Section 1
The Virasoro algebra

This first part of the course is devoted to the introduction of the Virasoro algebra, one of the key players in conformal field theory (CFT). I begin with a discussion of conformal symmetries in arbitrary dimensions. Next, I introduce the concept of an infinitesimal symmetry, to explain how the so-called Witt algebra is generated by the algebra of infinitesimal orientation preserving conformal transformations on the punctured complex plane. To motivate the concept of central extensions, I give a brief account of quantization, leading to the search for central extensions of the holomorphic Witt algebra by C. I prove that up to equivalence it allows a unique one-parameter family of such central extensions, the Virasoro algebra at central charge c with c an arbitrary complex number.