Stringy Things

# Notes on String Theory – Further Introduction to Operator Product Expansions

1. Generalising the Formula for OPEs

In the last post we continued a review of Chapter 2 in Polchinski, focusing on building understanding of conformal field theories from the perspective of local operator insertions. We finally also arrived at the basic formula for operator product expansions (OPEs). What follows in this post is a continuation of that discussion. That is to say, the following review will also necessarily reference equations in the previous entry. To avoid confusion, equation numbers from the last post will be explicitly stated.

Recall that, in an introduction to the basic formula for OPEs, it was mentioned that because it is an operator statement this means it holds inside a general expectation value. It follows that the operator equation of the form that we considered can have additional operator insertions. This implies that we may write the formula for OPEs in a more general way,

$\displaystyle \langle \mathcal{O}_{i}(z, \bar{z})\mathcal{O}_{j}(z^{\prime}, \bar{z}^{\prime}) ... \rangle = \sum_{k} C_{ij}^{k}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \langle \mathcal{O}_{k}(z^{\prime}, \bar{z}^{\prime}) ... \rangle \ \ (1)$

Where ‘…’ again denotes additional insertions and is often left implicit. One can also work out quite simply the equivalent description in the path integral formalism for ${n-1}$ fields.

1.1. OPEs – Generalise for an Infinite Set of Operators

There are a number of other caveats and subtleties about OPEs that we have not yet explored. It will be our aim to do so in this section by reviewing the remaining contents of section 2.2 in Polchinski’s textbook, before progressing toward more advanced topics that will then aid in our understanding of stringy CFTs and the procedure for how to compute OPEs.

Moreover, at this point in Polchinski’s introduction to OPEs, a number of results and definitions are given which may not make complete sense until later. This is because there are a number of key interrelated concepts that have not yet been formally introduced, such as radial ordering, Wick’s theorem, conformal invariance, and the necessary mode expansions that we must consider. These are important conceptual tools in establishing a wider understanding of CFTs and how we may think of OPEs in string theory. So what follows in this section may be considered more in the way of definition, introducing some ideas that relate to OPEs as we work toward more advanced topics that will clarify and enrich some of these ideas.

For instance, let us recall that in the last entry we discussed a normal ordered product that was defined in such a way that it satisfies the naive equation of motion [equation (17) from previous post]. What it is telling us is how the operator product is a harmonic function of ${(z_{1}, \bar{z}_{1})}$. This statement already offers a hint of what is to come both in this section and other future parts of our study on CFTs, particularly when we more explicitly discuss Wick’s theorem and mode expansions in relation to computing OPEs. For now, we may maintain an introductory tone and say that this statement leads us to an important insight early in Polchinski’s discussion in Section 2.2 of his textbook: notably that from the theory of complex variables a harmonic function may be decomposed locally as the sum of holomorphic and antiholomorphic functions. To begin to explain what this means, and to explain Polchinski’s discussion on pp.37-38 let us consider more deeply (17) from the last post. We can think of it this way,

$\displaystyle \bar{\partial}_{1} [\partial_{1} :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):] = 0$

$\displaystyle \bar{\partial}_{1} [:\partial_{1} X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):] = 0 \ \ (2)$

The point of (2) is to show that we now have a holomorphic derivative inside the normal ordering. But notice also that this holomorphic derivative will get annihilated by the antiholomorphic derivative acting on it. In other words, by the equation of motion mixed ${\partial \bar{\partial}}$ derivatives vanish. This is telling us something we may perhaps already know or suspect, namely as we continue to think in terms of operators ${:\partial_{1} X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):}$ is in fact a holomorphic function. Now, as Polchinski explains, from the theory of complex analysis it is within the rules that we can Taylor expand such holomorphic (and antiholomorphic) functions. This use of Taylor expansion may be considered one of the first tools in understanding how to compute OPEs. Consider, for example, only the holormorphic case. When we proceed with Taylor expansion in ${z_{12}}$ it is implied that we have nonsingularity as ${z_{1} \rightarrow z_{2}}$ and we obtain the following infinite series,

$\displaystyle :\partial_{1 \xi} X^{\mu}(z_{1} + \xi, \bar{z}_{1} + \xi)X^{\nu}(z_{2}, \bar{z}_{2}): = \sum_{k=1}^{\infty} \frac{\xi^{k}}{k!} :X^{\nu} \partial^{k}X^{\mu}: \ \ (3)$

Where ${\xi = z_{12}}$. We can rewrite (3) as follows, including also the antiholomorphic series,

$\displaystyle = \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}(z_{2}, \bar{z}_{2}): + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\bar{\partial}^{k}X^{\mu}(z_{2}, \bar{z}_{2}):] \ \ (4)$

Which is now written only as a function of ${z_{2}}$. What this is telling us is that if we have some normal ordered product, we may write more generally for this product,

$\displaystyle :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):$

$\displaystyle = :X^{\mu}(z_{2}, \bar{z}_{2})X^{\nu}(z_{2}, \bar{z}_{2}): + \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}: + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\bar{\partial}^{k}X^{\mu}:] \ \ (5)$

This is exactly the result that Polchinski describes in equation (2.2.4), with the exception that we have simplified the equation by dropping the ${\alpha^{\prime}}$ term. Keeping the ${\alpha^{\prime}}$ term explicit we arrive precisely at Polchinski’s equation,

$\displaystyle :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):$

$\displaystyle = - \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2} + :X^{\mu}(z_{2}, \bar{z}_{2})X^{\nu}(z_{2}, \bar{z}_{2}) + \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}: + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}:] \ \ (6)$

In which ${- \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2}}$ is the regular part of the OPE that one may remember from the two-point function ${\langle X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime}) \rangle}$. Again, this is something we will become more familiar with as we progress. Furthermore, notice in general that (6) looks very much like an OPE as given in (1). In fact, it will become increasingly clear, especially toward the end of our present study, that we may think of this as the free field OPE hence the inclusion of the regular piece. Later, we will show explicitly the computation to achieve this result. In the meantime, since it is simply given in Polchinski’s textbook, it has also been stated here with addition of a few more comments as follows.

Note that like its equation of motion, (6) is an operator statement. Secondly, as previously alluded, OPEs in quantum field theory are very much like the analogue of Taylor expansions in calculus. When Taylor expanding some general function ${\mathcal{G}(z_{1}, \bar{z}_{1}; z_{2}, \bar{z}_{2}) = :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):}$ as above, note that one will obtain terms of the form ${\partial^{k}:X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):}$ in which the derivative is outside the normal ordering as opposed to inside the normal ordering. But differentiation and normal ordering commute, which can be proven using some basic identities of functional derivatives, hence the structure of the normal ordering in the OPE (6). Also, for any arbitrary expectation value that involves some product ${X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2})}$ multiplied by a number of fields at other points, we have been building (and will continue to build) the intuition to understand exactly why the OPE describes the behaviour for when ${z_{1} \rightarrow z_{2}}$ as an infinite series. In the case of (6), as we deepen our study of CFTs we will come to understand more clearly why it has a radius of convergence in any given expectation value which is equal to the distance to the nearest other insertion in the path integral’ and why The operator product is harmonic except at the positions of operators’ (p.38).

Although how we arrive at (6) may not yet make complete sense, the key idea at this point in Polchinski’s discussion is simply that we have a product of two operators and we have described this product as an infinite sum of some coefficients ${C_{k}}$ of some basis operators ${A_{k}}$. As asymptotic expansions, we will come to write OPEs up to nonsingular terms.

