nLab
Lagrangian Grassmannian

Contents

Definition

For VV a symplectic vector space, its Lagrangian Grassmannian LGrass(V)LGrass(V) is the space of its Lagrangian (maximal isotropic) subspaces.

Properties

As a coset space

The group of symplectomorphisms of VV naturally acts on LGrass(V)LGrass(V).

The unitary group U J(V)U_J(V) associated to any fixed compatible complex structure JJ on VV acts transitively on LGrass(V)LGrass(V). In fact, LGrass(V)LGrass(V) is diffeomorphic to the coset space U(n)/O(n)U(n)/O(n) of the unitary group by the orthogonal group, where nn is the complex dimension of (V,J)(V,J) (so the real dimension of VV is 2n2n).

Cohomology and Maslov index

The first ordinary cohomology of the stable Lagrangian Grassmannian with integer coefficients is isomorphic to the integers

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

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

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

Given a Lagrangian submanifold YXY \hookrightarrow X of a symplectic manifold (X,ω)(X,\omega), its tangent bundle is classified by a function

i:YLGrass. i \;\colon\; Y \to LGrass \,.

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

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

References

  • Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf

  • Andrew Ranicki, The Maslov Index (pdf)

  • Esteban Andruchow, Gabriel Larotonda, Lagrangian Grassmannian in Infinite Dimension (arXiv:0808.2270)

  • J. Carrillo-Pacheco, F. Jarquín-Zárate, M. Velasco-Fuentes, F. Zaldívar, An explicit description in terms of Plücker coordinates of the Langrangian-Grassmannian, arxiv/1601.07501

Last revised on January 28, 2016 at 11:42:46. See the history of this page for a list of all contributions to it.