_{\mu} l^{\sigma} \partial^{\mu} l^{\lambda} + \mathcal{I}_{IJ}\mathcal{F}^{I}_{\mu \nu}\mathcal{F}^{J, \mu \nu} + \mathcal{R}_{IJ}\mathcal{F}^{I}_{\mu \nu} (\star \mathcal{F})^{J, \mu \nu}]  \ \ (3)

The structure of which can be read off beginning with metrics, {g_{ij}} and {h_{\sigma \lambda}}. In totality, the moduli space is split into vector multiplets and hypermultiplets, {\mathcal{M} = \mathcal{M}_{V} \times \mathcal{M}_{H}}. And so, as one would expect even notationally, these two metrics describe two separate manifolds. We are going to focus on the vector multiplets which span a special Kähler manifold, from which we can generalise for the hypermultiplets on the quaternionic Kähler manifold. What is important to note is the periodicity {\{X^{I}, F_{I} \}} for the multiplet field space, in which we are dealing with holomorphic functions of {t^{i}}.

Notice also the gauge kinetic functions, {\mathcal{R}} and {\mathcal{I}}. These both contain real and imaginary parts of a complex matrix.

4.2. Charge Vector and Kähler Potential

It was mentioned that we have D3-branes wrapping 3-cycles. When a certain D3-brane wraps the 3-cycles in the CY, this is labelled by a charge vector {q \in \mathbb{Z}} (of {\mathcal{O}(100)}). This charge vector is in fact a 100-dimensional vector, where each entry is some holomorphic function of the 100’s of scalar fields in our theory. The basic idea, to give some more intuition, is that once we know the charge vector we know the mass of the BPS state, which, again, are the charged states under the gauge fields. Study (4) below,

\displaystyle  m(\underline{q}) = \mid z(\underline{q}) \mid = \mid \frac{\underline{q \eta \underline{\prod}}(t)}{[i \underline{\prod}^{T}(t) \eta \bar{\prod}(t)]^{1/2}} \mid \ \ (4)

Where {\prod} is the period vector. Notice that in the denominator we have complex conjugation as the object {\underline{\prod}^{T}(t)} must be real. Furthermore, all of the geometry of the CY is captured in the period vector {\underline{\prod}(t)}. One can see that it is a function of {t^{i}}. This is because it is a 100-dimensional vector that is an arbitrary function of the complex structure moduli. We should also highlight, for pedagogical purposes, that the expressions for {\eta} and {\prod} are a local choice of basis on the moduli space. Without going into all of the details, the period vector {\prod} can be defined on a local coordinate basis such that,

\displaystyle \prod = \begin{bmatrix} X^{0} \\ x^{i} \\ F_{j} \\ F_{0} \\ \end{bmatrix} (5)

So that the electric index increases down the vector and the magnetic index increases from the bottom-up. The {eta} term in (4) is the natural symplectic form of this multiplet vector space, and so we may indeed construct the appropriate symplectic inner products.

It is not difficult to understand that the field space that we are working with is very complicated. In [6], the metric is given by the derivative of the Kähler potential (also note, much of the same notation and general construction is in this paper, which as with other points discussed can also read in ref. [1]),

\displaystyle  g_{t^{i} \bar{t}^{j}} = \partial_{t^{i}} \partial_{\bar{t}^{j}} K \ \ (6)

Where the {K} is the Kähler potential, {K = -log [i \prod^{T} \eta \bar{\prod}]}. In other words, we have the log of the period vector. This potential is actually very interesting, and one can derive it by considering in general a Kähler potential for some CY manifold, {Y_{D}}, of complex dimension {D} where the complex structure moduli is given by a {h^{D-1, 1} (Y_{D})}-dimensional Kähler manifold. The potential is then generally written as K = -log [-i^{D} \int_{Y_{D}} \omega \wedge \bar{\omega}] in which one finds metric components of the form above. Once one finds the appropriate integral basis, the potential above is found.

