Stringy Things

# Notes on String Theory: Conformal Field Theory – Massless Scalars in Flat 2-dimensions

In past entries we familiarised ourselves very briefly with conformal transformations and the 2-dimensional conformal algebra. To progress with our study of Chapter 2 in Polchinski, we need to equip ourselves with a number of other essential tools which will assist in building toward computing operator product expansions (OPEs). In this post, we will focus on notational conventions, transforming to complex coordinates, and utilising holomorphic and antiholomorphic functions. Then, in the next post, we will focus on the path integral and operator insertions, before turning attention to the general formula for OPEs.

To start, it will be beneficial if we define a toy theory of free massless scalars in 2-dimensions. For consistency, we will use the same toy theory that Polchinski describes on p.32. (Please also note, this theory is very similar to the theory on the string WS and this is why it will be useful for us, as it will support an introductory study within an applied setting).

In the context of our toy theory, the Polyakov action takes the form,

$\displaystyle S_{P} = \frac{1}{4\pi \alpha^{\prime}} \int d^{2}\sigma [\partial_{1}X^{\mu} \partial_{1}X^{\nu} + \partial_{2}X^{\mu}\partial_{2}X^{\nu}] \ \ (1)$

This is the Polyakov action with ${\gamma_{ab}}$ being replaced by a flat Euclidean metric ${\delta_{ab}}$ and with Wick rotation. What is the benefit of the Euclidean metric and what is meant by Wick rotation?

In general, a lot of calculation in string theory is performed on a Euclidean WS, in which case, for flat metrics, standard analytic continuation may be used to relate Euclidean and Minkowksi amplitudes. The benefit is that the Euclidean metric enables us to study ordinary geometry and to use conformal field theory on the string. But one will note that in previous constructions of the Polyakov action we used a Minkowski metric, and in past discussions we have also been using light-cone coordinates. So let’s consider transforming from a Minkowski measure to a Euclidean one. To achieve a flat Euclidean metric, the idea is simple: we use a Wick rotation to rewrite the Minkowski metric. Moreover, recall that in Euclidean space, namely the x-y plane, the infinitesimal measure is given by ${ds^{2} = dx^{2} + dy^{2}}$. Compare this with the Minkowski measure, which we may write in WS coordinates as ${ds^{2} = -d\tau^{2} + d\sigma^{2}}$. Notice that in the Euclidean picture all quantities are positive (or at least share the same sign). Now, by Wick rotation, we make a transformation on the time coordinate in the Minkowski measure such that ${\tau \rightarrow -i\tau}$. This means ${d\tau \rightarrow -id\tau}$ and from this it follows ${ds^{2} = - (-id\tau)^{2} + d\sigma^{2} = d\tau^{2} + d\sigma^{2}}$. This is a Euclidean metric.

Hence, by Wick rotation, we are working in imaginary time signature such that the new metric in Euclidean coordinates may be written as,

$\displaystyle \delta = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \ \ (2)$

As suggested, the goal is to end up with a Euclidean theory of massless scalars in flat 2-dimensions. Note, also, that as a result of Wick rotation,

$\displaystyle (\sigma_{0}, \sigma_{1}) \rightarrow (\sigma_{2}, \sigma_{1}) \ \ (3)$

Where we define ${\tau \equiv \sigma_{0}}$ and ${\sigma \equiv \sigma_{1}}$, and where ${\sigma_{2} \equiv i\sigma_{0}}$.

Moving forward, one should note that after Wick rotation from LC coordinates ${(+, -)}$, we enter into the use of complex coordinates ${(z, \bar{z})}$. We first observed these coordinates in the last section on the 2-dimensions conformal algebra. Further clarification may be offered. Most notably, the description of the WS is now performed using complex variables by defining these complex coordinates ${(z, \bar{z}}$ that are, in fact, a function of the variables ${(\tau, \sigma)}$ with which we have already grown accustomed. Hence, ${z = \tau + i\sigma}$ and ${\bar{z} = \tau - i\sigma}$. The benefit of setting up complex coordinates is that it enables us to employ holomorphic (left-moving) and antiholomorphic (right-moving) indices, where holomorphic = ${z}$ and antiholomorphic = ${\bar{z}}$ as also observed in our discussion on the conformal generators.

