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]].

Standard
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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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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Stringy Things

Notes on string theory: The closed string spectrum

[latexpage]

Following light cone quantisation of the bosonic string, in the last post we briefly reviewed the free open string spectrum. Now let’s take a look at the closed string spectrum.

***

For the closed string spectrum, let’s review from the start. The method of light cone quantisation is essentially the same. We impose the same gauge conditions for the closed-string as we did for the open string (p.25). The one nuance here, as Polchinski highlights, is that in the open string case the gauge was determined completely by the conditions imposed. This is not true for the closed string, as there is extra coordinate freedom. We may state this as,

[ sigma^{prime} = sigma + s(tau)mod l ]

Further review should be sought on p.25. Moving forward, one of the main differences between the open string case and the closed string case to be highlighted here concerns how we’re now working with the periodicity condition: $sigma sim sigma + 2pi$.

As the closed string forms, in a sense, a closed loop, we must also take into account our left and right-movers. Consider, for instance,

[X^{mu}(tau, sigma) = X_{L}^{mu}(tau, sigma) + X_{R}^{mu}(tau – sigma) ]

Where $u = tau + sigma$ and $v = tau – sigma$. We may also use the following shorthand notation,

[ X^{mu}(tau, sigma) = X_{L}^{mu}(u) + X_{R}^{nu}(v) ]

With the periodicity condition noted, the point remains to be stated explicitly that the left and right-movers are equivalent, which is to also say

[X^{mu}(tau, sigma + 2pi) = X^{mu}(tau, sigma) ]

This makes complete sense if you consider,

[X^{mu}(tau, sigma + 2pi) = X_{L}(u + 2pi) + X_{R}(v – 2pi) = X_{L}(u) + X_{L}(v) ]

[implies X_{L}(u + 2pi) – X_{L}(u) = X_{R}(v) – X_{R}(v – 2pi) (1) ]

Moreover, a key lesson or insight here is that the closed string is periodic. What is interesting, and what perhaps may have already ‘jumped out’ to the engaged reader, this statement about periodicity also implies that failure of the left-side to be periodic is equal to the right-side failing to be periodic. So, curiously, both sides are in a sense sharing information.

Now, before proceeding, we need to look at what it means to take the derivative of both $u$ and $v$:

[ frac{partial}{partial u} : X_{L}^{mu prime}(u + 2pi) – X_{L}^{mu prime}(u) = 0 ]

[ frac{partial}{partial v} : X_{R}^{mu prime}(v) – X_{R}^{mu prime}(v – 2pi) = 0 ]

It follows that for our primed left and right-movers, we have:

[ X_{L}^{mu prime}(u) = sqrt{frac{alpha^{prime}}{2}}sum_{n in Z} tilde{alpha}_{n}^{mu}e^{-inu} ]

[ X_{R}^{mu prime}(v) = sqrt{frac{alpha^{prime}}{2}}sum_{n in Z} alpha_{n}^{mu}e^{inv} ]

Notice that what we have is representative of a set of independent oscillators. Indeed, and moreover, notice that in the above we have essentially two copies of a product of an open string. When we integrate, we obtain what follows

[ X_{L}^{mu}(u) = x_{0}^{mu} + sqrt{frac{alpha^{prime}}{2}}tilde{alpha}_{0}^{mu}u + isqrt{frac{alpha^{prime}}{2}}sum_{n neq 0} frac{1}{n}tilde{alpha}_{n}^{mu}e^{-inu} ]

[ X_{R}^{mu}(v) = x_{0}^{mu} + sqrt{frac{alpha^{prime}}{2}}alpha_{0}^{mu}v + isqrt{frac{alpha^{prime}}{2}}sum_{n neq 0} frac{1}{n}alpha_{n}^{mu}e^{-inv} ]

This should look familiar, where we have our spacetime coordinate, momentum and on the far right-hand side the excitations. From $(1)$ also notice,

[ sqrt{frac{alpha^{prime}}{2}}tilde{alpha}_{0}^{mu}(2pi) = sqrt{frac{alpha^{prime}}{2}}alpha_{0}^{mu}(2pi) ]

[rightarrow tilde{alpha}_{0}^{mu} = alpha_{0}^{mu} ]

Which is to say that the periodicity condition that describes the waves moving left and right along the string, it basically constrains the system in such a way to show that the momenta on both sides are equal. Or, more accurately, there is only one momentum operator for the closed quantum string. This is rather neat.

