On the spectral sets of Inoue surfaces

TL;DR

This paper investigates the spectral properties of twisted Dirac operators on Inoue surfaces, revealing conditions for invertibility and spectral points, and establishing Fredholm properties on certain end-periodic manifolds.

Contribution

It characterizes the spectral points of twisted Dirac operators on Inoue surfaces and links these to Fredholm properties on end-periodic manifolds.

Findings

No spectral points inside a specific annulus in the complex plane.

Spectral points are located on the boundary of this annulus.

Twisted Dirac operators are Fredholm on certain end-periodic manifolds.

Abstract

The Inoue surfaces are certain non-Kaehler complex surfaces that have the structure of a $T^3$ bundle over the circle. We study the Inoue surfaces $S_M$ with the Tricerri metric and the canonical spin$^c$ structure, and the corresponding chiral Dirac operators twisted by a flat $\mathbb C^*$--connection. The twisting connection is determined by $z \in \mathbb C^*$, and the points for which the twisted Dirac operators $\mathcal D^{\pm}_z$ are not invertible are called spectral points. We show that there are no spectral points inside the annulus $\alpha^{-1/4} < |z| < \alpha^{1/4}$, where $\alpha >1$ is the only real eigenvalue of the matrix $M$ that determines $S_M$, and find the spectral points on its boundary. Via Taubes' theory of end-periodic operators, this implies that the corresponding Dirac operators are Fredholm on any end-periodic manifold whose end is modeled on $S_M$.