AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY


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1 1 AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY Andrew Ranicki and Daniele Sepe (Edinburgh) aar Maslov index seminar, 9 November 2009
2 The 1dimensional Lagrangians 2 Definition (i) Let R 2 have the symplectic form [, ] : R 2 R 2 R ; ((x 1, y 1 ), (x 2, y 2 )) x 1 y 2 x 2 y 1. (ii) A subspace L R 2 is Lagrangian of (R 2, [, ]) if L = L = {x R 2 [x, y] = 0 for all y L}. Proposition A subspace L R 2 is a Lagrangian of (R 2, [, ]) if and only if L is 1dimensional, Definition (i) The 1dimensional Lagrangian Grassmannian Λ(1) is the space of Lagrangians L (R 2, [, ]), i.e. the Grassmannian of 1dimensional subspaces L R 2. (ii) For θ R let L(θ) = {(rcos θ, rsin θ) r R} Λ(1) be the Lagrangian with gradient tan θ.
3 The topology of Λ(1) 3 Proposition The square function det 2 and the square root function : Λ(1) S 1 ; L(θ) e 2iθ ω : S 1 Λ(1) = R P 1 ; e 2iθ L(θ) are inverse diffeomorphisms, and π 1 (Λ(1)) = π 1 (S 1 ) = Z. Proof Every Lagrangian L in (R 2, [, ]) is of the type L(θ), and L(θ) = L(θ ) if and only if θ θ = kπ for some k Z. Thus there is a unique θ [0, π) such that L = L(θ). The loop ω : S 1 Λ(1) represents the generator ω = 1 π 1 (Λ(1)) = Z.
4 4 The Maslov index of a 1dimensional Lagrangian Definition The Maslov index of a Lagrangian L = L(θ) in (R 2, [, ]) is 1 2θ if 0 < θ < π τ(l) = π 0 if θ = 0 The Maslov index for Lagrangians in (R 2, [, ]) is reduced to the n special case n = 1 by the diagonalization of unitary matrices. The formula for τ(l) has featured in many guises besides the Maslov index (e.g. as assorted η, γ, ρinvariants and an L 2 signature) in the papers of Arnold (1967), Neumann (1978), Atiyah (1987), Cappell, Lee and Miller (1994), Bunke (1995), Nemethi (1995), Cochran, Orr and Teichner (2003),... See aar/maslov.htm for detailed references.
5 5 The Maslov index of a pair of 1dimensional Lagrangians Note The function τ : Λ(1) R is not continuous. Examples τ(l(0)) = τ(l( π 2 )) = 0, τ(l(π 4 )) = 1 2 R with L(0) = R 0, L( π 2 ) = 0 R, L(π ) = {(x, x) x R}. 4 Definition The Maslov index of a pair of Lagrangians (L 1, L 2 ) = (L(θ 1 ), L(θ 2 )) in (R 2, [, ]) is 1 2(θ 2 θ 1 ) if 0 θ 1 < θ 2 < π τ(l 1, L 2 ) = τ(l 2, L 1 ) = π 0 if θ 1 = θ 2. Examples τ(l) = τ(r 0, L), τ(l, L) = 0.
6 6 The Maslov index of a triple of 1dimensional Lagrangians Definition The Maslov index of a triple of Lagrangians in (R 2, [, ]) is (L 1, L 2, L 3 ) = (L(θ 1 ), L(θ 2 ), L(θ 3 )) τ(l 1, L 2, L 3 ) = τ(l 1, L 2 ) + τ(l 2, L 3 ) + τ(l 3, L 1 ) { 1, 0, 1} R. Example If 0 θ 1 < θ 2 < θ 3 < π then τ(l 1, L 2, L 3 ) = 1 Z. Example The Wall computation of the signature of C P 2 is given in terms of the Maslov index as σ(c P 2 ) = τ(l(0), L(π/4), L(π/2)) = τ(l(0), L(π/4)) + τ(l(π/4), L(π/2)) + τ(l(π/2), L(0)) = = 1 Z R.
