Physics 234A: String Theory I

Time and place for discussions: Thu, 3:40 pm - 4:30 pm, 402 Physics South. (If you are registered for the course but cannot make the regular discussion sessions in this time slot because of a time conflict, please contact me by email for individual arrangements.)

Instructor: Petr Hořava (email: horava@berkeley.edu)
Office: 401 Physics South.
Office hours: Tue, 2:30 pm - 3:30 pm.

This course will give a thorough introduction to modern string theory.

In the past three decades or so, string theory has become the dominant theoretical framework for addressing questions about the fundamental structure of matter in our Universe. It has produced many new ideas and insights in other fields of physics (and also mathematics), ranging from particle phenomenology to quantum gravity, cosmology, to condensed matter systems, and more recently to quantum information. Some of this surprising power of string theory is logically explained by the fact that string theory represents a natural extension or completion of the language of quantum many-body systems and quantum field theory, which makes it relevant in all those diverse areas where the methods of many-body physics play an important role. Basics of string theory should now be in every theorist's vocabulary, regardless of the specific field of physics.

Note that Physics 234A is a prerequisite for taking our String Theory II course, Physics 234B, taught in the Spring semesters, in which the more advanced aspects of quantum gravity and non-perturbative string theory are typically discussed. You should take 234A before you attempt to uderstand 234B.

Our main focus in the first semester will be on what I call the "alphabet" of string theory, and our goal will be two-fold: First, we will develop some practical understanding of the basic elements and techniques of string theory at the technical level. Simultaneously, we will also focus on understanding the "big picture" -- how string theory produces new ideas and techniques relevant to other areas of physics besides particle phenomenology and quantum gravity. Throughout the semester, I will also point out some of the open questions and puzzles of our field.

In this first semester, our 234A will specifically cover the following five areas:

I. Introduction: Why strings?
II. The bosonic string.
III. Superstrings, heterotic strings, supergravity.
IV. D-branes.
V. Intro to M-theory and nonperturbative string-string dualities.

The style will be somewhat similar to the style of my course taught in Fall 2008, Fall 2009, Fall 2011, Fall 2015, and Fall 2019. However, as the field is evolving, we will take a novel, fresh look at some of the basic questions and applications of string theory in the context relevant for current research. The main textbook is going to be

K. Becker, M. Becker and J.H. Schwarz, String Theory and M-Theory. A Modern Introduction (Cambridge University Press, 2006).

We will occasionally use other resources, including the two volumes of Joe Polchinski's book, the classic two volumes of Green-Schwarz-Witten, or the newer book by Elias Kiritsis (String Theory in a Nutshell), whose second and improved edition has been published recently (in 2019). Sometimes we will use review papers from the arXiv, especially for the more advanced topics.

Week 1: We only had our first lecture during this week, and there is so far no homework assignment. However, I do recommend a beautiful reference,
The Birth of String Theory, Editors: A. Capelli, E. Castellani, F. Colomo and P. DiVecchia (Cambridge University Press, 2012),
a collection of historical reminiscences from many of the founding fathers of string theory, covering the time period from before the beginnings in 1968 all the way to the first superstring revolution in 1986. Fascinating historical details can be found there, as well as a tapestry of the physics context driving the evolution of string theory in its earlier days. A great read! (Note that many of the individual contributions to this book have been posted by their authors in the arXiv.)

Week 2: As an introduction to the subject, we discussed the question: Why strings? We presented the five amazing "coincidences" that make strings special, in comparison to branes of any other dimension (or point-like particles). Recommended reading: Chapter 1 of [BBS] for a historical outline, and Pages 58-62 of Chapter 3 in [BBS] for the properties of conformal symmetries in various dimensions.

Week 3: At the beginning of the week, we first summarized the lessons from the five "coincidences" that make strings special among all branes, and then we started Part II of the course, the bosonic string. We are largely following Chapter 2 of [BBS] (you can also consult Ch. 2 of Green-Schwarz-Witten, or Ch. 1 of Polchinski). I particularly recommend going through Exercises 2.8 and 2.9 of [BBS].

HW 1 (due on Thu, Sept 16, in class): Problem 3.2 (on page 106) and Problem 2.2 (on page 54) from [BBS]. In addition, verify that the actions in Eqn. (2.4) and (2.5) on page 19 of [BBS] are invariant under infinitesimal worldline diffeomorphisms.

