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Conformal field theory in two dimensions

This page describes a graduate course of conformel 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

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  • Complex analysis: contour integrals of complex analytic functions on  .
  • Lie algebras and their representations.

Physical

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  • Notions of quantum mechanics and quantum field theory.

Course topics

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Conformal symmetry

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  • Scale invariance and conformal invariance.
  • Fixed points of the renormalization group.
  • Applications to statistical physics, high-energy physics, quantum gravity.

The bootstrap method

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  • Bootstrap vs Lagrangian.
  • Conformal symmetry in d dimensions.
  • Crossing symmetry.
  • Unitarity.

Conformal invariance in 2d

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  • Virasoro algebra and its representations.
  • Fields, energy-momentum tensor.
  • OPEs, fusion rules. Example: minimal models.
  • Correlation functions. Single-valuedness. Example: loop models.
  • Conformal blocks.

Analytic bootstrap in 2d

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  • Belavin-Polyakov-Zamolodchikov differential equations.
  • Shift equations for structure constants.
  • Double Gamma function and solutions of shift equations. Example: Liouville theory.

Additional symmetries

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  • Global vs local conformal symmetry.
  • Discrete symmetries: Ising and Potts models.
  • Chiral algebra: affine Lie algebras, W-algebras.
  • Interchiral symmetry.

CFTs based on affine Lie algebras

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  • 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).

Resources

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

Lecture notes on 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.
  • (E) TASI Lectures on the Conformal Bootstrap[2], by David Simmons-Duffin, 65 pages: from the basics of CFT in any dimension, to numerical bootstrap methods.

Lecture notes on 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.

Lecture notes on 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.
  • (E) Conformal Field Theory for 2d Statistical Mechanics[7], 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[8], 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[9], 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. ↑ Simmons-Duffin, David (2016). "TASI Lectures on the Conformal Bootstrap". arXiv.org. Retrieved 2024-08-31.
  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). "Exactly solvable conformal field theories". GitHub. Retrieved 2024-08-31.
  6. ↑ Ginsparg, Paul (1988). "Applied Conformal Field Theory". arXiv.org. Retrieved 2024-09-03.
  7. ↑ Estienne, Benoît; Ikhlef, Yacine (2023). "Conformal Field Theory for 2d Statistical Mechanics" (PDF). Retrieved 2024-08-31.
  8. ↑ Cardy, John (1996). Scaling and Renormalization in Statistical Physics. Cambridge University Press. doi:10.1017/cbo9781316036440. ISBN 978-0-521-49959-0. 
  9. ↑ Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997). Conformal Field Theory. New York Berlin Paris [etc.]: Springer Science & Business Media. ISBN 0-387-94785-X. 
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