Topological $\epsilon$-factors

The article describes a purely topological counterpart of the $\epsilon$-factorization of constants in the functional equations (which is a key ingredient in the interplay between L-functions and classical automorphic forms). We consider the determinant of the cohomology of a constructible sheaf F on a real analytic manifold X (or a bit more precise object, which is $R\Gamma(X,F)$ seen as a homotopy point of the K-theory spectrum), and show that it can be "computed" by means of a "spectral" version of the Dubson-Kashiwara formula, which yields, in particular, the $\epsilon$-factorization format. This picture may lead to a better understanding of a recent work of Bloch-Deligne-Esnault on the determinant of the period matrix.