Hassaan Saleem

This is a chapter-by-chapter review of volume 1 of Polchinski’s book on string theory. I will later add some pictures to this post to make it more reader-friendly.

A guide to this review
Polchinski’s two volumes on string theory are still considered to be the best and most comprehensive resources to learn string theory. Although they were written in the late 90s, they are still the first choice of many string theorists. This is a review and a reading guide for Polchinski’s first volume. It has been written for students who want to read Polchinski’s first volume and have a basic understanding of the prerequisites like GR and QFT.

You don’t need to know every word in this review. Some of the unfamiliar words in this review will become familiar when you read the book. You don’t need to understand every word to understand the structure of the book described in this review. I have put my heart into writing this review and I hope it is helpful to you. Let’s go.

Chapter 1
In the first chapter, Polchisnki builds some arguments to make it clear that string theory is an important subject to study. These reasons include the fact that string theory has no free parameters and the fact that string theory has gravity built into it. He then develops the action for the bosonic strings (which is also known as the Polyakov action) and solves the resulting equations of motion using a gauge condition called the lightcone gauge. At the end of the chapter, he talks about unoriented strings. Unoriented strings will be important to formulate type I superstring theory.

Chapter 2
In Chapter 2, he develops conformal field theory which is also known as CFT. CFT is very important to study string theory. This chapter can be built into a whole book and I would recommend reading Blumenhagen’s book or Ginsparg’s notes on conformal field theory to understand conformal field theory better. There is one thing that the reader should read from this chapter though. At the end of chapter 2, Polchinski describes a mnemonic to calculate zero-point energies that he will be using throughout his book. The reader should read this mnemonic from Chapter 2. Polchinski develops more sophisticated CFT techniques and results in chapter 15. We won’t be talking about this chapter later because that chapter is also about CFT.

Chapter 3
Chapter 3 of Polchinski’s book is really important and the reader is strongly recommended to read this chapter fully. In this chapter, the Polyakov action is gauge fixed, and the resulting ghosts are derived. These ghosts are called the b and c ghosts. After that, the Weyl anomaly is discussed. This anomaly is related to the breaking of the conformal symmetry when we work with spacetimes that aren’t flat. The guess for the expression of the string S matrix follows the Weyl invariance section. This guess will be refined in Chapter 5.

Afterward, Polchinski discusses some tools that we will need later to calculate string scattering cross sections. These tools are called vertex operators. At the end of the chapter, the worldsheet actions are derived. These actions are also called Sigma models. It is followed by the discussion of Spacetime action and the derivation of the equations of motion by imposing the condition that the Weyl symmetry is not broken. A small discussion of compactifications concludes this chapter.

Chapter 4
This chapter is rather short. It starts with the derivation of the lowest energy states of string theory. This is also known as the string spectrum. Some ad-hoc constraints are applied to get some results. These ad-hoc constraints are imposed in an approach called Old Covariant Quantization or OCQ.

Most of this chapter consists of deriving the string spectrum by another approach called the BRST quantization. If you haven’t learned BRST quantization before, then you should spend a lot of time understanding the basic philosophy of this approach. In the end, it is proven that the string spectrum will not have unphysical states known as ghosts by proving the “no-ghost theorem” and it is also proven that BRST and OCQ are equivalent.

Chapter 5
This chapter is all about refining the guess for the string S matrix that Polchisnki made in Chapter 3. To refine this guess, this chapter develops the concept of metric moduli and conformal killing vectors. Afterward, these concepts are used to derive the refined expression of the string S matrix. In this refined expression, some ghost insertions can be seen. This expression is written in a different way that will be useful in chapters 6 and 7.

The chapter ends with some more discussion on Riemann surfaces and about the integration measure appearing in the S matrix but these sections can be skipped in the first reading.

Chapter 6
Now we come to those chapters where we calculate the scattering amplitudes. In this chapter, only three surfaces are discussed. They are spheres, discs, and projective planes. Why these three? Well, because these three surfaces contribute to the tree-level string processes. The chapter starts with listing some properties of these three surfaces.

Afterward, a quite general CFT expectation value is calculated on these three surfaces. Calculations of the Expectation values including the ghosts follow. These results are used to calculate the three and four-point open string tree level amplitude. The four-point open string tree level amplitude is called the Veneziano amplitude.

This calculation is followed by a generalization of the open strings. This generalization is done by introducing new degrees of freedom on the endpoints of the open string called the Chan-Paton factors. Following this, The results for the open string amplitudes in the presence of Chan-Paton factors are quoted. It is followed by a small section on unoriented strings. Here, it is established that unoriented strings can only have two kinds of gauge groups i.e. So(n) or Sp(n).

The calculation for tree-level closed string amplitudes begin now which is also known as the Virasoro-Shapiro amplitude It is checked if the resulting amplitude is gauge invariant. Then in a small subsection, a discussion about strings on a disc and a projective plane is carried out. The last section of this chapter talks about some general results that were swept under the rug before. This section can be skipped in the first reading.

Chapter 7
In this chapter, Polchinski continues with the calculation of scattering amplitudes but now, we calculate the first-order loop contributions. four surfaces contribute here. They are the torus, the cylinder, the Klein bottle, and the Mobius strip.

