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Minimal lectures on two-dimensional conformal field theory

by Sylvain Ribault

This Submission thread is now published as SciPost Phys. Lect. Notes 1 (2018)

Submission summary

As Contributors: Sylvain Ribault
Arxiv Link: (pdf)
Date accepted: 2018-01-18
Date submitted: 2018-01-15 01:00
Submitted by: Ribault, Sylvain
Submitted to: SciPost Physics Lecture Notes
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We provide a brief but self-contained review of conformal field theory on the Riemann sphere. We first introduce general axioms such as local conformal invariance, and derive Ward identities and BPZ equations. We then define minimal models and Liouville theory by specific axioms on their spectrums and degenerate fields. We solve these theories by computing three- and four-point functions, and discuss their existence and uniqueness.

Published as SciPost Phys. Lect. Notes 1 (2018)

Author comments upon resubmission

I have made a number of relatively minor changes.

List of changes

Reply to Reviewer 1: The typo after (3.15) is now corrected.

Reply to Reviewer 3:

1. There was indeed a contradition because Axiom 3.4 was too strong. I
have modified that Axiom so that it applies only to fields that
correspond to states in the spectrum. Then OPEs of degenerate fields
are not covered, and should be discussed separately. I have added
such a discussion after Axiom 4.10.

2. I have added a paragraph at the very end of Section 5, in order to
make contact with the original definition (and naming) of Liouville
theory, and to refer the reader to ref. [3] for more details.

3. I have added a paragraph between eqs. (4.10) and (4.11) in order to
further discuss the guess for the lower bound, and to mention the
alternative guess that the lower bound could be zero.

4. Between eqs. (5.18) and (5.19), I have amended the discussion of
invariance under field renormalization: it is now stated that
invariant quantities are sufficient for checking crossing symmetry,
and explained why we may want to choose a normalization.

Additional changes:

1. At the end of the introduction, I have added a further motivation
for the minimalistic approach, namely that it can be useful in
research on other CFTs. Reference [6] is given as an example.

2. In Exercise 3.5, I have added substantial hints.

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