1.2. Subtractions and Cross-contractions

To conclude a review of Section 2.2 in Polchinski, let us consider another example where we have an arbitrary number of fields. As we discussed earlier, the sum then runs over all of the different ways we might choose pairs of fields from the product. We then replace each pair with the expectation value as mentioned in the description of the definition (16) in the last post – i.e., what we have also termed to be the regular part of the OPE. So, if for instance we have three fields, the computation generally takes the following form,

$\displaystyle :X^{\mu_{1}}(z_{1}, \bar{z}_{1})X^{\mu_{2}}(z_{2}, \bar{z}_{2})X^{\mu_{3}}(z_{3}, \bar{z}_{3}):$

$\displaystyle =X^{\mu_{1}}(z_{1}, \bar{z}_{1})X^{\mu_{2}}(z_{2}, \bar{z}_{2})X^{\mu_{3}}(z_{3}, \bar{z}_{3}) + (\frac{\alpha^{\prime}}{2} \eta^{\mu_{1} \mu_{2}} \ln \mid z_{12} \mid^{2} X^{\mu_{3}}(z_{3}, \bar{z}_{3}) + 2 \ \text{permutations}) \ \ (7)$

Now, consider again (16) from the previous entry. It can now be seen how we may write this definition in a more compact and general way. Consider, for instance, the arbitrary functional ${\mathcal{F} = \mathcal{F}[\partial X^{\mu_{1} ... \mu_{n}}]}$. The terms in brackets represent a combination of an arbitrary number of fields. If, as before, we Taylor expand and make this expression an expansion of polynomials of ${X}$, it follows that we may then write the normal ordering for each monomial. This leads directly to the equation (2.2.7) in Polchinski,

$\displaystyle :\mathcal{F}: = \exp (\frac{\alpha^{\prime}}{4} \int d^{2}z_{1}d^{2}z_{2} \ln \mid z_{12} \mid^{2} \frac{\delta}{\delta X^{\mu}(z_{1}, \bar{z}_{1})} \frac{\delta}{\delta X_{\mu}(z_{2}, \bar{z}_{2})}) \mathcal{F} \ \ (8)$

Where ${\mathcal{F}}$ is any functional of ${X}$. It can be shown that (8) is equivalent to (16) from the previous post. Again, this may not yet make complete sense. But for now notice that there is a double derivative in the exponent. This double derivative contracts each pair of fields. What this means is that, every time we compute the expansion we will effectively kill two ${X}$ terms. Instead of these ${X}$ terms, we then insert ${\ln \mid z_{12} \mid^{2}}$ which is, of course, the subtraction. Now, reversely, if we act with the inverse exponential, we obtain the opposite of a sum of subtractions in the form of a sum of contractions,

$\displaystyle \mathcal{F} = \exp (-\frac{\alpha^{\prime}}{4} \int d^{2}z_{1}d^{2}z_{2} \ln \mid z_{12} \mid^{2} \frac{\delta}{\delta X^{\mu}()z_{1}, \bar{z}_{1}} \frac{\delta}{\delta X_{\mu}(z_{2}, \bar{z}_{2})}) :\mathcal{F}:$

$\displaystyle = :\mathcal{F}: + \ \text{contractions} \ \ (9)$

As it will become increasingly clear when we compute some detailed examples, this means we are now summing over all of the ways of choosing pairs of fields from ${:\mathcal{F}:}$ instead of ${\mathcal{F}}$. We then replace each pair with the contraction ${-\frac{1}{2} \alpha^{\prime}\eta^{\mu_{i} \mu_{j}} \ln \mid z_{ij} \mid^{2}}$. It follows that for any pair of operators, we can generate the respective OPE

$\displaystyle :\mathcal{F}: :\mathcal{G}: = :\mathcal{F} \mathcal{G}: + \sum \ \text{cross-contractions} \ \ (10)$

What (10) is saying is that we are now summing over all of contracting pairs with one field in ${\mathcal{F}}$ and one field in ${\mathcal{G}}$, where, again, ${\mathcal{F}}$ and ${\mathcal{G}}$ are arbitrary functionals of ${X}$. It is this construction of the cross-contractions that enables the following formal expression,

$\displaystyle : \mathcal{F}: :\mathcal{G}: = \exp (-\frac{\alpha^{\prime}}{2} \int d^{2}z_{1}d^{2}z_{2} \ln \mid z_{12} \mid^{2} \frac{\delta}{\delta X_{F}^{\mu}(z_{1}, \bar{z}_{1})} \frac{\delta}{\delta X_{G \mu}(z_{2}, \bar{z}_{2})}) : \mathcal{F} \mathcal{G}: \ \ (11)$

In which the entire operation now acts on the normal ordering ${: \mathcal{F} \mathcal{G}:}$.

This concludes the opening discussion on OPEs in Polchinski’s textbook, from which he goes on to consider two examples of computing normal ordering (p.40) before focusing on the important study of Ward identities and Noether’s theorem. It will prove beneficial to review in the future the computation of the two examples that Polchinski offers (see the Appendix of this chapter). In the meantime, it may aid one’s understanding if we instead pause and first explore other concepts integral to stringy CFTs and their OPEs. This will enable us to introduce more notation and more deeply explicate mathematical procedure. Taking such an approach has its obvious advantages, but it also has its disadvantages. The way in which Chapter 2 is structured in Polchinski’s textbook means that, in a few instances, it will be required that we advance our study of CFTs to include a number of other key concepts before making better sense of what we have already discussed, particular in why OPEs have the structure that they do and how we may think about their computational procedure in a more exemplified way. So at this point we bracket the definitions given above to discuss other related topics, before ultimately returning specifically to the subject of OPEs and computing a number of different examples step by step.

References

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

Standard
Stringy Things

# Notes on String Theory: Conformal Field Theory – Local Operators, the String Propagator, and Operator Product Expansions

1. Local Operators

In the last entry we introduced a theory of free massless scalars in flat 2-dimensions (i.e., a free X-CFT). From this we also introduced new terms and established notation relevant to our ongoing study of CFTs in string theory (Chapter 2 in Polchinski). What we now want to do is proceed with a review of a number of interrelated topics at the heart of stringy CFTs: namely, local operators, techniques with path integrals, string propagators, and finally operator product expansions. Each of these topics has a number of parts, and so we shall need to work piece by piece and then stitch everything together.

To begin, we note that in string perturbation theory, one of the main objects of interest is the expectation value of the path integral of a product of local operators (Polchinski, p.36). This interest is our entry point, and it represents a primary theme for much of the following discussion. So our first step should be to define what we mean by local operators. These objects may also be described as fields; however, in the context of CFTs, the notion of a field carries a different meaning than, for instance, the definition of a field in quantum field theory. In our case, a field may be viewed generally as a local expression, which may be the generic field ${\phi}$ that enters the path integral in QFT, or as a composite operator ${e^{i\phi}}$ or as a derivative ${\partial^{n}\phi}$ (Tong, p.69). These are all different types of fields or local operators in the CFT dictionary.

