Physics 234B: String Theory II

Quantum Gravity and Unifications

Spring 2015

the list of reading assignments has been posted here!

shortcut to latest Miscellaneous References

Basic Info

Lectures: Tue and Thu, 12:40-2pm.
Discussions: will start after Spring break. The primary time for discussions will be Thu, 3:40-4:30pm. We will also have a few discussions in the secondary time slot, Fri 2:10-3pm. If anyone anticipates having time conflicts during these discussion times, please let me know by email.

Place: 402 Le Conte Hall (both lectures and discussions)

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

This year, our focus in the 234B course will be on gravity -- both classical and quantum, how it emerges from string and superstring theory, whether it can have an independent life outside of strings, how it enters into various dualities and unifications across different areas of physics (from supersymmetric unifications with the Standard Model of particle physics, to unifications with non-gravitational physics via the AdS/CFT correspondence and other dualities, to nonrelativistic Lifshitz gravity and its relation to the Causal Dynamical Triangulations approach to quantum gravity, to applications in condensed matter physics). We will also discuss some of the many fundamental open puzzles of gravity.

A slightly more detailed plan for this course is as follows:

1. Survey of (semi)classical gravity.
This Chapter will serve a two-fold purpose: First, it is an introduction into the subject, making sure that we all start on the same page. Secondly, it will give a first glimpse of what are some of the fundamental puzzling features of gravity, from the perspective of many-body physics and quantum field theory, especially from the viewpoint of the renormalization group.

2. Superstring theory as quantum theory of gravity.
Strings and M-theory give us a beautiful, self-consistent and very rich arena for understanding how gravity can be combined with quantum mechanics. We will discuss first the question "Why Strings?", and will see several independent reasons why strings are special among objects of all possible dimensions. Then we will focus on the spacetime supergravity description, and discuss various string-string dualities and unifications.

3. AdS/CFT correspondence and holography.
AdS/CFT correspondence is perhaps the most successful byproduct of string theory. It is a duality, relating a seemingly non-gravitational quantum field theory on one hand, to a system with gravity (such as string theory) on the other. It therefore represents another "unification", of gravity with field theories both relativistic (of interest to particle physicists) and nonrelativistic (of interest to condensed matter and statistical physics). It also represents a constructive example of the (still rather mysterious) "holographic behavior" of quantum gravity, predicted by robust arguments based on the thermodynamic properties of black holes.

4. Effective field theory of gravity.
We will examine relativistic gravity using the framework of low-energy effective field theory. In this picture, the cosmological constant problem arises as the most pressing of all Naturalness problems. I will review the concept of technical Naturalness. We will also discuss the Appelquist-Carazzone decoupling theorem (and whether it should apply to gravity), and take a look at various previous attempts how to solve the cosmological constant problem. Also to be included: the Weinberg-Witten theorem.

5. Keldysh-Schwinger formalism, QFT out of equilibrium.
One of the best guarded secrets in theoretical physics: Relativistic QFT, as it is usually taught in our high-energy theory courses, is only a fragment of the whole story! It is a simplification of the whole thing, resulting from the assumption that the system has a stable, static and eternal vacuum. However, this assumption -- while OK for scattering particles at colliders -- seems rather dubious for gravity, especially when we take into account that we live in an evolving cosmological universe. The whole story of QFT without the simplifying assumptions about the vacuum is known for historical reasons as the "Keldysh-Schwinger formalism" -- I will review its basics, and take a look at how it is slowly making its way into quantum gravity.

6. Nonrelativistic gravity (often now called "Hořava-Lifshitz gravity"), causal dynamical triangulations.
This is a rather novel approach to making sense of gravity in a quantum theory, and it has attracted a lot of attention. The basic idea is to take QFT and the renormalization group seriously, but Lorentz invariance not so seriously. I will review the basics, including some applications to nonrelativistic AdS/CFT correspondence, and the comparison to the Monte Carlo lattice approach to quantum gravity known as Causal Dynamical Triangulations.

7. Effective field theory of inflation.
Methods of effective field theory have now been successfully applied to inflationary cosmology, a setting in which gravity and its quantum features are at the forefront. We will review the basics of this approach, and point out its connections to some of the previous Chapters discussed in this course.

Clearly, this is a rather ambitious and long list, which means that we will only be spending about two weeks on each of the 7 Chapters. At that pace, I expect that we will have covered Chapters 1 through 4 around Spring break, leaving Chapters 5 through 7 for after the break.