4.3. Studying the Field Space

The discussion in this section is based almost entirely on [6], as well as parts of Palti’s summary in lecture 4 and his review in [1]. Additionally, we will be working with a number of very powerful mathematical theorems offered by Wilfried Schmid [7] building on Deligne’s work [8] in Hodge theory. (Please note, while we will not explore a detailed study / re-derivation of some of the theorems found in [7], I am very interested in this work and also in [6] which leverages Schmid’s nilpotent orbit theorem, so I will offer a detailed review in a future post).

In a schematic way, what we want to do is consider some point of infinite distance on this field space. Following Palti in his lecture, we shall label this point by the parameter {t} going to {+i \infty}. We now invoke the theorem that tells us that for such a point the period vector has a monodromy around it. In other words, if we send the real part of {t} to infinity, {\text{Re} t \rightarrow \text{Re} t + 1}, which, in a sense, is like encircling the point at infinity, we have a transformation of the period vector. In fact, we see that the period vector transforms by the action of a monodromy matrix, \prod (t) \rightarrow T_{i} \prod(t) . Then, due to properties studied in [6], we see that each {T_{i}} can be decomposed and, with the monodromy matrix massaged in a way that it only gives its infinite order part, we can define the log of this {T_{i}} in the form of a matrix equation,

\displaystyle  N_{i} = \log T^{u}_{i} = \sum_{k = 1}^{\infty} (-1)^{k + 1} \frac{1}{k} (T^{(u)}_{i} - Id)^{k} = \frac{1}{m_{i}} \log T^{mi}_{i} \ \ (7)

From this, we invoke the nilpotent orbit theorem [7]. With space limited the essentially idea may be summarised in the result that {N} is nilpotent. This means that if we take a high enough power we will get zero, {N^{n+1} = 0, \ n \leq 3}. Moreover, remember that we have sent {t} an infinite distance, and as things are currently constructed we need to know what this point looks like. What Schmid’s theorem in ref. [7] tells us is precisely what the period vector looks like around any point at infinite distance. In fact, it says that the period vector must look like,

\displaystyle  \prod (t) = \exp [t N](a_{0} (S) + \mathcal{O}(E^{2\pi i t})) \ \ (8)

What is this telling us exactly? It says that we have a parameter {t}, and as {t \rightarrow i \infty} we get exponentially small corrections. In other words, because {N} is nilpotent we see in (8) that we get some polynomial in {t}. The vector {a_{0}} depends on the other moduli, but not {t}, and as the exponential term may be neglected we see that we can know the form of the period vector around any point.

There is another theorem in [7], as Palti cites in his lecture, which, using again the nilpotent theorem, tells us if this point is indeed an infinite distance then it must be the matrix {[t N]} does not annihilate the vector {a_{0} (S)}. And so what we have, to be terse, is the following,

\displaystyle  \text{Infinite distance} \longleftrightarrow N^{d + 1} a_{0} \neq 0, \ d > 0 \ \ (9)

Now, all we need to do is take the period vector and this form {[t N]} and plug it into the formulae for the mass of the BPS states and for the metric on the moduli space. What we find is that we must have some local expression near any infinite locus in the moduli space. Schematically, from section 3.2 in ref. [6] we may write,

\displaystyle  g_{t \bar{t}} = \partial_{t} \partial_{\bar{t}} K = \frac{1}{4} \frac{d}{\text{Im} t^{2}} \ \ (10)

Where we have dropped the subleading terms. With the universal leading term only depending on degree {d}, quadratic {1 / \text{Im} t} it is found that the proper field distance is logarithmic when we send {t} to infinity,

\displaystyle  d_{\gamma}(P, Q) = \int_{Q}^{P} \sqrt{g_{t \bar{t}}} \mid dt \mid \sim \frac{\sqrt{d}}{2} \log (\text{Im} t) \ \ (11)

From which it is found that, in the case of a CY compactification that preserves {\mathcal{N} = 2} supersymmetry the BPS states become massless at the singularity point. More technically, in the paper these singular points have to do with what the author’s study as infinite quotient monodromy orbits. But for our purposes we note in particular for the mass,

\displaystyle  M_{q} \sim \frac{\sum_{j}\frac{1}{j!}(\text{Im} t)^{j} S_{j}(q, a_{0})}{(2^{d} / d!)^{1/2} (\text{Im} t)^{d/2}} \ \ (12)

