We introduce Thompson field theory, a class of toy models of conformal field theory in which Thompson's group T takes the role of a discrete analogue of the chiral conformal group. T and the related group F are discrete transformations of dyadic partitions of the circle and the unit interval, respectively. When vectors or tensors are associated with partitions, one can construct a direct limit Hilbert space, here called the semicontinuous limit, and F and T have unitary representations on this space. We give an abstract description of these representations following the work of Jones. We also show that T can be thought of as acting on the boundary of an equal-time Poincaré disk in AdS3. This defines a representation of T on the Hilbert space that contains all tree-like holographic states, as introduced by Pastawski, Yoshida, Harlow, and Preskill. It also establishes a bulk-boundary correspondence through Imbert's isomorphism between T and Penner's Ptolemy group. We further propose definitions of field operators and correlation functions for the discrete theory. Finally, we sketch new developments like particle creation and annihilation, as well as black holes and possible connections with topological quantum field theory.