With a definition of local operators in mind, we opened the discussion by mentioning the expectation value of the path integral as a primary object of interest. Let us now consider some general expectation value. Consider, for instance, ${\mathcal{A}_{i}}$ that is some basis for a set of local operators. We may write the general expectation value as follows,

$\displaystyle \langle \mathcal{A}_{i_{1}}(z_{1}, \bar{z}_{1}) \mathcal{A}_{i_{2}}(z_{2}, \bar{z}_{2}) ... \mathcal{A}_{i_{n}}(z_{n}, \bar{z}_{n}) \rangle \ \ (1)$

If the basic idea, as mentioned, is to compute the expectation value of the path integral, a more technical or detailed description of our overarching interest is to understand the behaviour of this expectation value (1) in the limit of two operators taken to approach one another (Polchinski, p. 36). The tool that we use for such analysis is the operator product expansion (OPE). Understanding the definition of OPEs and how to compute them is one of the ultimate aims of studying stringy CFTs, important for more advanced topics that we will consider throughout the remainder of this paper. But before formally defining OPEs, it is useful to first build a deeper sense of intuition about their meaning. To do this, let as briefly review some more basics.

2. The Path Integral and Arbitrary Operator Insertions

What do we mean by path integral? And how do we understand this idea of local operator insertions? Additionally, how do we construct the important operator equations required to build a picture of OPEs? Polchinski offers several valuable contributions to a definition of the path integral, including a lengthy treatment in the Appendix of Volume 1. For our purposes, we might first emphasise the QFT view of the path integral as an integral over fields,

$\displaystyle Z = \int [dX]e^{-S} \ \ (2)$

We may describe this as a partition function. Now, if what we want to know is the expectation value given some operator, this implies that we want to employ the path integral representation to derive operator equations. For instance, as we read in Polchinski (p.34), given some operator we may compute,

$\displaystyle \left\langle \mathcal{F}[X] \right\rangle = \int [dX]e^{-S}\mathcal{F}[X] \ \ (3)$

Where ${\mathcal{F}[X]}$ is some functional of X, typically a product of two operators, and where $\langle \mathcal{F}[X] \rangle = \langle 0 \mid \mathcal{F} \mid 0 \rangle$ . For multiple entries in the form,

$\displaystyle \mathcal{F}_{1}[X(z_{1}, \bar{z}_{1})] \mathcal{F}[X(z, \bar{z})] \mathcal{F}_{2}[X(z_{2}, \bar{z}_{2})]$

We may write,

$\displaystyle \langle 0 \mid \mathcal{F}_{1} \mathcal{F} \mathcal{F}_{2} \mid 0 \rangle =\int [dX]e^{-S} \mathcal{F}_{1}[X(z_{1}, \bar{z}_{1})] \mathcal{F}[X(z, \bar{z})] \mathcal{F}_{2}[X(z_{2}, \bar{z}_{2})] \ \ (4)$

There is a notion of time-ordering present in (4), which we will discuss later. For now, we should note that the path integral of a total derivative is always zero. This fact will prove useful in just a moment and on many other occasions in the future. As Polchinski reflects, This is true for ordinary bosonic path integrals, which can be regarded as the limit of an infinite number of ordinary integrals, as well as for more formal path integrals as with Grassmann variables’ (Polchinski, pp. 34-35). Hence eq.(2.1.15) in Polchinski (p.35), where he considers the path integral with the inclusion of Grassmann variables,

$\displaystyle 0 = \int [dX] \frac{\delta}{\delta X_{\mu}(z, \bar{z})} \exp (-S)$

$\displaystyle = - \int [dX] \exp (-S) \frac{\delta S}{\delta X_{\mu} (z, \bar{z})}$

$\displaystyle = - \int \bigg \langle \frac{\delta S}{\delta X_{\mu} (z, \bar{z})} \bigg \rangle$

$\displaystyle = \frac{1}{\pi \alpha^{\prime}} \langle \partial \bar{\partial} X^{\mu}(z, \bar{z}) \rangle \ \ (5)$

There is something interesting with this result. If we recall the action for the free X-CFT in the last post, remember that we found the classical EoM to be ${\partial \bar{\partial} X^{\mu}(z, \bar{z}) = 0}$. Notice, then, that the result (5) is the analogue statement in the quantum theory for the classical equations of motion. What is this telling us? Let us dig a bit deeper.

First, consider how the same calculation in (5) holds if we have arbitrary additional insertions …’ in the path integral. We already considered what multiple entries in the path integral in (4). But there is a caveat: namely, these additional insertions cannot also be at ${z}$ (something we will elaborate below). Second, in the case of multiple entries in the path integral, which implies that we may write something of the form ${\int [dX] \frac{\delta}{\delta X^{\mu}(z, \bar{z})}[e^{-S}\mathcal{F}(z, \bar{z})]}$, one can think of some of the insertions as preparing a state in the theory. In other words, we should note that these insertions prepare arbitrary initial and final states in the theory (Polchinski, p.35). These arbitrary initial and final states perform a similar role should we instead consider boundary conditions, except with the offered convenience that we may now write the following path integral statement,

$\displaystyle \left\langle \partial\bar{\partial}X^{\mu}(z, \bar{z}) ... \right\rangle = 0 \ \ (6)$

Now, if (5) is the analogous statement in the quantum theory for the classical equations of motion, look at (6). Notice, as an operator statement, it is the same as in the Hilbert space formalism,

$\displaystyle \partial\bar{\partial}\hat{X}(z, \bar{z}) = 0 \ \ (7)$

Polchinski describes (7) as holding for all matrix elements of the operator ${\hat{X}(z, \bar{z})}$, with all relations that hold (6) being operator equations (Polchinski, p.34). These two points are important. It should also be noted that (7) is Ehrenfest’s theorem, which makes a lot of sense because it is telling us something that we already know or suspect: namely, the expectation values of the operators obey the classical equations of motion. But, again, this proves true only when the additional insertions …’ in the path integral are located away from ${z}$. So let us now look into this subtlety. If, for example, addition insertions cannot be coincident at ${z}$, then let us consider what happens when we do indeed have coincident points at ${z}$! It follows,

$\displaystyle 0 = \int [dX] \frac{\delta}{\delta X_{\mu}(z, \bar{z})}[exp(-S)X^{\nu}(z^{\prime}, \bar{z^{\prime}})]$

$\displaystyle = \int d[X] exp(-S) [\eta^{\mu \nu}\delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) + \frac{1}{\pi\alpha^{\prime}}\partial_{z}\partial_{\bar{z}}X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime})]$

$\displaystyle = \eta^{\mu \nu} \langle \delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \rangle + \frac{1}{\pi\alpha^{\prime}}\partial_{z}\partial_{\bar{z}}\langle X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime})\rangle \ \ (8)$

Where the ${\delta^{2} (z^{\prime} - z, \bar{z}^{\prime} - \bar{z})}$ term comes from differentiating ${\frac{\delta X^{\mu}(z^{\prime}, \bar{z}^{\prime})}{\delta X^{\mu}(z, \bar{z})}}$ that appears in the computation. What we see in (8) is that at coincident points the classical equations of motion do not hold at the quantum level. This implies a few things. First, the good news is that we obtain our previous result that the EoM agrees as an operator statement of the ground state specifically under the conditions ${z \neq z^{\prime}}$. Second, the implication is clearly that with arbitrary additional insertions …’ in the path integral, so long that these are far away from ${z}$ and ${z^{\prime}}$, we may can rewrite (8) as,

$\displaystyle \frac{1}{\pi \alpha^{\prime}} \partial_{z}\partial_{\bar{z}} \langle X^{\mu}(z, \bar{z}) X^{\nu}(z^{\prime}, \bar{z}^{\prime}) \ ... \ \rangle = -\eta^{\mu \nu} \langle \delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \ ... \ \rangle \ \ (9)$

Where the ellipses are, again, the additional fields. Importantly, we may note that the following holds as an operator equation,

$\displaystyle \frac{1}{\pi \alpha^{\prime}} \partial_{z}\partial_{\bar{z}} X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime},\bar{z}^{\prime}) = - \eta^{\mu \nu} \delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \ \ (10)$

We are going to want to solve this equation in the future, because the solution will prove useful when computing OPEs. In the meantime, what should be understood is that what we have accomplished here is that we’ve modified the EoM to take into account that there is a collision between points at ${z}$ and ${z^{\prime}}$. And we have also found that this behaviour can be derived as an operator statement. The purpose and greater logic for such an exercise will become increasingly clear. Meanwhile, notice that we now have a product of operators. Although it will not be proven here, it follows that in the Hilbert space formalism this product in the path integral becomes time-ordered (Polchinski, p.36). We also see that the delta function appears when the derivatives act on the time-ordering.

