A correspondence linking regular holonomic systems of linear differential equations (or regular holonomic D-modules, i.e. linear connections with regular singularities) on complex manifolds with representations of the fundamental group (local systems), pairing analytic solution data with topological monodromy data.

Principle

The organizing idea is that analytic continuation of local solutions around singularities produces monodromy representations; the solution functor maps a regular holonomic D-module to a locally constant sheaf (or perverse sheaf) whose stalks carry the representation of the fundamental group, and this map is an equivalence in the regular holonomic category.

Demonstration

Consider a linear ordinary differential equation on the punctured complex plane with regular singularities; analytic continuation of a basis of local solutions around the puncture gives a representation of π1, and the correspondence assigns to the differential system the resulting monodromy local system.

Misapplication

Applying the correspondence to systems with irregular singularities or non-holonomic modules; in such cases additional data (Stokes phenomena) are needed and the naive equivalence between D-modules and local systems fails.

Consequence

One can classify regular linear differential equations by their monodromy representations and transfer problems between analytic differential equations and topological or representation-theoretic descriptions, enabling algebraic and categorical methods to study solutions.

Reversal

The inverse direction constructs a differential system (up to equivalence) from a given representation of the fundamental group by producing a flat connection or D-module whose horizontal sections realize that monodromy; for regular cases this yields an equivalence of categories.

Boundary

Requires the regular holonomic hypothesis (regular singularities, holonomicity) and is formulated in the complex-analytic or algebraic setting; it excludes irregular singularities unless augmented with Stokes data or other enhancement.

Semantic Tension

Tension exists with theories for irregular singularities (Stokes phenomena) and with microlocal or derived categorical refinements (perverse sheaves, derived Riemann–Hilbert), where the simple one-to-one correspondence must be enriched or replaced.

Synthesis

The Riemann-Hilbert Correspondence unifies analytic and topological perspectives by stating that, for regular holonomic systems, the analytic structure of differential equations (solutions and continuation) and the topological structure of local systems (monodromy representations) are equivalent descriptions.