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.