For smooth families of homology tori with odd determinant, the spin obstruction of the fiberwise tangent bundle is canonically isomorphic to that of the self-dual harmonic 2-forms bundle via an anti-linear Z/4-action on the determinant line bundle of the K-theoretic Seiberg-Witten invariant.
The D ehn twist on a connected sum of two homology tori
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A gerbe-like construction in gauge theory II: the case of homology tori
For smooth families of homology tori with odd determinant, the spin obstruction of the fiberwise tangent bundle is canonically isomorphic to that of the self-dual harmonic 2-forms bundle via an anti-linear Z/4-action on the determinant line bundle of the K-theoretic Seiberg-Witten invariant.