Math 2.0

Differential geometry Definition

Maslov index

Builds on Unitary groupOrthogonal groupWinding numberFundamental groupLagrangian submanifold

Definition

Identify ℝ2n\R^{2n} with ℂn\C^n by (x,y)↦x+iy(x, y) \mapsto x + iy, with the symplectic form ω=∑j=1ndxj∧dyj\omega = \sum_{j=1}^n dx_j \wedge dy_j. A linear subspace Λ⊂ℝ2n\Lambda \subset \R^{2n} is Lagrangian if dim⁡Λ=n\dim \Lambda = n and ω(u,v)=0\omega(u, v) = 0 for all u,v∈Λu, v \in \Lambda. Let Λ(n)\Lambda(n) be the set of Lagrangian subspaces, the Lagrangian Grassmannian.

Every Λ∈Λ(n)\Lambda \in \Lambda(n) is Λ=Uℝn\Lambda = U \R^n for some UU in the unitary group U(n)U(n). This UU is unique up to right multiplication by an element of the orthogonal group O(n)O(n). Hence the map Λ↦det⁡(U)2∈S1⊂ℂ\Lambda \mapsto \det(U)^2 \in S^1 \subset \C is well defined on Λ(n)\Lambda(n). Let γ ⁣:[0,1]→Λ(n)\gamma \colon [0, 1] \to \Lambda(n) be a continuous loop, so γ(0)=γ(1)\gamma(0) = \gamma(1). Its Maslov index is the winding number about 00 of the loop t↦det⁡(U(t))2t \mapsto \det(U(t))^2, where γ(t)=U(t)ℝn\gamma(t) = U(t) \R^n.

Remarks

Example: n=1n = 1 and γ(t)=eiπtℝ⊂ℂ\gamma(t) = e^{i\pi t} \R \subset \C for t∈[0,1]t \in [0, 1], the line through 00 turned by half a turn. Here U(t)=eiπtU(t) = e^{i\pi t} and det⁡(U(t))2=e2πit\det(U(t))^2 = e^{2\pi i t} winds once around 00, so the Maslov index is 11.

Non-example: in ℝ4=ℂ2\R^4 = \C^2, the plane P=ℂ×{0}P = \C \times \set{0} has dimension 22, but ω\omega pairs the vectors (1,0)(1, 0) and (i,0)(i, 0) of PP to 1≠01 \ne 0. So PP is not Lagrangian, and a loop of planes through PP is not a loop in Λ(2)\Lambda(2) and has no Maslov index.

The Maslov index depends only on the homotopy class of the loop, and it gives an isomorphism of the fundamental group π1(Λ(n))\pi_1(\Lambda(n)) with ℤ\Z. For a Lagrangian submanifold L⊂ℝ2nL \subset \R^{2n}, a loop in LL gives the loop of tangent spaces in Λ(n)\Lambda(n), and its Maslov index defines a class in H1(L;ℤ)H^1(L; \Z), the Maslov class.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Fundamental group, concept, no evidence levelFundamental groupconceptIdentity matrix, object, no evidence levelIdentity matrixobjectLagrangian submanifold, concept, no evidence levelLagrangian submanifoldconceptMatrix, definition, no evidence levelMatrixdefinitionWinding number, concept, no evidence levelWinding numberconceptTranspose, definition, no evidence levelTransposedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary group, object, no evidence levelUnitary groupobjectMaslov index, definition, no evidence levelMaslov indexdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Fundamental group concept
  2. Identity matrix object
  3. Lagrangian submanifold concept
  4. Matrix definition
  5. Winding number concept
  6. Transpose definition
  7. Conjugate transpose definition
  8. Orthogonal matrix definition
  9. Unitary matrix definition
  10. Orthogonal group object
  11. Unitary group object
  12. Maslov index definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Maslov_index
About this page
Tag28ADY9WTV3
URIm2://28ADY9WTV3
Kinddefinition
WritingW0 stub
MSC53D12
DefinesMaslov index
UsesUnitary group, Orthogonal group, Winding number, Fundamental group, Lagrangian submanifold

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