Maslov index

As a homotopy invariant

If ${\displaystyle \gamma \colon S^{1}\to \Lambda (V)}$ is a loop in the Lagrangian Grassmannian, its Maslov index ${\displaystyle \mu (\gamma )}$ is the integer corresponding to the homotopy class of ${\displaystyle \gamma }$ under the isomorphism

${\displaystyle \pi _{1}(\Lambda (V))\cong \mathbf {Z} .}$^[1][3]

This definition makes the Maslov index a topological invariant of loops of Lagrangian subspaces.^[1]

Via the Maslov cycle

Fix a Lagrangian subspace ${\displaystyle L_{0}\subseteq V}$. For a sufficiently generic path ${\displaystyle \gamma \colon [a,b]\to \Lambda (V)}$ whose endpoints are transverse to ${\displaystyle L_{0}}$, the Maslov index ${\displaystyle \mu _{L_{0}}(\gamma )}$ is defined as the signed intersection number of ${\displaystyle \gamma }$ with the Maslov cycle ${\displaystyle \Sigma (L_{0})}$.^[1][2]

At a transverse crossing time ${\displaystyle t_{*}}$ for which ${\displaystyle \gamma (t_{*})\cap L_{0}\neq 0}$, the sign is determined by the crossing form, a quadratic form on the intersection ${\displaystyle \gamma (t_{*})\cap L_{0}}$. The total Maslov index is the sum of the inertial index of these crossing forms over all crossings.^[2]

Via the Kashiwara form

Given a cycle of Lagrangian subspaces ${\displaystyle A_{1}\to A_{2}\dots \to A_{k}\to A_{1}}$, the Kashiwara form is the quadratic form on ${\displaystyle A_{1}\oplus A_{2}\oplus \cdots \oplus A_{k}}$ $${\displaystyle \kappa (a_{1},a_{2},\dots ,a_{k})=\omega (a_{1},a_{2})+\omega (a_{2},a_{3})+\cdots +\omega (a_{k-1},a_{k})+\omega (a_{k},a_{1}).}$$ The Maslov index is the inertial index of the form ${\displaystyle \kappa }$.^[4][7] When the subspaces are mutually transverse, the cycle ${\displaystyle A_{1},\dots ,A_{k},A_{1}}$ can be realized as a polygon in ${\displaystyle \Lambda _{n}}$, such that ${\displaystyle A_{k+1}}$ is the graph of a symmetric transformation ${\displaystyle S:A_{k}\to A_{k-1}}$, $${\displaystyle A_{k+1}=\{x+Sx\mid x\in A_{k}\}.}$$ Then the edge ${\displaystyle A_{k}A_{k+1}}$ in the polygon is parameterized by ${\displaystyle u\in [0,1]}$, ${\displaystyle \{x+uSx\mid x\in A_{k}\}}$.^[4]

For paths of symplectic matrices

The Maslov index can also be attached to a path in the symplectic group ${\displaystyle \mathrm {Sp} (2n,\mathbf {R} )}$. If ${\displaystyle \Phi _{t}}$ is a path of symplectic matrices and ${\displaystyle L}$ is a fixed Lagrangian subspace, then the path ${\displaystyle \Phi _{t}(L)}$ in the Lagrangian Grassmannian has a Maslov index. In this form the invariant is closely related to the Conley-Zehnder index and to stability questions for periodic orbits of Hamiltonian systems.^[2]

Geometric optics and caustics

In geometric optics, the eikonal equation determines a Lagrangian submanifold ${\displaystyle \Lambda \subset T^{*}X}$. Away from caustics, the projection ${\displaystyle \pi |_{\Lambda }:\Lambda \to X}$ is locally a diffeomorphism, so ${\displaystyle \Lambda }$ can be written locally as the graph of ${\displaystyle dS}$, and one obtains a WKB solution of the form $${\displaystyle u_{h}(x)\sim a(x)e^{iS(x)/h}.}$$ A caustic occurs when this projection ceases to be regular, so that the single-phase description breaks down.^[6][5]

In ray coordinates, the amplitude contains a Jacobian factor $${\displaystyle a(x)\propto \left|\det {\frac {\partial x}{\partial \alpha }}\right|^{-1/2},}$$ which becomes singular at a caustic. Near a generic fold caustic, the local model is given by the Airy function. Its oscillatory asymptotics, $${\displaystyle \operatorname {Ai} (-z)\sim {\frac {1}{{\sqrt {\pi }}\,z^{1/4}}}\sin \!\left({\frac {2}{3}}z^{3/2}+{\frac {\pi }{4}}\right),}$$ display the phase shift that is encoded globally by the Maslov index.^[6][8]

Fourier integral operators

The Maslov index also appears in the theory of Fourier integral operators. If an oscillatory integral is written using two different phase functions that parametrize the same Lagrangian submanifold, the method of stationary phase relates the two expressions. The transformation law contains the factor $${\displaystyle |\det \partial _{\theta \theta }^{2}\Phi |^{-1/2}e^{\,i\pi \,\operatorname {sgn} (\partial _{\theta \theta }^{2}\Phi )/4},}$$ so the signature of the Hessian contributes a phase defined modulo integer multiples of ${\displaystyle \pi /2}$.^[5][6]

Globally, these local phase ambiguities are encoded by the Keller–Maslov bundle, and the principal symbol of a Fourier integral operator is naturally a half-density twisted by that bundle.^[5][6]

Semiclassical analysis

The Maslov index appears in WKB theory and related semiclassical approximations as a phase correction. When an oscillatory solution passes through a caustic, the phase jumps by a multiple of ${\displaystyle \pi /2}$. This correction is encoded by the Maslov index.^[6][5]

For this reason the invariant is fundamental in the asymptotic study of wave propagation, Fourier integral operators, and quantization conditions in classical and quantum mechanics.^[1][6]

Microlocal analysis

In the theory of Fourier integral operators, phase functions may be replaced under changes of coordinates or generating functions. The resulting phase ambiguity is controlled by the Maslov index, which governs the correct choice of half-density and phase factor.^[5][6]

Symplectic topology

In Floer homology and the theory of pseudoholomorphic curves, Maslov-type indices determine gradings, expected dimensions of moduli spaces, and transversality formulas. For a Lagrangian submanifold, the Maslov class enters the definition of monotonicity and the construction of Lagrangian Floer homology.^[3]

Hamiltonian dynamics

For periodic orbits and linearized Hamiltonian flows, Maslov-type indices measure spectral flow and stability. In this setting they are closely related to the Morse index, the Conley-Zehnder index, and the theory of symplectic paths.^[2]

Infinite-dimensional generalizations

The Maslov index also has infinite-dimensional analogues, especially for paths in the Fredholm–Lagrangian Grassmannian of a symplectic Hilbert space. An early formulation in this direction was given by Nobukazu Otsuki and Kenro Furutani for Lagrangian intersections in real Hilbert spaces,^[9] and a widely cited functional-analytic definition for continuous curves in the Fredholm–Lagrangian Grassmannian was developed by Bernhelm Booss-Bavnbek and Kenro Furutani.^[10] In this setting the Maslov index is closely related to spectral flow and to index theory for families of self-adjoint Fredholm operators.^[7][10] A standard review is Furutani's article on the Fredholm–Lagrangian Grassmannian.^[11]