Abstract interlevel persistence for Morse-Novikov and Floer theory

We develop a general algebraic framework involving "Poincar\'e--Novikov structures" and "filtered matched pairs" to provide an abstract approach to the barcodes associated to the homologies of interlevel sets of $\mathbb{R}$- or $S^1$-valued Morse functions, which can then be applied to Floer-theoretic situations where no readily apparent analogue of an interlevel set is available. The resulting barcodes satisfy abstract versions of stability and duality theorems, and in the case of Morse or Novikov theory they coincide with the standard barcodes coming from interlevel persistence. In the case of Hamiltonian Floer theory, the lengths of the bars yield multiple quantities that are reminiscent of the spectral norm of a Hamiltonian diffeomorphism.