Week 4: We continued with the "Old Covariant Quantization" of the free bosonic string. Next, we will be discussing the absence of negative norm-squared states, and discuss the spectra of physical states in the critical dimension; to prepare for this, besides reading [BBS], you may want to read the first four-and-a-half pages of Chapter 4.1 of [Polchinski].

HW 2 (due on Thu, Sept 23, in class): Problems 2.5, 2.6, 2.9, 2.13 and 2.14 (on pages 55-7 of [BBS]).

Week 5: We contrasted the old covariant quantization and light-cone quantization of the free bosonic string, and started talking about the path-integral quantization that leads to the BRST formalism. We also studied the Hagedorn exponential growth of massive string states, and the path-integral origin of the string coupling constant from the worldsheet Einstein-Hilbert (Gauss-Bonnet) term in the Polyakov action. The Faddeev-Popov treatment of the worldsheet path integral can be found in Chapters 3.1-3.4 (and its extension to BRST in Chapters 4.2-3]) of [Polchinski, Vol. 1].

HW 3 (due on Thu, Sept 30, in class): Problems 2.8 and 2.15 (on pages 56 and 57 of [BBS]).

Week 6: We completed the discussion of the BRST quantization, both for general gauge theories with arbitrary Lie algebras of gauge symmetries, and for the string (as well as the pointlike particle).

HW 4 (due on Thu, Oct 7, in class):
Problem I: Show that the Faddeev-Popov determinant in the conformal gauge of the Polyakov path integral (as defined, for example, in Eqn. (3.3.11) of [Polchinski, Vol.1]) is gauge invariant.
Problem II: Show that the BRST transformation on the fields X and c, as defined in Eqn. (3.82) of [BBS], squares to zero.
Problem III: Show that the integrand in Eqn. (3.79) of [BBS] changes by a total derivative under the BRST transformations given in (3.82).
Starting in our next lecture (Tue, Oct 5), we will develop the important tool of 2d CFTs called the "operator product expansion" (or OPE for short). You can start practicing this technology by solving Problem 3.6 (on page 106 of [BBS]). If you are unfamiliar or uncomfortable with the OPE technique, you may want to postpone solving this problem until after Tuesday's lecture.

Week 7: We formulated abstract two-dimensional CFTs using the language of Operator Product Expansions (OPEs). Examples of more or less well-known CFTs have been given. It takes some practice to get used to this new, more abstract language and formulation! Today's homework will give you a good start on this :)

HW 5 (due on Thu, Oct 14, in class):
Problems 3.4, 3.7, 3.8, 3.10 and 3.11 (on pp. 106-7 of [BBS]).

Week 8: We further explored the landscape of 2d CFTs, and in particular discussed the full classification of all unitary CFTs with c=1. This classification leads to a surprising number of nontrivial features, including T-duality, an equivalence of fermionic systems to systems of bosons (known as "bosonization"), enhanced symmetry at special points of the moduli space, isolated CFTs without exactly marginal operators, and different branches of the moduli space connected to each other at special points (leading to a duality between two different target-space geometries). In this week's homework set, we will continue exploring the BRST formulation of the free strings using the abstract CFT techniques.

HW 6 (due on Thu, Oct 21, in class):
Problems 3.12, 3.13 and 3.14 (on pp. 107-8 of [BBS]).

Week 9: We applied our understanding of c=1 CFTs to bosonic string compactifications on a circle. We saw the emergence of T-duality, and the origin of the enhanced spacetime gauge symmetry at the self-dual radius of the compactification circle. We studied T-duality of more general sigma-model backgrounds, and discovered the Bucher rules (discussed in [BBS] in Chapter 9.4, for the closely related case of the bosonic "NS-NS" fields of the superstring). We formulated string scattering amplitudes at arbitrary genus, and studied the origin and structure of the moduli spaces of Riemann surfaces, and the associated symmetry of "modular invariance" (especially at genus one, the torus, where the condition of invariance of amplitudes under the SL(2,Z) modular group leads to powerful self-consistency constraints on string theory solutions).

HW 7 (due on Thu, Oct 28, in class):
Exercises 3.8 and 3.9 (on pp. 96-7 of [BBS]).
Next, Problem 6.5 (on p. 245 of [BBS]); WARNING: Eqn. (6.94) of [BBS], which is crucial for this computation, contains several typos -- there are three additive terms on line one, and four additive terms on line two of (6.94); the first and the third term on line two should each be multiplied by a factor of 2.
Finally, Problem 7.9 (on p. 293 of [BBS]).

