Zeta functions and topology of Heisenberg cycles for linear ergodic flows

TL;DR

This paper explores the spectral properties of Dirac-Schr"odinger operators along flows on manifolds, linking zeta functions to ergodic theory and extending the concept of Heisenberg cycles to irrational tori with applications in noncommutative geometry.

Contribution

It extends the analysis of Heisenberg cycles to irrational tori, proving meromorphic continuation of zeta functions for a broad class of elements and establishing a KK-duality in this setting.

Findings

Zeta functions have meromorphic continuation related to ergodic averages.

Extended the class of elements for which zeta functions are meromorphically extendable.

Established a KK-duality via a spectral cycle for certain noncommutative tori.

Abstract

Placing a Dirac-Schr\"odinger operator along the orbit of a flow on a compact manifold \(M\) defines an \(\R\)-equivariant spectral triple over the algebra of smooth functions on \(M\). We study some of the properties of these triples, especially their zeta functions, which have the form \(\trace (fH^{-s})\) with \(f\) the restriction to \(\R\) of a function on \(M\) and \(H = -\frac{\partial^2}{\partial x^2} + x^2\) the harmonic oscillator. The meromorphic continuation property and pole structure of these zeta functions is related to ergodic time averages in dynamics. The construction reproduces the `Heisenberg cycles' of Lesch and Moscovici, in the case of the periodic flow on the circle, where it produces a spectral triple over the smooth irrational torus in the irrational rotation algebra \(A_\h\). We strengthen a result of these authors, showing that the zeta function \(\trace (aH^{-s})\) extend mermomorphically for any element \(a\) of the C*-algebra \(A_\h\). Another variant of the construction produces a spectral cycle for \(A_\h\otimes A_{1/\h}\) and a spectral triple over a suitable subalgebra with the meromorphic continuation property if \(\h\) satisfies a Diophantine condition. The class of this cycle defines a fundamental class in the sense that it determines a KK-duality. We employ the Local Index Theorem of Connes and Moscovici in order to elaborate an index theorem of Connes for certain classes of differential operators on the line and compute the intersection form on K-theory induced by the fundamental class.