7 The Maslov index and the degree I. 7 A pair of 1dimensional Lagrangians (L 1, L 2 ) = (L(θ 1 ), L(θ 2 )) determines a path in Λ(1) from L 1 to L 2 ω 12 : I Λ(1) ; t L((1 t)θ 1 + tθ 2 ). For any L = L(θ) Λ(1)\{L 1, L 2 } (ω 12 ) 1 (L) = {t [0, 1] L((1 t)θ 1 + tθ 2 ) = L} = {t [0, 1] (1 t)θ 1 + tθ 2 = θ} { } θ θ1 if 0 < θ θ 1 < 1 = θ 2 θ 1 θ 2 θ 1 otherwise. The degree of a loop ω : S 1 Λ(1) = S 1 is the number of elements in ω 1 (L) for a generic L Λ(1). In the geometric applications the Maslov index counts the number of intersections of a curve in a Lagrangian manifold with the codimension 1 cycle of singular points.
8 8 The Maslov index and the degree II. Proposition A triple of Lagrangians (L 1, L 2, L 3 ) determines a loop in Λ(1) ω 123 = ω 12 ω 23 ω 31 : S 1 Λ(1) with homotopy class the Maslov index of the triple ω 123 = τ(l 1, L 2, L 3 ) { 1, 0, 1} π 1 (Λ(1)) = Z. Proof It is sufficient to consider the special case with 0 θ 1 < θ 2 < θ 3 < π, so that (L 1, L 2, L 3 ) = (L(θ 1 ), L(θ 2 ), L(θ 3 )) det 2 ω 123 = 1 : S 1 S 1, degree(det 2 ω 123 ) = 1 = τ(l 1, L 2, L 3 ) Z
9 9 The Euclidean structure on R 2n The phase space is the 2ndimensional Euclidean space R 2n, with preferred basis {p 1, p 2,..., p n, q 1, q 2,..., q n }. The 2ndimensional phase space carries 4 additional structures. Definition The Euclidean structure on R 2n is the positive definite symmetric form over R (, ) : R 2n R 2n R ; (v, v ) n x j x j + n y k y k, j=1 k=1 (v = n x j p j + n y k q k, v = n x j p j + n y k q k R 2n ). j=1 k=1 The automorphism group of (R 2n, (, )) is the orthogonal group O(2n) of invertible 2n 2n matrices A = (a jk ) (a jk R) such that A A = I 2n with A = (a kj ) the transpose. j=1 k=1
10 10 The complex structure on R 2n Definition The complex structure on R 2n is the linear map ı : R 2n R 2n ; n n x j p j + y k q k j=1 k=1 n n x j p j y k q k j=1 k=1 such that ı 2 = 1. Use ı to regard R 2n as an ndimensional complex vector space, with an isomorphism R 2n C n ; v (x 1 + iy 1, x 2 + iy 2,..., x n + iy n ). The automorphism group of (R 2n, ı) = C n is the complex general linear group GL(n, C) of invertible n n matrices (a jk ) (a jk C).
11 11 The symplectic structure on R 2n Definition The symplectic structure on R 2n is the symplectic form [, ] : R 2n R 2n R ; (v, v ) [v, v ] = (ıv, v ) = [v, v] = n (x j y j x j y j ). The automorphism group of (R 2n, [, ]) is the symplectic group Sp(n) of invertible 2n 2n matrices A = (a jk ) (a jk R) such that ( ) A 0 In A = I n 0 ( 0 ) In I n 0 j=1.
12 The ndimensional Lagrangians Definition Given a finitedimensional real vector space V with a nonsingular symplectic form [, ] : V V R let Λ(V ) be the set of Lagrangian subspaces L V, with L = L = {x V [x, y] = 0 R for all y L}. Terminology Λ(R 2n ) = Λ(n). The real and imaginary Lagrangians n n R n = { x j p j x j R n }, ır n = { y k q k y k R n } Λ(n) j=1 are complementary, with R 2n = R n ır n. Definition The graph of a symmetric form (R n, φ) is the Lagrangian n n n Γ (R n,φ) = {(x, φ(x)) x = x j p j, φ(x) = φ jk x j q k } Λ(n) j=1 k=1 j=1 k=1 complementary to ır n. Proposition Every Lagrangian complementary to ır n is a graph. 12
13 13 The hermitian structure on R 2n Definition The hermitian inner product on R 2n is defined by, : R 2n R 2n C ; (v, v ) v, v = (v, v ) + i[v, v ] = n j=1 (x j + iy j )(x j iy j ). or equivalently by, : C n C n C ; (z, z ) z, z = n z j z j. j=1 The automorphism group of (C n,, ) is the unitary group U(n) of invertible n n matrices A = (a jk ) (a jk C) such that AA = I n, with A = (a kj ) the conjugate transpose.
