The derived category of a generalized complex torus

Abstract

For any translation invariant generalized complex structure J on a torus we construct the derived category D(J) generalizing the derived category of coherent sheaves on a complex torus and the affine/convergent Fukaya category of a symplectic torus. The category is defined in terms of a non-commutative curved deformation of the Dolbeault algebra of a complex torus. We show that this category is independent of the chosen deformation path, which we prove using non-commutative Fourier-Mukai duality. We also prove that the action of O(2n,2n; Z) on the space of generalized complex structures leaves the category D(J) invariant. In particular, T-duality acts trivially on D(J). We construct concrete objects associated to affine Lagrangians in the derived category associated to a symplectic torus and show that their spaces of morphisms are given by Floer cohomology. We show that our construction is consistent with the homological mirror symmetry conjecture (including its non-commutative version).

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BibTeX RIS

BibTeX

@misc{antweiler2026derived,
  title = {{The derived category of a generalized complex torus}},
  author = {Patrick Antweiler and Vicente Cortés and Julian Holstein and Markus Röser and Aldo Witte},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21960},
  eprint = {2610.21960},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21960},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The derived category of a generalized complex torus
AU  - Patrick Antweiler
AU  - Vicente Cortés
AU  - Julian Holstein
AU  - Markus Röser
AU  - Aldo Witte
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21960
UR  - https://arxiv.org/abs/2610.21960
AB  - For any translation invariant generalized complex structure J on a torus we construct the derived category D(J) generalizing the derived category of coherent sheaves on a complex torus and the affine/convergent Fukaya category of a symplectic torus. The category is defined in terms of a non-commutative curved deformation of the Dolbeault algebra of a complex torus. We show that this category is independent of the chosen deformation path, which we prove using non-commutative Fourier-Mukai duality. We also prove that the action of O(2n,2n; Z) on the space of generalized complex structures leaves the category D(J) invariant. In particular, T-duality acts trivially on D(J). We construct concrete objects associated to affine Lagrangians in the derived category associated to a symplectic torus and show that their spaces of morphisms are given by Floer cohomology. We show that our construction is consistent with the homological mirror symmetry conjecture (including its non-commutative version).
ER  -

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