Complex generalized spin-c Killing spinors in low dimensions
Abstract
We classify all complex generalized spin-c Killing spinors with complex Killing function iμ−α on complete Riemannian surfaces, presenting an explicit list of all possible cases. We show that, contrary to previous statements in the literature, there exist complex generalized spin-c Killing spinors on Riemannian surfaces for which αμnot≡0. We illustrate the classification by providing several explicit local, compact, and complete examples, including a complete example for which αμ>0 everywhere. We then move to three dimensions, proving that a spin-c Riemannian three-manifold admits a complex generalized spin-c Killing spinor with complex Killing function iμ−α if and only if it is a μ-exact trans-Sasakian manifold of type (α,μ). In addition, we construct a geodesically incomplete Riemannian three-manifold equipped with a complex generalized spin-c Killing spinor for which neither μ nor α vanishes identically, and then prove that this is not possible if the characteristic vector field of the associated trans-Sasakian structure is complete. These results settle a conjecture by Große and Nakad (Results in Mathematics, 2015) stating that on two- and three-dimensional Riemannian manifolds the complex Killing function of any complex generalized spin-c Killing spinor must be either globally real or globally imaginary.
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BibTeX
@misc{jimenez2026complex,
title = {{Complex generalized spin-c Killing spinors in low dimensions}},
author = {José Luis Carmona Jiménez and Alejandro Gil-García and C. S. Shahbazi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20568},
eprint = {2610.20568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20568},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complex generalized spin-c Killing spinors in low dimensions
AU - José Luis Carmona Jiménez
AU - Alejandro Gil-García
AU - C. S. Shahbazi
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20568
UR - https://arxiv.org/abs/2610.20568
AB - We classify all complex generalized spin-c Killing spinors with complex Killing function iμ−α on complete Riemannian surfaces, presenting an explicit list of all possible cases. We show that, contrary to previous statements in the literature, there exist complex generalized spin-c Killing spinors on Riemannian surfaces for which αμnot≡0. We illustrate the classification by providing several explicit local, compact, and complete examples, including a complete example for which αμ>0 everywhere. We then move to three dimensions, proving that a spin-c Riemannian three-manifold admits a complex generalized spin-c Killing spinor with complex Killing function iμ−α if and only if it is a μ-exact trans-Sasakian manifold of type (α,μ). In addition, we construct a geodesically incomplete Riemannian three-manifold equipped with a complex generalized spin-c Killing spinor for which neither μ nor α vanishes identically, and then prove that this is not possible if the characteristic vector field of the associated trans-Sasakian structure is complete. These results settle a conjecture by Große and Nakad (Results in Mathematics, 2015) stating that on two- and three-dimensional Riemannian manifolds the complex Killing function of any complex generalized spin-c Killing spinor must be either globally real or globally imaginary.
ER -