Exact Results in Quantum Theory

Conformal field theories (CFT's), that is field theories with a non-anomalous conformal symmetry are ubiquitous in modern theoretical physics - suffice it to mention the minimal models that capture the dynamics of a large class of lattice theories at criticality, the gauged Wess-Zumino-Witten model that describes the continuum limit of a family of spin-chain models, the non-linear sigma model that defines a vacuum of the critical Polyakov string, or - in space-time dimensions higher than 2 - the supersymmetric gauge theories playing a central role in the AdS/CFT correspondence that provides us with precious insights into the physics of strongly-coupled condensed-matter systems.The above, in conjunction with the rich and much-explored structure, as well as a well-developed methodology of CFT, motivates taking a closer look at its generic features - a task to which the present lectures are devoted. In their course, I shall- introduce the conformal algebra and study its (quantum-)field-theoretic realisations;- indicate certain significant ramifications of conformal symmetry in the quantum theory, such as, e.g., the reconstruction of correlation functions;- briefly discuss the conformal (trace) anomaly.Once the general scheme has been laid out, I shall specialise to the particularly well-studied and important case - that of a CFT on a two-dimensional space-time. In this context, I shall- present the fundamental state-operator correspondence;- set up, using basic operator-algebraic techniques (such as, e.g., the OPE), a Hilbert-space description and embed it in the representation theory of the quantum symmetry algebra - the Virasoro algebra;- briefly discuss the concept of a current-algebra extension of the Virasoro algebra;- review chosen aspects of the conformal-bootstrap programme and address the issue of unitarity.I shall conclude the lectures by giving a long list of scandalous yet unavoidable omissions of topics from the forefront of the current research in the field.