To summarise, these last results signal what has already been alluded (however vaguely) about the definition of OPEs in the final paragraph of Section 1. If, moreover, the general theme is so far one of path integrals and local operator insertions, the picture we are ultimately constructing is one of such insertions inside time-ordered correlation functions. These correlation functions can then be held as operator statements.

3. Time-ordered Correlation Functions, Normal Ordering, and the String Propagator

Before formally introducing and defining OPEs, we should spend a few more moments developing the picture and building intuition. For example, when it comes to the idea of time-ordered correlation functions, we will learn that solving the operator equation (10) gives us,

$\displaystyle \langle X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime}) \rangle = - \eta^{\mu \nu} \frac{\alpha^{\prime}}{2} \ln \mid z - z^{\prime} \mid^{2} \ \ (11)$

The computation required to arrive at this result may not yet have much meaning and may be too forward thinking. We will come to understand it soon. What can be understood at this juncture are some of the pieces of this equation. The most important note is that (11) is the propagator of the theory of massless scalars that we have been working with in our study of CFTs (i.e., the free X-CFT). Notice, on the left-hand side of the equality, we a two-point correlation function. As it has been stated, correlation functions are time-ordered. Let us focus on this notion of time-ordering. For instance, consider a Wick expansion for ${X^{\mu}(z, \bar{z})}$,

Where we have indicated the use of contraction notation that will be defined later. The first observation is that we have a two-point correlation function, and we have some term ${T}$. We also have colons on the right-hand side. For the ${T}$ term, it indicates that the expression is time-ordered in the same way one will find in basic QFT (Polchinski, p.36). Writing ${T}$ in full we find,

$\displaystyle T (X^{\mu} (z, \bar{z}), X^{\nu} (z^{\prime}, \bar{z}^{\prime}))$

$\displaystyle = X^{\mu} (z, \bar{z}) X^{\nu} (z^{\prime}, \bar{z}^{\prime}) \theta(z - z^{\prime}) + X^{\nu}(z^{\prime}, \bar{z}^{\prime})X^{\mu}(z, \bar{z})\theta(z^{\prime} - z) \ \ (13)$

Now, looking again at (13), it is worth pointing out a few other things. Firstly, what we will learn in the future, particularly as we advance our discussion on CFTs, is that this time-ordering will prove very useful. Eventually we are going to want to make conformal transformations from an infinite cylinder to the complex plane, and we will learn that time-ordering on the cylinder corresponds to radially ordering on the complex plane. Reversely, we will see that radial ordering on the complex plane corresponds with time-ordering in the path integral. This is a featured point of study in Section 2.6 of Polchinski and it is something we will discuss later. Secondly, for the colons on the right-hand side, they indicate normal ordering. We saw normal ordering in the past discussion on the free string string spectrum using light-cone gauge quantisation. Notice, then, that on the far right-hand side we have a normal ordered product. The definition of normal ordered operators follows as (Polchinski, p.36),

$\displaystyle :X^{\mu}(z, \bar{z}): = X^{\mu}(z, \bar{z}) \ \ (14)$

And for the normal ordered product we have,

$\displaystyle :X^{\mu}(z_{1}, \bar{z}_{1}), X^{\nu}(z_{2}, \bar{z}_{2}): = X^{\mu}(z_{1}, \bar{z}_{1}) X^{\nu}(z_{2}, \bar{z}_{2}) + \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2} \ \ (15)$

Where ${z_{ij} = z_{i} - z_{j}}$. Furthermore, for arbitrary numbers of fields, the normal ordered product may be written as,

$\displaystyle :X^{\mu_{1}} (z_{1}, \bar{z}_{1}) ... X^{\mu_{n}}(z_{n}, \bar{z}_{n}): = X^{\mu_{1}}(z_{1}, \bar{z}_{1}) ... X^{\mu_{n}}(z_{n}, \bar{z}_{n}) + \sum \text{subtractions} \ \ (16)$

Where, for the subtractions, we sum the pairs of fields from the product and then replace each pair with its expectation value ${\frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{ij} \mid^{2}}$. We will elaborate more on (16) later. Meanwhile, consider again the operator equation (10). If what we want to do is define a product of operators that would satisfy the classical EoM, then from (16) and using (10) we can compute,

$\displaystyle \partial_{z} \partial_{\bar{z}} :X^{\mu}(z_{1}, \bar{z}_{1}) X^{\nu}(z_{2}, \bar{z}_{2}): = \partial_{z} \partial_{\bar{z}} X^{\mu}(z_{1}, \bar{z}_{1}) X^{\nu}(z_{2}, \bar{z}_{2}) + \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \partial_{z} \partial_{\bar{z}} \ln \mid z_{12} \mid^{2}$

$\displaystyle = - \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2}(z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2}) + \frac{\alpha^{\prime}}{2} \eta^{\mu \nu} \partial_{z}\partial_{\bar{z}} \ln \mid z_{12} \mid^2$

$\displaystyle = - \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2}(z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2}) + \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2} (z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2}) = 0 \ \ (17)$

Where, for the last line in the computation, we used the standard result,

$\displaystyle \partial \bar{\partial} \ln \mid z \mid^{2} = 2\pi \delta^{2}(z, \bar{z}) \ \ (18)$

Which is derived from an application of Stokes’ theorem.

Importantly, (17) is precisely the property that Polchinski highlights in equation (2.1.23) on p.36 of his textbook. What (17) is telling us is that, on the last line, ${- \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2}(z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2})}$ are the quantum corrections to the classical EoM. It is also telling us that, as we want to define a product of operators that satisfy the classical EoM, we must necessarily induce normal ordering.

So what does this all mean? In order to further extend the picture being developed here, we are lead directly to a definition of OPEs.

4. Operator Product Expansions

We may now define operator product expansions. The definition follows (pp. 37-38) directly from the intuition and logic that we have so far established, notably that OPEs may be considered a direct statement about the behaviour of local operators as they approach one another. The formula for OPEs is as follows,

$\displaystyle \langle \mathcal{O}_{i}(z, \bar{z})\mathcal{O}_{j}(z^{\prime}, \bar{z}^{\prime}) \rangle = \sum_{k} C_{ij}^{k}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime})\langle \mathcal{O}_{k}(z^{\prime}, \bar{z}^{\prime}) \rangle \ \ (19)$

Which is, again, an operator statement. This means that it also holds inside a general expectation value. Saving the general formula for OPEs until later, note that in (19) the ${C_{ij}^{k}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime})}$ should be considered as a set of functions that depend only on the separation between the two operators (i.e., there is translational invariance).