Miscellaneous References

Here I will be posting various comments, references and extra reading material options, on a weekly basis as we go through the material.

Week 1. Before we can quantize gravity we need to understand its basics classically. You can review classical General Relativity by leafing through Sean Carroll's book

S.M. Carroll, Spacetime and Geometry. An Introduction to General Relativity, Addison Wesley, 2004.

This book is now a classic, and your first reference if you need to refresh your memory on anything in classical GR.

Week 2. I mentioned in lecture that relativistic gravity with the general R-squared Lagrangian in 3+1 dimensions is formally asymptotically free, but when interpreted in Minkowski signature it suffers (at least on first inspection) from the presence of propagating ghosts and hence apparent violations of unitarity. Per audience request, here is a reference that discusses renormalization of the higher-derivative gravity and contains further useful references:
R. Percacci, A Short Introduction to Asymptotic Safety, arXiv:1110.6389.

Week 3. Here are some useful or interesting references related to the miscellaneous (semi)classical aspects of gravity reviewed in the lectures. Witten's spinor-based proof of the positive energy theorem in asymptotically flat spacetimes is in
E. Witten, A New Proof of the Positive Energy Theorem, Comm. Math. Phys. 80 (1981) 381.
The closest partial analog in de Sitter space holds only perturbatively, and with stringent boundary conditions at the cosmological horizon, as shown by
L.F. Abbott and S. Deser, Stability of Gravity with a Cosmological Constant, Nucl. Phys. B195 (1982) 76.
A very interesting discussion of some issues (related to the absence of a proper positive energy theorem) in quantum gravity in de Sitter space are in Witten's talk from the Strings 2001 in Mumbai,
E. Witten, Quantum Gravity in De Sitter Space, arxiv:hep-th/0106109.
The Chern-Simons approach to quantum gravity in 2+1 dimensions is discussed in some detail in the review paper
S. Carlip, Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe, Living Rev. Rel. 8:1 (2005), arxiv:gr-qc/0409039.
A fascinating extension and revision of the older results on quantum gravity in 2+1 dimensions, in view of AdS/CFT, can be found in
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, arxiv:0712.0155,
and are further extended in various follow-up papers by Alex Maloney and his group.
The generalization of Penrose's notion of conformal infinity of spacetime, required for more complicated asymptotics such as that of nonrelativistic AdS/CFT dualities, was presented in
P. Hořava and C.M. Melby-Thompson, Anisotropic Conformal Inifinity, arXiv:0909.2841.

Week 4. A very careful treatment of the proper (IR and UV) regularization and renormalization of the free scalar field theory in 1+1 dimensions can be found in Chapter 17 of
M. Stone, The Physics of Quantum Fields (Springer, 1999).
The large N limit is discussed in many textbooks on QFT and stat-mech, including Chapter VII.4 of Tony Zee's book Quantum Field Theory in a Nutshell. The 1+1 dimensional scalar field theory with O(N) symmetry (discussed in class) is solved in the large N limit for example on pages 463-5 of Peskin-Schroeder.
A rather prehistoric (written 15 years before AdS/CFT!), yet illuminating review paper of the idea that large N is actually a sort of classical limit:
L.G. Yaffe, Large N Limits as Classical Mechanics, Rev. Mod. Phys. 54 (1982) 407.
A great comprehensive reference on matrix models, their large N expansion and relation to the branch of string theory known as "noncritical string theory" is in Yu Nakayama Master's Thesis,
Y. Nakayama, Liouville Field Theory -- A Decade after the Revolution, arXiv:hep-th/0402009;
(warning: that review contains much much more than just the parts relevant for our lectures, so please read at your own risk!).
Attempts to generalize from matrices to higher-rank tensors have finally started producing somewhat intriguing results in recent years, under the name of "tensor models" (and to some degree, the closely related area of so-called "group field theory" pioneered by V. Rivasseau and his school); see for example the review
R. Gurau, A Review of the Large N Limit of Tensor Models, arXiv:1209.4295.

