The Maslov phase of maximal totally real subspaces #
Let E be a complex vector space of dimension n, and call a real subspace L of E
maximal totally real when it is complementary to i L (TauCeti.IsMaximalTotallyReal for the
real-linear map J = i •). Such an L is a real form of E: a real basis of L is a complex
basis of E. Consequently the complex-linear automorphisms of E act transitively on maximal
totally real subspaces, and a complex-linear automorphism C preserving L is the
complexification of its restriction to L, so that det C is real.
These two facts make the Maslov phase
ρ(L₀, A L₀) = det A / conj (det A) = det A ^ 2 / |det A| ^ 2
well defined: its value does not depend on the automorphism A taking L₀ to L = A L₀,
because two choices differ by an automorphism preserving L₀, whose determinant is real. This is
the map GL(n, ℂ) / GL(n, ℝ) → S¹ through which the Maslov index of a loop of totally real
subspaces is defined as a winding number, and which, for the boundary values of a
Cauchy--Riemann operator with totally real boundary conditions, enters the Riemann--Roch formula
for its Fredholm index. For Lagrangian subspaces of ℂⁿ one may take A unitary, recovering
the classical formula ρ(Λ) = det (U) ^ 2 for Λ = U ℝⁿ.
The phase is taken relative to a reference subspace L₀ rather than to a fixed ℝⁿ ⊆ ℂⁿ, so
it is coordinate-free; it is multiplicative along chains
(TauCeti.IsMaximalTotallyReal.maslovPhase_mul_maslovPhase) and invariant under simultaneous
complex-linear changes of coordinates (TauCeti.IsMaximalTotallyReal.maslovPhase_map_map).
Main declarations #
TauCeti.IsMaximalTotallyReal.maslovPhase: the Maslov phaseρ(L₀, L).TauCeti.IsMaximalTotallyReal.maslovPhase_map:ρ(L₀, A L₀) = det A / conj (det A).TauCeti.IsMaximalTotallyReal.norm_maslovPhase: the Maslov phase has modulus one.TauCeti.IsMaximalTotallyReal.maslovPhase_map_lsmul: rotating byzhas phase(z / conj z) ^ n, soe^{iθ} L₀has phasee^{2inθ}.
References #
- D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., AMS Colloquium Publications 52, 2012, Appendix C.3 (the boundary Maslov index of a bundle pair with totally real boundary condition).
- D. McDuff and D. Salamon, Introduction to Symplectic Topology, Section 2.3 (the map
ρ(U ℝⁿ) = det (U) ^ 2on the Lagrangian Grassmannian).
The Maslov phase ρ(L₀, L) of two maximal totally real subspaces of a complex module.
For L = A L₀ it is det A / conj (det A); the value is independent of the choice of A.
Instances For
The Maslov phase has modulus one.
The Maslov phase is nonzero.
Rewrites the second subspace of a Maslov phase along an equality of subspaces.
Rewrites the first subspace of a Maslov phase along an equality of subspaces.
The Maslov phase of A L relative to a maximal totally real L is the determinant phase
det A / conj (det A) of the complex-linear automorphism A.
A maximal totally real subspace has Maslov phase one relative to itself.
The Maslov phase is multiplicative along chains of maximal totally real subspaces:
ρ(L₀, L₁) ρ(L₁, L₂) = ρ(L₀, L₂).
The Maslov phase of two subspaces is the quotient of their phases relative to a common
reference: ρ(L, L₂) / ρ(L, L₁) = ρ(L₁, L₂).
Exchanging the two subspaces inverts the Maslov phase.
The Maslov phase is invariant under a simultaneous complex-linear change of coordinates.
Rotating a maximal totally real subspace L of an n-dimensional complex module by a nonzero
scalar z has Maslov phase (z / conj z) ^ n; for z = e^{iθ} this is e^{2inθ}.
The image i L of a maximal totally real subspace L of an n-dimensional complex module
has Maslov phase (-1) ^ n relative to L.