Physics 234B -- String Theory II:

Beyond the Alphabet

Spring 2016

the list of reading assignments has been posted!
shortcut to Miscellaneous References

Basic Info

Lectures: Tue and Thu, 12:40-2pm, 402 Le Conte Hall.
Discussions: will start on April 1, and will contain students' presentations of their selected reading assignments.
Time and place: Thursdays 4:10-5pm, Fridays 1:10-2pm, 402 Le Conte. We will alternate between the two time slots, according to the exact schedule of the discussion sessions as posted here.

Instructor: Petr Hořava (email:
Office: 401 Le Conte Hall.

Having learned the basic ingredients of the "string-theory alphabet" in an introductory course String Theory I, our focus in the second semester will be on exciting advanced topics in string theory, in which the various ingredients come together (and where intriguing open challenges remain). After polling the students in my Fall 2015 234A class, I have decided that we shall focus on four major themes, divided into four relatively equally important parts:

Part I. Holography: AdS/CFT Correspondence and Other Holographic Dualities.

Part II. String Geometry: Topological Strings, Compactifications, Elements of Mirror Symmetry.

Part III. Cosmological Inflation: From the Effective Field Theory Approach, to String Inflation.

Part IV. Open challenges of quantum gravity: Strings and Gravity out of Equilibrium; Keldysh-Schwinger Formalism, ... (additional possible topics in Part IV will be determined interactively based on the interests of the students in the class).

Some interesting textbooks that we may use (to varying degree) are:

K. Becker, M. Becker and J.H. Schwarz, String Theory and M-Theory. A Modern Introduction (Cambridge University Press, 2006).
[We used this book for our String Theory I last semester, and it will be our go-to for the basic ingredients of the theory.]

E. Kiritsis, String Theory in a Nutshell (Princeton University Press, 2007).
[Another good one-volume survey of the modern aspects of string theory.]

M. Ammon and J. Erdmenger, Gauge/Gravity Duality. Foundations and Applications (Cambridge University Press, 2015).
[A recent survey of the basics of AdS/CFT correspondence, a good reference text for AdS/CFT basics.]

D. Baumann and L. McAllister, Inflation and String Theory (Cambridge University Press, 2015).
[This will be our go-to text for our critical survey of inflation.]

J. Zaanen, Y.-W. Sun, Y. Liu and K. Schalm, Holographic Duality in Condensed Matter Physics (Cambridge University Press, 2015).
[I am very excited about this book, although I haven't seen much of it yet -- it could be brilliant! Certainly a good source for the first few weeks of the semester.]

Grading, Homeworks, Reading Assignments

There will be no homework sets for this course (only recommended reading materials, for those interested in something extra beyond the lectures). Instead of homeworks, there is a list of reading assignments. Each student will choose one from this list of important or otherwise interesting string-related research papers. During the second half of the semester, the students will present their summary of the ideas and results of their chosen paper during our Discussion Sessions, according to the schedule posted with the List. The final grade will be based on this presentation. There will be no final exam.

Miscellaneous References & Reading Suggestions

Here I intend to post various comments, references and extra reading material options, on a weekly basis as we go through the material.

Week 1. A very interesting review of the basics of perturbative string theory is the first half of Joe Polchinski's 1994 TASI lectures:
J. Polchinski, What is String Theory?, arXiv:hep-th/9411028.
This paper is particularly interesting, as it was written just before the "2nd Superstring Revolution" (which happened in 1995), and there are probably useful lessons to learn from this paper in this day as well.
Reference [1] of Polchinski's TASI lectures is Ken Wilson's 1982 Nobel lecture, which should also be a mandatory reading for everyone who wishes to study QFT or many-body physics (and, by extension, string theory) -- it represents a fascinating brief account of, and introduction into, the Wilsonian renormalization group paradigm, the most important conceptual platform for almost everything we will do this semester:
K.G. Wilson, The Renormalization Group and Critical Phenomena, Rev. Mod. Phys. 55 (1983) 583.
And finally (at least for the references to our first lecture), a brief early history of string theory can be found in
J.H. Schwarz, The Early History of String Theory and Supersymmetry, arXiv:1201.0981.
The large-N expansion: The elegant large-N solution of QCD in two dimensions using light-cone gauge is in the short paper (only ten pages long!):
G. 't Hooft, A Two-Dimensional Model for Mesons, Nucl. Phys. B75 (1974) 461.

Week 2. G. 't Hooft's 2002 review of large N is here:
G. 't Hooft, Large N, arXiv:hep-th/0204069.
A nice "classic" review of the large N limit as a classical limit is in:
L.G. Yaffe, Large N Limits as Classical Mechanics, Rev. Mod. Phys. 54 (1982) 407.
As mentioned in class, with a few relatively uninteresting exceptions, scale invariance in a relativistic QFT implies full conformal invariance, but the precise statement is still work in progress. For an interesting recent contribution to this subject, with references to relevant older papers, see for example:
A. Dymarsky and A. Zhiboedov, Scale-Invariant Breaking of Conformal Symmetry, arXiv:1505.01152.

Week 3. The proof that N=4 super Yang-Mills is UV finite, and hence a CFT, was given by our very own Stanley Mandelstam :) in:
S. Mandelstam, Light-cone Superspace and the Ultraviolet Finiteness of the N=4 Model, Nucl. Phys. B213 (1983) 149.
The method he used? You guessed it ("whenever in doubt about a new, complicated theory ...") -- he used a light-cone version of superspace! Well, it also says that in the title ...
The groundbreaking paper that gave rise to the entire field of AdS/CFT correspondence is of course
J.M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, arXiv:hep-th/9711200.
So far, this paper has over 11,000 citations, and counting.
The two groundbreaking papers which presented the idea of gravitational holography are:
G. 't Hooft, Dimensional Reduction in Quantum Gravity, arXiv:gr-qc/9310026,
L. Susskind, The World as a Hologram, arXiv:hep-th/9409089.

Week 4. The lecture on Tuesday, February 9 will be given by a guest speaker, Stefan Leichenauer. He will talk about the recent results on the quantum version of the null energy condition, from the perspective of holography and large N.

Week 5. The basic dictionary of AdS/CFT correspondence, and its basic checks, are summarized in a very readable form in:
J.M. Maldacena, TASI 2003 Lectures on AdS/CFT, arXiv:hep-th/0309246.
Holographic renormalization is discussed in some detail in [Ammon&Erdmenger] (as cited above), Chapter 9; a detailed review is in
K. Skenderis, Lecture Notes on Holographic Renormalization, arXiv:hep-th/0209067;
see also Chapter 5 of [Zaanen et al] (as cited above).
We talked about two major examples of holography that preceded AdS/CFT correspondence. A state-of-the-art review of old matrix models, their dualities and also their modern interpretation in terms of open-closed string duality, D0-branes and tachyon condensation, can be found in:
Y. Nakayama, Liouville Field Theory: A Decade after the Revolution, hep-th/0402009.
The multi-layered history of M(atrix)-theory is beautifully explained in the short and very readable review
J. Polchinski, M-Theory and the Light Cone, hep-th/9903165.

Week 6. Both lectures in Week 6 will be given by a guest speaker, Misha Smolkin, who will give an overview of entanglement entropy in field theory, quantum gravity and holography.

Week 7. A comprehensive review of applications of M(atrix) theory to processes in M-theory is in:
W. Taylor, M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory, arXiv:hep-th/0101126.
The basics of BRST quantization are explained very clearly in Joe Polchinski's String Theory book, Volume 1, Chapter 4.2. The procedure is based on the Faddeev-Popov treatment of the gauge fixing steps during quantization of gauge theories, as reviewed in many basic QFT texts; see for example Weinberg's Quantum Theory of Fields, or Peskin-Schroeder (or Polchinski Volume 1 for the special case of the gauge symmetries on the string worldsheet).
The Topological Quantum Field Theory revolution was initiated by E. Witten in 1988 in a series of papers in which theories of the "cohomological" type were introduced:
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353,
E. Witten, Topological Sigma Models, Commun. Math. Phys. 118 (1988) 411,
E. Witten, Topological Gravity, Phys. Lett. B206 (1988) 601.
The connection between topological gravity in two dimensions and the matrix-model definition of non-critical strings in D<2 dimensions was first explored in
E. Witten, On the Structure of the Topological Phase of Two-Dimensional Gravity, Nucl. Phys. B340 (1990) 281.

Week 8. A very nice and clear early review of the basic logic of Topological QFT is in
E. Witten, Introduction to Cohomological Field Theories, Int. J. Mod. Phys. A6 (1991) 2775.
A very detailed, comprehensive review of supersymmetric & topological sigma models, the topological twist, the A- and B-models, in the context of mirror symmetry can be found in:
Mirror Symmetry (C. Vafa and E. Zaslow, editors), Clay Mathematics Monographs, Volume 1 (2003); or go here for the pdf file of this book.

Week 9. No new material, besides the topological and mirror symmetry references from last week -- but our List of Reading Assignments has been posted!

Week 10. Spring break.

Weeks 11 & 12. For more on the A- and B-model, see also the original paper
E. Witten, Mirror Manifolds and Topological Field Theory, hep-th/9112056.
Witten's open string field theory was introduced in
E. Witten, Non-commutative Geometry and String Field Theory, Nucl. Phys. B268 (1986) 253,
way before topological field theories. The relationship between topological strings of the A and B model, string field theory and Chern-Simons gauge theories was established in
E. Witten, Chern-Simons Gauge Theory as a String Theory, hep-th/9207094;
see also
M. Marino, Les Houches Lectures on Matrix Models and Topological Strings, hep-th/0410165.
Besides the book on String Inflation referenced above, there are many really good review papers on inflation. Two that are particularly notable are:
D. Baumann, TASI Lectures on Inflation, arXiv:0907.5424,
D. Baumann, The Physics of Inflation, Cambridge U. lectures.

Week 13. For a nice introductory review of Effective Field Theory and its applications to gravity (before we go back to cosmology), see the Petropolis lectures by John Donoghue,
J.F. Donoghue, The Effective Field Theory Treatment of Quantum Gravity, arXiv:1209.3511.
And for some more advanced material on this subject, see the references therein.

Week 14. A nice critical review of problems, challenges and difficulties of String Inflation is this very readable paper (with virtually no equations!):
C.P. Burgess and L. McAlister, Challenges for String Cosmology, arXiv:1108.2660.
We will not be able to discuss this paper in any detail in lectures, but I do recommend it for private reading.
In the Discussion Session on Thursday this week, the question of Green-Schwarz formalism, kappa-symmetry and their relation to superspace cohomology has come up. The story is quite beautiful and now well-understood; a really elegant review of these facts and relations can be found in
J.A. Azcarraga nd J.M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics (Cambridge U., 1995).

Week 15. In the past few weeks, we've talked separately about inflation, and about the Keldysh-Schwinger formalism (also referred to as the "in-in" formalism) which is the proper way how to treat quantum field theory in time-dependent situations. An excellent source for how these two topics are brought together is:
S. Weinberg, Quantum Contributions to Cosmological Correlations, arXiv:hep-th/0506236.
This beautiful paper shows how to properly do calculations in inflation using the KS formalism. Specifically, the Appendix derives the path-integral representation of the KS formalism from scratch, in the context appropriate for cosmology. Definitely worth reading!
A very nice overview of the KS formalism (including its path-integral formulation), mostly in the context of cond-mat physics (but not discussing cosmology) is:
A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge U. Press, 2011).

Week 16. This week is the Review Week, and there will be no official lectures or discussions. In the final weeks of the semester, we essentially covered the materials in Chapters 1 through 4 of the Baumann-McAllister book, plus some of the materials in the Appendices. This leaves Chapter 5, which represents a long detailed laundry list of various (attempts at) implementations of inflation in string theory. If interested in this topic, the students should now have all the background needed, to read this Chapter 5 as independent reading. This may be a good way how to spend the Review Week :)


Basics of General Relativity at the level of 231 and of QFT at the level of 232A would be great. Some basic knowledge of the "string alphabet", rougly at the level of my String Theory I course, are required, but having completed the official 234A course is not strictly a prerequisite -- I encourage all students interested in the material to sign up for this course, regardless of prior background. If you are interested in this course and have questions about having all the prerequisites, feel free to talk to me (in person or via email); I will try to be flexible, my intention is to accommodate as many students in this course as possible, even those who come from various diverse backgrounds such as particle phenomenology, cosmology, condensed matter, or pure math.