Week 5. For the basics of critical strings and superstrings, the book by K. Becker, M. Becker and J.H. Schwarz, String Theory and M-Theory. A Modern Introduction (Cambridge, 2007) represents a great reference. Compared to some other textbooks, it contains many of the more modern developments, including string dualities, M-theory, AdS/CFT, and supersymmetric quantum black hole/brane physics.
A nice outline of Friedan's derivation of the beta-functions in the nonlinear sigma model in 1+1 dimensions is presented in Ch. 3.4 of Volume 1 of the timeless classic Superstring Theory by M.B. Green, J.H. Schwarz and E. Witten (Cambridge, 1987).

Week 6. A comprehensive review of the first years of supergravity is
P. van Nieuwenhuizen, Supergravity, Phys. Rept. 68 (1981) 189.
A classic review of the first decade of supegravity in higher dimensions, and its compactifications to N>1 supergravities in 3+1 dimensions, is
M.J. Duff, B.E.W. Nilsson and C.N. Pope, Kaluza-Klein Supergavity, Phys. Rept. 130 (1986) 1.
At that point, supergravity was taken over completely by string theorists, and the more recent literature is then a part of the string theory literature (see, e.g., the book by Becker, Becker, Schwarz cited above).
There has been a lot of excitement and a lot of work recently on the possibility that N=8 supergravity in 3+1 dimensions may be finite, or at least finite up to some really high order in perturbation theory. Some interesting reviews of this subject are
R. Kallosh, An Update on Perturbative N=8 Supergravity, arXiv:1412.7117,
Z. Bern et al, Amplitudes and Ultraviolet Behavior of N=8 Supergravity, arXiv:1103.1848,
L.J. Dixon, Ultraviolet Behavior of N=8 Supergravity, arXiv:1005.2703,
Z. Bern, J.J.M. Carrasco and H. Johansson, Progress on Ultraviolet Finiteness of Supergravity, arXiv:0902.3765.
The subject is still quite controversial, see for example
T. Banks, Arguments Against a Finite N=8 Supergravity, arXiv:1205.5768.

Week 7. Among the many papers reviewing various aspects AdS/CFT correspondence, there are two really excellent starting points:
J.M. Maldacena, TASI 2003 Lectures on AdS/CFT, arXiv:hep-th/0309246;
G.T. Horowitz and J. Polchinski, Gauge/Gravity Duality, arXiv:gr-qc/0602037.
This second review is particularly well-matched to this course, as it is aimed at the gravitational audience. Both papers are a lot of fun to read.
It is also fascinating to compare the notion of holography, as it has developed in the past two decades under the influence of the explicit set of AdS/CFT and related examples, to the original notion of holography as first proposed (essentially for gravity in asymptotically flat spacetimes) by 't Hooft and Susskind in the two groundbreaking papers predating AdS/CFT:
G. 't Hooft, Dimensional Reduction in Quantum Gravity, arXiv:gr-qc/9310026;
L. Susskind, The World as a Hologram, arXiv:hep-th/9409089.

Week 8. In the week of March 9-13, we will have two guest speakers (both are our own BCTP postdocs): Stefan Leichenauer on Tuesday, and Misha Smolkin on Thursday. They will provide their own recommendations for reading material during their lectures.

Week 9. A good starting point for the effective field theory approach to quantum gravity is the review
J.F. Donoghue, The Effective Field Theory Treatment of Quantum Gravity, arXiv:1209.3511,
and references therein. (In particular, References [1-6] for more on EFT and its applications to gravity, and Refs. [33-39] on various aspects of the controversy surrounding the concept of a gravitational contribution to the running of the couplings in the Standard Model.)
A great pedagogical review of the basics of EFT, from one of its most avid practitioners, is
H. Georgi, Effective Field Theory, Ann. Rev. Nucl. Part. Phys. 43 (1993) 209.
(Just be aware: The correct spelling of Carazzone is Carazzone.)
The Weinberg-Witten theorems were proven in
S. Weinberg and E. Witten, Limits on Massless Particles, Phys. Lett. B96 (1980) 59. The Appelquist-Carazzone theorem goes back to
T. Appelquist and J. Carazzone, Infrared Singularities and Massive Fields, Phys. Rev. D11 (1975) 2856.
A good modern summary of EFT (including applications to gravity) is in Ch. 22 (especially 22.4) of
M.D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge U.P., 2013).
See also the relatively recent historical review
S. Weinberg, Effective Field Theory, Past and Future, arXiv:0908.1964.

Week 10. Spring break.