Regarding the matter of coordinates, for the full string closed string coordinate we may write,

[X^{mu}(tau, sigma) = frac{1}{2}(x_{0}^{L mu} + x_{0}^{Rmu}) + sqrt{2alpha^{prime}}alpha_{0}^{mu}tau + isqrt{frac{alpha^{prime}}{2}} sum_{n neq 0}^{infty} frac{e^{-intau}}{n}(alpha_{n}^{mu}e^{insigma} + tilde{alpha}^{mu}_{n}e^{-insigma}) ]

Recall, also, that the momentum density of the string is always,

[P^{tau mu} = frac{1}{2pi alpha^{prime}}frac{partial X^{mu}}{partial tau} ]

And so,

[P^{mu} = int_{0}^{pi} P^{tau mu} dsigma = sqrt{frac{2}{alpha^{prime}}}alpha_{0}^{mu} ]

Where the $alpha$ term can be seen to be proportional to the spacetime momentum of the string.

As a next step in working our way toward the closed string spectrum, it may be helpful to record the $tau$ and $sigma$ derivatives of the coordinate, recalling from before $X^{mu} = X_{L}^{mu}(tau + sigma) + X_{R}^{mu}(tau – sigma)$. And so,

[dot{X}^{mu} = X_{L}^{prime mu}(tau + sigma) + X_{R}^{prime mu}(tau – sigma) ]

[X^{prime mu} = X_{L}^{prime mu}(tau + sigma) – X_{R}^{prime mu}(tau – sigma) ]

When we add and subtract these we find “that the barred oscillators do not mix with the unbarred oscillators in these combinations of derivatives” (Zwiebach, p.284). Ultimately, as we have something analogous to the open string expansion, it follows that for the commutators of the closed string we don’t have to perform any new calculations (Polchinski, pp. 26-27).

As review, recall that we have the commutators,

[ [X^{I}(tau, sigma), P^{IJ}(tau, sigma^{prime}] = idelta(sigma – sigma^{prime})eta^{IJ} ]

[ [alpha_{m}^{I}, alpha_{n}^{J}] = [tilde{alpha}_{m}^{I}, tilde{alpha}_{n}^{J}] = mdelta_{m + n, 0}delta^{IJ} ]

[ [alpha_{m}^{I}, tilde{alpha}_{n}^{J}] = 0 ]

Following this, we may now turn our attention to the Virasoro operators for the closed string. For the open string, we have come to understand that for the LC coordinate $X^{-}$ the transverse Virasoro operators are the modes $alpha_{n}^{-}$. But, in the closed string case, we have both barred and unbarred modes (I have been using tildes to highlight this fact) using the $X^{-}$ coordinates.

How do the $X^{-}$ coordinates relate the transverse Virasoro operators in the closed string case? From the study of the LC equations of motion we have,

[ dot{X}^{-} pm X^{prime -} = frac{1}{beta alpha^{prime}}frac{1}{2p^{+}}(dot{X}^{I} pm X^{prime I})^{2} ]

Except, in our present case we set $beta = 1$. In short, we find that as $X^{-}(tau, sigma) = X_{0}^{-}, P^{-}$ it follows that $P^{-} rightarrow alpha_{0}^{-}$ or $tilde{alpha}_{0}^{-}$ in which it also follows that we have $L_{0}^{perp}$ or $tilde{L}_{0}^{perp}$. And so, in full,

[ L_{0}^{perp} = frac{alpha^{prime}}{4}p^{I}p^{I} + sum_{p=1}^{infty} alpha_{-p}^{I}alpha_{p}^{I} = frac{alpha^{prime}}{4}p^{I}p^{I} + N^{perp} ]

[tilde{L}_{0}^{perp} = frac{alpha^{prime}}{4}p^{I}p^{I} + sum_{p=1}^{infty} tilde{alpha}_{-p}^{I}alpha_{p}^{I} = frac{alpha^{prime}}{4}p^{I}p^{I} + tilde{N}^{perp} ]

Now, an interesting question about the above expressions concerns how there does not seem to be a way to set them equal. What is going on? Well, one way to handle this is to enforce the following constraint (though still quite complicated):

[ (L_{0}^{perp} – tilde{L}_{0}^{perp}) |psi > = 0 ]

[ (N_{perp} – tilde{N}_{perp}) | psi > = 0 ]

For the first, we have the closed string state. For the second, this kills the states.

After some computation and lengthy consideration (Zwiebach, pp. 287-288), particularly concerning the constant ambiguities due to the ordering of our operators, we arrive at the following equations for $n = 0$,

[sqrt{2alpha^{prime}}bar{alpha}_{0}^{-} = frac{2}{p^{+}}(tilde{L}_{0}^{perp} – 1), sqrt{2alpha^{prime}}alpha_{0}^{-} = frac{2}{p^{+}}(L_{0}^{perp} – 1) ]

Which, in terms of the closed string Hamiltonian we may write compactly as,

[ H = L_{0}^{perp} + tilde{L}_{0}^{perp} – 2 ]

We can also calculate the mass squared,

[M^{2} = frac{2}{alpha^{prime}} (N^{perp} + tilde{N}^{perp} – 2) ]

At this point we should note that, if not already clear, $L_{0}^{perp} + tilde{L}_{0}^{perp}$ generate $tau$ translations. So, it can also be said $L_{0}^{perp} + tilde{L}_{0}^{perp} sim H$ generates $tau$ evolution. For sigma translations we have, $[L_{0}^{perp} – tilde{L}_{0}^{perp}, X^{I}(tau, sigma)] = ifrac{partial X^{I}}{partial sigma}$.

