Monodromy

In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity.

As the name implies, the fundamental meaning of monodromy comes from "running round singly".

It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we "run round" a path encircling a singularity.

The failure of monodromy can be measured by defining a monodromy group: a group of transformations acting on the data that encodes what happens as we "run round" in one dimension.

For a loop γ: [0, 1] → X based at x, denote a lift under the covering map, starting at a point

, that is, an element [γ] fixes a point in F if and only if it is represented by the image of a loop in

For example, take Then analytic continuation anti-clockwise round the circle will result in the return not to F(z) but to In this case the monodromy group is the infinite cyclic group, and the covering space is the universal cover of the punctured complex plane.

The covering map is a vertical projection, in a sense collapsing the spiral in the obvious way to get a punctured plane.

Linear differential equations defined in an open, connected set S in the complex plane have a monodromy group, which (more precisely) is a linear representation of the fundamental group of S, summarising all the analytic continuations round loops within S. The inverse problem, of constructing the equation (with regular singularities), given a representation, is a Riemann–Hilbert problem.

For a regular (and in particular Fuchsian) linear system one usually chooses as generators of the monodromy group the operators Mj corresponding to loops each of which circumvents just one of the poles of the system counterclockwise.

If the indices j are chosen in such a way that they increase from 1 to p + 1 when one circumvents the base point clockwise, then the only relation between the generators is the equality

The Deligne–Simpson problem is the following realisation problem: For which tuples of conjugacy classes in GL(n, C) do there exist irreducible tuples of matrices Mj from these classes satisfying the above relation?

The problem has been formulated by Pierre Deligne and Carlos Simpson was the first to obtain results towards its resolution.

An additive version of the problem about residua of Fuchsian systems has been formulated and explored by Vladimir Kostov.

The problem has been considered by other authors for matrix groups other than GL(n, C) as well.

[2] In the case of a covering map, we look at it as a special case of a fibration, and use the homotopy lifting property to "follow" paths on the base space X (we assume it path-connected for simplicity) as they are lifted up into the cover C. If we follow round a loop based at x in X, which we lift to start at c above x, we'll end at some c* again above x; it is quite possible that c ≠ c*, and to code this one considers the action of the fundamental group π1(X, x) as a permutation group on the set of all c, as a monodromy group in this context.

In differential geometry, an analogous role is played by parallel transport.

In a principal bundle B over a smooth manifold M, a connection allows "horizontal" movement from fibers above m in M to adjacent ones.

The effect when applied to loops based at m is to define a holonomy group of translations of the fiber at m; if the structure group of B is G, it is a subgroup of G that measures the deviation of B from the product bundle M × G. Analogous to the fundamental groupoid it is possible to get rid of the choice of a base point and to define a monodromy groupoid.

Here we consider (homotopy classes of) lifts of paths in the base space X of a fibration

we can consider its induced diffeomorphism on local transversal sections through the endpoints.

In this way it also becomes independent of the path (between fixed endpoints) within a simply connected chart and is therefore invariant under homotopy.

This has connections with the Galois theory of covering spaces leading to the Riemann existence theorem.