Types of quantum field thories
Recall that a Lagrangean submanifold is a smooth submanifold of dimension whose tangent spaces at all points are Lagrangean subspaces, 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 Maslov index of is the universal Maslov index pulled back along this map
The index first appears maybe in
Its cohomological interpretation as a universal characteristic class was explained in
A review in the context of geometric quantization (Maslov correction) is in
The interpretation of the Maslov index as a quadratic space is due to
and this definition and basic examples are briefly collected in
G. Lion, Michele Vergne, The Weil representation, Maslov index and theta series, Progress in Math. 6, Birkhäuser 1980 (Rus. transl. Mir 1983).
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.
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. 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
Paolo Piccione, Daniel Victor Tausk, A student’s guide to symplectic spaces, Grassmannians and Maslov index, IMPA 2011, pdf
Application in the theory of Schroedinger operators: