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
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'' 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 ...
:
, and
– 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
is
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
:
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
:
of the distinguished generator of
:
.
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