Mod 3 Congruence Families for twisted signatures of string manifolds in dimensions up to twenty-eight
Abstract
In the paper, we establish congruences for twisted Dirac indices and signatures on closed string manifolds by specializing the two-parameter modular forms of Liu and Wang to the tangent bundle. Fourier coefficient comparisons yield uniform congruences modulo $3$ in dimensions 4k, 1≤ k≤7. Suitable parameter choices give explicit formulas involving only genuine twisting bundles in dimensions $8,12,16,20,24,$ and $28$, and recover the comparisons of Chen and Han in dimension $24$. The level-one Witten genus further strengthens selected Dirac congruences to moduli $729,27,27,$ and $9$ in dimensions $16,20,24,$ and $28$, respectively. The proofs use Fourier expansions in reduced K-theory and integer identities obtained by clearing powers of two from the denominators before reduction modulo $3$. We also construct non-string spin counterexamples in each of the six listed dimensions and identify the additional coefficient that obstructs extending the uniform argument to dimension $32$ without further information.
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BibTeX
@misc{wu2026congruence,
title = {{Mod 3 Congruence Families for twisted signatures of string manifolds in dimensions up to twenty-eight}},
author = {Tong Wu and Yong Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21959},
eprint = {2610.21959},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21959},
}
RIS
TY - UNPB
TI - Mod 3 Congruence Families for twisted signatures of string manifolds in dimensions up to twenty-eight
AU - Tong Wu
AU - Yong Wang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21959
UR - https://arxiv.org/abs/2610.21959
AB - In the paper, we establish congruences for twisted Dirac indices and signatures on closed string manifolds by specializing the two-parameter modular forms of Liu and Wang to the tangent bundle. Fourier coefficient comparisons yield uniform congruences modulo $3$ in dimensions 4k, 1≤ k≤7. Suitable parameter choices give explicit formulas involving only genuine twisting bundles in dimensions $8,12,16,20,24,$ and $28$, and recover the comparisons of Chen and Han in dimension $24$. The level-one Witten genus further strengthens selected Dirac congruences to moduli $729,27,27,$ and $9$ in dimensions $16,20,24,$ and $28$, respectively. The proofs use Fourier expansions in reduced K-theory and integer identities obtained by clearing powers of two from the denominators before reduction modulo $3$. We also construct non-string spin counterexamples in each of the six listed dimensions and identify the additional coefficient that obstructs extending the uniform argument to dimension $32$ without further information.
ER -