14 14 The general linear, orthogonal and unitary groups Proposition (Arnold, 1967) (i) The automorphism groups of R 2n with respect to the various structures are related by O(2n) GL(n, C) = GL(n, C) Sp(n) = Sp(n) O(2n) = U(n). (ii) The determinant map det : U(n) S 1 is the projection of a fibre bundle SU(n) U(n) S 1. (iii) Every A U(n) sends the standard Lagrangian ır n of (R 2n, [, ]) to a Lagrangian A(ıR n ). The unitary matrix A = (a jk ) is such that A(ıR n ) = ır n if and only if each a jk R C, with O(n) = {A U(n) A(ıR n ) = ır n } U(n).
15 15 The Lagrangian Grassmannian Λ(n) I. Λ(n) is the space of all Lagrangians L (R 2n, [, ]). Proposition (Arnold, 1967) The function is a diffeomorphism. U(n)/O(n) Λ(n) ; A A(ıR n ) Λ(n) is a compact manifold of dimension dim Λ(n) = dim U(n) dim O(n) = n 2 n(n 1) 2 The graphs {Γ (R n,φ) φ = φ M n (R)} Λ(n) define a chart at R n Λ(n). Example (Arnold and Givental, 1985) = n(n + 1) 2. Λ(2) 3 = {[x, y, z, u, v] R P 4 x 2 + y 2 + z 2 = u 2 + v 2 } = S 2 S 1 /{(x, y) ( x, y)}.
16 The Lagrangian Grassmannian Λ(n) II. 16 In view of the fibration Λ(n) = U(n)/O(n) BO(n) BU(n) Λ(n) classifies real nplane bundles β with a trivialisation δβ : C β = ɛ n of the complex nplane bundle C β. The canonical real nplane bundle η over Λ(n) is The complex nplane bundle C η E(η) = {(L, l) L Λ(n), l L}. E(C η) = {(L, l C ) L Λ(n), l C C R L} is equipped with the canonical trivialisation δη : C η = ɛ n defined by δη : E(C η) = E(ɛ n ) = Λ(n) C n ; (L, l C ) (L, (p, q)) if l C = (p, q) C R L = L il = C n.
17 The fundamental group π 1 (Λ(n)) I. 17 Theorem (Arnold, 1967) The square of the determinant function det 2 : Λ(n) S 1 ; L = A(ıR n ) det(a) 2 induces an isomorphism det 2 = : π 1 (Λ(n)) π 1 (S 1 ) = Z. Proof By the homotopy exact sequence of the commutative diagram of fibre bundles det SO(n) O(n) O(1) = S 0 SU(n) SΛ(n) U(n) Λ(n) det U(1) = S 1 z z 2 det 2 Λ(1) = S 1
18 18 Corollary 1 The function The fundamental group π 1 (Λ(n)) = Z II. π 1 (Λ(n)) Z ; (ω : S 1 Λ(n)) degree( S 1 ω Λ(n) det2 S 1 ) is an isomorphism. Corollary 2 The cohomology class α H 1 (Λ(n)) = Hom(π 1 (Λ(n)), Z) characterized by α(ω) = degree( S 1 ω Λ(n) det2 S 1 ) Z is a generator α H 1 (Λ(n)) = Z.
