Introduction to Conformal Field Theory in Two Dimensions

Annotation

Ideas and methods of conformal field theory are operational in different contexts of modern theoretical physics like string theory, statistical physics, theory of critical phenomena, etc. Its basic ingredient  — conformal symmetry — has a rich mathematical structure and imposes  severe restrictions on relevant physical quantities like spectrum and interactions. Though in higher dimensions conformal symmetry is finite-dimensional, in two dimensions it enjoys an infinite-dimensional extension that leads to additional interesting implications and finally paves the road to exactly solvable models.

This course provides an introduction to two-dimensional conformal field theory. It describes basic concepts like primary operators and their conformal families, correlations functions and Ward identities, mode expansions, Virasoro algebra and its Verma modules. Particular attention is paid to operator product expansions, conformal block decompositions of the correlation functions and bootstrap program.

Course plan

1. Symmetries in classical and quantum field theory. Conformal group and algebra, conformal Killing equations.
2. Primary and secondary operators. Correlation functions of two and three primary operators. 
3. (Anti)holomorphic representation for conformal Ward identities.
4. Radial quantization and radial ordering. States of conformal field theory. Operator-state correspondence.
5. Two-dimensional free scalar as conformal field theory.
6. Transformation properties of the stress tensor, central charge. Mode expansions. Virasoro algebra.
7. Verma modules for sl(2) and Virasoro algebras. 
8. Kac determinant and unitary bounds.
9. Operator product expansion (OPE). 
10. Chain equations and OPE coefficients.
11. Conformal block decompositions.  
12. Crossing symmetry and bootstrap. Ising model.

Literature

1. P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory, Springer 1997
2. R. Blumenhagen,  E. Plauschinn, Introduction to conformal field theory (with applications to string theory), Springer 2009