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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# 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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# 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.

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Stringy Things

# Notes on String Theory: Conformal Group in 2-dimensions

1. Introduction: Conformal Group in 2-dimensions

Following our previous study of the d-dimensional conformal group and the generators of conformal transformations, we now turn our attention to the study of the conformal group in 2-dimensions. Although we have taken some time to considered the d-dimensional conformal algebra, it should already be clear from past discussions that our interest is particularly in 2-dimensions. To begin our study of the 2-dimensional conformal algebra, where ${d = 2}$, note that we’re now employing a 2-dimensional Euclidean metric such that ${g_{\mu \nu} = \delta_{\mu \nu}}$. The first task is to construct the generators. Moreover, it can be found when studying the conserved currents on the WS (substituting for the Euclidean metric, see the last post),

$\displaystyle \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} = (\partial \cdot \epsilon)\delta_{\mu \nu} \ \ (1)$

When we take the coordinates ${(x^1, x^2)}$ and as we calculate for different values of ${\mu}$ and ${\nu}$, the above equation reduces rather nicely:

For ${\mu = \nu = 1}$, we arrive at ${2\partial_{1}\epsilon_{1} = \partial_{1}\epsilon_{1} + \partial_{2}\epsilon_{2} \implies \partial_{1}\epsilon_{1} = \partial_{2}\epsilon_{2}}$.

For ${\mu = \nu = 2}$, we arrive reversely at ${2\partial_{2}\epsilon_{2} = \partial_{1}\epsilon_{1} + \partial_{2}\epsilon_{2} \implies \partial_{2}\epsilon_{2} = \partial_{1}\epsilon_{1}}$.

Now, for the symmetric case where ${\mu = 1}$ and ${\nu = 2}$ (and, equivalently by symmetry, ${\mu = 2}$ and ${\nu = 1}$), we arrive ${\partial_{1}\epsilon_{2} + \partial_{2}\epsilon_{1} = 0}$. It follows, ${\partial_{1}\epsilon_{2} = -\partial_{2}\epsilon_{1}}$.

Notice, from these results, we have two distinguishable equations:

$\displaystyle \partial_{1}\epsilon_{1} = \partial_{2}\epsilon_{2} \ \ (2)$

$\displaystyle \partial_{1}\epsilon_{2} = -\partial_{2}\epsilon_{1} \ \ (3)$

If it is not obvious to the reader, it can be explicitly stated that these are nothing other than the Cauchy-Riemann equations. What this means, firstly, is that the conformal Killing equations reduce to the Cauchy-Riemann equations. Secondly, in 2-dimensions the infinitesimal conformal transformations that are of primary focus obey these equations.

Why is this notable? We know that in the theory of complex variables we’re working with analytic functions. As Polchinski explicitly communicates (p.34), the advantage here is that in working with analytical functions we can employ the coordinate convention ${(z, \bar{z})}$. This means, firstly, that conformal transformations correspond with holomorphic and antiholomorphic coordinate transformations. These coordinate transformations are given by,

$\displaystyle z \rightarrow f(z), \ \ \ \bar{z} \rightarrow \bar{f}(\bar{z}) \ \ (4)$

Following Polchinski (pp.33-34), we are working with complex coordinates ${z = \sigma + i\sigma^2}$ and ${\bar{z} = \sigma - i\sigma^2}$. It is also the case that ${d^{2}x = dx^{0}dx^{1} = \frac{1}{2}dzd\bar{z}}$. More will be said about this in the next section. Meanwhile, we may also define in the Euclidean signature and with complex variables,

$\displaystyle \epsilon^{z} = \epsilon^0 + i\epsilon^1, \ \ \bar{\epsilon}^{\bar{z}} = \epsilon^0 - i\epsilon^1 \ \ (5)$

In which ${\epsilon}$ and ${\bar{\epsilon}}$ are infinitesimal conformal transformations. This implies that ${\partial_{z}\epsilon = 0}$ and ${\partial_{z}\bar{\epsilon} = 0}$.

