The Maslov index of a loop of totally real subspaces #
Let E be a finite-dimensional complex normed space and Λ a loop of maximal totally real
subspaces of E (TauCeti.TotallyRealLoop): a family Λ : [0, 1] → Submodule ℝ E with
Λ 0 = Λ 1 of the form Λ t = A t L₀, where L₀ is maximal totally real and A is a continuous
path of complex-linear automorphisms of E.
Its Maslov index μ(Λ) (TauCeti.TotallyRealLoop.maslovIndex) is the degree
(Circle.degree) of the loop of Maslov phases t ↦ ρ(Λ 0, Λ t) of the circle, read as a path
(TauCeti.TotallyRealLoop.maslovPhasePath), where
ρ(L₀, A L₀) = det A / conj (det A) is TauCeti.IsMaximalTotallyReal.maslovPhase.
- It does not depend on the reference subspace: for any maximal totally real
L₀the phasesρ(L₀, Λ t)differ fromρ(Λ 0, Λ t)by the constant factorρ(L₀, Λ 0), soμ(Λ)is the degree oft ↦ ρ(L₀, Λ t)(TauCeti.TotallyRealLoop.maslovIndex_eq_degree), and every continuous angle functionθof it satisfiesθ 1 - θ 0 = 2π μ(Λ)(TauCeti.TotallyRealLoop.sub_eq_maslovIndex_mul). - It counts half-turns of the determinant of a frame: if
Λ t = A t L₀anddet (A t) = |det (A t)| e^{i φ t}withφcontinuous, thenφ 1 - φ 0 = π μ(Λ)(TauCeti.TotallyRealLoop.sub_eq_maslovIndex_mul_pi). - It is invariant under homotopies of loops given by continuous two-parameter frames
(
TauCeti.TotallyRealLoop.maslovIndex_eq_of_homotopy). - The Maslov phase relative to a fixed maximal totally real subspace varies continuously along
any continuous frame over a topological space
(
TauCeti.IsMaximalTotallyReal.continuous_maslovPhase_of_frame), in particular along a loop (TauCeti.TotallyRealLoop.continuous_maslovPhase), which is whymaslovPhasePathis well defined. - It is normalized: the half-turn
t ↦ e^{iπt} L₀, which closes up since-L₀ = L₀, has Maslov indexn = dim_ℂ E(TauCeti.TotallyRealLoop.maslovIndex_rotation); forE = ℂandL₀ = ℝthis is the loope^{iπt} ℝof Maslov index one. More generally thek-fold half-turnt ↦ e^{iπkt} L₀has Maslov indexk nfor every integerk(TauCeti.TotallyRealLoop.maslovIndex_rotationZPow). The constant loop has Maslov index zero (TauCeti.TotallyRealLoop.maslovIndex_const). - It changes sign when the loop is traversed backwards
(
TauCeti.TotallyRealLoop.maslovIndex_symm).
The boundary Maslov index of a bundle pair, which enters the Riemann--Roch formula for Cauchy--Riemann operators with totally real boundary conditions, is the Maslov index of the loop of boundary subspaces in a trivialization.
References #
- D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., AMS Colloquium Publications 52, 2012, Appendix C.3 (the Maslov index of a loop of totally real subspaces and the boundary Maslov index of a bundle pair).
- D. McDuff and D. Salamon, Introduction to Symplectic Topology, Section 2.3 (the Maslov index
of loops of Lagrangian subspaces as the degree of
det² : Λ(n) → S¹).
Along a loop of maximal totally real subspaces, the Maslov phase relative to any fixed maximal totally real subspace varies continuously.
The loop of Maslov phases t ↦ ρ(Λ 0, Λ t), as a loop in the circle based at 1.
Instances For
The Maslov phase of Λ t relative to Λ 0, read as a complex number.
The Maslov index of a loop of maximal totally real subspaces: the degree of its loop of
Maslov phases t ↦ ρ(Λ 0, Λ t) in the circle.
Instances For
The Maslov index read off a continuous angle function of the Maslov phases relative to an
arbitrary maximal totally real reference subspace L₀: if e^{i θ t} = ρ(L₀, Λ t), then
θ 1 - θ 0 = 2π μ(Λ).
The Maslov index is determined by any one continuous angle function of the Maslov phases relative to a maximal totally real reference subspace.
The Maslov index is the degree of the loop of Maslov phases t ↦ ρ(L₀, Λ t) relative to any
maximal totally real reference subspace L₀.
Homotopy invariance of the Maslov index. Let A (s, t) be a continuous two-parameter
family of complex-linear automorphisms and L₀ a maximal totally real subspace such that every
t ↦ A (s, t) L₀ is a closed loop. Then the loops Λ₀ t = A (0, t) L₀ and Λ₁ t = A (1, t) L₀
have the same Maslov index.
The Maslov index counts half-turns of the determinant of a frame: if Λ t = A t L₀ and
det (A t) = |det (A t)| e^{i φ t} with φ continuous, then φ 1 - φ 0 = π μ(Λ).
The constant loop has Maslov index zero.
Normalization of the Maslov index, k-fold version. The k-fold half-turn
t ↦ e^{iπkt} L₀ of a maximal totally real subspace of an n-dimensional complex space has
Maslov index k n, for every integer k.
Normalization of the Maslov index. The half-turn t ↦ e^{iπt} L₀ of a maximal totally
real subspace of an n-dimensional complex space has Maslov index n.
Reversing a loop negates its Maslov index: traversing the boundary the other way turns the loop of Maslov phases around.