Non-Abelian Zeta Function, Fokker-Planck Equation and Projectively Flat Connection

TL;DR

This paper explores the deep connections between non-abelian zeta functions, Fokker-Planck equations, and geometric structures on moduli spaces, proposing a novel framework linking zeros of zeta functions to differential equations and flat connections.

Contribution

It introduces an infinite dimensional Hilbert bundle structure associated with non-abelian zeta zeros and conjectures a natural projectively flat connection underlying Fokker-Planck equations.

Findings

Zeros of non-abelian zeta functions relate to averaged Fokker-Planck equations.

Constructs an infinite dimensional Hilbert bundle parametrized by zeta zeros.

Proposes a conjecture on an essential projectively flat connection in this setting.

Abstract

Over the moduli space of rank $n$ semi-stable lattices is a universal family of tori. Along the fibers, there are natural differential operators and differential equations, particularly, the heat equations and the Fokker-Planck equations in statistical mechanics. In this paper, we explain why, by taking averages over the moduli spaces, all these are connected with the zeros of rank $n$ non-abelian zeta functions of the field of rationals, which are known lie on the central line except a finitely many if $n\geq 2$. Certainly, when $n=1$, our current work recovers that of Armitage, which from the beginning motivates ours. However, we reverse the order of the results and the hypothesis in their works, i.e. we construct averaged versions of Fokker-Planck equations using the above structure of non-abelian zeta zeros. This then leads to an infinite dimensional Hilbert vector bundle with smooth sections parametrized by non-abelian zeta zeros. We conjectures that the above structure of Fokker-Planck equation comes naturally from an \lq essential projectively flat connection' of the infinite dimensional Hilbert bundle and the above family of smooth sections are its \lq essential pro-flat sections'.