Let g be a simple Lie algebra over the complex numbers and let (?, p) be an s-pointed curve (for any 1 ? s). We fix a positive integer c called the level or central charge. Let D = D(c) be the set of dominant integral weights of g of level at most c and let ? = (?(1), ..., ?(s)) be an s-tuple of weights with each ?(i) in D attached to the points p. To this data, there is associated the space of vacua (also called the space of conformal blocks) and its dual space, the space of covacua (or the space of dual conformal blocks), which are fundamental objects of this book. It is shown that these spaces are finite dimensional (the dimensions of which are given by the Verlinde dimension formula, stated and proved in Chapter 4). We prove the propagation of vacua, which shows that the space of vacua does not change by adding additional smooth points on the curve if we attach zero weight to these points. Then, we study in detail the spaces of vacua for ? the projective line.

The KZ equations are a fundamental mathematical structure related to hypergeometric functions. Solutions of all versions of KZ equations are given by multidimensional hypergeometric integrals. The semi-classical limit of KZ equations leads to basic quantum chain models of mathematical physics and representation theory. In this chapter we describe the main examples of the KZ equations (rational, trigonometric, elliptic, differential or difference) with integral hypergeometric solutions. We also describe the semi-classical limit of KZ equations and associated Bethe ansatz method as the semi-classical limit of the hypergeometric solutions.