Speaker
Description
Two-dimensional conformal field theories (CFT) play an important role in the study of statistical mechanics, condensed matter systems and worldsheet descriptions of string theory. Even though two-dimensional CFTs possess an infinite dimensional symmetry algebra (Virasoro algebra), it is generically not possible to solve the theory exactly. Exceptions are so-called rational CFTs which have extended symmetry algebras that allow for complete solvability in all known examples. Therefore it is a natural problem to determine their distribution in the moduli space of all two-dimensional CFTs. For those rational CFTs for which one can construct a Hodge structure, there is a conjectured equivalence between the presence of an extended symmetry algebra of the CFT and existence of an enlarged endomorphism algebra of the Hodge structure (Hodge structures of complex multiplication type). If this relation turns out to be true, one could formulate the distribution of rational CFTs in terms of the distribution of Hodge structures of complex multiplication type. In this talk I will provide further evidence for the conjectured relation between rational CFTs and Hodge structures of complex multiplication type. In particular I will show how the Galois group naturally associated to any rational CFT provides under some technical assumptions an enlarged endomorphism algebra of the associated Hodge structure. This construction uses only CFT data and does not rely on a geometric realization of the associated Hodge structure.
This is based on upcoming work with Hans Jockers and Pyry Kuusela.
