The Spectral Flow of the Odd Signature operator and Higher Massey Products

We show how to compute the spectral flow of the odd signature operator $\pm d_{a_t}-d_{a_t}$alongananalyticpathofflatconnections$a_t$ on a bundle over a closed odd-dimensional manifold in terms of Massey products in the DGLA of bundle-valued differential forms. To obtain this information, we set up a sequence of cochain complexes $\{{\text{calg}}_n^{\ast },{\delta }_n\}$, for $n=0,1,2,\dots$ and Hermitian forms $$Q_n:{\text{calg}}_n\times {\text{calg}}_n\text{ra}\text{bbbC}$$ whose signatures determine the spectral flow through $t=0$. The complexes and Hermitian forms are constructed using Massey products.