And so, in terms of infinitesimal conformal transformations, we may write a change of holomorphic and antiholomorphic coordinates in an infinitesimal form,

$\displaystyle z \rightarrow z^{\prime} = z^{\prime} + \epsilon(z), \ \ \ \bar{z} \rightarrow \bar{z}^{\prime} = \bar{z}^{\prime} + \bar{\epsilon}(\bar{z}) \ \ (6)$

2. Generators of the 2-dimensional Conformal Group

What we want to do is obtain the basis of generators that produce the algebra of infinitesimal conformal transformations. To do so, we expand ${\epsilon}$ and ${\bar{\epsilon}}$ in a Laurent series obtaining the result,

$\displaystyle \epsilon(z) = \sum_{n \in \mathbb{Z}} \epsilon_{n} z^{n+1} \ \ (7)$

And,

$\displaystyle \bar{\epsilon}(\bar{z}) = \sum_{n \in \mathbb{Z}} \epsilon_{n} \bar{z}^{n+1} \ \ (8)$

With the basis of generators that generate the infinitesimal conformal transformations given by,

$\displaystyle l_{n} = -z^{n+1}\partial_{z} \ \ (9)$

And,

$\displaystyle \bar{l}_{n} = -\bar{z}^{n+1}\partial_{\bar{z}} \ \ (10)$

Classically, the above generators satisfy the Virasoro algebra. Moreover, it follows that these generators form the set ${\{l_{n},\bar{l}_{n}\}}$ and this set becomes the algebra of infinitesimal conformal transformations for ${n \in \mathbb{Z}}$. The algebraic structure is given by the commutation relations,

$\displaystyle [l_m, l_n] = (m - n)l_{m + n} \ \ (11)$

$\displaystyle [\bar{l_m}, \bar{l_n}] = (m - n)\bar{l}_{m + n} \ \ (12)$

$\displaystyle [l_m, \bar{l_n}] = 0 \ \ (13)$

Importantly, the preceding generators obey the Witt algebra (Weigand, p.68). Also important, the generators that we’ve derived come into contact with the Möbius group (Weigand, p.69). To show this, we note the special case in which ${l_{0, \pm 1}}$ and ${\bar{l}_{0, \pm 1}}$. Considering an infinitesimal coordinate transformation, we find in the following cases:

* ${l_{-1} = -\partial_z}$ generates rigid translations of the form ${z^{\prime} = z - \epsilon}$;

* ${l_{0} = z -\epsilon z}$ generates dilatations;

* ${l_{1} = z - \epsilon z^2}$ generates special conformal transformations.

When we collect these we can describe globally defined conformal diffeomorphisms as stated below, which give the Möbius transformation,

$\displaystyle z \rightarrow \frac{az + b}{cz + d}$

Where ${ad - bc = 1}$. A list of other constraints can be reviewed (Weigand, p.69).

References

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

Joshua D. Qualls. (2016). “Lectures on Conformal Field Theory” [lecture notes].

Timo Weigand. (2015/16). “Introduction to String Theory” [lecture notes].

Kevin Wray. (2009). “An Introduction to String Theory” [lecture notes].

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Stringy Things

# Notes on string theory: Generators of conformal transformations

1. Infinitesimal Generators of the Conformal Group

In the last post, we considered a brief introduction to conformal field theory in string theory. We also began to study the d-dimensional conformal group, as described in equations (1) and (2). What we want to do now is study the infinitesimal generators of the d-dimensional conformal group, and in doing so we will refer back to these equations.

In other words, if we assume the background is flat, such that ${g_{\mu \nu} = \eta_{\mu \nu}}$, the essential point of interest here concerns the infinitesimal transformation of the coordinates. Returning to (2) in the last post, infinitesimal coordinate transformations may be considered generally in the form ${x^{\mu} \rightarrow x^{\prime \mu} = x^{\mu} + \epsilon^{\mu}(x) + \mathcal{O}(\epsilon^{2})}$. For the scaling factor ${\Omega (x)}$ we have ${\Omega (x) = e^{\omega(x)} = 1 + \omega(x) + [...]}$.

