Conformal Field Theory and Strings (16 lectures)

Conformal field theories are Euclidean quantum field theories that are characterised by the property that their symmetry group contains, in addition to the Euclidean symmetries, local conformal transformations, i.e. transformations that preserve angles but not lengths. They are relevant in (at least) three different areas of modern theoretical physics: they provide toy models for interacting quantum field theories, they describe two-dimensional critical phenomena and they play a central role in string theory, at present the most promising candidate for a unifying theory of all forces. The local conformal symmetry is of special importance in two dimensions since the corresponding symmetry algebra is infinite-dimensional in this case. As a consequence, two-dimensional conformal field theories have an infinite number of conserved quantities, and are completely solvable by symmetry considerations alone.

This course will introduce the basic concepts and methods of conformal field theory in a pedestrian fashion, with emphasis on the very special case of two dimensions, with aview to using these methods to derive some of the fundamental early results in bosonic string theory.

Outline of course

Conformal transformations: definition in d dimensions; conformal Killing equation; generators and their algebra when d is not 2; solutions to the 2d conformal Killing equation; Mobius transformations; algebra of local 2d conformal transformations (Witt algebra).

Conformal invariance in classical field theory: transformation of scalar fields; tracelessness of the energy-momentum tensor; examples of classically conformally invariant theories.

Conformal invariance in QFT: transformation of Green's functions and Ward identities; constraints on Green's functions from conformal invariance. The 2d case: quantum conformal generators; quasi-primary, primary and secondary fields; chiral fields; state-operator mapping; vertex operator algebra; OPE vs operator formalism; examples: minimal models, free boson, free fermion.

CFT methods in string theory: conformal transformations and conformal ghosts; derivation of the bosonic critical dimension d=26. Vertex operators and tree amplitudes: BRST charge, bosonization of ghosts; string scattering amplitudes.

Book for the Course

P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory (Springer, 1997)