Maslov index

Maslov index
Name	Maslov index
Field	Symplectic geometry, Mathematical physics
Introduced	1960s
Inventor	V. P. Maslov

Maslov index is an integer-valued topological invariant associated with paths of Lagrangian subspaces in a symplectic vector space and with families of phase functions in semiclassical analysis. It measures the signed count of certain singularities or intersections encountered along a path, linking ideas from Arnold's theory of Lagrangian manifolds, Maslov's semiclassical quantization, and later developments by Raoul Bott, Hörmander, and Bismut. The index appears in asymptotic formulas for oscillatory integrals, spectral counting, and in the study of periodic orbits in dynamical systems connected to KAM theory.

Definition and basic properties

The Maslov index assigns an integer to a continuous path of Lagrangian subspaces in a symplectic vector space such as (R^{2n}, ω) used in constructions by Kolmogorov and Arnold. It is homotopy invariant under endpoint-fixing deformations and satisfies additivity under concatenation of paths, properties reminiscent of the fundamental group and the Atiyah–Singer index theorem context studied by Atiyah and Singer. For closed loops the Maslov index defines a map from π1 of the Lagrangian Grassmannian to Z, linking to computations by Bott in homotopy theory and to the Maslov class appearing in the work of Fulton and Harris on characteristic classes.

Historical context and contributors

Origins lie in the 1960s semiclassical work of Maslov on asymptotic methods and the WKB approximation used by researchers influenced by Dirac and Schrödinger. Further formalization came from symplectic and microlocal analysts such as Hörmander, who connected the index with Fourier integral operators, and Arnold, who developed Lagrangian singularity theory and catastrophe theory with links to Thom. Subsequent contributions by Bott, Loomis, Bismut, and Weinstein embedded the invariant within global analysis, while modern formulations draw on work by Eliashberg, McDuff, and Salamon in symplectic topology.

Maslov index in symplectic geometry and topology

In symplectic topology the Maslov index characterizes loops in the Lagrangian Grassmannian Lag(n) and interacts with invariants studied by Seidel and Eliashberg. It appears in Floer homology calculations by Floer and graded Lagrangian submanifold theory developed by Fukaya, Seidel, and Abouzaid. The Maslov class gives an obstruction to exactness for Lagrangian embeddings considered by Gromov in his h-principle work and by Eliashberg in contact topology. Relations with characteristic classes connect it to classical sources such as Weyl and modern treatments by Bott and Morrey.

Analytic and algebraic formulations

Analytically the Maslov index is defined via spectral flow of a family of self-adjoint Fredholm operators in frameworks used by Atiyah and Singer, or via jumps in phase of oscillatory integrals as in work by Hörmander and Maslov. Algebraic descriptions use the properties of the metaplectic group and the double cover of the symplectic group explored by Weil and Schwartz. In the categorical setting the index appears in grading data for Fukaya categories studied by Kontsevich and in equivalences conjectured in homological mirror symmetry between Kontsevich and Bondal frameworks.

Calculations and examples

Explicit computations occur for standard models such as the one-dimensional WKB turning point problems treated by Jeffreys and Kramers, and for periodic orbits in classical systems like the harmonic oscillator analyzed by Arnold and Maslov. In symplectic linear algebra examples link to intersections with the Maslov cycle inside the Lagrangian Grassmannian; such intersections were tabulated in classical homotopy computations by Bott and employed in spectral asymptotics by Ivrii and Vasiliev.

Applications in physics and partial differential equations

The Maslov index corrects phase shifts in semiclassical quantization rules in the works of Maslov, influencing computations in quantum mechanics by practitioners such as Dirac and in molecular dynamics modeled using techniques advanced by Karplus. It appears in trace formulae related to periodic orbit theory developed by Gutzwiller and in spectral asymptotics underlying results of Ivrii. In PDEs the index governs propagation of singularities in microlocal analysis by Hörmander and enters boundary value problems studied by Tarski-adjacent analysts and modern researchers in scattering theory.

Generalizations and related invariants

Generalizations include the Conley–Zehnder index introduced by Conley and Zehnder for paths of symplectic matrices, the Robbin–Salamon index by Robbin and Salamon for non-transverse intersections, and various Maslov-type indices in contact and relative settings studied by Seidel and Eliashberg. Connections to the η-invariant of Atiyah, spectral flow studied by Epstein and Atiyah, and to categories in homological mirror symmetry by Kontsevich position the Maslov index as a central bridge between topology, analysis, and quantum theory.

Category:Symplectic topology