Now, the question remains: in the case of an infinitesimal transformation, what happens to the metric? It turns out that the metric is left unchanged. To consider why this is the case, we may consider (1) from the linked discussion. Moreover, if, as above, we take an infinitesimal coordinate transformation then we have

$\displaystyle g_{\mu \nu}^{\prime} (x^{\mu} + \epsilon^{\mu}) = g_{\mu \nu} + (\partial_{\mu}\epsilon^{\mu} + \partial_{\nu}\epsilon^{\nu})g_{\mu \nu}$

$\displaystyle = g_{\mu \nu} + \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} \ \ (3)$

However, to satisfy the condition of a conformal transformation, (3) must be equal to (1). So we equate (3) and (1),

$\displaystyle \omega(x)g_{\mu \nu}(x) = g_{\mu \nu} + \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} \ \ (4)$

Where, ${\omega(x)}$ is just an arbitrary function denoting a very small deviation from identity. Thus, we may also write ${\omega(x) = \omega(x) - 1}$ which then gives,

$\displaystyle (\omega(x) - 1)g_{\mu \nu} = \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} \ \ (5)$

For this to make sense, we must find some expression for the scaling term ${\omega(x) - 1}$. One way to proceed is to multiply both sides of (5) by ${g^{\mu \nu}}$. As we are working in ${d}$ spacetime dimensions, it follows ${g_{\mu \nu}g^{\mu \nu} = d}$. Hence,

$\displaystyle (\omega(x) - 1)g_{\mu \nu}g^{\mu \nu} = (\partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu})g^{\mu \nu}$

$\displaystyle (\omega(x) - 1)d = g^{\mu \nu}\partial_{\mu}\epsilon_{\nu} + g^{\mu \nu}\partial_{\nu}\epsilon_{\mu} \ \ (6)$

The left-hand side of (6) is simple to manage. Focusing on the right-hand side, we raise indices and relabel. This gives us a usual factor of ${2}$. Hence, for the RHS of (6),

$\displaystyle = \partial_{\mu}\epsilon^{\nu} + \partial_{\nu}\epsilon^{\mu}$

$\displaystyle = 2 \partial_{\mu}\epsilon^{\nu} \ \ (7)$

Therefore, substituting (7) into the RHS of (6) we get,

$\displaystyle (\omega(x) - 1)d = 2 \partial_{\mu}\epsilon^{\nu} \ \ (8)$

Now, if we divide both sides by ${d}$ and simplify, we end up with

$\displaystyle (\omega(x) - 1) = \frac{2}{d} \partial_{\mu}\epsilon^{\nu} = \frac{2}{d} (\partial \cdot \epsilon) \ \ (9)$

For which we may note the substitution,

$\displaystyle \frac{2}{d}(\partial \cdot \epsilon) = \partial_{\mu}\epsilon^{\nu} + \partial_{\nu}\epsilon^{\mu} \ \ (10)$

Where we can see that the infinitesimal conformal transformation, ${\epsilon}$, obeys the above equation. What is significant about this equation? It is the conformal Killing equation. And, it turns out, solutions to the above correspond to infinitesimal conformal transformations. Let us now study these solutions.

To simplify things, notice firstly that we can define ${\partial_{\mu}\epsilon^{\mu} = \Box}$. Taking the derivative of the left and right-hand sides of the conformal Killing equation we obtain the following,

LHS:

$\displaystyle = \partial^{\mu}(\frac{2}{d}(\partial \cdot \epsilon))$

RHS:

$\displaystyle \partial^{\mu}\partial_{\mu}\epsilon_{\nu} + \partial^{\mu}\partial_{\nu}\epsilon_{\mu} = \Box\epsilon_{\nu} + \partial_{\nu}(\partial \cdot \epsilon)$

Putting everything together, equating both sides, and then rearranging terms we find,

$\displaystyle \Box\epsilon_{\nu} + (1 - \frac{2}{d})\partial_{\nu}(\partial \cdot \epsilon) = 0 \ \ (11)$

It is clear that when ${d = 2}$, our first equation may be written as

$\displaystyle \Box\epsilon_{\nu} = 0 \ \ (12)$

For ${d > 2}$, we arrive at the following commonly cited equations that one will find in most texts:

1) ${\epsilon^{\mu} = a^{\mu}}$ which represents a translation (${a^{\mu}}$ is a constant).

