Generalized cycles on spectral curves

Abstract

Generalized cycles can be thought of as the extension of form-cycle duality between holomorphic forms and cycles to meromorphic forms and generalized cycles.
They appeared as a ubiquitous tool in the study of spectral curves and integrable systems in the topological recursion approach. They parametrize deformations, implementing the special geometry, where moduli are periods, and derivatives with respect to moduli are other periods, or more generally “integrals”, whence the name “generalized cycles”. They appeared over the years in various works, each time in specific applied frameworks, and here, we provide a comprehensive self-contained corpus of definitions and properties for a very general setting. The geometry of generalized cycles is also fascinating in itself.