To summarise, if OPEs describe what happens when local operators approach one another other, we have already developed a sense of technical intuition for why the key idea is one of having two local operators inserted in such a way that they are situated close to one another but not at coincident points. As we have already discussed, upon insertion of local operators at ${z_{1}}$ and ${z_{2}}$ for example, we obtain some normal ordered product. Then, what we can do is compute their approximation by way of a string of operators at only one of the insertion points (Tong, p.69). There can be any number of operator insertions, which is of course why we have included …’ in the general formula for the OPE (19); it denotes insertions that are not coincident at ${z}$. (From this point forward, the ellipse will be removed and the following statement will be implied). This leads us directly to an illustration of OPEs as provided in Polchinski’s textbook.

In figure 4.1, we see that we have a number of local operator insertions, ${z_{1}}$ to ${z_{4}}$, hence what we would be computing is the expectation of 4 local operators. Given that the OPE describes the limiting behaviour of ${z_{1} \rightarrow z_{2}}$ as a series, where the pair of operators are replaced by a single operator at ${z_{2}}$, one way to think about this is analogous to the Taylor series in calculus (i.e., the OPE plays a similar role in quantum field theory). In fact, the analogue of computing a Taylor series is apt, as we will see when we start computing OPEs.

Another thing to note is that the circle in the picture illustrates the radius of convergence, such that this radius is computed as the distance to the nearest other operator positioned on the circle. In CFTs, OPEs have a finite radius of convergence.

Now, from our previous discussions, and from the formal definition of OPEs, we can see quite clearly why they are always to be understood as statements which hold as operator insertions inside time-ordered correlation functions. Should one ask, ‘what are the observables in string theory?’, the answer is that we compute a set of correlation functions of local/composite operators at their insertion points. So, should we take for example the Polyakov action, ${S_{P}}$, and compute the correlation functions for the CFT, one motivation is to show the correlation function to be related to the scattering amplitude in 26-dimensional spacetime (in the case of the bosonic string). So, in perturbative string theory, we look at the critical theory – that is, the critical coefficients and components of the correlation function,

$\displaystyle \langle A_{ij}(z_{i}\bar{z}_{j}) ... A_{ij}(z_{n}, \bar{z}_{n})) \rangle \ \ (20)$

Where we are interested in the singular behaviour. Moreover, recall the definition of the normal ordered product (15). Notice that we have very interesting log behaviour. If what we want to know exactly is what will happen with the product of the two operators as ${z_{1} \rightarrow z_{2}}$, this implies that we have an operator singularity. As we start computing OPEs and moving forward in our study of string CFTs, it will become very clear why this singular behaviour is actually the only thing we care about.

In the next post, we will extend our discussion of OPEs. Following that, we will look to derive the Ward Identities and then turn our attention to the Virasoro algebra among other important topics in Chapter 2 of Polchinski’s textbook.

References

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

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

Standard
Stringy Things

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

The following collection of notes is based on a series of lectures that I attended by Eran Palti at SiftS 2019 at Universidad Autonoma de Madrid. The theme of the lecture series was ‘String Theory and the Swampland’. Palti’s five lectures were supported by his most recent and impressive 200 page review paper on the Swampland, which includes over 600 references [arXiv: 1903.06239 [hep-th]]. The reader is directed to this paper in addition to supplementary references that I also provide at the end of each set of notes.

In the following entry, the notes presented follow the second lecture of Palti’s series.

1. Introduction

In this collection of notes, we continue to build toward the view of why it is valid to think of the Weak Gravity Conjecture (WGC) and the Distance Conjecture (DC) as being almost axiomatic to the Swampland programme. In other words, we are working toward the understanding of how and why these two conjectures are two fundamental pillars of the Swampland programme, from which every other conjecture is related or connected in some way.

In the last post, one will recall that we began by considering a general introduction to the Swampland programme in the context of arguments about constraining effective field theories. We also considered a very basic introduction to the Magnetic Weak Gravity Conjecture (WGC) and reviewed how, if we have some U(1) gauge field with a gauge coupling ${g}$, then as ${\Lambda \sim M \sim gM_{p}}$ we should have an infinite tower of states [1]. This infinite tower of states was found to have a mass scale ${M}$ set by the product of the Planck scale and the gauge coupling.

In the present entry, we will continue our review of the Swampland, turning our attention to the WGC in the context of the 10-dimensional superstring. Following this, we will also outline a basic introduction to the DC. Then, to close, we will perform our first tests of the DC and discuss how this conjecture relates to the parameter space of M-theory.

2. An Infinite Tower of States: From the Weak Gravity Conjecture to the Distance Conjecture

In this section, we will work toward a gentle (and generally informal) introduction to the DC (much like we did for the WGC in the first entry) following Palti’s second lecture at SiftS 2019. To begin, let’s recall some basic facts about the mass scale in bosonic string theory. We start with the following spacetime action for the low-energy effective theory,

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

This is the low-energy effective action of the bosonic string. For review of the construction of this action, see Section 3.7 in [2]. What is important to note is that, neglecting the tachyon mode, this action contains the massless spectrum. But we are not particularly interested in the massless spectrum, as Palti emphasised in his talk. Instead, our present interest has to do with the massive modes. When we study the string massive modes, it can be reviewed in [2,3] and in other textbooks that we have for the nth harmonic of the string ${M^{2} = \frac{4}{\alpha^{\prime}} (N_{\perp} - 1)}$, where ${N_{\perp} = \sum_{n =1}^{\infty} : \alpha^{\dagger}_{-n} \alpha_{n} :}$. In other words, if we increase the internal excitations ${N_{\perp}}$ along the free bosonic string, we increase the mass with the mass of the string set by the string scale ${M_{s} \sim \frac{1}{\sqrt{\alpha^{\prime}}}}$.

What we learn in performing such a study of the bosonic string is that string theory has an infinite tower of massive states. One can observe this infinite tower quite quickly through light-cone gauge quantisation of the Polyakov action, providing direct access to the study of the string spectrum. The key message emphasised at this point in Palti’s lecture is how, in this theory of massive states, the mass scale of the tower is ${M \sim M_{s}}$. Crucially, this is UV data.

Rather than spending more time reviewing the bosonic string, what we want to do is investigate how this mass scale behaves when we vary the parameters of the low-energy theory. In particular, we want to know how this infinite tower of massive states behaves when we vary the string coupling ${g_{s}}$. To this end, let us for the moment introduce some more basic ideas by considering a generic 10-dimensional string from the action (1), which may be written as follows,

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

We are currently not worried about the extra terms …’ in (2). As Palti explains, what we really want to focus on is the relation between the string scale and the Planck scale. Recall the d-dimensional Planck mass, ${M_{p}^{d}}$. In the Swampland programme we often want to work in Planck units (a point which is emphasised in lecture 2), and it is useful to fix the Planck mass such that ${M_{p}^{d} = 1}$. Furthermore, we should also note that the d-dimensional Planck mass from the effective string action is defined as, ${\frac{M_{P}^{d - 2}}{2}R \equiv 2\pi M_{s}^{D-2}}$. To convert the string scale to the Planck scale for the action (2), we look at the Ricci scalar pre-factor and consider dimensional reduction. In 10-dimensions, it is fairly trivial to see that we have (remembering that ${g_{s} \sim e^{\Phi}}$ and so ${e^{-2\Phi} \sim 1 / g_{s}^{2}}$),

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

Notice, upon rewritting the string scale in terms of the Planck scale, in the weak coupling limit where ${g_{s} \rightarrow 0}$ we find ${\frac{M_{s}}{M_{P}} \rightarrow 0}$, which implies that we have an infinite tower of states that become light relative to the Planck mass. In another way, ${\frac{M_{oscillator}}{M_{P}} \rightarrow 0}$ as ${g_{s} \rightarrow 0}$. What is curious about this is that it is very reminiscent of the WGC that we discussed in the last entry, where we take a U(1) gauge field and make it weakly coupled to a find a light tower of states.

