Abstract
We provide a brief but self-contained review of conformal field theory on the Riemann sphere. We first introduce general axioms such as local conformal invariance, and derive Ward identities and BPZ equations. We then define minimal models and Liouville theory by specific axioms on their spectrums and degenerate fields. We solve these theories by computing three- and four-point functions, and discuss their existence and uniqueness.
Highlights
Since the time of Euclid, mathematical objects are defined by axioms
Axiomatic definitions focus on the basic features of the defined objects, thereby avoiding alternative constructions that may be less fundamental
That these lectures are minimal does not just mean that they are relatively brief. This means that they omit many concepts and assumptions that are usually introduced in two-dimensional conformal field theory, but that are not necessary for our purposes. Among these concepts and assumptions, let us mention the existence of a vacuum state, the notion of unitarity, the construction of fields as operators on the space of states, radial quantization, and consistency on Riemann surfaces other than the sphere
Summary
Since the time of Euclid, mathematical objects are defined by axioms. Axiomatic definitions focus on the basic features of the defined objects, thereby avoiding alternative constructions that may be less fundamental. We check that these uniquely defined theories do exist, by studying their four-point functions It is the success of such checks, much more than a priori considerations, that justifies our choices of axioms. That these lectures are minimal does not just mean that they are relatively brief This means that they omit many concepts and assumptions that are usually introduced in two-dimensional conformal field theory, but that are not necessary for our purposes. Among these concepts and assumptions, let us mention the existence of a vacuum state, the notion of unitarity, the construction of fields as operators on the space of states, radial quantization, and consistency on Riemann surfaces other than the sphere. Minimalism is valuable for pedagogy, and for research: we will follow shortcuts that may have been hard to see when originally solving Liouville theory and minimal models, but that can be used when exploring other theories, for example non-diagonal theories [6]
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