Week 10: We wrapped up the discussion of string scattering amplitudes at genus g, and moved on from the bosonic string to the superstring (defined as the string theory with worldsheet supersymmetry). We revisited many of the features of the bosonic string in the supersymmetric case, and found many similarities and some important differences. The need for combining appropriately the Neveu-Schwarz and Ramond sectors (with their appropriate truncation) was explained by modular invariance, leading to Type 0A,B, Type I and Type IIA,B superstrings.

HW 8 (due on Thu, Nov 4, in class):
Problems 4.2, 4.6, 4.10, 4.11 and 4.13 (on pp. 144-6 of [BBS]).

Week 11: We discussed Type II and Type I superstrings in detail. Studied the modular invariance of the one-loop partition function, and how it determines the need for the "GSO projection" of the naive NS and R sectors. The massless spectrum contains various RR bosonic fields, which can couple electrically or magnetically to D-branes. We also touched on the existence of the NS5-brane, and discussed the importance of the BPS condition in supersymmetric theories. The appropriate D-branes and NS-branes will not only carry the corresponding form charges, they will be be half-BPS.

HW 9 (CHANGE of due date: since it turns out that Thu, Nov 11, is an academic holiday, there will be no lecture and no discussion that day. This HW is now DUE on FRIDAY, Nov 12, by 12 noon in my office or by email):
Problems 4.5, 4.7 and 4.14 (on pp. 145-7 of [BBS]); Exercises 8.1 and 8.2 (on pp.298-9 of [BBS]).

Week 12: We studied the formulation of the spacetime-supersymmetric solutions of the superstring using the Green-Schwarz formalism. We discussed the origin of the kappa-symmetry in the Goldstone theorem, in combination with the fact that the stable strings (and branes) of interest break only half of the spacetime supersymmetry.

HW 10 (due on Thu, Nov 18, in class):
Problems 5.1, 5.3 and 5.7 (on pp. 184-5 of [BBS]).

Week 13: We studied the free heterotic string, first in the fermionic formulation, and then we switched to the bosonic representation. FYI, the full classification of all ten-dimensional, Poincare-invariant, modular invariant solutions of superstring and heterotic string theories -- spacetime supersymmetric or not -- was first given in
H. Kawai, D.C. Lewellen and S.-H. H. Tye, Classification of Closed-Fermionic-String Models, Phys. Rev. D34 (1986) 3794;
see, in particular, Table I of that paper (note: what they called "Type A" and "Type B" we now call "Type 0A" and "Type 0B").

HW 11 (due IN TWO WEEKS, on Thu, Dec 2, in class):
Problems 7.3, 7.4, 7.5 and 7.7 (on pp. 292-3 of [BBS]).

Week 14: We introduced M-theory, described at low energies by supergravity in eleven dimensions, and studied its duality to Type IIA string theory upon compactification on a circle. We found the non-perturbative S-duality of Type IIB theory in ten uncompactified dimensions. For the continuation of this story of T-, S-, and U-dualities upon further compactification on tori, see Section 11.6 of E. Kiritsis's book String Theory in a Nutshell.

HW 12 (due on Thu, Dec 9, in class): This is our final official HW set of this semester! :) Problems 7.10 and 7.14 (on pp. 293-5 of [BBS]), and Problems 8.3 and 8.12 (on pp. 351 and 353 of [BBS]).

Week 15: We studied more examples of string and M-theory dualities, including the heterotic/Type-I SO(32) duality, the strong coupling M-theory limit of the E8xE8 heterotic string. Then we studied various half-BPS M-branes and D-branes as solutions of supergravity equations of motion (i.e., as "black branes"), we explained the M-theory origin of the D6-branes from the smooth Taub-NUT solution in M-theory. Finally, we discussed the Green-Schwarz anomaly cancellation mechanism in string theory, and its further refinement required in heterotic M-theory, with E8 super-Yang-Mills on each spacetime boundary component. Recommended exercises or problems to look at: For anomalies and their cancellation, you can read Exercises 5.9 and 5.10 in [BBS], and if you wish to practice some numerology of anomalies I recommend Problems 5.10-5.15 from [BBS]. For black holes and black branes, a good starting point would be Chapter 11 of [BBS], in particular Problems 11.2-11.5 (for starters). Of course, these topics would all be normally covered in Physics 234B: String Theory II ... :)

horava@berkeley.edu