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On the spectral sets of Inoue surfaces

### Daniel Ruberman and Nikolai Saveliev

Vol. 5 (2022), No. 1, 285–297
##### 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 ${\mathsc{𝒟}}_{z}^{±}$ 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 ${S}_{M}$.

##### Keywords
Dirac operator, Seiberg–Witten, Inoue surface, Tricerri metric
##### Mathematical Subject Classification
Primary: 32J15, 53C55
Secondary: 57R57, 58J50