Monodromy

Monodromy is the study of how objects obtained by analytic continuation along a path behave when the path forms a loop. The term derives from Greek monodromos meaning “going around once.” In complex analysis and differential equations, it captures how a function or a set of solutions transforms after being continued around a closed loop encircling a singularity.

If one considers a linear differential equation on a domain X, the local solutions at a base point form a vector space. Analytically continuing these solutions along loops based at that point yields a linear automorphism of the solution space. Repeating for all loops gives a homomorphism from the fundamental group π1(X) to GL(n, C), called the monodromy representation; its image is the monodromy group. The nature of the singularities influences the monodromy, and special cases include regular singular points and Fuchsian equations.

In algebraic geometry and topology, monodromy describes how fibers of a family vary when parameters trace loops around critical values. This leads to a local system or flat connection (Gauss–Manin). Monodromy actions also arise in Picard–Lefschetz theory, where looping around a critical value induces a transformation on homology given by a reflection in vanishing cycles.

A familiar illustration is the multivalued complex logarithm: analytic continuation around the origin increases by 2πi, corresponding to the monodromy around the branch point; similarly, branched coverings and Riemann surfaces encode monodromy as permutations of sheets.

Monodromy provides a bridge between analysis, topology, and algebraic geometry, encoding how local data fail to extend globally and revealing the symmetry underlying a family of objects.