Abstract

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 ${ℂ}^{\ast }$-connection. The twisting connection is determined by $z\in {ℂ}^{\ast }$, and the points for which the twisted Dirac operators ${\mathcal{𝒟}}_z^{\pm }$ are not invertible are called spectral points. We show that there are no spectral points inside the annulus ${\alpha }^{-1∕4}<\left|z\right|<{\alpha }^{1∕4}$, 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 $S_M$.

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