Mathematics

 

Two-dimensional conformal field theory (CFT) is a tool which is used in many branches of mathematical or condensed matter physics. One classical application of CFT is the worldsheet theory in the string theory. Another appearance is the second-order phase transitions in 2d systems, namely, it turns out that all critical points of such systems are described by some CFTs, and moreover, in  any cases they can be classified by CFT methods. The most recent application of CFT are non-linear equations of Painlev´e type, or more generally, isomonodromic deformations. The aim of this course is to give some basic knowledge of conformal field theory and consider some number of systems that allow solutions more explicit than in the generic CFT.

 

Basic knowledge of complex analysis, classical mechanics (or better classical field theory), quantum mechanics (or representation theory).

 

1. Symmetries and integrals of motion in classical field theory.
2. Conformal symmetry and other coordinate-dependent symmetries.
3. Introduction to quantum field theory, path integral, operator formalism. Example of the free boson and free fermions.
4. Symmetries in the quantum field theory, Ward identities, conformal Ward identities.
5. Operator product expansions, their relation to commutators, energy-momentum tensor.
6. Virasoro algebra and its Verma modules. Operator-state correspondence.
7. Expression of n-point functions in terms of 3-point, conformal blocks.
8. Liouville theory: quantum theory and its classical limit.
9. Reducible Verma modules. Degenerate fields and equations they satisfy.
10. Screening operators and integral representations.
11. CFT on a cylinder, correlation lengths and conformal dimensions.
12. Exact solution of the critical Ising model and its identification with CFT.
13. Boson-fermion correspondence. Equivalence between Thirring model and sin-Gordon.
14. Twist-field construction of conformal blocks, exact conformal blocks in the Ashkin-Teller model, twist fields for entanglement entropy.
15. Other conformal algebras: Kac-Moody, W-algebras, super-Virasoro. Their free-fermionic realizations, vertex operators, conformal blocks.
16. Free fermions and integrability. Fermionic twist fields.
17. Monodromies of degenerate fields/free fermions. Isomonodromic deformations, conformal field theory of Painlev´e VI.
18. Crossing symmetry. Fusion transformation of conformal blocks. Explicit fusion kernel at c = 1 from isomonodromic deformations.

Last topics can be adjusted according to the desires of participants.

 

One has to know the basics of CFT (points 1-7) and some of the advanced topics.

 

If you cannot find some of the books, ask me by e-mail, and I can provide them.

It is probably better to start reading from the shorter texts.

1. John Cardy, Conformal Field Theory and Statistical Mechanics, 2008
2. Paul Ginsparg, Applied Conformal Field Theory, https://arxiv.org/abs/hep-th/9108028v1
3. Zamolodchikov A. B., Zamolodchikov Al. B., Conformal field theory and critical phenomena in two-dimensional systems
4. Philippe Di Francesco, Pierre Mathieu, David S´en´echal, Conformal field theory
5. John Cardy, Conformal Invariance and Statistical Mechanics, 1988
6. Alexey Litvinov’s lecture notes on CFT, http://strings.itp.ac.ru/Lecture-Notes/CFT2022.pdf
7. Alexei Zamolodchikov and Alexander Zamolodchikov, Lectures on Liouville Theory and Matrix Models
8. C.J. Efthimiou, D.A. Spector, A Collection of Exercises in Two-Dimensional Physics, Part 1, https://arxiv.org/abs/hep-th/0003190