19 π 1 (Λ(1)) = Z in terms of line bundles Example The universal real line bundle η : S 1 BO(1) = R P is the infinite Möbius band over S 1 E(η) = R [0, π]/{(x, 0) ( x, π)} S 1 = [0, π]/(0 π). The complexification C η : S 1 BU(1) = C P is the trivial complex line bundle over S 1 E(C η) = C [0, π]/{(z, 0) ( z, π)} S 1 = [0, π]/(0 π). The canonical trivialisation δη : C η = ɛ is defined by = δη : E(C η) E(ɛ) = C [0, π]/{(z, 0) (z, π)} = C S 1 ; [z, θ] [ze iθ, θ]. The canonical pair (δη, η) represents (δη, η) = 1 π 1 (Λ(1)) = π 1 (U(1)/O(1)) = Z, corresponding to the loop ω : S 1 Λ(1); e 2iθ L(θ). 19
20 The Maslov index for ndimensional Lagrangians Unitary matrices can be diagonalized. For every A U(n) there exists B U(n) such that BAB 1 = D(e iθ 1, e iθ 2,..., e iθn ) is the diagonal matrix, with e iθ j S 1 the eigenvalues, i.e. the roots of the characteristic polynomial n ch z (A) = det(zi n A) = (z e iθ j ) C[z]. j=1 Definition The Maslov index of L Λ(n) is n τ(l) = (1 2θ j /π) R j=1 with θ 1, θ 2,..., θ n [0, π) such that ±e iθ 1, ±e iθ 2,..., ±e iθn are the eigenvalues of any A U(n) such that A(ıR n ) = L. There are similar definitions of τ(l, L ) R and τ(l, L, L ) Z. 20
21 21 The Maslov cycle Corollary 2 above shows that the Maslov index α H 1 (Λ(n)) is the pullback along det 2 : Λ(n) S 1 of the standard form dθ on S 1. Definition The Poincaré dual of α is the Maslov cycle m = {L Λ(n) L ır n {0}}. In his 1967 paper Arnold constructs this cycle following ideas that generalise the standard constructions on Schubert varieties in the Grassmannians of linear subspaces of Euclidean spaces, and proves that [m] H (n+2)(n 1)/2 (Λ(n)) is the Poincaré dual of α H 1 (Λ(n)).
22 Lagrangian submanifolds of R 2n I. 22 An ndimensional submanifold M n R 2n is Lagrangian if for each x M the tangent nplane T x (M) T x (R 2n ) = R 2n is a Lagrangian subspace of (R 2n, [, ]). The tangent bundle T M : M BO(n) is the pullback of the canonical real nplane bundle over Λ(n) E(η) = {(λ, l) λ Λ(n), l λ} along the classifying map ζ : M Λ(n); x T x (M), with T M : M ζ η Λ(n) BO(n). The complex nplane bundle C T M = T C n M has a canonical trivialisation. Definition The Maslov index of a Lagrangian submanifold M n R 2n is the pullback of the generator α H 1 (Λ(n)) τ(m) = ζ (α) H 1 (M).
23 Lagrangian submanifolds of R 2n II. Given a Lagrangian submanifold M n R 2n let π = proj : M n R n ; (p, q) p be the restriction of the projection R 2n = R n ır n R n. The differentials of π are the differentiable maps dπ x : T x (M) T π(x) (R n ) = R n ; l p if l = (p, q) T x (M) R 2n with ker(dπ x ) = T x (M) ır n ır n. If π is a local diffeomorphism then each dπ x is an isomorphism of real vector spaces, with kernel T x (M) ır n = {0}. Definition The Maslov cycle of a Lagrangian submanifold M R 2n is m = {x M T x (M) ır n {0}}. Theorem (Arnold, 1967) Generically, m M is a union of open submanifolds of codimension 1. The homology class [m] H n 1 (M) is the Poincaré dual of the Maslov index class τ(m) H 1 (M), measuring the failure of π : M R n to be a local diffeomorphism. 23
24 24 Symplectic manifolds Definition A manifold N is symplectic if it admits a 2form ω which is closed and nondegenerate. Examples S 2, R 2n = C n = T R n ; the cotangent bundle T B of any manifold B with a canonical symplectic form. Definition A submanifold M N of a symplectic manifold is Lagrangian if ω M = 0 and each T x M T x N (x M) is a Lagrangian subspace. Remark A symplectic manifold N is necessarily even dimensional (because of the nondegeneracy of ω) and the dimension of a Lagrangian submanifold M is necessarily half the dimension of the ambient manifold. Examples If N is two dimensional, any one dimensional submanifold is Lagrangian (why?). If N = T B for some manifold B with the canonical symplectic form, then each cotangent space T b B is a Lagrangian submanifold.
