Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lecture notes on conformal field theory

Pavel Mnev

TL;DR

These notes establish a rigorous, math-focused account of two-dimensional conformal field theory through Segal’s functorial QFT lens, linking cobordism-based axioms to conformal geometry and operator structure. They articulate how conformal invariance, modularity, and conformal anomalies arise from projective representations and Virasoro symmetry, and they connect path-integral intuition to the sewing axioms and Ward identities. A broad spectrum of topics is surveyed, including genus-one modular invariance, the role of the moduli space of Riemann surfaces, and the representation-theoretic underpinnings via Virasoro and Kac–Moody algebras, plus links to string theory and 3-manifold invariants. The framework emphasizes the correlator-centric view of CFT, the OPE structure, and the geometric/topological context provided by Teichmüller theory and mapping class groups, making it a substantial mathematical foundation for further development and collaboration in CFT research.

Abstract

These are the notes on two-dimensional conformal field theory, based on a lecture course for graduate math students, given by P.M. in fall 2022 at the University of Notre Dame. These notes are intended to be substantially reworked and expanded in coauthorship with Nicolai Reshetikhin.

Lecture notes on conformal field theory

TL;DR

These notes establish a rigorous, math-focused account of two-dimensional conformal field theory through Segal’s functorial QFT lens, linking cobordism-based axioms to conformal geometry and operator structure. They articulate how conformal invariance, modularity, and conformal anomalies arise from projective representations and Virasoro symmetry, and they connect path-integral intuition to the sewing axioms and Ward identities. A broad spectrum of topics is surveyed, including genus-one modular invariance, the role of the moduli space of Riemann surfaces, and the representation-theoretic underpinnings via Virasoro and Kac–Moody algebras, plus links to string theory and 3-manifold invariants. The framework emphasizes the correlator-centric view of CFT, the OPE structure, and the geometric/topological context provided by Teichmüller theory and mapping class groups, making it a substantial mathematical foundation for further development and collaboration in CFT research.

Abstract

These are the notes on two-dimensional conformal field theory, based on a lecture course for graduate math students, given by P.M. in fall 2022 at the University of Notre Dame. These notes are intended to be substantially reworked and expanded in coauthorship with Nicolai Reshetikhin.
Paper Structure (263 sections, 76 theorems, 1236 equations, 43 figures)

This paper contains 263 sections, 76 theorems, 1236 equations, 43 figures.

Table of Contents

  1. Abstract
  2. Acknowledgements
  3. A long introduction: functorial picture of 2d conformal field theory
  4. Functorial framework for quantum field theory
  5. The definition of QFT
  6. Remarks.
  7. Unitarity (and its Euclidean counterpart)
  8. Examples of functorial QFTs
  9. TQFTs and a silly example
  10. $D=1$ Riemannian QFT -- quantum mechanics
  11. Quantum observables in the language of functorial QFT (the idea)
  12. Example: point observables in quantum mechanics
  13. 2d conformal field theory as a functorial QFT
  14. Genus one partition function, modular invariance
  15. Correction to the picture: conformal anomaly
  16. ...and 248 more sections

Key Result

Lemma 5.2.7

Let $D\subset \mathbb{R}^2$ be an open set. For a smooth map $\phi\colon D\rightarrow \mathbb{R}^2$ the following statements are equivalent:

Figures (43)

  • Figure 3: Sewing.
  • Figure 4: $\epsilon$-thickenings.
  • Figure 5: Point observable in quantum mechanics.
  • Figure 6: Correlator of several point observables in quantum mechanics.
  • Figure 7: (a) a generic cobordism in 2d CFT and some relevand cobordisms embedded in $\mathbb{C}$ -- (b) annulus (coformally equivalent to a cylinder) and (c) a 2d equivalent of Figure \ref{['l2_fig6']} (corresponding to several point observables).
  • ...and 38 more figures

Theorems & Definitions (360)

  • Remark 4.1.1
  • Remark 4.1.2
  • Remark 4.1.3
  • Remark 4.1.4
  • Remark 4.1.5
  • Remark 4.1.6
  • Remark 4.2.1
  • Remark 4.2.2
  • Remark 4.2.3
  • Example 4.2.4: Quantum mechanics of a free particle on a circle
  • ...and 350 more