Contents

Overview

Consider a symplectic manifold (representing say a phase space of a physical system) of dimension $2n$ .

Recall that a Lagrangian submanifold is a smooth submanifold of dimension $n$ whose tangent spaces at all points are Lagrangian subspaces, i.e. maximal isotropic subspaces with respect to the symplectic form. Lagrangian submanifold describes the phase of short-wave oscillations.

The Maslov index is an invariant of a smooth path in a Lagrangian submanifold.

The Maslov index can be reinterpreted as a characteristic class of theories of Lagrangian and Legendrean cobordism.

Definition

As a universal characteristic class

The first ordinary cohomology of the stable Lagrangian Grassmannian with integer coefficients is isomorphic to the integers

The generator of this cohomology group is called the universal Maslov index

Since $LGrass$ is a classifying space for tangent bundles of Lagrangian submanifolds, this is a universal characteristic class for Lagrangian submanifolds.

Specifically, given a Lagrangian submanifold $Y \hookrightarrow X$ of a symplectic manifold $(X,\omega)$, its tangent bundle is classified by a function

The Maslov index of $Y$ is the universal Maslov index pulled back along this map

References and links

The index first appears maybe in

• Victor Maslov, Théorie des perturbations et méthodes asymptotiques. 1972

Its cohomological interpretation as a universal characteristic class was explained in

• Vladimir Arnold, Characteristic class entering in quantization conditions, Funct. Anal. its Appl. 1967, 1:1, 1–13, doi (В. И. Арнольд, “О характеристическом классе, входящем в условия квантования”, Функц. анализ и его прил., 1:1 (1967), 1–14, pdf)

A review in the context of geometric quantization (Maslov correction) is in

• Sean Bates, Alan Weinstein, section 4.2 of Lectures on the geometry of quantization, pdf

The interpretation of the Maslov index as a quadratic space is due to

• T. Thomas. The Maslov index as a quadratic space. Math. Res. Lett. 13 no. 6 (2006), 985–999

and this definition and basic examples are briefly collected in

• Andrew Ranicki, The Maslov Index, seminar notes 2010/11 (pdf)

See also

• G. Lion, Michele Vergne, The Weil representation, Maslov index and theta series, Progress in Math. 6, Birkhäuser 1980 (Rus. transl. Mir 1983).

• Alan Weinstein, The Maslov gerbe, Lett. Math. Phys. 69, 1-3, July, 2004, doi. (arXiv:0312274)

• Jean Leray, Lagrangian analysis and quantum mechanics. A mathematical structure related to asymptotic expansions and the Maslov index, (trans. from French), MIT Press 1981. xvii+271 pp.

• Victor Guillemin, Shlomo Sternberg, Geometric asymptotics, AMS 1977, online; Semi-classical analysis, 499 pages, pdf

• J. J. Duistermaat, On the Morse index in variational calculus, Adv. Math. 21 (1976), 2, 173–195, pdf.

• D. Salamon, E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), no. 10, 1303–1360, doi

• Joel Robbin, Dietmar Salamon, The Maslov index for paths, Topology 32 (1993), no. 4, 827–844, (doi; preprint version pdf); The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), no. 1, 1–33 (doi)

• A. B. Givental’, Global properties of the Maslov index and Morse theory, Funct. Anal. Its. Appl. 22, 2, 1988, doi (Rus. orig: функц. анализ и его приложения 22, 1988, вып. 2, 69—70: pdf)

• A. B. Giventalʹ, The nonlinear Maslov index, in “Geometry of low-dimensional manifolds” vol. 2 (Durham, 1989), 35–43, London Math. Soc. Lec. Note Ser. 151, Cambridge Univ. Press 1990.

• Maurice A. de Gosson, Maslov classes, metaplectic representation and Lagrangian quantization, Math. Research 95, Akademie-Verlag, Berlin, 1997. 186 pp.; Symplectic geometry, Wigner-Weyl-Moyal calculus, and quantum mechanics in phase space, 385 pp. pdf

• Leo T. Butler, The Maslov cocycle, smooth structures and real-analytic complete integrability, arxiv/0708.3157

• S. Merigon, L’indice de Maslov en dimension infinie, J. Lie Theory 18 (2008), no. 1, 161–180.

• S. E. Cappell, R. Lee, E. Y. Miller, On the Maslov index, Comm. Pure Appl. Math. 47 (1994), no. 2, 121–186.

• K. Furutani, Fredholm–Lagrangian–Grassmannian and the Maslov index, J. Geom. Phys. 51, 3, July 2004, 269–331, doi

• Many links are at Andrew Ranicki‘s Maslov index seminar page.

• Paolo Piccione, Daniel Victor Tausk, A student’s guide to symplectic spaces, Grassmannians and Maslov index, IMPA 2011, pdf

• M. V. Finkelberg, Orthogonal Maslov index, Funct. Anal. Appl. 29(1) 72–74 (1995) doi

• Alan Weinstein, The Maslov cycle as a Legendre singularity and projection of a wavefront set, Bull. Braz. Math. Soc., N.S. 44, 593–610 (2013) doi

Application in the theory of Schroedinger operators:

• Yuri Latushkin, Alim Sukhtayev, Selim Sukhtaiev, The Morse and Maslov indices for Schrödinger operators, arxiv/1411.1656; Yuri Latushkin, Alim Sukhtayev, Hadamard-type formulas via the Maslov form, arxiv/1601.07509