nLab Lagrangian Grassmannian

Definition

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

Properties

As a coset space

The symplectic group of $V$ naturally acts on $LGrass(V)$.

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

Cohomology and Maslov index

The first ordinary cohomology of the stable Lagrangian Grassmannian

$$LGrass = \lim_{n \to \infty} LGrass(\mathbb{C}^n)$$

with integer coefficients is isomorphic to the integers

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

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

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

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

$$i \;\colon\; Y \to LGrass \,.$$

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

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

Unstably, a generator of

$$H^1(\LGrass(\mathbb{C}^n),\mathbb{Z}) \cong [U(n)/O(n), U(1)]$$

is given by the map

$$\mathrm{det}^2 \colon U(n) \to U(1)$$

This map passes to the quotient $U(n)/O(n)$ because the determinant of an element of $O(n)$ can only be $\pm 1$.

Related concepts

• Grassmannian

  • isotropic Grassmannian

  • Lagrangian Grassmannian

• Maslov index

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