2) ${e^{\mu} = \lambda x^{\mu}}$ which represents a scale transformation. Note, this corresponds to an infinitesimal Poincaré transformation.

3) ${\epsilon^{\mu} = w^{\mu}_{\nu}x^{\nu}}$ which represents a rotation, where ${w^{\mu}_{\nu}x^{\nu}}$ is an antisymmetric tensor. Note, this antisymmetric tensor also acts as the generator of the Lorentz group. Also note, this corresponds to an infinitesimal Poincaré transformation.

4) ${\epsilon^{\mu} = b^{\mu}x^{2} - 2x^{\mu}(b \cdot x)}$ which represents a special conformal transformation.

From these equations, and with the inclusion of the Poincaré group, we have the collection of transformations known as the conformal group in d-dimensions. This group is isomorphic to SO(2,d).

To complete our discussion, we may note that generally we may also incorporate the following generators and thus the conformal group has the following representation:

1) ${P_{\mu} = -\partial_{\mu}}$, which generates translations and is from the Poincaré group.

2) ${D = -ix \cdot \partial}$, which generates scale transformations.

3) ${J_{\mu} = i(x_{\mu}\partial_{\nu} - x_{\nu}\partial_{\mu})}$, which generates rotations.

4) ${K_{\mu} = i(x^2\partial_{\mu} - 2x_{\mu}(x \cdot \partial))}$, which generates special conformal transformations.

This completes our review of the d-dimensional conformal group and its algebra. In the next post, we will study the conformal group in 2-dimensions.

References

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

Timo Weigand. (2015/16). “Introduction to String Theory” [lecture notes].

Kevin Wray. (2009). “An Introduction to String Theory” [lecture notes].

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Stringy Things

# Notes on string theory: Introduction to Conformal Field Theory

1. Introduction

The aim of this post is to introduce the topic of Conformal Field Theories (CFTs) in string theory. In general, CFTs allow us to describe a number of systems in different areas of physics. To list one example, conformal invariance plays an important role in condensed matter physics, particularly in the context of second order phase transitions in which the critical behaviour of systems may be described. But as we are focused on the stringy case, we may motivate the study of CFTs as follows: 2-dimensional CFTs prove very important when it comes to the study of the physical dynamics of the worldsheet.

In past posts we already observed, for instance, how the internal modes along the string relate to conformal transformations. Indeed, upon fixing the worldsheet diffeomorphism plus Weyl symmetries, the result is precisely a CFT. There are many other topics that leverage the conformal symmetry of the worldsheet theory, including how we describe string-on-string interactions and how we compute scattering amplitudes. But perhaps one of the ultimate motivational factors is that, as an essential tool in perturbative string theory, CFTs enable the study of the quantum field theory of the worldsheet. There is also the added benefit that many CFTs are completely solvable.

2. Conformal Group in d-dimensions

Before we proceed with a study of conformal field theories (beginning with Chapter 2 in Polchinski), it is useful to first think generally about the conformal group and its algebra.

Formally, a CFT is a quantum field theory that is invariant under the conformal group. To give some geometric intuition, the conformal group may be described as follows: it is the set of transformations that preserve local angles but not necessarily distances. This may also be thought of as invariance under scaling, with a conformal mapping being quite simply a biholomorphic mapping.

We may give further intuition about the conformal group by revisiting a more familiar symmetry group. Recall in previous chapters a discussion about the Poincaré group. One will remember that transformations under the D-dimensional Poincaré group combine translations and Lorentz transformations. These may be thought of as symmetries of flat spacetime, such that the flat metric is left invariant.

