Conformal field theory in two dimensions

A graduate course on w:conformal field theory (CFT), with 18 hours lectures and 18 hours tutorials.

We sketch the main motivations of CFT, including its applications to statistical physics, high-energy physics, and quantum gravity. We introduce CFT in the bootstrap approach, an axiomatic approach that starts from symmetry and consistency conditions for quantum fields, and deduces crossing symmetry equations for correlation functions.

For most of the course, we specialize to 2 dimensions, where the existence of infinitely many conformal transformations leads to exact solutions of a number of nontrivial CFTs. We study the relevant technical constructions, from the Virasoro algebra to conformal blocks. Solving crossing symmetry and other constraints or assumptions, we obtain CFTs such as minimal models, Liouville theory and loop models. We also introduce CFTs that have extra symmetries beyond conformal symmetry, such as free bosons and Wess-Zumino-Witten models.

## Prerequisites

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### Mathematical prerequisites

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* Complex analysis: contour integrals of complex analytic functions on  C {\displaystyle \mathbb {C} } .
* Lie algebras and their representations.

### Physical prerequisites

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* Notions of quantum mechanics and quantum field theory.

## An introduction to CFT in general dimension

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### Conformal symmetry

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* Conformal transformations
* Conformal symmetry and gravitation
* Scale invariance and conformal invariance.

### Bootstrap approach

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* Bootstrap vs functional integral.
* Conformal symmetry, operator product expansions, crossing symmetry.
* Assumptions on extra symmetries and on the spectrum.
* Unitarity.

## Two-dimensional CFT

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### Fundamental structures

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* The Virasoro algebra and its representations.
* Fields, OPEs.
* Fusion rules. Examples: (generalized) minimal models.
* Correlation functions and conformal blocks.

### Exactly solvable theories

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* Spectrums.
* Loop CFTs.
* Liouville theory.
* Non-diagonal minimal models.

### Analytic bootstrap

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* Belavin-Polyakov-Zamolodchikov differential equations.
* Shift equations for structure constants.
* Double Gamma function and solutions of shift equations.

### Additional symmetries

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* Discrete symmetries: Ising and Potts models.
* Interchiral symmetry.
* Chiral algebras: affine Lie algebras, W-algebras.
* Free bosons, WZW models.

## Other topics

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These topics are a priori not covered in detail, but they could be mentioned, or be the subjects of student projects:

* Boundary CFT, defects. CFT on the cylinder.
* Entanglement entropy.
* Modular invariance.
* Numerical bootstrap.
* Coulomb gas approach.
* Fusing matrix (= fusion kernel).
* Conformal perturbation theory.
* Renormalization group flows between CFTs.

## Relevant Wikipedia articles

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The following Wikipedia articles are particularly relevant to this course. Consulting them can be helpful for seeing the relations of CFT with other subjects, and for finding relevant references. Moreover, student projects may involve criticizing these articles and improving them (see tutorial).

* w: Conformal field theory.
* w: Two-dimensional conformal field theory.
* w: Virasoro algebra.
* w: Minimal model (physics).
* w: Liouville theory.

## Resources

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(E) = texts with exercises, (ES) = texts with exercises and their solutions.

### CFT and bootstrap in general dimensions

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* EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions[1], by Slava Rychkov, 68 pages: an introduction to CFT that starts with a discussion of the history and ideas, and provides a guide to some of the relevant literature.
* The Conformal Bootstrap: Theory, Numerical Techniques, and Applications[2], by David Poland, Slava Rychkov and Alessandro Vichi, 81 pages: a review article that has much to say on the applications to 3d CFTs.

### 2d CFT in the bootstrap approach

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* (ES) Minimal lectures on two-dimensional conformal field theory[3], by Sylvain Ribault, 37 pages: a concise introduction to 2d CFT in the bootstrap approach.
* (E) Conformal field theory on the plane[4], by Sylvain Ribault, 145 pages: an introduction to 2d CFT in the bootstrap approach, including a chapter on affine symmetry.
* Exactly solvable conformal field theories[5], by Sylvain Ribault, 85 pages: an introduction to 2d CFT with an emphasis on exact solvability and on loop models.

### 2d CFT from other points of view

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* Applied Conformal Field Theory[6], by Paul Ginsparg, 178 pages: an early review that can still be useful, in particular for its treatment of free fermions and bosons, orbifolds thereof, and CFT on a torus.
* Conformal Field Theory and Statistical Mechanics[7], by John Cardy, 37 pages: a concise introduction to 2d CFT from the point of view of statistical mechanics.
* (E) Conformal Field Theory for 2d Statistical Mechanics[8], by Benoît Estienne and Yacine Ikhlef, 150 pages: a course that insists on statistical physics motivations and applications.

### Wider horizons

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* (E) Scaling and Renormalization in Statistical Physics[9], by John Cardy, 238 pages: an excellent text for understanding the role of CFT in statistical physics, although CFT is not its main subject.
* (E) Conformal Field Theory[10], by Philippe di Francesco, Pierre Mathieu and David Sénéchal, 890 pages: the Big Yellow Book on CFT, mostly in 2d, with an in-depth treatment of minimal models and Wess-Zumino-Witten models.

## References

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1. ↑ Rychkov, Slava (2016). "EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions". SpringerBriefs in Physics. doi:10.1007/978-3-319-43626-5. ISBN 978-3-319-43625-8. https://arxiv.org/abs/1601.05000.
2. ↑ Poland, David; Rychkov, Slava; Vichi, Alessandro (2019-01-11). "The conformal bootstrap: Theory, numerical techniques, and applications". Reviews of Modern Physics (American Physical Society) 91 (1). doi:10.1103/revmodphys.91.015002. ISSN 0034-6861. https://arxiv.org/abs/1805.04405.
3. ↑ Ribault, Sylvain (2018). "Minimal lectures on two-dimensional conformal field theory". SciPost Physics Lecture Notes (Stichting SciPost). doi:10.21468/scipostphyslectnotes.1. https://arxiv.org/abs/1609.09523.
4. ↑ Ribault, Sylvain (2014). "Conformal field theory on the plane". arXiv.org. Retrieved 2024-08-31.
5. ↑ Ribault, Sylvain (2024-11-26). "Exactly solvable conformal field theories". arXiv.org. Retrieved 2024-11-27.
6. ↑ Ginsparg, Paul (1988). "Applied Conformal Field Theory". arXiv.org. Retrieved 2024-09-03.
7. ↑ Cardy, John (2008-07-22). "Conformal Field Theory and Statistical Mechanics". arXiv.org. Retrieved 2024-11-12.
8. ↑ Estienne, Benoît; Ikhlef, Yacine (2023). "Conformal Field Theory for 2d Statistical Mechanics" (PDF). Retrieved 2024-08-31.
9. ↑ Cardy, John (1996). Scaling and Renormalization in Statistical Physics. Cambridge University Press. doi:10.1017/cbo9781316036440. ISBN 978-0-521-49959-0.
10. ↑ di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997). Conformal Field Theory. Springer. ISBN 0-387-94785-X.