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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 thesezero-modes. We show that the mode-shell formalism provides a general frameworkunifying important results of topological physics, such as the bulk-edgecorrespondence, higher-order topological insulators, but also the Atiyah-Singerand the Callias index theories. In this paper, we discuss the already richphenomenology of chiral symmetric Hamiltonians where the topological quantityis the chiral number of zero-dimensionial zero-energy modes. We explain how, ina lot of cases, the shell-invariant has a semi-classical limit expressed as ageneralised winding number on the shell, which makes it accessible toanalytical computations.

Mode-Shell correspondence, a unifying theory in topological physics -- Part I: Chiral number of zero-modes