The conformal group includes the Poincaré group, with the addition of extra spacetime symmetries. It has already been alluded, for example, that a type of conformal transformation is a scale transformation, in which we may act by zooming in and out of some region of spacetime. This extra spacetime symmetry is an act of rescaling.

More precisely, the conformal group may be thought of as the subgroup of the group of general coordinate transformations (or diffeomorphisms). Consider the following. If one has a metric ${g_{ab}(x)}$ (which is a 2-tensor) in d-dimensional spacetime, it follows that under the change of coordinates ${x \rightarrow x^{\prime}}$, we have a transformation of the general form

$\displaystyle g_{\mu \nu}(x) \rightarrow g^{\prime}_{\mu \nu}(x^{\prime}) = \frac{\partial x^{a}}{\partial x^{\prime \mu}}\frac{\partial x^{b}}{\partial x^{\prime \nu}} \ g_{ab} \ \ (1)$

Now, let us consider some function ${\Omega(x)}$ of the spacetime coordinates. If a conformal transformation is a change of coordinates such that the metric changes by an overall factor, then we may consider how the metric transforms as

$\displaystyle g_{\mu \nu}(\sigma) \rightarrow g^{\prime}_{\mu \nu}(\sigma^{\prime}) = \Omega (\sigma)g_{ab}(\sigma) \ \ (2)$

For some scaling factor ${\Omega(x)}$. This is a conformal transformation of the metric. Hence why there is preservation of angles but not lengths. As this particular subgroup of coordinate transformations preserve angles while distorting lengths, in studying how to construct conformally invariant theories we will learn that conformal systems do not possess definitions of scale with respect to intrinsic length, mass or energy. For these reasons one might say the working physics is somewhat constrained or confined, such that there is no induction of a reference scale in the purest sense of the word. This is also why, in our case, CFTs prove interesting: they lend themselves quite naturally to the study of massless excitations.

Now, in thinking again of the conformal transformation described in (4.2), another important and directly related point concerns a description of the metric. It is common in the literature that the background is flat. It also turns out – and this will become more apparent later on – the background metric can either be fixed or dynamical (Tong, p.61). In the future, as we work in the Polyakov formalism, the metric is dynamical and, in this case, the transformation is a diffeomorphism – not just a gauge symmetry, but a residual gauge symmetry which, we will learn, can be undone by a Weyl transformation. But before that, in simpler examples, the background metric will be fixed and so the transformation will be representative of a global symmetry. In this case of a fixed metric, the transformation should be thought of as a genuine physical symmetry, and this global symmetry contains corresponding conserved currents. The corresponding charges for these currents are the Virasoro generators, which is something we will study later on.

References

Katrin Becker, Melanie Becker, John H. Schwarz. (2006). “String Theory and M-Theory: A Modern Introduction”.

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

David Tong. (2009). “String Theory” [lecture notes].

Kevin Wray. (2009). “An Introduction to String Theory” [lecture notes].

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Physics Diary

# Papers: Holography and the cosmological constant problem, plus a new publication from Zwiebach

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Non-perturbative de Sitter vacua via $alpha^{prime}$ corrections – Barton Zwiebach

Barton Zwiebach has a new paper out. It’s rather lovely.

The paper was written for, and submitted to, the Gravity Research Foundation 2019 Awards for Essays on Gravitation. Zwiebach’s contribution was awarded second place. However, given the content and general parameters of the first place paper, I believe Zwiebach’s contribution could have easily been given the top prize. (Granted, I have my stringy biases).

As for the paper itself, the premise is both subtle and interesting. The stage is set in the context of two-derivative supergravity theories, in which a large – indeed, infinite – number of higher-derivative corrections correlate with the parameter $alpha^{prime}$. One should note that $alpha^{prime}$ is indeed the dimensionful parameter in string theory.