More pointedly, great emphasis is placed at this juncture about what is generally an interesting feature of string theory. If we make our theory weakly coupled, we obtain an infinite tower of light states relative to ${M_{P}}$. That is to say, weakly coupled string theory doesn’t have light states close to the Planck scale as one may expect or anticipate. In fact, what we see is that in weakly coupled string theory we have states arbitrarily lower than ${M_{P}}$. From a traditional effective theory point of view, this is quite striking behaviour. So let’s think about this more deeply.

To begin with, it should be highlighted that the string coupling ${g_{s}}$ is a scalar field in the theory. More generally, we should remember that there are no coupling constants in string theory such that they are in fact expectation values of fields. As a quick review, remember that the expectation value of the massless scalar field, ${\Phi}$, which is the dilaton, controls the string coupling. This can be explained a bit more eloquently. Consider ${\Phi (X) = \lambda}$, where ${\lambda}$ is some constant. The dilaton coupling reduces to ${\lambda \chi}$, where ${\chi}$ is the Euler characteristic of the string worldsheet. The lesson we learn, often first in the study of the bosonic string, is how the constant dilaton mode taken to be the asymptotic value ${\Phi = \lim_{X \rightarrow \infty} \Phi (X)}$ determines the string coupling constant, ${g_{s}}$, such that we find ${g_{s} \sim e^{\Phi}}$, corresponding to the amplitude to emit a closed string [2].

All of that is to say that we should remember that the string coupling, ${g_{s}}$, is a dynamical parameter – i.e., a field – that is determined by the dilaton. In this light, the previous statement about how we obtain a light tower of states relative to ${M_{P}}$ implies that (3) can be rewritten in the following way,

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

Where ${\alpha}$ is some constant such that ${\alpha > 0}$ and ${\alpha \sim \mathcal{O}(1)}$. Following Palti, notice how if we send ${\phi \rightarrow -\infty}$, we get a light tower of states. In studying this behaviour, it is nice to reflect back on the recent reference made to the WGC; however, this behaviour implies an encounter with another Swampland Conjecture, namely the Distance Conjecture.

The main segment of lecture 2 was structured on this idea: namely, one will find that the simplest example of the DC is by going to weak coupling in string theory, As we have hinted so far, this implies going to large distances in the dilaton field space in which one obtains, indeed, a light tower of states. To put it another way, the DC states that this kind of behaviour – where we obtain an infinite tower of light states – is universal. This means that whenever we give a value to a scalar field that is very large, we obtain a light tower of states that is, as Palti put it, exponential in the expectation value of the scalar field [1,5].

We have already sent ${\Phi \rightarrow -\infty}$. An interesting question now is to ask: what happens when we send ${\Phi \rightarrow + \infty}$? This implies the strong coupling limit, which raises many curiosities. We will explore more deeply and define this limit more carefully in time. For now, as a point of introduction, we note that the DC states: given a scalar field ${\phi}$, there is an infinite tower of states whose mass relative to ${M_{P}}$ may be written,

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

Let’s now test this conjecture with a simple example.

3. Testing the Distance Conjecture: Compactification of String Theory on a Circle

In the entry on Palti’s first lecture, it was mentioned that one approach to the study of the Swampland Conjectures is by way of a direct study of certain deep patterns to have emerged in string theory over time. What we will see, now that a gentle introduction to the WGC and the DC is out of the way, is that both of these conjectures are increasingly general. Indeed, we will see that they describe very general and deep patterns in string theory. But are they fundamentally true? The argument is that we may take the persistence of the sort of behaviour we find in string theory as evidence that these conjectures are true (or seem to be increasingly true). This is one of the main messages to come from Palti’s second lecture, wherein we consider a first basic test of the DC.

We construct this basic test by seeking to study the symmetry of the massive spectrum when string theory is compactified on a circle. We take the approach of first studying field theory compactified on a circle and then focus on the string case.

3.1. Field Theory Compactified on a Circle

Consider, for instance, ${D = d + 1}$ spacetime. As before, we are working in Planck units where ${M_{P}^{d}}$ is the d-dimensional Planck mass. For the circle there is also of course a periodic identification of the form ${X^{d} = X^{d} + 1}$. We must also be mindful of notation when working in the higher dimensional and lower dimensional space. For the higher D-dimensional spacetime we have the following product metric,

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

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

The circumference of the circle on which we will be compactifying our theory is given by,

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

Where we can see quite explicitly the relation between ${\phi}$ and the radius of the circle. Crucially, the radius of the circle becomes a dynamical field in d-dimensions. As it is a dynamical field, we will want to study the behaviour of the d-dimensional theory when we vary the expectation value of ${\phi}$. Also important is that, when we reduce the higher D-dimensional Ricci scalar, ${R}$, we obtain something in the Einstein frame in lower dimensions,

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

Moreover, to obtain (8) we have decomposed the Ricci scalar ${R^{D}}$ on the left-hand side of the equality for the metric (6). To do this, we take the metric ansatz and plug it into the higher dimensional Ricci scalar, which gives us a lower dimensional Ricci scalar ${R^{d}}$ from restricting the higher dimensional indices to lower dimensional indices. From one’s knowledge of scalar curviture, it can also be seen that the higher dimensional Ricci scalar is a two derivative object. This means that those derivatives act on the field ${\phi}$; however, the choice for ${\alpha}$ and ${\beta}$ ensure no ${\phi}$ factorises in front of ${R^{d}}$ (hence the chosen definitions of ${\alpha}$ and ${\beta}$). One can see that, after all this, we end up with a kinetic term that is canonically normalised.

Now that some notation has been established and we have dimensionally reduced to a circle, the idea is to consider a massless D-dimensional scalar field,

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

Where we have performed a Fourier expansion of the higher dimensional field in terms of the lower dimensional modes along the circle. Note, ${\Psi}$ is made to be periodic because it depends here on the lower dth dimension, hence the decomposition already implicit in (9). Moreover, notice the coefficients depend on the external spacetime (lower dimensional coordinates). This means they are like lower dimensional fields. To word it another way, a higher dimensional field gives an infinite number of lower dimensional fields. The ${\psi_{n}}$ modes are d-dimensional scalar fields, where ${\psi_{0}}$ is the zero mode of ${\Psi}$ and ${\psi_{n}}$ are the nth Kaluza-Klein (KK) modes of the higher dimensional field.

Another point worth highlighting as a natural consequence of compactification concerns how we also see that the ${n}$ in the exponential is quantised. This means it should be an integer, since we should have periodicity. This indicates that the momentum of the lower dimensional fields is quantised in the compact direction allowing us to write,

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

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

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

Where ${\partial^{M}\partial_{M} \Psi = 0}$ is just the Klein-Gordon equation. When we expand this equation we obtain (by restricting the ${M}$ indices to be external indices plus the inverse metric) what is written to the right of the first equality. From this, we can look at the equations of motion for each of the ${\psi_{n}}$ modes,

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

The question that is raised: what is this lower dimensional equation for each of the KK modes? It is a Klein-Gordon equation for a massive field. But what is the mass of this field? Quite simply, it is set by the radius of the circle, ${R}$, and the KK number. So the mass of the nth KK mode is given by,

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

What is this telling us? It says that when we dimensionally reduce on a circle, as we have done, we obtain something similar to the string (which we will look at in a moment). Notice, moreover, that we have a lower dimensional theory and that theory has an infinite number of massive states. What we have found, as Palti emphasises in his lecture, is that in the lower d-dimensional theory the KK modes are a massive tower of states. The masses here are increasing. Why is this so? Recall that the radius of the circle, ${R}$, is a dynamical field in the lower dimensional theory. As such, the mass of the infinite tower of states that we observe depends on the expectation value of the field in the lower dimensional theory.

