Physics 234B -- String Theory II

Spring 2019

link to the list of Reading Assignments
shortcut to Miscellaneous References

Basic Info

Instructor: Petr Hořava (email: horava@berkeley.edu)
Office: 401 Le Conte Hall.

Having learned the basic ingredients of the bosonic "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). The topics planned to be covered in our 234B include:

Part I. Preamble: Why strings? (a concise introduction to string theory).

Part II. Superstrings, M-theory and their dualities.

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

Part IV. Open questions in string theory, with particular focus on: Cosmology, time dependence and strings out of equilibrium, and puzzles of quantum gravity. [The exact topics covered in Part IV will be determined interactively depending on the wishes and needs of the audience, so any suggestions and/or preferences are welcomed.]

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).
[I use this book whenever I teach String Theory I, and it will be our go-to for the basic ingredients of the theory in the first parts of this semester.]

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).
[An entertaining introduction into modern applications of string theory, holography and AdS/CFT correspondence to condensed matter.]

Prerequisites

Grading, Homeworks, Reading Assignments

There will likely be no homework sets for this course (only recommended reading materials, for those interested in something extra beyond the lectures). Instead of homeworks, there will be 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

Week 1. As our informal warm-up Homework Assignment for Week 1, we had to look up a good proof of the Nambu-Golstone theorem. My favorite proof is that of
G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Broken Symmetries and the Goldstone Theorem, from 1968.
The same proof is explained very concisely in Ch 5.3 of:
T.-P. Cheng and L.-F. Li, Gauge Theory of Elementary Particle Physics, Oxford, 1984.
For a very interesting historical note from G. Guralnik, see:
G.S. Guralnik, arXiv:0907.3466.

Week 2. Another great and thorough -- and more mathematically rigorous -- source of information on the Goldstone theorem and spontaneous symmetry breaking is:
F. Strocchi, Symmetry Breaking (Lecture Notes in Physics 732, Springer, 2008).
Definitely a good read!

Week 3. The idea of a large N expansion in relativistic quantum field theory (for QCD and similar theories) was presented in
G. 't Hooft, A Planar Diagram Theory for Strong Interactions, N ucl. Phys. B72 (1974) 461.
Over the next 1/4 century, this idea culminated in the celebrated AdS/CFT corre spondence, which we will discuss in Part III at great length. For now, you can consult the introduction about large N and how it leads to string theory in the opening paragraphs of:
J. Maldacena, TASI 2003 Lectures on AdS/CFT, arXiv:hep-th/0309246.
In stat-mech, the large N idea was applied very successfully earlier,
H.E . Stanley, Dependence of Critical Properties on the Dimensionality of Spins, Phys. Rev. Lett. 20 (1968) 589.
A good pedagogical review of the large N approach to QCD is:
A. Manohar, Large N QCD, arXiv:hep-ph/9802419.
That same review also contains a detailed discussion of the large N expansion of the Gross-Neveu model, which we discussed in class.
Another great paper to read, from the founding father of this field, is
G. 't Hooft, Large N, arXiv:hep-th/0204069.

Now that we see why strings are special and unique among all p-branes, we will begin Part II of the course, starting with the systematic construction of the worldsheet picture of the superstring, following Chapter 4 of [BBS].

Week 4. We shall continue the study of the worldsheet formalism for the superstring following [BBS], with Professor Ori Ganor as the guest speaker.

Week 5. The careful treatment of two-dimensional free scalar field theory (including the correct IR and UV regularization) is discussed nicely in Ch. 17 of:
M. Stone, The Physics of Quantum Fields (Springer, 2000).
The role of the logarithmic propagator of the two-dimensional scalar field in string theory cannot be overstated; some related implications for the behavior of the fundamental string in spacetime can be found in
M. Karliner, I. Klebanov and L. Susskind, Size and Shape of Strings, Int. J. Mod. Phys. A3 (1988) 1981.

Weeks 6 and 7. We continued with the story of the superstring in the NSR and then GS formulations, following Chapters 4 and 5 of [BBS]. Here are some additional references to the relevant material:
Modular invariance as a crucial self-consistency constraint is discussed in Ch. 7.2 of {Polchinski, Vol.1]. How modular invariance applies to the superstring (and how it makes a GSO projection mandatory) is in Ch.10.7 of [Polchinski, Vol.2]. This section also contains a discussion of the spacetime-nonsupersymmetric Type 0A and 0B solutions of superstring theory.
More general structure of multiloop string amplitudes (including the details of the structure of the moduli spaces and modular symmetries, both for bosonic strings and for superstrings) are covered in the classic review:
E. D'Hoker and D.H. Phong, The Geometry of String Perturbation Theory, Rev. Mod. Phys. 60 (1988) 917.
This detailed paper represented the state-of-the-art of the subject for several decades. The subject has been revived somewhat in recent years; for some of the more recent progress from this decade, see e.g.:
E. Witten, Superstring Perturbation Theory Revisited, arXiv:1209.5461,
and
E. Witten, More on Superstring Perturbation Theory: An Overview of Superstring Perturbation Theory via Super Riemann Surfaces, arXiv:1304.2832.
For a geometric understanding of Wess-Zumino terms in quantum field theory, and their origin in Lie algebra cohomology, see:
J.A. de Azcarraga and J.M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics (Cambrdige U.P., 1995),
especially Chapter 8 of that book. In particular, the Wess-Zumino terms relevant for the Green-Schwarz superstring, and more generally for other supersymmetric extended objects, are discussed in detail in Sections 8.7 and 8.8 there.