Now that our field theory has been sketched, and complex coordinates have been formally established, to understand how to transform these coordinates we must understand how to compute the derivatives. The first step is to invert the coordinates and then we will differentiate,

$\displaystyle \tau = \frac{z + \bar{z}}{2}, \ \ \sigma = \frac{z - \bar{z}}{2i} \ \ (4)$

Differentiating with respect to ${z}$ and ${\bar{z}}$ coordinates we obtain the following,

$\displaystyle \frac{\partial \tau}{\partial z} = \frac{\partial \tau}{\partial \bar{z}} = \frac{1}{2} \ \ (5)$

And,

$\displaystyle \frac{\partial \sigma}{\partial z} = \frac{1}{2i}, \ \ \ \frac{\partial \sigma}{\partial \bar{z}} = -\frac{1}{2i} \ \ (6)$

With these results we can then compute for the holomorphic coordinates,

$\displaystyle \frac{\partial}{\partial z} = \frac{\partial \tau}{\partial z}\frac{\partial}{\partial \tau} + \frac{\partial \sigma}{\partial}\frac{\partial}{\partial \sigma} = \frac{1}{2}\frac{\partial}{\partial \tau} + \frac{1}{2i}\frac{\partial}{\partial \sigma} = \frac{1}{2}(\frac{\partial}{\partial \tau} - i\frac{\partial}{\partial \sigma}) \ \ (7)$

One can also repeat the same steps for the antiholomorphic case ${\bar{z}}$,

$\displaystyle \frac{\partial}{\partial \bar{z}} = \frac{\partial \tau}{\partial \bar{z}}\frac{\partial}{\partial\tau} + \frac{\partial \sigma}{\partial \bar{z}}\frac{\partial}{\partial \sigma} = \frac{1}{2}\frac{\partial}{\partial \tau} - \frac{1}{2i}\frac{\partial}{\partial \sigma} = \frac{1}{2}(\partial_{\tau} + i\partial_{\sigma}) \ \ (8)$

Hence, the shorthand notation as read in Polchinksi and which we will use from this point forward,

$\displaystyle \partial \equiv \partial_{z} = \frac{1}{2}(\partial_1 - i \partial_2) \ \ (9)$

$\displaystyle \bar{\partial} \equiv \partial_{\bar{z}} = \frac{1}{2}(\partial_1 + i\partial_2) \ \ (10)$

Where ${\partial_zz = 1}$ and ${\partial_{\bar{z}}z = 0}$.

To continue setting things up, we must now also register that we may set ${\sigma = (\sigma^{1},\sigma^{2})}$ and ${\sigma^{z} = \sigma^{1} + i\sigma^{2}}$ and ${\sigma^{\bar{z}} = \sigma^{1} - i\sigma^{2}}$. The reason for this will become clear in a moment. For the metric, given the above ${\gamma_{ab} \rightarrow \delta_{ab} \rightarrow g_{ab}}$,

$\displaystyle g_{ab} = \begin{bmatrix} g_{zz} & g_{z\bar{z}} \\ g_{\bar{z}z} & g_{\bar{z}\bar{z}} \\ \end{bmatrix} (11)$

From this we can also say that ${\det g = \sqrt{g} = \frac{1}{2}}$, which is true for Minkowski and indeed ${\delta_{ab}}$ since we have Wick rotated. When we raise indices a factor of ${2}$ is returned: ${g^{z\bar{z}} = g^{\bar{z}z} = 2}$. Lastly, the area element transforms as ${d\sigma^{1}d\sigma^{2} \equiv d^{2}\sigma \equiv 2 d\sigma^{1}d\sigma^{2}}$. So, we see, ${d^{2}z \sqrt{g} \equiv d^{2}\sigma}$.

We now need to study how the delta function transforms. Given ${\int d^{2}\sigma \delta^{2}(\sigma_{1},\sigma_{2}) = \int d^{2}\sigma \delta(\sigma_{1})\delta(\sigma_{2}) = 1}$, we find that in our new coordinates:

$\displaystyle \int d^{2}z \delta^{2}(z,\bar{z}) = 1 \implies \delta^{2}(z,\bar{z}) = \frac{1}{2} \delta^{2}(\sigma_{1},\sigma_{2}) \ \ (12)$

We can continue establishing relevant notation by focusing on how we may rewrite the Polyakov action. In the notation that we’ve constructed we find,

$\displaystyle S_{P} = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu} \bar{\partial}X_{\mu} \ \ (13)$

Where ${d^{2}z = dzd\bar{z}}$. Using identities constructed throughout this post, the tools are available to see how we arrive at this simpler form of the action. The task now is to see what returns when we vary (13).

