AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY

1 AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY Andrew Ranicki and Daniele Sepe (Edinburgh) http://www.maths.ed.ac.uk/ aar Maslov index seminar, 9 November 2009

The 1-dimensional 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 1-dimensional, Definition (i) The 1-dimensional Lagrangian Grassmannian Λ(1) is the space of Lagrangians L (R 2, [, ]), i.e. the Grassmannian of 1-dimensional 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.

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 The Maslov index of a 1-dimensional 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-, ρ-invariants 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 http://www.maths.ed.ac.uk/ aar/maslov.htm for detailed references.

5 The Maslov index of a pair of 1-dimensional 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 The Maslov index of a triple of 1-dimensional 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 2 + 1 2 + 0 = 1 Z R.

The Maslov index and the degree I. 7 A pair of 1-dimensional 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 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 The Euclidean structure on R 2n The phase space is the 2n-dimensional Euclidean space R 2n, with preferred basis {p 1, p 2,..., p n, q 1, q 2,..., q n }. The 2n-dimensional 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 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 n-dimensional 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 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.

The n-dimensional Lagrangians Definition Given a finite-dimensional 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 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 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 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)}.

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 n-plane bundles β with a trivialisation δβ : C β = ɛ n of the complex n-plane bundle C β. The canonical real n-plane bundle η over Λ(n) is The complex n-plane 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.

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 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.

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

The Maslov index for n-dimensional 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 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)).

Lagrangian submanifolds of R 2n I. 22 An n-dimensional submanifold M n R 2n is Lagrangian if for each x M the tangent n-plane 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 n-plane 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 n-plane 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).

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 Symplectic manifolds Definition A manifold N is symplectic if it admits a 2-form ω which is closed and non-degenerate. 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 non-degeneracy 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.

The Λ(n)-affine structure of Λ(V ) Let V be a 2n-dimensional 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 An application of the Maslov index I. For any n-dimensional manifold B the tangent bundle of the cotangent bundle E = T(T B) T B has each fibre a 2n-dimensional 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 2n-dimensional symplectic vector space T x T B.

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 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).

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 n-plane 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 non-zero then the projection π M : M B fails to be a local diffeomorphism.

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 n-dimensional manifold which is topologically T n, and which admits a geometrically semi-simple convex Hamiltonian, then Σ has the standard differentiable structure.

31 References V.I.Arnold, On a characteristic class entering into conditions of quantisation, Functional Analysis and Its Applications 1, 1-14 (1967) V.I.Arnold and A.B.Givental, Symplectic geometry, in Dynamical Systems IV, Encycl. of Math. Sciences 4, Springer, 1-136 (1990) L.Butler, The Maslov cocycle, smooth structures and real-analytic complete integrability, http://uk.arxiv.org/abs/0708.3157, to appear in Amer. J. Maths. V.Guillemin and S.Sternberg, Symplectic geometry, Chapter IV of Geometric Optics, Math. Surv. and Mon. 14, AMS, 109-202 (1990)