The hyperbolic class and vanishing laws in a TMF-valued four-manifold invariant
Abstract
Under explicit hypotheses on the spectral Looijenga construction, we compute the hyperbolic-plane value of the zero-section invariant of Gukov, Krushkal, Meier, and Pei in periodic topological modular forms. The value is the Hopf element eta, and adjoining a hyperbolic plane acts by multiplication by this element. Earlier work obtains the hyperbolic value under an additional cobordism-duality assumption; here we derive it directly from the Looijenga restriction maps without that assumption. Together with annihilation and vanishing results for definite lattices, this gives a complete value formula on smooth closed simply connected spin four-manifolds. In particular, nonzero signature forces the invariant to vanish, so both orientations of a K3 surface have value zero.
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BibTeX
@misc{li2026hyperbolic,
title = {{The hyperbolic class and vanishing laws in a TMF-valued four-manifold invariant}},
author = {Yuqi Li and Hao-Yu Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09819},
eprint = {2609.09819},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09819},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The hyperbolic class and vanishing laws in a TMF-valued four-manifold invariant
AU - Yuqi Li
AU - Hao-Yu Sun
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09819
UR - https://arxiv.org/abs/2609.09819
AB - Under explicit hypotheses on the spectral Looijenga construction, we compute the hyperbolic-plane value of the zero-section invariant of Gukov, Krushkal, Meier, and Pei in periodic topological modular forms. The value is the Hopf element eta, and adjoining a hyperbolic plane acts by multiplication by this element. Earlier work obtains the hyperbolic value under an additional cobordism-duality assumption; here we derive it directly from the Looijenga restriction maps without that assumption. Together with annihilation and vanishing results for definite lattices, this gives a complete value formula on smooth closed simply connected spin four-manifolds. In particular, nonzero signature forces the invariant to vanish, so both orientations of a K3 surface have value zero.
ER -