25 The Λ(n)affine structure of Λ(V ) Let V be a 2ndimensional real vector space with nonsingular symplectic form [, ] : V V R and a complex structure ı : V V such that V V R; (v, w) [ıv, w] is an inner product. Let U(V ) be the group of automorphisms A : (V, [, ], ı) (V, [, ], ı). For L Λ(V ) let O(L) = {A U(V ) A(L) = L}. Proposition (Guillemin and Sternberg) (i) The function U(V )/O(L) Λ(V ) ; A A(L) is a diffeomorphism. (ii) There exists an isomorphism f L : (R 2n, [, ], ı) = (V, [, ], ı) such that f L (ır n ) = L. (iii) For any f L as in (ii) the diffeomorphism f L : Λ(n) = U(n)/O(n) Λ(V ) = U(V )/O(L) ; λ f L (λ) only depends on L, and not on the choice of f L. Thus Λ(V ) has a Λ(n)affine structure: for any L 1, L 2 Λ(V ) there is defined a difference element (L 1, L 2 ) Λ(n), with (L 1, L 1 ) = ır n R 2n 25
26 26 An application of the Maslov index I. For any ndimensional manifold B the tangent bundle of the cotangent bundle E = T(T B) T B has each fibre a 2ndimensional symplectic vector space. The symplectic form is given in canonical (or Darboux) coordinates as ω = dp dq (q B, p T qb). The bundle of Lagrangian planes Π : Λ(E) T B is the fibre bundle with fibres the affine Λ(n)sets Π 1 (x) = Λ(T x T B) of Lagrangians in the 2ndimensional symplectic vector space T x T B.
27 27 An application of the Maslov index II. The projection π : T B B induces the vertical subbundle V = ker dπ T(T B), whose fibres are Lagrangian subspaces. The map s : T B Λ(E) ; x = (p, q) V x = ker(dπ x : T x (T B) T π(x) (B)) is a section of the bundle of Lagrangian planes. For any other section r : T B Λ(E) use the Λ(n)affine structures of the fibres Π 1 (x) to define a difference map ζ = (r, s) : T B Λ(n) ; x (r(x), s(x))
28 28 An application of the Maslov index III. Let M be a Lagrangian submanifold of T B, dim M = dim B = n. If π M : M B is a local diffeomorphism then each d(π M ) x : T x (M) T π(x) (B) is an isomorphism of vector bundles, with kernel T x (M) V x = {0}. The Maslov index measures the failure of π M : M B to be a local diffeomorphism, in general. We let E M = TM T(T B), Λ(E M ) = Π 1 (M) Λ(E) so that Π : Λ(E M ) M is a fibre bundle with the fibre over x M the Λ(n)affine set (Π ) 1 (x) = Λ(T x M).
29 An application of the Maslov index IV. 29 The maps r, s M : M Λ(E M ) defined by r(x) = T x M, s M (x) = V x T x (T B) are sections of Π. The difference map ζ = (r, s M ) : M Λ(n) classifies the real nplane bundle over M with complex trivialisation in which the fibre over x M is the Lagrangian (T x M, V x ) Λ(n). Definition The Maslov index class is the pullback along ζ of the generator α = 1 H 1 (Λ(n)) = Z α M = ζ α H 1 (M). Proposition The homology class [m] H n 1 (M) of the Maslov cycle m = {x M T x M V x {0}}, is the Poincaré dual of the Maslov index class α M H 1 (M). If the class [m] = α M H n 1 (M) = H 1 (M) is nonzero then the projection π M : M B fails to be a local diffeomorphism.
30 30 An application of the Maslov index V. (Butler, 2007) This formulation of the Maslov index can be used to prove that if Σ is an ndimensional manifold which is topologically T n, and which admits a geometrically semisimple convex Hamiltonian, then Σ has the standard differentiable structure.
31 31 References V.I.Arnold, On a characteristic class entering into conditions of quantisation, Functional Analysis and Its Applications 1, 114 (1967) V.I.Arnold and A.B.Givental, Symplectic geometry, in Dynamical Systems IV, Encycl. of Math. Sciences 4, Springer, (1990) L.Butler, The Maslov cocycle, smooth structures and realanalytic complete integrability, to appear in Amer. J. Maths. V.Guillemin and S.Sternberg, Symplectic geometry, Chapter IV of Geometric Optics, Math. Surv. and Mon. 14, AMS, (1990)
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