nLab
Maslov index

Overview

Consider a symplectic manifold representing say a phase space of a physical theory of dimension $2n$.

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

The Maslov index is an invariant of a smooth path in a Lagrangean submanifold. The existence of such an invariant is related to the universal Maslov index which is a generator of the first integral cohomology of the Langrangean Grassmanian (the space of $n$-dimensional Lagrangean subspaces in ${ℝ}^{2n}$, cf. wikipedia: Lagrangian Grassmannian).

The Maslov index can be reinterpreted as a characteristic class of theories of Lagrangean and Legendrean cobordisms.

References and links

• S. Bates, A. Weinstein, Lectures on the geometry of quantization, pdf

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

• A. Weinstein, The Maslov gerbe, Lett. Math. Phys. 69, 1-3, July, 2004, doi.

• 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.

• V. Guillemin, S. Sternberg, Geometric asymptotics, AMS 1977, online

• V. I. Arnol’d, 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)

• 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 de Gosson, Maslov classes, metaplectic representation and Lagrangian quantization, Math. Research 95, Akademie-Verlag, Berlin, 1997. 186 pp.

• 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 Ranicki’s Maslov index seminar page.