We extend the “miraculous cancellation” formulas of Alvarez–Gaumé, Witten and Kefeng Liu to a twisted version where an extra complex line bundle is involved. We apply our result to discuss intrinsic relations between the higher dimensional Rokhlin type congruences due to Ochanine, Finashin and Zhang. In particular, an analytic proof of the Finashin congruence is given.
"Modular Invariance Characteristic Numbers and η Invariants." J. Differential Geom. 67 (2) 257 - 288, June 2004. https://doi.org/10.4310/jdg/1102536202