In other words, as Palti motivates it, we see that the D3-branes become massless as the imaginary part goes to infinity. The behaviour of the mass is argued to be universal for any massless BPS states. Furthermore, what is observed is the presence of a power law in {t} whilst the proper distance is logarithmic in {t}. If we consider some path, {\gamma}, as implied in (11), the effective theory at two points (P, Q) in the moduli space approach singularity. The mass of the BPS states decreases exponentially fast in the proper distance. And so, in a schematic way in these notes, we may describe this in the form of {\Delta \phi \sim \log t} and M \sim \frac{1}{t^{\alpha}} \sim e^{-\alpha \Delta \phi}, which is just the Distance Conjecture and the Weak Gravity Conjecture at work.

We have of course been crude in our description, and there is a subtlety about the state not necessarily being confirmed in the theory, with the need remaining that one must show the BPS states being in the spectrum. Perhaps a detailed individual post would be beneficial in the future. For now, we can say that in [6] the case is shown for when {d = 3}. For our current purposes, the result is notable it shows that the DC and WGC hold for any CY compactification for Type IIB string theory. And this result should not in any way be understated. Altgough we are dealing with a very complicated 100-dimensional field space, the fact the it can be proven mathematically that both of these first-class Swampland conjectures hold for any CY compactification – and that very powerful mathematical theorems tell us this is necessarily true – we are driven directly toward the suggestion of some deeply general physics.

5. de Sitter Conjecture

5.1. Introduction

To conclude this series of notes, and to celebrate what has been a fairly lengthy and detailed engagement with Palti’s lectures at IFT this past summer, we turn our attention to a brief discussion on some of the cosmological implications of the Swampland. We will not discuss things like tensors modes in inflation or other topics covered in the lectures, which can be easily reviewed in [1]. Instead, we begin with a brief review of the de Sitter Conjecture, which states that the gradient of the potential is bounded,

\displaystyle  \mid \nabla V \mid \geq \frac{c}{M_{P}} V \ \ (13)

In other words, the scalar potential of the theory must satisfy (13) or the refined version below,

\displaystyle  \text{min} (\nabla_{i} \nabla_{j} V) \leq - \frac{c^{\prime}}{M_{P}^{2}}V \ \ (14)

Where this second condition is based on or motivated by entropy arguments. There are a number of connections between the de Sitter conjectures and ongoing experiments, including dark energy constraints and constraints from inflation. Interaction with experimental observation is quite active here, as Palti summarises. What we shall focus on is what motivates the de Sitter conjecture from string theory.

5.2. Evidence of the de Sitter Conjecture – Type IIA on CY with Flux

What follows is based on a simplified version of the more general study in ref. [8], where flux compactifications of Type IIA string theory are considered and the author’s study the classical stabilisation of geometric moduli. The main idea that we consider in general is that we want to switch on the fluxes for the background CY and then we study them from the perspective of the 4-dimensional effective theory. That is to say, we study the potential from the fluxes in the 4-dimensional theory. In the referenced study there are two fields in the low-energy effective theory. More precisely, there are two moduli fields that parameterise the geometry of the CY, \rho = (vol)^{1/3} , which is the volume of the CY and another field, \tau = e^{-\phi} (vol)^{1/2} , which is the string coupling times the volume of the CY. As a result of the flux being switched on, these two fields will have some potential.

Now let us consider the canonically normalised fields,

\displaystyle  \hat{\rho} = \sqrt{\frac{3}{2}} M_{P} \ln e \ \ (15)

\displaystyle  \hat{\tau} = \sqrt{2} M_{P} \ln \tau \ \ (16)

As these fields are canonically normalised, we may write the following Lagrangian in the Einstein frame,

\displaystyle \mathcal{L} = \frac{M_{P}^{2}}{2} R - \frac{1}{2} (\partial \hat{\rho})^{2} - \frac{1}{2} (\partial \hat{\tau}) + ... V(\rho, \tau) \ \ (17)

Now, the featured point here is that the potential is of course quite complicated. We can include any number of things to generate the potential – for example, we can turn off and on certain RR-fluxes or a combination of fluxes. What is interesting is that, in playing with different scenarios, a number of general properties are found. For instance, consider the case of turning on only certain RR-fluxes, where we have an expectation value for the p-form field strength, and also the H-flux which is the field strength of the NS sector,