Moreover, in producing $sigma$ translations the states are also killed. That is, the statement $L_{0}^{perp} – tilde{L}_{0}^{perp}$ generates $sigma$ translations but kills all states, what this says in other words is that all physical states are invariant under $sigma$ translations. Thus, too, we are led to the fact that no one has been able to figure out how to select a special point in a closed string (at least not that I know of, and was also highlighted in a past lecture series by Zwiebach). That is, at what point might one determine the position of $sigma = 0$ for the closed string? It’s quite interesting to think about and explore.

With all of that out of the way, we can now focus on the state space for the closed string. For the ground states we have $|p^{+},p_{tau}>$ in which,

[ | lambda, tilde{lambda}> equiv [prod_{n=1}^{infty} prod_{I=2}^{D-1}(alpha_{n}^{I})^{dagger lambda n, I}][prod_{m=1}^{infty} prod_{J=2}^{D-1}(tilde{alpha}_{m}^{J})^{dagger tilde{lambda} n, J}] | p^{+},p_{tau}> ]

We may also note that,

[N^{perp} = sum_{n, I} nlambda_{n,I} ]

[tilde{N}^{perp} = sum_{n, I}ntilde{lambda}_{n,I} ]

And so,

[ N^{perp} = tilde{N}^{perp} ]

Now, for $N^{perp} = tilde{N}^{perp} = 0$ and $M^{2} = -frac{4}{alpha^{prime}}$, we have a scalar field – a tachyon.

A few words were already shared about the tachyon in the study of the open string. It is a subject that deserves an entire paper in itself. But, for the sake of maintaining focus, let’s consider the case where $N^{perp} = tilde{N}^{perp} = 1$ and $M^{2} = 0$. We also have $R_{IJ} = S_{IJ} + A_{IJ} = tilde{S}_{IJ} + S^{prime}delta_{IJ} + A_{IJ}$, in which $tilde{S}_{IJ}$ is symmetric traceless. It follows, after computation,

[R_{IJ} (alpha_{1}^{I})^{dagger} – (tilde{alpha}_{1}^{J})^{dagger} | p^{+},p_{tau}> ]

[tilde{S}_{IJ} (alpha_{1}^{I})^{dagger} – (tilde{alpha}_{1}^{J})^{dagger} |p^{+},p_{tau}> graviton states ]

[A_{IJ} (alpha_{1}^{I})^{dagger} – (tilde{alpha}_{1}^{J})^{dagger} | p^{+},p_{tau}> Kalb-Ramond states ]

[S^{prime}_{IJ} (alpha_{1}^{I})^{dagger} – (tilde{alpha}_{1}^{J})^{dagger} | p^{+},p_{tau}> dilaton scalar ]

Note: the KR field has to do with charge in ST. The dilaton is an important field that we will come into contact with again in the future, as it controls the string coupling.

References

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

Kevin Wray. (2009). “An Introduction to String Theory”.

Barton Zwiebach. (2009). “A First Course in String Theory”, 2nd ed.

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Stringy Things

Notes on string theory: Internal excitations of a free open bosonic string

[latexpage]

After reviewing some of the particulars that comprise Chapter 1 in Polchinski’s book (I have mostly kept to key concepts in this blog series, with a more thorough review to be found in my lecture notes), we now arrive at the culmination of the first stage in our study.

We will first study the free open string spectrum. Then, in a following post, we will review the closed string spectrum. This will also allow us the opportunity to explore some basic closed string physics. After these two entries, we will then switch focus and engage in a thorough study of Conformal Field Theory, as directed by Polchinski in Chapter 2.

***

So far a lot of what we have been constructing, especially in relation to the quantum string, is premised on the idea of wanting to obtain quick and direct access to the study of the physical states of the bosonic string. One may recall that this was the motivation to invoke light cone quantisation in the first place. Ultimately, we will want to work toward path integral quantisation of the string; but for now, let’s see what we find.

Recall the mass-shell condition from previous entries. For the open string spectrum, the mass shell condition becomes

[ M^{2} = frac{1}{alpha^{prime}} (N^{perp} – 1) ]

Where $N^{perp}$ is the number operator and $N^{perp} = sum_{k=1}^{infty} k (alpha_{k}^{I})^{dagger}alpha_{k}^{I}$. The commutator can be written as $[N^{perp}, (alpha_{k}^{I})^{dagger}] = k(alpha_{k}^{I})^{dagger}$.