Proposition: We vary the action (13) and find the EoM to be ${\partial\bar{\partial}X^{\mu} = 0}$.

Proof: The string coordinate field, if not obvious, is now ${X(z, \bar{z}) = X(z) + \bar{X}(\bar{z})}$. It will become clear in the following discussion that we want to compute a quantity without linear dependence on ${\tau}$. To that end we use the derivative of the coordinate field ${\partial X(z)}$ and ${\bar{\partial}\bar{X}(\bar{z})}$.

Now, when we vary the simplified action we find,

$\displaystyle 0 = \frac{\delta S}{\delta X^{\mu}} = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu}\bar{\partial}(X_{\mu} + \delta X_{\mu})$

$\displaystyle = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu} (\bar{\partial}X_{\mu} + \bar{\partial}\delta X_{\mu})$

$\displaystyle = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu}\bar{\partial}X_{\mu} + \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu}(\bar{\partial}\delta X_{\mu}) \ \ (14)$

Continuing with the conventional procedure, where we now integrate by parts (and for convenience discard the boundary terms), we find the EoM to be

$\displaystyle \delta S = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu}(\bar{\partial}\delta X_{\mu})$

$\displaystyle = - \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial\bar{\partial}X^{\mu} (\delta X_{\mu}) = 0$

$\displaystyle \implies \partial\bar{\partial}X^{\mu} (z, \bar{z}) = 0 \ \ (15)$

$\Box$

Using the fact that partial derivatives commute. This completes the proof. The classical solution can be solved by, or in other words it decomposes as, ${X(z) + \bar{X}(z)}$. And we should also note, for pedagogical purposes, that we may write the EoM as $\partial (\bar{\partial} X^{\mu}) = \bar{\partial} (\partial X^{\mu}) = 0$ such that

$\displaystyle \partial X^{\mu} = \partial X^{\mu}(z) \ \ \ \text{holomorphic function}$

$\displaystyle \bar{\partial}X^{\mu} = \bar{\partial}X^{\mu} (\bar{z}) \ \ \ \text{antiholomorphic function}$

References

Joseph Polchinski. (2005). ‘String Theory: An Introduction to the Bosonic String’, Vol. 1.

David Tong. (2009). ‘String Theory’ [lecture notes].

Standard
Stringy Things

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

1. Introduction

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

2. Review – Effective Field Theory

2.1. Schematic Overview

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

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

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

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

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

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

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

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

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

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

2.2. Alternative Approach to EFTs

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

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

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

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

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

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

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

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

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

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

Where we have some operators suppressed by the energy scale.

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

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

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

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

2.3. EFT Coupled to Gravity

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

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

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

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

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

3. EFTs and the Swampland

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

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

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

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

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

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

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

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

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

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

4. From EFTs to the Rules of the Swampland

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

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

5. Weak Gravity Conjecture (Magnetic)

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

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

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

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

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

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

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

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

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

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

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

6. Summary

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

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

References

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

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

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

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

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

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

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

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Mathematics

# Jensen Polynomials, the Riemann-zeta Function, and SYK

A new paper by Michael Griffin, Ken Ono, Larry Rolen, and Don Zagier appears to have a made some intriguing steps when it comes to the Riemann Hypothesis (RH). The paper is titled, ‘Jensen polynomials for the Riemann zeta function and other sequences’. The preprint originally appeared in arXiv [arXiv:1902.07321 [math.NT]] in February 2019. It was one of a long list of papers that I wanted to read over the summer. And with the final version now in the Proceedings of the National Academy of Sciences (PNAS), I would like to discuss a bit about the author’s work and one way in which it relates to my own research.

First, the regular reader will recall that in a past post on string mass and the Riemann-zeta function, we discussed the RH very briefly, including the late Sir Michael Atiyah’s claim to have solved it, and finally the separate idea of a stringy proof. The status of Atiyah’s claim still seems unclear, though I mentioned previously that it doesn’t look like it will hold. The idea of a stringy proof also remains a distant dream. But we may at least recall from this earlier post some basic properties of the RH.

What is very interesting about the Griffin et al paper is that it returns to a rather old approach to the RH, based on George Pólya’s research in 1927. The authors also build on the work of Johan Jensen. The connection is as follows. It was the former, Pólya, a Hungarian mathematician, who proved that, for the Riemann-zeta function $\zeta{s}$ at its point of symmetry, the RH is equivalent to the hyperbolicity of Jensen polynomials. For the inquisitive reader, as an entry I recommend this 1990 article in the Journal of Mathematical Analysis and Applications by George Csordas, Rirchard S. Varga, and Istvan Vincze titled, ‘Jensen polynomials with applications to the Riemann zeta-function’.

Pólya’s work is generally very interesting, something I have been familiarising myself with in relation to the Sachdev-Ye-Kitaev model (more on this later) and quantum gravity. When it comes to the RH, his approach was left mostly abandoned for decades. But Griffin et al formulate what is basically a new general framework, leveraging Pólya’s insights, and in the process proving a few new theorems and even proving criterion pertaining to the RH.

1. Hyperbolicity of Polynomials

I won’t discuss their paper in length, instead focusing on a particular section of the work. But as a short entry to their study, Griffin et al pick up from the work of Pólya, summarising his result about how the RH is equivalent to the hyperbolicity of all Jensen polynomials associated with a particular sequence of Taylor coefficients,

$\displaystyle (-1 + 4z^{2}) \Lambda(\frac{1}{2} + z) = \sum_{n=0}^{\infty} \frac{\gamma (n)}{n!} \cdot z^{2n} \ \ (1)$

Where ${\Lambda(s) = \pi^{-s/2} \Gamma (s/2)\zeta{s} = \Lambda (1 - s)}$, as stated in the paper. Now, if I am not mistaken, the sequence of Taylor coefficients belongs to what is called the Laguerre-Pólya class, in which case if there is some function ${f(x)}$ that belongs to this class, the function satisfies the Laguerre inequalities.

Additionally,  the Jensen polynomial can be seen in (1). Written generally, a Jensen polynomial is of the form ${g_{n}(t) := \sum_{k = 0}^{n} {n \choose k} \gamma_{k}t^{k}}$, where ${\gamma_{k}}$‘s are positive and they satisfy the Turán inequalities ${\gamma_{k}^{2} - \gamma_{k - 1} \gamma_{k + 1} \geq 0}$.

Now, given that a polynomial with real coefficients is hyperbolic if all of its zeros are real, where read in Griffin et al how the Jensen polynomial of degree ${d}$ and shift ${n}$ in the arbitrary sequence of real numbers ${\{ \alpha (0), \alpha (1), ... \}}$ is the following polynomial,

$\displaystyle J_{\alpha}^{d,n} (X) := \sum_{j = 0}^{d} {d \choose j} \alpha (n + j)X^{j} \ \ (2)$

Where ${n}$ and ${d}$ are the non-negative integers and where, I think, ${J_{\alpha}^{d,n} (X)}$ is the hyperbolicity of polynomials. Now, recall that we have our previous Taylor coefficients ${\gamma}$. From the above result, the following statement is given that the RH is equivalent to ${J_{\gamma}^{d,n}(X)}$ – the hyperbolicity of polynomials – for all non-negative integers. What is very curious, and what I would like to look into a bit more, is how this conditions holds under differentiation. In any case, as the authors point out, one can prove the RH by showing hyperbolicity for ${J_{\alpha}^{d,n} (X)}$; but proving the RH is of course notoriously difficult!

Alternatively, another path may be chosen. My understanding is that Griffin-Ono-Rolen-Zagier use shifts in ${n}$ for small ${d}$, because, from what I understand about hyperbolic polynomials, one wants to limit the hyperbolicity in the ${d}$ direction. Then the idea, should I not be corrected, is to study the asymptotic behaviour of ${\gamma(n)}$.

This is the general entry, from which the authors then go on to consider a number of theorems. I won’t go through all of the theorems. One can just as well read the paper and the proofs. What I want to do is focus particularly on Theorem 3.

2. Theorem 3

Aside from the more general considerations and potential breakthroughs with respect to the RH, one of my interests triggered in the Griffin-Ono-Rolen-Zagier paper has to do with my ongoing studies concerning Gaussian Unitary Ensembles (GUE) and Random Matrix Theory (RMT) in the context of the Sachdev-Ye-Kitaev (SYK) model (plus similar models) and quantum gravity. Moreover, RMT has become an interest in relation to chaos and complexity, not least because in SYK and similar models we consider late-time behaviour of quantum black holes in relation to theories of quantum chaos and random matrices.

But for now, one thing that is quite fascinating about Jensen polynomials for the Riemann-zeta function is the proof in Griffin et al of the GUE random matrix model prediction. That is, the derivative aspect GUE random matrix model prediction for the zeros of Jensen polynomials. One of the claims here is that the GUE and the RH are satisfied by the symmetric version of the zeta function. To quote in length,

‘To make this precise, recall that Dyson, Montgomery, and Odlyzko [9, 10, 11] conjecture that the nontrivial zeros of the Riemann zeta function are distributed like the eigenvalues of random Hermitian matrices. These eigenvalues satisfy Wigner’s Semicircular Law, as do the roots of the Hermite polynomials ${H_{d}(X)}$, when suitably normalized, as ${d \rightarrow +\infty}$ (see Chapter 3 of [12]). The roots of ${J){\gamma}^{d,0} (X)}$, as ${d \rightarrow +\infty}$ approximate the zeros of ${\Lambda (\frac{1}{2} + z)}$ (see [1] or Lemma 2.2 of [13]), and so GUE predicts that these roots also obey the Semicircular Law. Since the derivatives of ${\Lambda (\frac{1}{2} + z)}$ are also predicted to satisfy GUE, it is natural to consider the limiting behavior of ${J_{\gamma}^{d,n}(X)}$ as ${n \rightarrow +\infty}$. The work here proves that these derivative aspect limits are the Hermite polynomials ${H_{d}(X)}$, which, as mentioned above, satisfy GUE in degree aspect.’

I think Theorem 3 raises some very interesting, albeit searching questions. I also think it possibly raises or inspires (even if naively) some course of thought about the connection of insights being made in SYK and SYK-like models, RMT more generally, and even studies of the zeros of the Riemann-zeta function in relation to quantum black holes. In my own mind, I also immediately think of the Hilbert-Polya hypothesis and the Jensen polynomials in this context, as well as ideas pertaining to the eigenvalues of Hamiltonians in different random matrix models of quantum chaos. There is connection and certainly also an interesting analogy here. To what degree? It is not entirely clear, from my current vantage. There are also some differences that need to be considered in all of these areas. But it may not be naive to ask, in relation to some developing inclinations in SYK and other tensor models, about how GUE random matrices and local Riemann zeros are or may be connected.

Perhaps I should save such considerations for a separate article.

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Stringy Things

# Pure Spinor Formalism

In recent days, pure spinors have become my life. And that is by no means a bad thing.

I don’t want to divulge too much at this time. The short of it is that I’ve been looking into the pure spinor formalism for a possible research project. Whether the project comes to fruition or not has yet to be determined. Regardless of the outcome, the time will have been well spent as I’ve immensely enjoyed learning the topic. What is intriguing is the power of the formalism when studying superstrings on different curved backgrounds. It is also useful when studying multiloop amplitudes. More personally, I have also found it nice to work through and think about because there is some connection with my interests in twistor theory, among other things.

As it is quite a rich area, there is a lot to comment on. Given time, I will type and upload my own notes as a sort of tour through the formalism. For now I’ve put together a select list of preprint papers that give an overview, organised by date. I haven’t listed everything, and the reader may find other works that adequately study pure spinors. For me, I found it useful to simultaneously read [2, 3, 6] as review, having then marched on from there.

[1] N. Berkovits, ‘Super-Poincare Covariant Quantization of the Superstring’, (2000) preprint in arXiv [arXiv:hep-th/0001035 [hep-th]].

[2] N. Berkovits, ‘ICTP Lectures on Covariant Quantization of the Superstring’ [lecture notes], (2002) preprint in arXiv [arXiv:hep-th/0209059 [hep-th]].

[3] N. Berkovits and D. Z. Marchioro, ‘Relating the Green-Schwarz and Pure Spinor Formalisms for the Superstring’, (2004) preprint in arXiv [arXiv:hep-th/0412198 [hep-th]].

[4] N.I. Farahat and H.A. Elegla, ‘Path Integral Quantization of Brink-Schwarz Superparticle’, EJTP 5, No. 19 (2008) 57–64.

[5] C.R. Mafra, ‘Superstring Scattering Amplitudes with the Pure Spinor Formalism’, (2008) preprint in arXiv [arXiv:0902.1552v3 [hep-th]].

[6] O. A. Bedoya and N. Berkovits, ‘GGI Lectures on the Pure Spinor Formalism of the Superstring’, (2009) preprint in arXiv [arXiv:0910.2254v1 [hep-th]].

[7] T. Adamo and E. Casali,’Scattering equations, supergravity integrands, and pure spinors’, (2015) preprint in arXiv [arXiv:1502.06826v2 [hep-th]].

[8] N. Berkovits, ‘Untwisting the Pure Spinor Formalism to the RNS and Twistor String in a Flat and $AdS_5 \times S^5$ Background’, (2015) preprint in arXiv [arXiv:1604.04617v2 [hep-th]].

[9] N. Berkovits, ‘Origin of the Pure Spinor and Green-Schwarz Formalisms’, (2015) preprint in arXiv [arXiv:1503.03080 [hep-th]]

*Image: A 2005 poster by the IHES promoting a pure spinor workshop.

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Physics Diary

# SiftS 2019

SiftS 2019 concluded on Friday. It was an enjoyable two weeks of study and discussion on topics in string theory and holography. Eran Palti and Kyriakos Papadodimas were for me the highlight of the event. This is not meant to take away from others, it is just that Palti and Papadodimas were one of the main reasons for my attending SiftS. I could sit and listen to Papadodimas talk physics for hours. And Palti’s lectures on the Swampland were outstanding, as expected.

If I had one minor personal grief about the summer school as a whole, it’s that there wasn’t enough pure string theory. But it is very likely that I would say this at a number of different engagements, with the exception perhaps of Strings 2019 and String-Math 2019, two of the main string conferences. So it is unfair to make any such complaint formal, and one must also be mindful that while string theory was the theme, the engagement wasn’t necessarily meant to serve pure stringy discussion.

All of this is to say that I am both thrilled and honoured to have had the privilege of attending SiftS 2019. To mark its conclusion, I want to take a moment to congratulate the SiftS organisers for putting together a terrific summer school. I also want to take a moment to thank everyone at the Universidad de Autonomous Madrid for their hospitality and support throughout my stay. My impression of the university before arriving was that it was one of the best in Europe, and I left the campus and the Instituto de Física Teórica UAM/CSIC with the same view. I can say with honesty that I very much look forward to my return at some point in the future.

***

Now that SiftS is over for the year, and with the conclusion of my admittedly brief holiday during the weekend, I have returned to my research and studies at the University of Nottingham. There is a lot to discuss and catch up on with Prof. Padilla, with a number of possibly interesting ideas percolating. My return to Nottingham also means that I will start actively blogging again. In addition to covering some interesting topics from SiftS 2019, I am also working on a number of research projects which will be nice to write about in the coming days, weeks, and months. I will also be continuing my series of string notes, where the reader and I are on our way to covering the whole of textbook bosonic and superstring theory. We will start from where we left off, namely an introduction to conformal field theory. (In the background, I am going to continue working on my blog to fix the LaTeX of older posts as a result the move).

With regards to SiftS 2019 in particular. I will not write about all of the lecture series and topics covered. Instead, I will focus on sharing my notes and thoughts from the lectures by Palti on the Swampland and by Papadodimas on the Black Hole Interior. This will serve as a nice opportunity to also reflect on some of their respective papers, and to summarise key arguments.

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Philosophy and General Reading

# New Home, New LaTeX Problems

It has been two weeks since my last post. At the time, I was preparing to travel to Universidad Autonoma de Madrid for SiftS 2019. In my note I also mentioned that I was enjoying a brief pause from active blogging, mostly to take some time to revise, edit and reorganise the collection of string theory notes that I had already uploaded. I also mentioned that I wanted to take some time to reflect more generally on my first few months as a blogger, particularly about what I might change and improve.

The regular reader will have noticed that a lot has indeed changed. For one, my blog has moved to new home.

One problem that I was having with the old website concerned how it was configured in the backend. It was incredibly inefficient to upload LaTeX, which proved a hindrance considering most posts that I write use LaTeX. This is actually one reason why, before leaving for my summer string theory and holography engagement in Spain, I was not posting regularly; between finishing my paper and my ongoing research, it took too much time to transfer work from my usual LaTeX environment to my blogging environment each time I wanted to post something.

While away in Spain, I have since restructured everything. In addition to the new backend configuration, I am now also using LaTeX to WordPress in conjunction with Python. This software should hopefully enable me to transfer work directly and efficiently from my everyday LaTeX environment to my blogging environment.

Unfortunately, the new set-up also means that all of my old posts using LaTeX are broken and need to be re-written / re-uploaded one-by-one. This includes all of my string notes.