One of the challenges since the 1980s has been to achieve a complete description of these higher-derivatives. It turns out a rather interesting approach may be taken by way of duality covariant ‘stringy’ field variables, as opposed to directly supergravity field variables.

From a quick readthrough, the idea presented in Zwiebach’s paper is to drop all dependence on spatial coordinates, such that only time dependence remains. The time dependent ansatz enables a general analysis on cosmological, purely time-dependent backgrounds on which all the duality invariant corrections relevant to these backgrounds may be classified. Leveraging the duality group $O(d,d, R)$, an $O(d,d, R)$ invariant action is constructed of the form

[ I_{0} equiv int dt e^{-Phi}frac{1}{n}(-dot{Phi}^{2} – frac{1}{8}tr(dot{S}^{2}) ]

Then, Zwiebach includes in the two-derivative action all of the $alpha^{prime}$ corrections. Around this point, the work of Meissner is referenced. Admittedly, I will have to review Meissner’s work moving future. Meanwhile, I rewrite the general duality invariant action from the paper as it is quite nice to look at,

[ I_{0} equiv int dt e^{-Phi}frac{1}{n}(-dot{Phi}^{2} + sum_{k=1}^{infty}(alpha^{prime})^{k+1}c_{k}tr(dot{S}^{2k}) + multi-traces ]

This action, it turns out, encodes the complete $alpha^{prime}$ corrections. Moreover, these corrections are for cosmological backgrounds. The aim is thus to classify the higher-derivative interactions, and then derive general cosmological equations to order of $alpha^{prime}$. Upon deriving these equations of motions, the key point to highlight is how one can then go on to construct non-perturbative de Sitter solutions.

Perhaps it remains to be said that de Sitter vacua are a hot topic in string theory. In the end, Zwiebach finds a non-perturbative de Sitter solution – that is, non-perturbative in $alpha^{prime}$, which is very cool.

After only a few minutes with this paper, I am eager to dig a bit deeper. Needless to say, it is intriguing to think more about the string landscape in the context of its study.

This 2011 paper by Kyriakos Papadodimas is on the cosmological constant problem in the context of the dual conformal field theory. The preprint version linked is several years old, and so I should preface what follows by stating that I have not yet had a chance to review all of the papers that directly proceed it. Indeed, this paper ends with the allusion of a follow-up, and I look forward to reading it.

Another comment for the sake of historical context, this paper on the CC problem precedes by almost two-years the publication of two important papers by Papadodimas and Suvrat Raju, the focus of which is on the black hole interior and AdS/CFT.

Now, as for the paper linked above, one of the key questions entertained concerns whether holography theory can teach us anything new about the cosmological constant problem. Of course the CC problem – or what is sometimes dramatically referred to as the vacuum catastrophe – represents one of the big questions for 21st century physics. It is one which Prof. Padilla and I sometimes talk about.

In approaching Papadodimas’ paper, the following question may be offered: is it possible that Nature’s mechanism for apparent fine-tuning may reside in a fundamental theory of emergent fields?

In the first section or two, Papadodimas builds an analogue of c.c. fine-tuning in terms of 4-point functions. We come to understand that, in the CFT, 4-point functions can more easily be translated into correlation functions. Using Witt diagrams, which are like Feynman diagrams, one takes the external points to infinity – or, in this case, to the conformal boundary of AdS. (The basics of this approach will most certainly be studied in my series of blog posts on string theory, beginning with the collection of notes on Conformal Field Theory). The advantage, moreover, is that the 4-point functions become Witt diagrams, and, then, through the AdS/CFT correspondence, one can relate these to the correlators on the boundary.

So this is the plan, which, Papadodimas argues, will enable the study of fine-tuning in the bulk in terms of fine-tuning within the dual CFT.

In proceeding his study, Papadodimas gives quite a few comments as to how we might think of fine-tuning as it manifests in the dual CFT. The picture here is really quite interesting; because if, as noted above, we can express fine-tuning in terms of graviton correlation functions, what I have so far failed to mention is that these correlation functions of gravitons are duel correlation functions of the stress-energy tensor on the boundary.

