Title: Quadratic reciprocity from a family of adelic conformal field theories

Abstract: We consider a deformation of the 2d free scalar field action by raising the Laplacian to a positive real power. It turns out that the resulting non-local generalized free action is invariant under two commuting actions of the global conformal symmetry algebra, although it’s no longer invariant under the local conformal symmetry algebra. Furthermore, there is an adelic version of this family of global conformal field theories, parametrized by the choice of a number field, together with a Hecke character. Tate’s thesis plays an important role here in calculating Green’s functions of these theories, and in ensuring the adelic compatibility of these theories. In particular, the local L-factors contribute to prefactors of these Green’s functions. We shall try to see quadratic reciprocity from this context, as a consequence of an adelic version of holomorphic factorization of these theories. This is work in progress with B. Stoica and X. Zhong.