As I work on this issue, I am eager to return to active blogging. There is much to write about and discuss from SiftS 2019, and I am also eager to write about more pure string theory matters that I have been researching of late. I also will close by saying that I want to continue sharing more string notes over the course of the summer, hopefully up to and through the textbook contents of superstring theory. How it all gets organised on the new site, however, remains an open question.

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Philosophy and General Reading

# Learning to Write for Others: String Notes and Future Posts

It is fitting that I write this post on National Writing Day. A few months ago, following some encouraging nudges by others, I decided to commit to active blogging. I already had this website designed and online, but it was mostly left idle. The odd time I would post a note, or share a short essay. But for the most part, my struggle with writing was so much that it took too much energy to find the courage to write on a regular basis. Besides, as much as I genuinely enjoy the idea of writing, in practice I find it uncomfortable.

In reflection, these may seem like odd things to say because over the years I have written a lot of things. Essays, articles, and academic papers. Two years ago I also had a book published. But as a writer I have never been more than barely functional, driven primarily by my enthusiasm and penchant for the written word. Writing, not to mention communication in general, has never come naturally to me. As much as I read and spend my days studying, method in writing is a completely different matter.

To be a good writer, I think one needs to have access to understanding the reader. Empathy, in other words, seems important. In the world of Asperger’s, we talk about “strategies”. And there are a lot of support organisations and groups, including at my university, who help people with Asperger’s like myself learn effective strategies to be a good writer. Part of the strategy for me involves developing a consistent sense of structure – a logically ordered and intelligible set of rules – that assist in navigating the writing process, such that the communication extended to the other person is received as lucid, apprehensible and overall pedagogically cogent. Or, that is the idea and what I am trying to attain. In general, subjective concepts and practices are difficult for me to understand. What defines the objective criteria in which one may assess the art of a piece of writing? What makes a good piece of writing? Or, perhaps more pointedly, what makes for effective writing and communication? As a person with Asperger’s, writing is very much like floundering in the dark.

Since developing this little space on the internet – and with much encouragement – I have been working hard on trying to understand how to write for others. It’s like with my maths/physics – I have become aware since joining university that I am no longer writing my maths/physics for myself, where I can do as I please with my notation. On my whiteboard at home, or on the back of a scrap piece of paper, I may write my equations as I please, as long as I am following the rules. When I write my maths/physics for myself, I don’t need to explain or communicate anyone else. I can change my indices or for convenience suddenly switch notation, without needing to contemplate the legibility for another set of eyes. But to another person, it might not communicate or it might be difficult to follow. And so, to function as a professional physicist, it means I must learn to write my maths/physics for another and thus to ensure consistency and effective communication for the student or external reader. It is the same when it comes to the writing process. Like with the rules of mathematics and, indeed, the basic set of rules that we practice in mathematical physics, it is not so much a problem in the sense of grammar, syntax and spelling. It is more about what makes for good transmission – how people communicate, with use of signposting and other mechanisms.

In the few months since actively maintaining this blog, I’ve learned a lot about writing, and I have started to make small steps in understanding in my own way how to think about communicating through the written word, thanks to feedback from others and from my own support networks. There is still a lot of work to be done. But never before have I felt more confident that I am on track to obtaining a clear sense of consistent structure when communicating through writing – how to think about referencing for the reader, being more concise when explaining an idea, and how to convey a building of logic for another person. I find it exciting, and for once I look forward to writing more and to continued practise on my blog.

Having said that, I would like to highlight that over the coming weeks I may not publish anything new. The reason for this is because I want to take some time to revise, edit and perhaps even reorganise the collection of string theory notes I have already posted. It will also be a time for me to reflect. In some cases, I may even perform a complete rewrite of a particular article in my string theory series (each reviewed article will be timestamped from this day forward). I want to make sure these articles are clear and communicate the inspiring and wondrous ideas of string theory, both technically and pedagogically.

Another reason this blog will be quiet for the next month or so is because I will be away for three weeks in July, visiting the Universidad Autonoma de Madrid for a string theory and holography engagement (SiftS 2019).

But while it may be a few weeks until my next post appears, I very much look forward to regularly contributing to this blog and to continuing to share with whoever may visit this space. In addition to the continuation of my string theory notes, and other writings in fundamental physics, I would also like to think about writing a few new essays – perhaps even a paper or two – on history and other subjects.

Until then, thanks for reading.

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