In mathematics, the Lagrangian Grassmannian is the

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

of Lagrangian subspaces of a real

symplectic vector space In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...

''V''. Its dimension is ''n''(''n'' + 1) (where the dimension of ''V'' is ''2n''). It may be identified with the homogeneous space :, where is the

unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...

and the orthogonal group. Following

Vladimir Arnold Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...

it is denoted by Λ(''n''). The Lagrangian Grassmannian is a submanifold of the ordinary

Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...

of V. A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex

''V'' of dimension 2''n''. It may be identified with the homogeneous space of complex dimension ''n''(''n'' + 1) :, where is the compact symplectic group.

Topology

The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the

Bott periodicity theorem In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comp ...

\Omega(\mathrm/\mathrm U) \simeq \mathrm U/\mathrm O

, and

\Omega(\mathrm U/ \mathrm O) \simeq \mathbb\times \mathrm

– they are thus exactly the homotopy groups of the stable orthogonal group, up to a shift in indexing (dimension). In particular, the fundamental group of

U/O

infinite cyclic In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...

, with a distinguished generator given by the square of the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...

of a

unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...

, as a mapping to the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

. Its first

homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...

is therefore also infinite cyclic, as is its first

cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

. Arnold showed that this leads to a description of the Maslov index, introduced by V. P. Maslov. For a

Lagrangian submanifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...

''M'' of ''V'', in fact, there is a mapping :

M\to\Lambda(n)

which classifies its

tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...

at each point (cf.

Gauss map In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ' ...

). The Maslov index is the pullback via this mapping, in :

H^1(M, \mathbb)

of the distinguished generator of :

H^1(\Lambda(n), \mathbb)

Maslov index

A path of

symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...

s of a symplectic vector space may be assigned a Maslov index, named after V. P. Maslov; it will be an integer if the path is a loop, and a half-integer in general. If this path arises from trivializing the symplectic vector bundle over a periodic orbit of a

Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is ...

on a symplectic manifold or the Reeb vector field on a contact manifold, it is known as the Conley–Zehnder index. It computes the spectral flow of the Cauchy–Riemann-type operators that arise in

Floer homology In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer in ...

. It appeared originally in the study of the

WKB approximation In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ...

and appears frequently in the study of quantization,

quantum chaos Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mech ...

trace formulas, and in symplectic geometry and topology. It can be described as above in terms of a Maslov index for linear Lagrangian submanifolds.

References

*V. I. Arnold, '' Characteristic class entering in quantization conditions'', Funktsional'nyi Analiz i Ego Prilozheniya, 1967, 1,1, 1-14, . * V. P. Maslov, ''Théorie des perturbations et méthodes asymptotiques''. 1972 *{{citation , url=http://www.maths.ed.ac.uk/~aar/maslov.htm , title=The Maslov index home page , first=Andrew , last=Ranicki , access-date=2009-10-23 , archive-url=https://web.archive.org/web/20151201193450/http://www.maths.ed.ac.uk/~aar/maslov.htm , archive-date=2015-12-01 , url-status=dead Assorted source material relating to the Maslov index. Symplectic geometry Topology of homogeneous spaces Mathematical quantization