Week 8. For the classification of all ten-dimensional solutions of supertring theory consistent with modular invariance, see:
H. Kawai, D.C. Lewellen and S.-H.H. Tye, Classification of Closed Fermionic String Models, Phys. Rev. D34 (1986) 3794,
and also
L.J. Dixon and J.A. Harvey, String Theories in Ten Dimensions without Spacetime Supersymmetry, Nucl. Phys. B274 (1986) 93.

Week 9. Our discussion of the heterotic string followed mostly the logic of Chapter 7 of [BBS]. We concluded the week with the discussion of the spectrum of supersymmetric D-branes in ten-dimensional superstrings, following Chapter 6 of [BBS].
An excellent resource on the mathematics of lattices in diverse dimensions, and their application to miscellaneous other problems, is:
J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups (Springer, 3rd edition 1999).
I find the connection to error-correcting codes particularly fascinating! It might be a fun read for Spring break :)

Week 10. Spring break.

Weeks 11 and 12. We are taking the spacetime perspective, and studying string theory (and M-theory) using the spacetime effective supergravity. The main reference that we are more-or-less following is Ch. 8 of [BBS]. Another good reference on the various black-brane solutions of supergravity theories (and their additional properties, such as thermodynamics away from extremality etc) is Clifford Johnson's book entitled simply D-Branes (Cambridge University Press, 2003).
The spacetime description is an excellent starting point for discovering various nonperturbative string dualities and M-theory.

Week 13. We continued with our analysis of string and M-theory in the spacetime effective-action picture, taking a closer look at nonperturbative dualities in 10 and 11 dimensions, our discussion paralleled that of Ch. 8 of [BBS]. The famous Green-Schwarz anomaly-cancellation mechanism in ten dimensional supergravities is discussed in Ch. 5.4 of [BBS], and in many other places. Its refinement to heterotic M-theory was developed in:
P. Hořava and E. Witten, Heterotic and Type I String Dynamics from Eleven Dimensions, arXiv:hep-th/9510209,
and in further detail in
P. Hořava and E. Witten, Eleven-Dimensional Supergravity on a Manifold with Boundary, arXiv:hep-th/9603142.
A very elegant and concise explanation of the cohomological approach to gauge anomalies (including the Wess-Zumino consistency conditions and the Stora-Zumino descent equations) is in Ch. 22.6 of Weinberg's Quantum Theory of Fields (Vol. 2).

Week 14. We wrapped up our discussion of miscellaneous string/string, string/M-theory and F-theory dualities, and returned to the topic of AdS/CFT correspondence.

Week 15. An excellent intoduction into, and an overview of, AdS/CFT holographic duality is in Maldacena's lectures cited above in Week 3, when we were talking about large-N expansions:
J. Maldacena, TASI 2003 Lectures on AdS/CFT, arXiv:hep-th/0309246,
and in the more recent book
M. Ammon and J. Erdmenger, Gauge/Gravity Duality. Foundations and Applications (Cambridge U.P., 2015).

Week 16. We continued our lectures in RRR week, with some selected "open questions" in string theory. The selection is of course highly personal. I focused on three topics: Holographic renormalization and nonrelativistic systems (how much of the "nonrelativistic realm" of string theory are we missing?), puzzles of naturalness in Nature, in particular: the Higgs mass hierarchy puzzle, the cosmological constant puzzle, the eta problem of effective field theory of inflation (can string theory help?), and systems out of equilibrium (why is string theory so good at describing static, highly supersymmetric configurations, but apparently not so good for systems out of equilibrium, including our own cosmologically evolving universe?).
Here are some relevant references:
Holographic renormalization in the context of AdS/CFT correspondence is reviewe\ d in
K. Skenderis, Lecture Notes on Holographic Renormalization, arXiv:hep-th/0209067,
and in
J. de Boer, The Holographic Renormalization Group, arXiv:hep-th/0101026.
Nonequilibrium systems are described in QFT by the Keldysh-Schwinger formalism. An excellent review of KS formalism, in the modern path-integral language, is:
A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge U. Press, 2011).
Applications of the KS formalism (referred to in this paper as the "in-in" formalism) to effective theory of cosmological inflation are brilliantly explained in:
S. Weinberg, Quantum Contributions to Cosmological Correlations, arXiv:hep-th/0506236.
In particular, the Appendix derives the path-integral representation of the KS formalism from scratch, in the context appropriate for cosmology. Definitely worth reading!
Gerard 't Hooft's influential paper on Technical Naturalness can be found here:
G. 't Hooft, Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking, in: Recent Developments in Gauge Theories (Springer, 1980).
An interesting read about puzzles of Naturalness is:
M. Dine, Naturalness Under Stress, arXiv:1501.01035.
For puzzles of Naturalness viewed from the perspective of nonrelativistic systems, you can also take a look at my own paper
P. Hořava, Surprises with Nonrelativistic Naturalness, arXiv:1608.06287.

horava@berkeley.edu