nLab Maslov index

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

$$H^1(LGrass, \mathbb{Z}) \simeq \mathbb{Z} \,.$$

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

$$u \in H^1(LGrass, \mathbb{Z}) \,.$$

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

$$i \;\colon\; Y \to LGrass \,.$$

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

$$i^\ast u \in H^1(Y,\mathbb{Z}) \,.$$

Literature

Original article:

• 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)

See also:

• Wikipedia, Maslov index

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 V. Tausk: A student’s guide to symplectic spaces, Grassmannians and Maslov index, Instituto de Matemática Pura e Aplicada (2011) [pdf, 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

On stability and Maslov indices of geodesics in (semi-)Riemannian manifolds:

• Paolo Piccione, Alessandro Portaluri, Daniel V. Tausk: Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics, Annals of Global Analysis and Geometry 25 (2004) 121–149 [doi:10.1023/B:AGAG.0000018558.65790.db]