AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY Andrew - PowerPoint PPT Presentation

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

2 The 1-dimensional Lagrangians ◮ 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 ) = { ( r cos θ, r sin θ ) | r ∈ R } ∈ Λ(1) be the Lagrangian with gradient tan θ .

3 The topology of Λ(1) ◮ Proposition The square function det 2 : Λ(1) → S 1 ; L ( θ ) �→ e 2 i θ and the square root function ω : S 1 → Λ(1) = R P 1 ; e 2 i θ �→ 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 = k π 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 − 2 θ  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 4 ) = { ( x , x ) | x ∈ R } . ◮ 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 π τ ( L 1 , L 2 ) = − τ ( L 2 , L 1 ) =   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 ( L 1 , L 2 , L 3 ) = ( L ( θ 1 ) , L ( θ 2 ) , L ( θ 3 )) in ( R 2 , [ , ]) is 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 .

7 The Maslov index and the degree I. ◮ 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 + t θ 2 ) . ◮ For any L = L ( θ ) ∈ Λ(1) \{ L 1 , L 2 } ( ω 12 ) − 1 ( L ) = { t ∈ [0 , 1] | L ((1 − t ) θ 1 + t θ 2 ) = L } = { t ∈ [0 , 1] | (1 − t ) θ 1 + t θ 2 = θ } � θ − θ 1 �  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 ( L 1 , L 2 , L 3 ) = ( L ( θ 1 ) , L ( θ 2 ) , L ( θ 3 )) with 0 � θ 1 < θ 2 < θ 3 < π , so that 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 2 n ◮ The phase space is the 2 n -dimensional Euclidean space R 2 n , with preferred basis { p 1 , p 2 , . . . , p n , q 1 , q 2 , . . . , q n } . ◮ The 2 n -dimensional phase space carries 4 additional structures. ◮ Definition The Euclidean structure on R 2 n is the positive definite symmetric form over R � n � n ( , ) : R 2 n × R 2 n → R ; ( v , v ′ ) �→ x j x ′ y k y ′ j + k , j =1 k =1 � n � n � n � n y k q k , v ′ = x ′ y ′ k q k ∈ R 2 n ) . ( v = x j p j + j p j + j =1 k =1 j =1 k =1 ◮ The automorphism group of ( R 2 n , ( , )) is the orthogonal group O (2 n ) of invertible 2 n × 2 n matrices A = ( a jk ) ( a jk ∈ R ) such that A ∗ A = I 2 n with A ∗ = ( a kj ) the transpose.

10 The complex structure on R 2 n ◮ Definition The complex structure on R 2 n is the linear map n n n n � � � � ı : R 2 n → R 2 n ; x j p j + y k q k �→ x j p j − y k q k j =1 k =1 j =1 k =1 such that ı 2 = − 1. Use ı to regard R 2 n as an n -dimensional complex vector space, with an isomorphism R 2 n → C n ; v �→ ( x 1 + iy 1 , x 2 + iy 2 , . . . , x n + iy n ) . ◮ The automorphism group of ( R 2 n , 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 2 n ◮ Definition The symplectic structure on R 2 n is the symplectic form [ , ] : R 2 n × R 2 n → R ; � n ( v , v ′ ) �→ [ v , v ′ ] = ( ı v , v ′ ) = − [ v ′ , v ] = ( x ′ j y j − x j y ′ j ) . j =1 ◮ The automorphism group of ( R 2 n , [ , ]) is the symplectic group Sp ( n ) of invertible 2 n × 2 n matrices A = ( a jk ) ( a jk ∈ R ) such that � 0 � � 0 � I n I n A ∗ A = . − I n − I n 0 0

12 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 2 n ) = Λ( n ). The real and imaginary Lagrangians n n � � R n = { x j p j | x j ∈ R n } , ı R n = { y k q k | y k ∈ R n } ∈ Λ( n ) j =1 k =1 are complementary, with R 2 n = R n ⊕ ı R 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 = φ jk x j q k } ∈ Λ( n ) x j p j , φ ( x ) = j =1 j =1 k =1 complementary to ı R n . ◮ Proposition Every Lagrangian complementary to ı R n is a graph.

13 The hermitian structure on R 2 n ◮ Definition The hermitian inner product on R 2 n is defined by � , � : R 2 n × R 2 n → C ; � n ( v , v ′ ) �→ � v , v ′ � = ( v , v ′ ) + i [ v , v ′ ] = ( x j + iy j )( x ′ j − iy ′ j ) . j =1 or equivalently by � n � , � : C n × C n → C ; ( z , z ′ ) �→ � z , z ′ � = 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 2 n with respect to the various structures are related by O (2 n ) ∩ GL ( n , C ) = GL ( n , C ) ∩ Sp ( n ) = Sp ( n ) ∩ O (2 n ) = 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 ı R n of ( R 2 n , [ , ]) to a Lagrangian A ( ı R n ). The unitary matrix A = ( a jk ) is such that A ( ı R n ) = ı R n if and only if each a jk ∈ R ⊂ C , with O ( n ) = { A ∈ U ( n ) | A ( ı R n ) = ı R n } ⊂ U ( n ) .

15 The Lagrangian Grassmannian Λ( n ) I. ◮ Λ( n ) is the space of all Lagrangians L ⊂ ( R 2 n , [ , ]). ◮ Proposition (Arnold, 1967) The function U ( n ) / O ( n ) → Λ( n ) ; A �→ A ( ı R n ) is a diffeomorphism. ◮ Λ( n ) is a compact manifold of dimension dim Λ( n ) = dim U ( n ) − dim O ( n ) = n 2 − n ( n − 1) = n ( n + 1) . 2 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) = { [ x , y , z , u , v ] ∈ R P 4 | x 2 + y 2 + z 2 = u 2 + v 2 } Λ(2) 3 = S 2 × S 1 / { ( x , y ) ∼ ( − x , − y ) } .