However, this isn’t quite the spectrum of string theory on a circle. We have so far only been considering field theory compactified on a circle. What we observe is thus the massive spectrum of Einstein gravity. For the complete string spectrum on a circle we need to add another important piece to the picture. So let us go to the string theory picture, and then connect the results.

3.2. Compactification of String Theory on a Circle

In this section we consider generally the propagation of a string in spacetime in which one spatial dimension is curled up into a circle. One can review the full procedure in section 2.2.2 in [1]. For further review on compactifying on a circle, see [2,4]. To save space, and in following Palti’s lecture, we move directly toward the main point of focus: namely, when we compactify a dimension we modify the string mass spectrum. And, indeed, much like before it is the massive spectrum that we are interested in studying.

Working in 10-dimensions, as we have been, one will find that when compactifying the 10th dimension we obtain for the compactified direction,

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

Where one will notice that we now have winding states. In (14), ${\omega}$ is the winding number such that ${\omega \in \mathbb{Z}}$. This comes from the fact that the string can wind around the circle ${\omega}$ times. We can also define the winding as ${n = \frac{\omega R}{\alpha^{\prime}}}$. As we will discuss in a moment, the winding ${n}$ is actually a type of momenta. In review of the mode expansions, one will find both left and right-moving modes, which, together, for the compact direction may be written as,

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

The total center of mass momentum is therefore ${p^{d} = p_{L}^{d} + p_{R}^{d}}$. Importantly, when we compactify a dimension, the center of mass momentum is quantised along that direction. Moreover, it turns out that along the circle the string acts like a D0-brane, i.e. a particle with quantised momentum ${p^{d} = \frac{n}{R}}$. This ${n}$ term is, in fact, the Kaluza-Klein excitation number. And what we observe is how, in (15), we have the momentum mode in the form of ${(p_{L}^{d} + p_{R}^{d})}$ and another form of momentum in the form of ${(p_{L}^{d} - p_{R}^{d})}$, which is the winding mode of the string satisfying,

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

To realign with Palti’s talk, notice that we now have additional states that we must consider: i.e., when we compactify on a circle there are also winding modes. We will talk more about these in a moment. For now, we should remember that the entire point of the exercise is to look at the massive spectrum. If we go to the target space light-cone gauge, the mass spectrum of the string reads as,

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

If one were to look deeper it is not too difficult to prove (17) and see why the level matching condition no longer holds. Indeed, we find ${N_{\perp} - \tilde{N}_{\perp} = n \omega}$. And, if we drop the excited oscillators, for the mass formula we have,

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

Following Palti,the task in these notes is to now think of this result (which is standard and can be reviewed in any string textbook) in relation to what we found in (13). This involves changing to the Einstein frame (6). In changing from the string frame to the Einstein frame, Palti explains how the massive spectrum which now includes both the KK number and winding number matches the field theory result for the KK masses (13),

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

3.3. Testing the Distance Conjecture

What we now want to do is test the DC by studying the d-dimensional effective theory, with the action (8) and the mass spectrum (19). One can, and perhaps should, anticipate a discussion on T-duality. Although it has not yet been introduced, its presence is ubiquitous.

Looking at (8) and (19) recall the fact that we have a scalar field ${\phi}$ in our theory. As has so far been described, this scalar field gives the radius of the circle. So a natural question to ask is, what happens when we change the expectation value of ${\phi}$? Do we obtain exponentially light states?

As Palti highlights in his lecture, we see that this is precisely the case. Recall how the exponential of $\phi$ goes like $R$ in (7). The mass of the state parallel in $R$ in (14) will grow exponential in $\phi$. So, when considering the DC, we see that for $\phi$ (size of the circle) there is an infinite tower of KK modes that go something like the inverse power of $R$ in (14),

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

And we also have the winding mode tower,

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

Where, ${\gamma = \sqrt{2}(\frac{d - 1}{d - 2})^{1/2}}$. What we see is that, if we make the circle very big we obtain an infinite tower of states that becomes very light. These are the KK modes. Reversely, if we make the circle very small we obtain an infinite tower of states that becomes very light. These are the winding modes. What is going on here? A discussion on T-duality is well anticipated. But another way to visualise this behaviour is first by reviewing the following log plot, where the mass scale for the KK and winding modes are plotted as a function of the expectation value of the scalar field ${\phi}$.

The slope is $\gamma$, while the ${\mathcal{Z}_{2}}$ symmetry is indeed an expression of T-duality.

4. Lessons about the Distance Conjecture and T-duality

What did we learn in our first test of the DC? Several lessons can be gleaned, which then set-up for more advanced discussion:

1) We learn, for example, that the conjecture is deeply string theoretic. The presence of winding modes means we are learning about very stringy behaviour.

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

3) Think for instance of the case when ${\phi \rightarrow - \infty}$. In this limit the effective theory breaks down. Why? Notice that when we send ${\phi}$ to negative infinity, this pulls down an infinite number of modes below the cutoff scale (i.e., an infinite tower of light KK modes). The implication is that we have new modes now appearing in the theory. Moreover, as discussed in the previous section, whenever we set the scalar field ${\phi}$ to have a very large expectation value, what we obtain is an infinite tower of light states and new description of the physics, which in this case is the higher dimensional theory.

4) What about the limit ${\phi \rightarrow \infty}$? The effective theory still breaks down. Just as in 3), we obtain an infinite tower of lights states (winding modes). What about the description of the physics? Is there a new description in this limit? The answer is that it is, again, a D-dimensional theory because of T-duality.

To offer an example, consider the mass of the spectrum in the string frame,

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

The spectrum is invariant under the symmetry,

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

Which is T-duality. All that we are doing, as Palti puts it, is rearranging our degrees of freedom. To word this differently, T-duality is simply a special type of symmetry that allows us to relate our theory at a short distance with our theory at a long distance. They are the same theory, except from the vantage that we are viewing that theory from different perspectives: i,e., T-duality allows us to transform between small and large distance scales. In the case of compactification of some spatial dimension to a circle of radius ${R}$, as we have been considering throughout these notes, the simple idea to begin with is that we may transform the original radius ${R}$ to a larger (or smaller) radius ${R^{\prime}}$, such that ${R^{\prime} \leftrightarrow \frac{\alpha^{\prime}}{R}}$. One can then see that with such a transformation we must also transform the winding states, such that ${n \leftrightarrow w}$. The main premise is that high-momentum states in the one theory is exchanged for the winding number in the other (and vice versa). Under this transformation the whole theory stays the same (T-duality invariance), including the spectrum, it is just that we are transforming from the KK modes to the winding modes (and vice versa).

This is why, for instance, in the limit ${\phi \rightarrow \infty}$ the circle becomes very small and the winding modes become very light; but the physics in this limit is the same as in the case when the circle is very big.