One will recall the mass operator $M^{2}$ from previous discussions. The calculation to get to this point is straightforward, and can be reviewed in many textbooks.

For the state space of the theory, we should take note of: $x_{0}^{I}, p^{I}, p^{+}, alpha_{n}^{I}, (alpha_{n}^{I})^{dagger}$. $P_{T}$ is the collection of all $p^{I}$’s.

We introduce a vacuum – that is, a ground state of the string. This is like the ground state of all harmonic oscillators of every single mode. For each spatial direction there are infinitely many modes.

$| p^{+}, p_{T}>$ is killed by all $alpha_{n}^{I}$

And so the important thing is that the oscillators act on this state. Hence, as written a few sections earlier, a general state can be built by acting on the vacuum,

[ | N, k> = prod_{i=2}^{D-1} prod_{n=1}^{infty} frac{(a_{-n}^{i})^{N_{in}}}{(n^{N_{in}N_{in}!})^{frac{1}{2}}} | 0, k> ]

So, for our catalogue of states, we can now explicitly compute:

For $N^{perp} = 0$,

[ M^{2} |p^{+}, p_{T}> = frac{1}{alpha^{prime}}(-1 + N^{perp}) |p^{+}, p_{T}> ]

[= -frac{1}{alpha^{prime}} |p^{+}, p_{T}> ]

Notice something important: The ground state in free open bosonic string theory has negative mass squared. This gives us our first encounter with the tachyon.

The tachyon is often described as physically unrealistic, as, in addition to it having negative mass squared, it also travels faster than the speed of light.

For a long time, it was seen as a pathology of bosonic string theory. But fear of the tachyon has diminished over time. Luboš Motl offers a terrific summary of the history behind the study of the tachyon, including some of the sociology as well as a wonderful overview of Ashoke Sen’s timely interventions. A discussion of Sen’s contributions would extend beyond the limits of this particular article; but a single comment would point out that Sen developed the conjecture about the tachyon potential, as he pushed forward the study of brane/anti-brane physics. From his study it emerged that the instability of the tachyon corresponds to the instability of the D-brane to which the ends of the open string are attached. As there is a D-brane present in the physics of the string, the decay or disintegration of the D-brane relates to the potential – that is, the local minimum – of the tachyon.

I will save a deeper and more detailed discussion about this for a separate entry. Meanwhile, in that one will frequently read a description of the tachyon as being unphysical, a more pedagogically focused introduction would emphasise that as the tachyon is highly unstable, this relates to one of the issues with bosonic string theory: namely, the vacuum is unstable. In general, the situation we are left with is that for the consistency condition of bosonic string theory to be satisfied, which requires $a = 1$, this means the tachyon cannot be simply removed from the theory. That said, and as alluded, removal of the open string tachyon has been studied as it relates to the decay of D-branes into closed string radiation. But the closed string tachyon, which we will meet soon, has not yet been resolved.

Of course, all of this leaves one to say that one of the highly publicised benefits of superstring theory is that, by way of the GSO projection, we are able to construct a consistent theory that does not include tachyons at all. This will be a topic for a later time.

In general, before we get to the other masses, it should be pointed out that the tachyon is interesting for a number of reasons. The prospect of a non-perturbative study raises many intriguing questions. There is also some suspicion and some speculation in the literature, particularly in the case of the closed string case, that the instability of the tachyon – which we have already noted correlates with the instability of the D-brane – could therefore relate to the instability of spacetime itself (i.e., spacetime could disintegrate). That is, a tachyon fluctuation creates the spacetime and even annihilates it, something Zwiebach commented on in his 2012/13 lecture series.

Another interesting point is that, at the time of writing, it does not seem that anyone has been able to calculate the tachyon potential. In general, the study of the tachyon is intriguing and worth exploring, but deeper discussion shall be reserved for another time.

Now, in the case for $N^{perp} = 1 implies M^{2} = 0$, the mass for the first excited state can be found as

[(alpha_{1}^{I})^{dagger} |p^{+},p_{T}> ]

This is the Maxwell field, or one photon states. Again, we have a massless state with $D − 2$ components, with transverse spin.

As expected, we can continue up the scale of states. In the case of $N^{perp} = 2$, there are a total of 324 states. But they are all one particle states.

In the next post, we will study the closed string spectrum. Here we will go over some basic closed string physics, and, in the study of the spectrum, we will also arrive at our first encounter with the graviton and the dilaton.

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.

Kevin Wray. (2009). “An Introduction to String Theory”.

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