We show how to compute the spectral flow of the odd signature operator $\pm *d_{ a_t}-d_{ a_t}*$ along an analytic path of flat connections $a_t$ 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 $\{ \calg^*_n,\delta_n\}$, for $n=0,1,2,\ldots$ and Hermitian forms $$Q_n:\calg_n\times\calg_n\ra \bbbC$$ whose signatures determine the spectral flow through $t=0$. The complexes and Hermitian forms are constructed using Massey products.
Categories: dg-ga math.DG