Just like Chapter 6, this chapter starts with some basic facts about these surfaces and then, derives the expectation values on them. As a special interesting case, the partition function on the torus is given special importance because it does have a lot of importance in the study on CFT on the torus. Similar calculations are done for expectations values containing the bc ghosts and a small subsection talks about the torus partition function for general CFTs. This is followed by a subsection that lists the properties of special functions called theta functions. These functions are very relevant when studying torus partition functions. The reader is advised to go through this section very comprehensively because these properties will be used in chapters 10 and 13 of volume 2 as well.

After the theta function properties, the torus amplitude is calculated and it takes some time to come up with the correct integration measure because of all the modular invariance stuff. This is followed by the proof that torus doesn’t give massless divergences but it does give tachyonic divergence. This fact will be very important later.

This is followed by a small subsection named Physics of the vacuum amplitude. This subsection seems insignificant at first, but equation 7.3.24 from this subsection will be very important when you will be deriving the partition function of type II string theories in chapter 10.

In the last section of the chapter, the amplitudes for the cylinder, Klein bottle, and Mobius strip are calculated and it is shown that all of them have massless divergences. At the end of the chapter, it is shown that if we add these massless divergences together, then they vanish only if the gauge group of the open strings is SO(8192).

Chapter 8
Here we come to the first chapter where we start to talk about the compactifications. This chapter starts a small review of what happens when we compactify one dimension in a gravitational theory. It gives us the electromagnetic field for free. The next section talks about such a compactification in the context of closed string theory and the concept of winding strings comes out which was not present in the field theory. Such a compactification has a U(1) X U(1) symmetry.

The next section talks about the cases where the symmetry enhances to an SU(2) symmetry. It also proposes a Higgs-like mechanism where the particles acquire mass just because the radius of the compact dimension changes. The next subsection talks about a very important duality found in string theory called the T duality. This duality relates small-radius physics to large-radius physics,

The next section talks about the compactification of several dimensions into a torus. These compactifications are therefore called toroidal compactifications. The expressions derived in the case of a single compact dimension are generalized to several compact dimensions cases and a description to deal with these compactifications is also introduced. This description is called Narain compactifications.

In the next section, orbifolds are introduced which are really important for string compactifications. The introduction of orbifolds introduces a whole new set of states into the game known as the twisted states. Therefore, there is a whole section on the concept of twisting. The reader might be tempted to skip this subsection but this subsection is really important in chapter 11 where nonsupersymmetric heterotic string theories are constructed.

In the next section, the case of open strings is discussed. Open strings carry gauge degrees of freedom on their endpoints. Therefore, the concept of Wilson lines comes into play. Afterward, using t duality, it is shown that we should have more objects in string theory called the D branes. Open strings excitations are seen as excitations of the D branes. Afterward, some features of the D branes are discussed which include the curvature of the D branes and the non-commutativity of spacetime that is suggested by the existence of d branes.

In the next section, D brane action is developed which is known as the DBI action. A recurrence relation for the tension of D branes with different dimensions is also worked out. The hardest part of this section is the calculation of the amplitude of the closed string exchange between two D branes. This amplitude can be calculated in two different ways and the comparison of these two ways gives us the relation between the open string and closed string coupling constant. Therefore, these coupling constants are not independent. Later, in chapter 13, the closed string coupling constant is defined in terms of the ratio of the strings and the 1-dimensional D brane.

In the last section of the chapter, the T duality arguments are applied to the unoriented string theories to argue the existence of new objects called orientifold planes. A sketchy derivation of The effective action of an orientifold plane is given to end the chapter.

Chapter 9
This chapter can be skipped in the first reading. This chapter mainly deals with higher-order amplitudes regarding their consistency, convergence, and unitarity. Since it has to deal with higher-order Riemann surfaces, it has to talk about the relevant mathematics. Relevant mathematics includes topics like Schottky groups and period matrices. These topics are discussed in Chapter 9.

The following sections talk about the procedure using which we can cut and sew Reimann surfaces together. The case where these Riemann surfaces have CFTs defined on them is also discussed. The discussion of cutting and sewing the surfaces such that CFTs on the resulting surfaces make sense is also carried out.

The following section talks about string field theory, which is a really hard topic to study, even for string theorists. The last few sections talk about the scattering amplitudes in the high-temperature limit and about the noncritical strings in low dimensions. Nonbosonic critical strings are strings that live in less than 26 dimensions. These strings are studied using a particular CFT known as the linear dilaton theory which was introduced in Chapter 2.

Appendix
In the appendix, Polchinski talks about the basics of path integral with bosonic and fermionic fields (i.e. Grassman numbers). Any student who has studied path integrals in QFT before should be able to read the appendix with ease. However, one thing that may be novel for some students is the fermionic two-state system.

Please spend some time understanding this part of the appendix because if you don’t understand this part, then you will have great difficulty in understanding the behavior of the bc ghost system (for example, equation 2.7.18). You will also struggle in section 10.4 of volume 2 where Polchinski derives the vertex operators of the superconformal ghosts. It is tricky to understand the difference between the ground state and the NS ground state without understanding the fermionic two-state system.

References
Polchisnki collects all the chapter-wise references for further reading at the end. Please check out this part of the book if you are looking for resources for further reading.