  We propose a theory, that we call the \textit{mode-shell correspondence},
which relates the topological zero-modes localised in phase space to a
\textit{shell} invariant defined on the surface forming a shell enclosing these
zero-modes. We show that the mode-shell formalism provides a general framework
unifying important results of topological physics, such as the bulk-edge
correspondence, higher-order topological insulators, but also the Atiyah-Singer
and the Callias index theories. In this paper, we discuss the already rich
phenomenology of chiral symmetric Hamiltonians where the topological quantity
is the chiral number of zero-dimensionial zero-energy modes. We explain how, in
a lot of cases, the shell-invariant has a semi-classical limit expressed as a
generalised winding number on the shell, which makes it accessible to
analytical computations.
