Ph229ab Advanced Mathematical Methods of Physics:
Conformal Field Theory (2017)

Course Description

Quantum field theories fall into roughly two classes: "gapped" theories (where excited states are separated from the vacuum by an energy gap) and gapless theories. In gapped theories, correlations fall off exponentially with distance, so gapped theories are "almost trivial" at long distances. However, gapless theories have nontrivial correlations at long distances and display complicated and rich physics. In many cases (from boiling water to magnets to theories of quantum gravity), gapless theories have enhanced conformal symmetry and are described by Conformal Field Theory (CFT).

This course will cover nonperturbative methods in CFT, with applications to statistical physics, condensed matter physics, particle physics, and quantum gravity. We will use a combination of analytical and numerical techniques. For example, students will learn how to use computer algebra to study conformal correlation functions, and how to use convex optimization to derive bounds on critical exponents.

Course Information

Instructor: David Simmons-Duffin, Lauritsen 442, email: dsd.

Offered: Winter and spring terms, 2017-2018.

Class meetings: Tuesday and Thursday 9-10:30am in Lauritsen 469. My office hour is 10:30-11:30 (after class) on Tuesdays.

Grading and homework: This course will be pass/fail. There will be a few problem sets each quarter, which will be posted on this website. Students are encouraged to work together, but should write up their own solutions. Please use LaTeX. Instead of an exam, there will be final presentations at the end of each term.

Prerequisites: Students should have some background in quantum field theory. Come talk to me if you aren't sure whether your background is appropriate.

References

• Early lectures will follow some discussion in John McGreevy's notes on QFT
• Our main reference for axiomatic CFT will be my TASI Lectures on the Conformal Bootstrap.
• Another good reference for axiomatic CFT is Slava Rychkov's EPFL lectures.
• Textbooks on CFT and phase transitions include: Scaling and Renormalization in Statistical Physics by Cardy, and Conformal Field Theory by Di Francesco, Mathieu, and Senechal.
• Time permitting, we will introduce AdS/CFT following the approach of Jared Kaplan's notes and Joao Penedones' notes.
• A reference for scale vs. conformal invariance and renormalization group flows is Zohar Komargodski's lecture notes.
• The Bootstrap 2017 School has lots of great lecture notes and videos.

Lecture notes

Lecture notes can be found here. They are a work in progress (apologies if there are missing figures/references/sentences). I will be updating them as the course goes along.

If you notice any mistakes, or want to add anything, add an issue on the issue tracker, or even make a pull request!.

Assignments

1. Problem set 1, due February 1, 2018
2. Problem set 2, due February 20, 2018
3. Problem set 3, due March 8, 2018
4. Problem set 4, due May 8, 2018. The Mathematica notebook for problem 5 is here.
5. Problem set 5, due June 12, 2018.

Please choose a final presentation topic for winter quarter by March 1, 2018. Final presentations will be March 13th at 9am in Lauritsen 469.

Please choose a final presentation topic for spring quarter by June 2, 2018. Final presentations will be June 7, 2018 and June 12, 2018.

Course outline

Because this is the first iteration of this course, I don't know how much material we will cover. Here is a preliminary list of topics.

Introduction

• Path integrals, quantization, and the "quantum/classical correspondence"
• The renormalization group, fixed-points, and universality
• Scaling and critical exponents
• Scale vs. conformal invariance
• Examples of conformally-invariant theories

Axiomatic CFT

• Symmetries in QFT
• Conformal symmetry
• Primary operators
• Conformal correlators
• The embedding-space formalism
• Radial quantization and the state-operator correspondence
• Reflection positivity and unitarity bounds
• The operator product expansion
• Conformal blocks
• The bootstrap equations
• Weyl and 't Hooft anomalies

The conformal bootstrap in 2d

• Virasoro symmetry in 2d
• The 2d Ising model
• Modular invariance

Numerical methods

• Numerical bootstrap bounds
• Semidefinite programming
• Computing conformal blocks
• The 3d Ising model and the making of this video
• The modular bootstrap

Optional topics (depending on time and student preferences)

• Introduction to AdS/CFT
• Conformal perturbation theory
• Superconformal theories
• Large-spin perturbation theory
• OPE inversion formulae
• The SYK model
• Defects and boundaries
• Entanglement in CFT
• Monotonicity theorems
• CFTs at finite temperature