\displaystyle \text{RR-flux:} \ V_{p} \sim \rho^{3 - p} \tau^{-4} \ \ (18)

\displaystyle \text{H-flux:} \ V_{3} \sim \rho^{-3} \tau^{-2} \ \ (19)

And with these contributions, we can also have in this case D6-branes and 06-branes that contribute to the potential,

\displaystyle V_{D6} \sim \tau^{-3} \sim V_{06} \ \ (20)

It turns out that, completely generally (regardless of the fluxes we switch on or off, their combination, and the branes we choose), the potential always takes the form,

\displaystyle V = \frac{A_{3} (\phi^{i})}{\rho^{3} \tau^{2}} + \sum_{p} \frac{A_{p} (\phi^{i})}{\rho^{3 - p} \tau^{4}} + \frac{A_{}}{\tau^{3}} \ \ (21)

Where in the first two terms in the equality we have in the numerator some function of the other fields included in our theory over the contribution from the H-flux and RR-flux, respectively. In the last term, there is a contribution from localised sources in the numerator over the brane contribution. This is the most general form the potential can take, even when we consider the inclusion of hundreds of other fields.

Inspecting the general form of the potential (21), we may consider the following combination of derivatives,

\displaystyle -\rho \frac{\partial V}{\partial \rho} - 3\tau \frac{\partial V}{\partial \tau} \ \ (22)

It turns out that, in fact,

\displaystyle -\rho \frac{\partial V}{\partial \rho} - 3\tau \frac{\partial V}{\partial \tau} = 9V + \sum_{p} pVp \ \ (23)

Where {pVp} are positive components of the potential and so the following statement is made that, {9V + \sum_{p} pVp \geq 9V}. But what does this mean? Well, if we write this in terms of the canonically normalised fields,

\displaystyle M_{P} \mid \sqrt{\frac{3}{2}} \frac{\partial V}{\partial \hat{p}} + 3\sqrt{2} \frac{\partial V}{\partial \hat{\tau}} \mid \ \geq 9V \ \ (24)

We notice something striking. If, moreover, we consider the gradient of the potential as it also pertains to the statement made in the de Sitter Conjecture, notice that after some work we can go from a completely general statement to the below,

\displaystyle M_{P} \mid \nabla V \mid \geq M_{P} \mid \frac{\partial V}{\partial \hat{p}} +\frac{\partial V}{\partial \hat{\tau}} \mid \geq \frac{27}{13} V, \ \ \nabla V > 0 \ \ (25)

Where we see that the de Sitter Conjecture has been satisfied. As it is a completely general result for any choice of fluxes and any choice of branes for the given compactification, this result is quite striking. In other words, regardless of the complexity of the potential, there is also a lower bound to it.

6. 11-dimensional Supergravity

But what about other scenarios? Let us consider one last example, namely 11-dimensional supergravity and quickly think about what sort of potentials may be generated.

We start by noting the Maldecena-Nunez no-go theorem, which tells us that there is no de Sitter vacua in compactifications of 11-dimensional SUGRA down to any dimension. Moreover, it is shown in [10] that for 11-dimensional SUGRA on a smooth manifold compactified down to d-dimensions there is once again a lower bound which may be written as follows,

\displaystyle \frac{\mid \nabla \mid}{V} \geq \frac{6}{\sqrt{(d-2)(11-d)}} \ \ (26)

This is consistent with the de Sitter conjecture. But there are caveats, such as when orientifolds are present, as once again summarised [1]. The main point, with (13), (14), and (26) in mind, is that it is very difficult, if not somewhat extraordinary, to evade these constraints. The statement here is not that it is impossible, but that it is very difficult. Most notably, one is required to use stringy ingredients. For instance to violate these constraints you can include,

* Orientifolds (without D-branes and so where charges cannot be cancelled locally) – i.e., `naked’.

