## Lecture course "Conformal Field Theory", WS 2014/2015

Conformal field theory is a quantum field theory that is invariant under all conformal transformations of space-time, including scale transformations. In 1+1 or 2 dimensions the conformal symmetry implies the existence of an infinite number of conserved charges which allows for an exact and non-perturbative determination of spectra and correlation functions. In contrast to the perturbative approach to quantum field theory, the focus here is on the formulation and solution of consistency conditions based on the symmetries of the theory.

Over the years, conformal field theory has developed into a powerful tool with applications to critical systems (in condensed matter theory & statistical physics), string theory and probability theory. It also exhibits crosslinks to various topics of modern mathematics such as knot theory and quantum groups.

In the lecture we will discuss the fundamental principles and the mathematical framework of conformal field theory. In addition, we intend to cover a few of the many concrete applications in physics.

Monday 12:00-13:30 (Seminarraum Theoretische Physik), Friday 10:00-11:30 (Container). The lecture on Wednesday will be split into lecture and on-the-spot exercises.

**Outline of topics:**

- Motivation: Scale invariance in critical systems
- Conformal transformations
- The Virasoro algebra and its representations
- Free bosons and fermions
- Minimal models
- WZW models
- From null vectors to differential equations for correlations functions
- Modular invariance
- Applications (depending on the background and interest of the participants)

**Prerequisites:**

- Complex Analysis (holomorphic functions & residue theorem)
- Quantum Mechanics
- Quantum Field Theory I is not mandatory but very helpful, in particular the notion of path integral is frequently used in CFT

**Literature:**

- Di Francesco, Mathieu and Senechal: "Conformal Field Theory", Springer Verlag
- Mussardo: "Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics", Oxford University Press
- Ginsparg: "Applied Conformal Field Theory" (Link)
- Cardy: "Conformal Field Theory and Statistical Mechanics" (Link)

**Schedule**

- Lecture 01: Introduction and motivation (Notes)
- Lecture 02: The philosophy of CFT, infinitesimal conformal transformations (Notes)
- Lecture 03: The Lie algebra of conformal transformations, finite transformations, representations of the conformal group, correlation functions in higher-dimensional CFTs (Notes)
- Lecture 04: Exercise session (Exercises and solutions)
- Lecture 05: Basics of conformal invariance in 2D, i.e. relation to holomorphicity, Moebius transformations, Witt algebra, primary and quasi-primary fields, form of correlation functions (Notes)
- Lecture 06: The energy momentum tensor and its properties, conformal Ward identities + Exercise session (Notes, Exercises and solutions)
- Lecture 07: More on the energy momentum tensor, the central charge c, radial quantization (Notes)
- Lecture 08: Mode expansions, from OPEs to commutation relations, the Virasoro algebra + Exercise session (Notes, Exercises and solutions)
- Lecture 09: Descendant fields and conformal families, correlation functions for descendants, more on the state-field correspondence (Notes)
- Lecture 10: Exercise session (Exercises and solutions)
- Lecture 11: The massless free boson as a conformal field theory, U(1) currents, vertex operators (Notes)
- Lecture 12: Exercise session (Exercises and solutions)
- Lecture 13: Representations of the U(1) chiral algebra and their characters, the spectrum of the free boson theory (compactified and uncompactified), the extended chiral algebra U(1)_k (Notes)
- Lecture 14: Exercise session (Exercises and solutions)
- Lecture 15: The massless free fermion as a free field theory (Notes)
- Lecture 16: Exercise session (Exercises and solutions)
- Lecture 17: Verma modules over su(2) and over the Virasoro algebra, null vectors and how to find them using the matrix of scalar products (Notes)
- Lecture 18: Exercise session (Exercises and solutions)
- Lecture 19: The Kac determinant, the Kac table and its symmetries, the minimal series M(p,p') (Notes)
- Lecture 20: Irreducible characters for the Virasoro minimal models, unitarity, from null vectors to correlation functions (Notes)
- Lecture 21: The spectrum of conformal field theories: The idea of the conformal bootstrap, modular invariance of the torus partition function (Notes)
- Lecture 22: The associativity of the operator algebra, crossing symmetry, fusion rules and their properties, the Verlinde formula (Notes)
- Lecture 23: Entanglement and finite size scaling (Notes)
- Lecture 24: The transverse field Ising model (Notes)
- Lecture 25: Exercise session (Mathematica notebook)
- Lecture 26: Bosonization of the Dirac fermion (Notes)
- Lecture 27: Exercise session (see previous Mathematica file)
- Lecture 28: Bosonization of the Dirac fermion: Equivalence of partition functions (Notes)
- Lecture 29: Exercise session (Exercises)
- Lecture 30: The fractional quantum Hall effect, braid group statistics in 2D, Laughlin's wavefunction (Notes)
- Lecture 31: The Moore-Read state, non-abelian statistics and topological order (Notes)