Week 11. Weinberg's extremely inspiring and timeless review paper on the cosmological constant problem:
S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys. 61 (1989) 1.
Witten's intriguing paper on preserving the role of unbroken SUSY in attempting to solve the cosmological constant problem:
E. Witten, The Cosmological Constant from the Viewpoint of String Theory, arXiv:hep-th/0002297.
The holographic explanation (and prediction!) of the smallness of the cosmological constant was discussed in
P. Hořava, M-Theory as a Holographic Field Theory, arXiv:hep-th/9712130.
See also the summary in
P. Hořava and D. Minic, Probable Values of the Cosmological Constant in a Holographic Theory, arXiv:hep-th/0001145.
A good review of the basics of the Keldysh-Schwinger formalism can be found in
A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge U.P., 2011).

Week 12. For another good review of the Keldysh-Schwinger formalism, see also Kamenev's lectures
A. Kamenev, Many-Body Theory of Non-Equilibrium Systems, arXiv:cond-mat/0412296.
Another book on nonequilibrium QFT, with applications to BEC, RHIC and early-universe cosmology, is
E.A. Calzetta and B.-L.B. Hu, Nonequilibrium Quantum Field Theory (Cambridge U.P., 2008).
This book exists also in an e-book form, available to Berkeley people (thank you Zach for pointing out this link): Calzetta & Hu, e-book.
Weinberg's very influential paper on the use of the Keldysh-Schwinger formalism (or, as he calls it, the "in-in" formalism) applied to inflationary cosmology is
S. Weinberg, Quantum Contributions to Cosmological Correlations, arXiv:hep-th/0506236.

Week 13. Some other notable review papers on the KS formalism are:
J. Rammer and H. Smith, Quantum Field-Theoretical Methods in Transport Theory of Metals, Rev. Mod. Phys. 58 (1986) 323,
G.A. Vilkovisky, Expectation Values and Vacuum Currents of Quantum Fields, arXiv:0712.3379.
Quantum gravity with anisotropic scaling was first proposed in
P. Hořava, Membranes at Quantum Criticality, arXiv:0812.4287
P. Hořava, Quantum Gravity at a Lifshitz Point, arXiv:0901.3775.
See also a brief review in
P. Hořava, General Covariance in Gravity at a Lifshitz Point, arXiv:1101.1081,
and a review of some cosmological aspects in
S. Mukohyama, Hořava-Lifshitz Cosmology: A Review, arXiv:1007.5199.
Here is the pdf file of my talk about Lifshitz Holography at the 2nd BCTP Summit at Lake Tahoe.

Week 14. In preparation for the chapter on the effective field theory of inflation, you may want to consult some basic reviews of inflation: A good modern review is in the 2009 TASI lectures
D. Baumann, TASI Lectures on Inflation, arXiv:0907.5424.
This review actually includes some applications of the "in-in" formalism to inflation. If you wish to consult a more introductory text on inflation, look at the 2008 TASI lectures
W.H. Kinney, TASI Lectures on Inflation, arXiv:0902.1529.
A nice physical picture of the basics of inflation is developed in the following book,
V. Mukhanov, Physical Foundations of Cosmology (Cambridge U.P., 2005).

Week 15. Other excellent resources for the effective field theory of inflation are:
D. Baumann, The Physics of Inflation, Cambridge lectures,
the lighter and more introductory notes
L. Senatore, TASI 2012 Lectures on Inflation,
and finally
C.P. Burgess, H.M. Lee ad M. Trott, Power-counting and the Validity of the Classical Approximation During Inflation, arXiv:0902.4465.

Week 16. No lectures -- review week.


On the gravity side, basics of General Relativity at the level of 231 are required. On the quantum side, basics of QFT at the level of 232A would be a good start, and will effectively be required.

On the other hand, basics of Yang-Mills gauge theories (either 233A or 232B) would also be useful, but will not be strictly required for students who wish to sign up for this course. Same with the basics of String Theory -- it would be useful and nice to have seen strings (at the level of 234A), but it will not be formally required in order to take this course. 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, so that we can 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.

Homeworks and Reading Assignments

There will be no homework sets for this course. Instead, there is a list of reading assignments, posted here, where each student gets to choose one from the list of 14 important or otherwise interesting gravity-related research papers. After Spring break, the students will present their summary of the ideas and results of their chosen paper during our Discussion Sessions. The final grade will be based on this presentation. There will be no final exam.