* Higher derivative corrections

* Type IIA with orientifolds / something not CY

* Quantum corrections – i.e., quantum vacuum (large, like KKLT)

But these all imply a level of great difficulty, pertaining to the use of stringy ingredients of which we do not yet have a great understanding. So this is one problem, which already requires great consideration. But there is another, which refers to the Dine-Seiberg problem [11], and when combined with the first means one has to work doubly hard. The basic idea with the latter is that the source of the potential vanishes when {g_{s} \rightarrow 0}. Moreover, it says in the weakly coupled regime there is a non-interacting theory, and so any fluxes etc. vanish. This is a very generic statement; it applies to any point in the Hilbert space where many possible light tower of states may dominate. Consider, for example, a potential subject to the above statement regarding the string coupling in some expansion,

\displaystyle V \sim g_{s}^{n} + \sum_{k=1}^{\infty} g_{s}^{n+k}C_{k}

Now, imagine the expansion is controlled. To leading order,

\displaystyle V \sim g_{s}^{n} \sim e^{-n\phi} + \text{small corrections} \ \ (27)

With only small corrections in the well controlled limit such that {g_{s} << 1}. If the potential looks like {e^{-n\phi}} then one can quickly work out,

\displaystyle  \mid \partial_{\phi} V \mid \sim nV \ \ (28)

Which satisfies the conjecture. But as Palti points out, one can always fight this with coefficients, say, for instance, with some potential,

\displaystyle  V = Ag_{s} + B g_{s}^{2} + cg_{s}^{2} + ..., \ \ g_{s} << 1 \ \ (29)

Which is what people do when performing flux compactifications. As we know, we can always play with the fluxes and other things which corresponds in the above expansion to playing with the coefficients. So we can consider A and B and chose that {\frac{B}{A} > \frac{1}{g_{s}}} for which it is possible to then have these fields in minimum balance against each other. But then what of the C coefficient? One must ensure that this doesn’t takeover, so we could say {c \sim B}. But what the Dine-Seiberg argument says that if {A \sim B \sim C \sim O(1)} then we will never find the minimum to the potential, because {Ag_{s}} must be the leading term and we end up with a runaway direction in the field space. That is why for flux compactifications a general approach is to balance the coefficients by playing with the fluxes so that we can get a minimum for the potential.

We can see clearly that the situation is one where we have to overcome both problems, the no-go and the Dine-Seiberg problem, in order to show a de Sitter vacuum in string theory. One interpretation is that both the Maldecena-Nunez no-go theorem and the Dine-Sieberg problem motivates the de Sitter conjecture: i.e., string theory does not foster or does not like de Sitter vacua. But another, perfectly legitimate interpretation is that all that these two accounts are saying is that we just have to work very hard to obtain a de Sitter vacuum in string theory. For the no-go theorem, for example, to evade it requires working with stringy ingredients that we do not yet have much understanding of – such as working with naked orientifolds or in the case of higher derivative corrections. And so maybe the reality of the situation is not best described by the de Sitter Conjecture but instead motivates the need for even deeper thinking in string theory. In time, which of these interpretations is correct will likely clarify.

References

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[3] A. Ceresole, R. D’Auria, and S. Ferrara, The Symplectic structure of N=2 supergravity and its central extension, Nucl. Phys. Proc. Suppl. 46 (1996) 67–74, [hep-th/9509160 [hep-th]].

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[5] J. Polchinski, ‘String Theory: Superstring Theory and Beyond’, Vol. 2. (2005).

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[7] W. Schmid, ‘Variation of Hodge structure: the singularities of the period mapping’, Invent.
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[8] P. Deligne, Theorie de Hodge: III, Publications Mathematiques de l’IHES´ 44 (1974) 5–77.

[9] O. DeWolfe, A. Giryavets, S. Kachru, and W. Taylor, ‘Type IIA moduli stabilization’, JHEP 07 (2005) 066, [hep-th/0505160 [hep-th]].

[10] J. M. Maldacena and C. Nunez, ‘Supergravity description of field theories on curved manifolds and a no go theorem’, Int. J. Mod. Phys. A16 (2001) 822–855, [hep-th/0007018 [hep-th]].

[11] M. Dine and N. Seiberg, ‘Is the Superstring Weakly Coupled?’, Phys. Lett. 162B (1985) 299–302.