THE MASLOV INDEX, THE SIGNATURE AND BAGELS Andrew Ranicki - PowerPoint PPT Presentation

1 THE MASLOV INDEX, THE SIGNATURE AND BAGELS Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ � aar G¨ ottingen, 22 December 2009

2 Introduction ◮ The original Maslov index appeared in the early 1960’s work of the Russian mathematical physicist V.P.Maslov on the quantum mechanics of nanostructures and lasers; he has also worked on the tokamak (= magnetic field bagel with plasma filling). The Maslov index also appeared in the early 1960’s work of J.B.Keller and H.M.Edwards. ◮ V.I.Arnold (1967) put the Maslov index on a mathematical footing, in terms of the intersections of paths in the space of lagrangian subspaces of a symplectic form . ◮ The Maslov index is the generic name for a very large number of inter-related invariants which arise in the topology of manifolds, symplectic geometry, mathematical physics, index theory, L 2 -cohomology, surgery theory, knot theory, singularity theory, differential equations, group theory, representation theory, as well as the algebraic theory of quadratic forms and their automorphisms. ◮ Maslov index : 387 entries on Mathematical Reviews, 27,100 entries on Google Scholar, 45,000 entries on Google.

3 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 a 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 .

4 The topology of Λ(1) ◮ Proposition The square function 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 .

5 The real Maslov index of a 1-dimensional lagrangian I. ◮ Definition The real-valued Maslov index of a lagrangian L = L ( θ ) in ( R 2 , [ , ]) is  1 − 2 θ  if 0 < θ < π τ ( L ( θ )) = π ∈ R .  0 if θ = 0 ◮ Examples τ ( L (0)) = τ ( L ( π/ 2)) = 0 , τ ( L ( π/ 4)) = 1 / 2 , τ ( L (3 π/ 4)) = − 1 / 2 . ◮ For 0 < θ < π = 1 − 2 θ/π τ ( L ( θ )) = − 1 + 2( π − θ ) /π = − τ ( L ( π − θ )) ∈ R .

6 The real Maslov index of a 1-dimensional lagrangian II. ◮ Motivation in terms of the L 2 -signature for Z , with 0 < θ < π ∫ 1 sgn((1 − ω ) e i θ + (1 − ω ) e − i θ ) d ω τ ( L ( θ )) = 2 π ω ∈ S 1 2 π ∫ 1 sgn(sin( ψ/ 2)sin( ψ/ 2 + θ )) d ψ ( ω = e i ψ ) = 2 π ψ =0 2 π − 2 θ 2 π ∫ ∫ 1 = 2 π ( d ψ − d ψ ) ψ =0 ψ =2 π − 2 θ 1 = 2 π (2 π − 2 θ − 2 θ ) = 1 − 2 θ π ∈ R .

7 The real Maslov index ◮ Many other motivations! ◮ The real Maslov index formula τ ( L ( θ )) = 1 − 2 θ π ∈ R has featured in many guises (e.g. as assorted η -, γ -, ρ -invariants and an L 2 -signature) in the papers of Arnold (1967), Atiyah, Patodi and Singer (1975), Neumann (1978), Atiyah (1987), Cappell, Lee and Miller (1994), Bunke (1995), Nemethi (1995), Cochran, Orr and Teichner (2003), . . . ◮ Can be traced back to the failure of the Hirzebruch signature theorem and the Atiyah-Singer index theorem for manifolds with boundary. ◮ See http://www.maths.ed.ac.uk/˜aar/maslov.htm for detailed references.

8 The real Maslov index of a pair of 1-dimensional lagrangians ◮ Definition The Maslov index of a pair of lagrangians in ( R 2 , [ , ]) ( L 1 , L 2 ) = ( L ( θ 1 ) , L ( θ 2 )) is = τ ( L ( θ 2 − θ 1 )) τ ( L 1 , L 2 )  1 − 2( θ 2 − θ 1 )   if 0 � θ 1 < θ 2 < π ∈ R   π   − 1 + 2( θ 1 − θ 2 ) = if 0 � θ 2 < θ 1 < π   π     0 if θ 1 = θ 2 . ◮ τ ( L 1 , L 2 ) = − τ ( L 2 , L 1 ) ∈ R . ◮ Examples τ ( L ) = τ ( R ⊕ 0 , L ), τ ( L , L ) = 0.

9 The integral 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 .

10 The integral 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.

11 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

12 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.

13 The complex structure on R 2 n ◮ Definition The complex structure on R 2 n is the linear map n n n n ∑ ∑ ∑ ∑ J : 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 J 2 = − 1 : R 2 n → R 2 n . Use J 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 , J ) = C n is the complex general linear group GL ( n , C ) of invertible n × n matrices ( a jk ) ( a jk ∈ C ).

14 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 ′ ] = ( Jv , v ′ ) = − [ v ′ , v ] = ( x ′ j y j − x j y ′ j ) j =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 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 0 − I n 0

15 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 ). ◮ Proposition Every lagrangian L ∈ Λ( n ) has a canonical complement JL ∈ Λ( n ), with L ⊕ JL = R 2 n . ◮ Example R n and J R n are lagrangian complements, with R 2 n = R n ⊕ J 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 J R n . ◮ Proposition Every lagrangian complementary to J R n is a graph.