SOME NUMERICAL FUNCTIONS ASSOCIATED TO THE MASLOV INDEX Andrew - PowerPoint PPT Presentation

1 SOME NUMERICAL FUNCTIONS ASSOCIATED TO THE MASLOV INDEX Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ � aar The Many Facets of the Maslov Index American Institute of Mathematics, Palo Alto 11th April, 2014

2 Introduction ◮ Want to describe some numerical functions associated to the Maslov index (= nonadditivity invariant) of three lagrangians L 1 , L 2 , L 3 in a symplectic form ( K , φ ), particularly in the case � 0 � 1 ( K , φ ) = H − ( R ) = ( R ⊕ R , ) . − 1 0 ◮ There is a whole zoo of such functions in the literature: τ ( x 1 , x 2 , x 3 ) , [ x ] , { x } , (( x )) , µ ( x ) , η ( x ) , E ( x ) , log z . . . related to Dedekind sums, Rademacher functions, . . .

3 The space Λ(1) of lagrangians in H − ( R ) . ◮ The lagrangians of the symplectic form � 0 � 1 ( K , φ ) = H − ( R ) = ( R ⊕ R , ) − 1 0 are just the 1-dimensional subspaces L ( θ ) = { ( r cos θ, r sin θ ) | r ∈ R } ⊂ K = R ⊕ R for θ ∈ R , with L ( θ ) = L ( θ ′ ) if and only if θ ′ − θ ∈ π Z ⊂ R . ◮ The function S 1 → Λ(1) ; z = e i ψ �→ √ z = L ( ψ/ 2) is a homeomorphism. ◮ Λ(1) may seem a very trivial example, but . . .

4 From Auguries of innocence To see a world in a grain of sand And a heaven in a wild flower, Hold infinity in the palm of your hand, And eternity in an hour. William Blake

5 Nonadditivity, jumps, and signs ◮ The nonadditivity of a function f : R → R is the function R × R → R ; ( x , y ) �→ f ( x ) + f ( y ) − f ( x + y ) . ◮ The jump of a function f : R → R at x ∈ R is j ( x ) = lim ( f ( x + ǫ ) − f ( x − ǫ )) ∈ R . − → ǫ ◮ The sign of x ∈ R is   +1 if x > 0  sgn( x ) = 0 if x = 0   − 1 if x < 0 .

6 The whole and the part ◮ Given a real number x ∈ R let [ x ] ∈ Z be the integer part and let { x } ∈ [0 , 1) be the fractional part, so that x = [ x ] + { x } ∈ R . ◮ Many interesting algebraic and number theoretic properties of the Maslov index can be traced to the jumps and nonadditivity of the functions R → Z ⊂ R ; x �→ [ x ] , R → [0 , 1) ⊂ R ; x �→ { x } . ◮ First appeared in the context of algebraic topology of manifolds in the 1960’s calculations by Hirzebruch of the signatures of manifolds bounding exotic spheres in general (Brieskorn varieties), torus knots in particular,

7 The nonadditivity of [ x ] and { x } ◮ Proposition The functions [ ] : R → Z ; x �→ [ x ] , { } : R → [0 , 1) ; x �→ { x } = x − [ x ] have the following jump and nonadditive properties: 1. { } is continuous on R \ Z , with a jump − 1 at each x ∈ Z . � 0 if 0 � { x } + { y } < 1 2. { x } + { y }−{ x + y } = [ x + y ] − [ x ] − [ y ] = 1 if 1 � { x } + { y } < 2. 3. { x + 1 } = { x } . � 1 if x ∈ R \ Z 4. { x } + {− x } = { x } + { 1 − x } = 0 if x ∈ Z . � 1 / 2 if 0 � { x } < 1 / 2 5. { x + 1 / 2 } − { x } = − 1 / 2 if 1 / 2 � { x } < 1.

8 The triple signature in general ◮ Write the signature of a symmetric form ( L , Φ) as σ ( L , Φ) ∈ Z . ◮ Definition (Wall, Leray, Kashiwara, . . . 1970’s) The Maslov index (aka the triple signature ) of an ordered triple of lagrangians L 1 , L 2 , L 3 in a nonsingular symplectic form ( K , φ ) over R is the signature τ ( L 1 , L 2 , L 3 ) = σ ( L 1 ⊕ L 2 ⊕ L 3 , Φ 123 ) ∈ Z of the symmetric form   0 φ 12 φ 13   Φ 123 = φ 21 0 φ 23 φ 31 φ 32 0 with φ ij : L i × L j → R ; ( x i , x j ) �→ φ ( x i , x j ) .

