Spaces of Quantum Field Theories

The concept of a "space of quantum field theories" or "theory space" was set out in the 1970's in work of Wilson, Friedan and others. This structure should play an important role in organizing and classifying QFTs, and in the study of the string landscape, allowing us to say when two theories are connected by finite variations of the couplings or by RG flows, when a sequence of QFTs converges to another QFT, and bounding the amount of information needed to uniquely specify a QFT, enabling us to estimate their number. As yet we do not have any definition of theory space which can be used to make such arguments. In this talk, we will describe various concepts and tools which should be developed for this purpose, inspired by the analogous mathematical problem of studying the space of Riemannian manifolds. We state two general conjectures about the space of two-dimensional conformal field theories, and we define a distance function on this space, which gives a distance between any pair of theories, whether or not they are connected by varying moduli. Based on talks given at QTS6 (University of Kentucky), Erice, Texas A& M, and Northwestern University. To appear in the proceedings of QTS6.