5. The Dilaton Revisited in Type IIA String Theory

From the test of the DC by studying field theory compactified on a circle, we have already gained some interesting insights. We have observed that, in the case of a scalar field, we may go to opposite limits, ${\phi \rightarrow \infty}$ or ${\phi \rightarrow - \infty}$. In both cases we obtain an infinite tower of light states.

Now, recall that in the much earlier example of the dilaton, where we considered the string coupling, we only studied one limit: ${g_{s} \rightarrow 0}$ (see section 2). That is, we only considered what happens in the weakly coupled theory. Let’s now revisit this example, and consider what happens in the strong coupling limit where ${g_{s} \rightarrow \infty}$.

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

In short, if we send ${\phi \rightarrow \infty}$, we are lead to believe by the logic of the DC that in this strong coupling limit we should obtain a light tower of states. Is this true?

To think of the strongly coupled theory, let’s go to the superstring theory. Consider, for instance, the Type IIA string. This also has a massive spectrum, which we may consider. There is the universal Neveu-Schwarz sector in brackets ${\{...\}}$ and then also the Ramond-Ramond sector, which contains all odd-dimensional anti-symmetric forms,

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

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

With the presence of these anti-symmetric forms, we can study what sort of objects are in our theory. Moreover, recall that if we have anti-symmetric forms, this means we have some object that couples to it. We may restate this fact as follows,

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

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

*${C_{MN \rho}}$ is a 3-form and it couples to a membrane (D2-brane).

Notice the pattern that, as we have only odd-ranked anti-symmetric forms in our theory, this means we have only even ranked Dp-branes. Just as the fundamental string is an object that exists in our theory, branes are legitimate objects in our theory.

Let’s focus for a moment on the D0-branes and think about computing the mass of these particles. Given that Dp-branes have a mass/tension, we can write this in general in the following way

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

Which tells us the mass, because, for a D0-brane, we simply have ${M_{D0} \sim \frac{M_{s}}{g_{s}}}$. Now, at this point in Palti’s talk, we should take notice of something interesting about this object. If we send ${g_{s} \rightarrow 0}$, Dp-branes become very heavy. When this happens, the Dp-branes decouple from our theory. So in the weakly coupling limit, we see that relative to the strings (${M_{s}}$), these extended objects are actually unseen. This is precisely why in perturbative string theory, one cannot see Dp-branes. It also implies that to see these objects, what is required is strong coupling and non-perturbative limits.

So, in considering D0-branes, whenever we consider the masses of objects in the Swampland programme, Palti makes the point to emphasise that for these reasons we always go to the Einstein frame (remember, in the Swampland programme, we’re always thinking in terms of the Planck scale),

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

Now we see something interesting. Up to this point, the leading question concerns what happens when we send ${\phi \rightarrow \infty}$. We know that we have a strongly coupled regime. As ${g_{s} \rightarrow \infty}$ it follows that ${\phi \rightarrow \infty}$ and ${M_{D0} \rightarrow 0}$. So the mass of the D0-brane goes to zero.

But, one might ask, can we trust this regime (namely, strongly coupled string theory)? In general, the strongly coupled limit sets off various alarms of concern. But, by extending much of the same logic displayed throughout this entire discussion, the answer is that we can trust it. Why? Notice that the present example is very similar to the previous one, where we made the circle very small and the description of the physics was of the higher dimensional theory. We know that string theory can handle such limits because of T-duality. And, moreover, in the above limit, we know that we can trust the regime because we can see that as we obtain an infinite tower of light modes that are bound states of branes, strongly coupled Type IIA at low-energies is nothing but 11-dimensional supergravity (SUGRA).

6. Parameter Space of M-theory

Just as in the case when we obtained a description of the physics of the higher dimensional theory, so, too, in our present example, have we obtained a higher dimensional description. The point of emphasis is how this is T-duality in practice, and it leads us directly to a picture of M-theory.

To summarise, in the figure above we begin with a point in parameter space. As an example, we begin with Type IIA string theory that we just considered in Section 5. And then we consider another point, which is 11-dimensional supergravity. What we have found, or at least reviewed, is how we can move between these two theories depending on the string coupling limit. If we go to the weak coupling limit ${g_{s} \rightarrow 0}$ (or when the dilaton has a large negative expectation value), then we go to a perturbative Type IIA string theory and we obtain light states (from the light oscillator modes). On the other hand, when we go to the strong coupling limit ${g_{s} \rightarrow \infty}$, we have strongly coupled Type IIA string theory and, in this case, we should transform to a description of SUGRA, in which, again, there is an infinite tower of light states.

There are also other ways we can transform in parameter space. In another example we consider Type IIA string theory on a circle. So consider another direction in parameter space, governed by the size of the circle. In the limit of Type IIA / ${S^{1}}$ when the circle is very big, such that ${R \rightarrow \infty}$, we obtain a 10-dimensional Type IIA stirng theory (where from the 9-dimensional perspective we have a tower of states that are the KK modes). There is also T-duality, where ${\frac{IIA}{R} \longleftrightarrow \frac{IIB}{1/R}}$. That means, we can also go the other direction in parameter space and send ${R \rightarrow 0}$. We can see that this is tantamount to sending ${R \rightarrow \infty}$ in Type IIB string theory. So in Type IIA from the 9-dimensional perspective, this corresponds to the circle becoming very small and gives Type IIB on a circle that is very big, which is IIB string theory in 10-dimensions.

7. Summary

To conclude, what we see in these results is that the DC is an incredibly strong and powerful, if not a deeply insightful conjecture, that describes a provocative picture of the parameter space of M-theory. What we see moreover is how, when we look at the parameter space in string theory, those parameters are scalar fields. As we have been experimenting, we can give these scalar fields large expectation values, which then moves us to the limits of the parameters where we obtain an infinite tower of light states. These towers of states can offer us a different description of the physics in a new regime. To put it more concisely, the different limits correspond to the 5 string theories and 11-dimensional supergravity. All of the string theories are linked by dualities describing different parameterisations of the same theory, M-theory. Each of these string theories have their own unique characteristics, offering descriptions in their respective corners of parameter space.

In the next collection of notes, we will review the third lecture in Palti’s series and consider a more formal definition of the WGC. We will then look to perform a deeper test of the WGC than in previous discussions, focusing particularly in the context of the heterotic string.

References

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

[2] J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string’. Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2007.

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

[4] K. Becker, M. Becker, J.H. Schwarz, String Theory and M-Theory: A Modern Introduction’, 2006.

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

Standard
Stringy Things

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

And,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Now, when we vary the simplified action we find,

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

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

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

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

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

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

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

$\Box$

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

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

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

References

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

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

Standard
Stringy Things

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

1. Introduction

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

2. Review – Effective Field Theory

2.1. Schematic Overview

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

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

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

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

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

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

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

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

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

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

2.2. Alternative Approach to EFTs

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

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

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

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

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

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

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

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

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

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

Where we have some operators suppressed by the energy scale.

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

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

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

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

2.3. EFT Coupled to Gravity

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

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

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

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

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

3. EFTs and the Swampland

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

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

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

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

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

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

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

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

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

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

4. From EFTs to the Rules of the Swampland

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

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

5. Weak Gravity Conjecture (Magnetic)

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

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

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

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

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

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

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

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

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

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

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

6. Summary

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

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

References

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

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

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

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

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

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

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

Standard
Mathematics

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

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

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

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

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

1. Hyperbolicity of Polynomials

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

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

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

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

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

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

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

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

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

2. Theorem 3

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

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

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

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

Perhaps I should save such considerations for a separate article.

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.

Standard