Quadratic reciprocity from a family of adelic conformal field theories

TL;DR

This paper explores a family of adelic conformal field theories deformed by fractional Laplacians, revealing a connection between quadratic reciprocity and adelic holomorphic factorization through Green's functions and number field parameters.

Contribution

It introduces an adelic family of conformal field theories linked to number fields and Hecke characters, demonstrating how quadratic reciprocity emerges from their adelic holomorphic factorization.

Findings

Green's functions satisfy adelic product formulas

Quadratic reciprocity arises from adelic holomorphic factorization

Local L-factors influence Green's function prefactors

Abstract

We consider a deformation of the two-dimensional free scalar field theory 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 is no longer invariant under the full Witt algebra. Furthermore, there is an adelic version of this family of conformal field theories, parameterized by the choice of a number field, together with a Hecke character. Tate's thesis gives the Green's functions of these theories, and ensures that these Green's functions satisfy an adelic product formula. In particular, the local $L$-factors contribute to the prefactors of these Green's functions. Quadratic reciprocity turns out to be a consequence of an adelic version of a holomorphic factorization property of this family of theories on a quadratic extension of $\mathbb{Q}$. We explain that at the Archimedean place, the desired holomorphic factorization follows from the global conformal symmetry.