| This thesis examines a number of aspects of two-dimensional conformally invariant quantum field theories. In chapter two, consider conformal field theories with generalized blocks which satisfy Ising-type algebras. By requiring the correct analytic properties of the correlators, we find constraints on the central charge, operator product coefficients and conformal dimensions of the fields. In chapter three, we use the Coulomb-gas approach to construct the minimal model conformal blocks on higher-genus Riemann surfaces. We define the higher-genus blocks by sewing, and write them in terms of the rational blocks of a compactified scalar field. We show that spurious states decouple, which implies that the blocks degenerate correctly. As an example, we compute the genus-two partition function, and verify modular invariance for the subset of minimal models which only require one type of screening charge. In chapter four, we show that the differential equations for the partition function of two-dimensional quantum gravity coupled to the minimal models follows from the extremization of certain actions. The result is first illustrated for the Ising and tricritical Ising models. We also demonstrate consistency with generalized KdV flows and describe a technique for computing the actions. |
