This paper is concerned with the existence and compactness problems of perturbed Dirac-harmonic maps into flat tori. We consider two types of asymptotically linear perturbations, non-resonance and resonance cases. For the non-resonance case, we show that the set of solutions in a given free homotopy class is compact and there exist at least $(n+1)$-distinct solutions in each free homotopy class, where n is the dimension of the tori. For the resonance case, we prove a similar existence result under Landsman-Lazer type conditions.

中文翻译：

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渐近线性狄拉克-谐波映射成平坦托里

本文关注的是将狄拉克-调和映射扰动为平坦圆托的存在性和紧致性问题。我们考虑两种类型的渐近线性摄动，非共振和共振情况。对于非共振情况，我们证明给定的自由同伦类中的解集是紧的，并且至少存在$（ñ+\mathrm{1个}）$每个自由同伦类中的-distinct解，其中n是花托的维数。对于共振情况，我们证明在Landsman-Lazer型条件下存在相似的结果。