The greater goal – or ideal outcome – would be to study fine-tuning on the boundary, particularly how it is resolved. What we come to learn is how the large N expansion, the concept of which one may be familiar from their study of QFT, plays an important role. Moreover, one of the interesting suggestions is how the $frac{1}{N}$ suppression of correlators, which relates to the large N gauge theories in the t’Hooft limit, gives us early hints of fine-tuning.

Now, one of the questions I had early on pertained to the comment that it would not be difficult to split up a correlator into a sum of terms, and then expose this broken correlator to some interpretation of fine-tuning. That is to say, nothing is stopping us from breaking up the correlator into a sum of terms and then pointing at this object and saying, ‘see we have fine-tuning!’. But Papadodimas is a smart guy, and he already knows this.

To be clear, and to summarise what I am talking about, part of the strategy explained by the author early on includes expanding “the correlators into sums of terms, each of which corresponds to the exchange of certain gauge invariant operators in intermediate channels i.e. into a sum of conformal blocks. While the sum of all these contributions is suppressed by the expected power of 1/N, individual terms in the sum can be parametrically larger, as long as they cancel among themselves” (p.2). These cancelations are what we’re looking for, because, the interpretation of this paper is that they are in a way an expression of bulk fine-tuning in the dual CFT.

Papadodimas quickly extinguished my one concern, however; because we are not arbitrarily breaking up the correlator. He recognised that one may perform such an act artificially, which wouldn’t have much meaning. Instead, he searches for and performs an canonical procedure. I found this very interesting to follow. What we end up with is an expansion in conformal blocks that looks like this,

[ langle tilde{T}(x_{1})tilde{T}(x_{2})tilde{T}(x_{3})tilde{T}(x_{4})rangle_{con} = sum_{A} mid C_{TT}^{A}mid^{2}G_{A}(x_{1}, x_{2}, x_{3}, x_{4}) ]

Then, following a brief review of holography, some comments on hierarchy vs. fine-tuning, and also a study of important cutoffs, the idea of a sort of dual picture emerges, in which the physical description may appear natural in one picture – such as in the $frac{1}{c}$ expansions discussed in the paper – and then finely tuned in the second picture.

There is too much to summarise in such a small space. So I will just pick up on how, in the dual CFT, I found it intriguing that, should the arguments hold, the double operator product expansion of the correlator written above displays partial sums over the conformal blocks that cancel. As alluded early, in the context of the particular CFT, the interpretation is that the conformal blocks appear fine-tuned. But the mechanism behind this exhibited fine-tuning is not fine-tuning as I read it, such that it becomes clear that in large N gauge theories this expansion of the conformal block is natural. The naturalness of the expansion is consistent with what we know of the large $frac{1}{N}$ expansion in the t’Hooft limit.

Papadodimas closes with the explanation that in the context of a CFT with a large central charge, there is a notable c.c. problem in the holographic dual. As I focused on, it is found in this CFT a semblance of fine-tuning in the correlator blocks of the $frac{1}{N}$ expansion. However this may be interpreted, the interesting idea is that: “this fine-tuning may be visible when the correlators are expressed in terms of the exchange of conformal primaries, it may disappear if the correlators are written in terms of the fundamental fields of the underlying QFT” (p.32).

If this is in any way true, it raises some very intriguing questions and raises some very interesting potential pathways for future study, the likes of which Papadodimas spells out in explicit terms. The suggestion of the use of a toy theory, as opposed to a non-perturbative study, makes me think of the SYK model. It will be curious to follow-up on the papers that directly follow.

One last closing comment: I am excited to say that Papadodimas will be one of the lecturers at a special engagement I will be attending this summer at the University of Madrid. The engagement, SIFTS 2019, comprises of two weeks of discussion, review, study and brainstorming on issues in string theory (and fundamental physics broadly). Papadodimas will be lecturing on the black hole interior and AdS/CFT. It shall be brilliant.

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