9 The Maslov index τ ( θ 1 , θ 2 , θ 3 ) I. ◮ Definition The Maslov index of θ 1 , θ 2 , θ 3 ∈ R is τ ( θ 1 , θ 2 , θ 3 ) = τ ( L ( θ 1 ) , L ( θ 2 ) , L ( θ 3 )) ∈ Z , the triple signature of the lagrangians L ( θ 1 ) , L ( θ 2 ) , L ( θ 3 ) in H − ( R ). ◮ From the definition τ ( θ 1 , θ 2 , θ 3 ) = σ ( R ⊕ R ⊕ R , Φ 123 ) with   0 sin ( θ 1 − θ 2 ) sin ( θ 2 − θ 3 )   Φ 123 = sin ( θ 1 − θ 2 ) 0 sin ( θ 3 − θ 1 ) sin ( θ 2 − θ 3 ) sin ( θ 3 − θ 1 ) 0

10 The Maslov index τ ( θ 1 , θ 2 , θ 3 ) II. ◮ The signature of a symmetric matrix is the number of changes of sign in the minors. ◮ The matrix Φ 123 has minors 0 , − sin 2 ( θ 1 − θ 2 ) , sin ( θ 1 − θ 2 )sin ( θ 2 − θ 3 )sin ( θ 3 − θ 1 ) so that τ ( θ 1 , θ 2 , θ 3 ) = sgn(sin ( θ 2 − θ 1 )sin ( θ 3 − θ 2 )sin ( θ 3 − θ 1 ))   sgn( σ ) if { θ 1 / 2 π } , { θ 2 / 2 π } , { θ 3 / 2 π } ∈ [0 , 1) are distinct     with σ ∈ Σ 3 the permutation such that =  { θ σ (1) / 2 π } < { θ σ (2) / 2 π } < { θ σ (3) / 2 π }     0 otherwise ∈ {− 1 , 0 , 1 } ⊂ Z .

11 The Maslov index τ ( θ 1 , θ 2 , θ 3 ) III. ◮ Geometrically: 1 (resp. -1) if e 2 π i θ 1 , e 2 π i θ 2 , e 2 π i θ 3 ∈ S 1 arranged clockwise (resp. counterclockwise) around S 1 , and 0 if any coincidence. ◮ e 2 π i θ 2 e 2 π i θ 3 • • • e 2 π i θ 1 • e 2 π i θ 1 τ ( θ 1 , θ 2 , θ 3 ) = 1 τ ( θ 1 , θ 2 , θ 3 ) = − 1 • • e 2 π i θ 3 e 2 π i θ 2

12 The Maslov index τ ( θ 1 , θ 2 , θ 3 ) IV. ◮ In view of the identity sin ( θ 2 − θ 1 )sin ( θ 3 − θ 2 )sin ( θ 3 − θ 1 ) = (sin 2( θ 2 − θ 1 ) + sin 2( θ 3 − θ 2 ) + sin 2( θ 1 − θ 3 )) / 4 can also write τ ( θ 1 , θ 2 , θ 3 ) = sgn(sin 2( θ 2 − θ 1 ) + sin 2( θ 3 − θ 2 ) + sin 2( θ 1 − θ 3 )) .

13 The sawtooth function (( x )) I. ◮ The sawtooth function (( x )) : R → [ − 1 / 2 , 0) is defined by � { x } − 1 / 2 if x ∈ R \ Z (( x )) = 0 if x ∈ Z with { x } ∈ [0 , 1) the fractional part of x ∈ R . Nonadditive:   − 1 / 2 if 0 < { x } + { y } < 1  (( x ))+(( y )) − (( x + y )) = 1 / 2 if 1 < { x } + { y } < 2   0 if x ∈ Z or y ∈ Z or x + y ∈ Z . ✵ ✿ ✺ ✎ ✎ ✎ ✎ ✎ ✎ ✎ � ✷ � ✶ ✵ ✶ ✷ ✸ ✹ ① � ✵ ✿ ✺ ✶

14 The origin of the sawtooth function (( x )) ◮ Used by Dedekind (1876) in his commentary on the Riemann Nachlass to count ± 2 π i = ± 4( π/ 2) i jumps in the imaginary part of the complex logarithm log( re i θ ) = log( r ) + i ( θ + 2 n π ) ∈ C ( n ∈ Z ) . ◮ From Dedekind’s commentary:

15 Dedekind sums and signatures ◮ Eisenstein’s formula for x = p / q ∈ Q q − 1 � cot π j i q e 2 π ijx . (( x )) = 2 q j =1 ◮ The Dedekind sum for a , c ∈ Z with c � = 0 is | c |− 1 | c |− 1 � � �� k ���� ka �� � k π � � ka π � 1 s ( a , c ) = = cot cot ∈ Q . c c 4 | c | c c k =1 k =1 ◮ Feature prominently in work of Hirzebruch and Zagier. ◮ Barge and Ghys, Cocycles d’Euler et de Maslov (1992) use E ( x ) and Dedekind sums in the hyperbolic geometry interpretation of the Maslov index, related to the action of SL 2 ( Z ) on the upper half plane. ◮ Also Kirby and Melvin, Dedekind sums, µ -invariants and the signature cocycle (1994)

16 The sawtooth function (( x )) II. ◮ Proposition The sawtooth function has the following jumping and nonadditivity properties: 1. (( )) is continuous on R \ Z , with a jump − 1 at each x ∈ Z . 2. ((0)) = ((1 / 2)) = 0. 3. (( x + 1)) = (( x )), (( − x )) = − (( x )). 4. (( x )) = x + ([ − x ] − [ x ]) / 2 = ( { x } − {− x } ) / 2.   0 if x ∈ Z or y ∈ Z or x + y ∈ Z  5. (( x )) + (( y )) − (( x + y )) = − 1 / 2 if 0 < { x } + { y } < 1   1 / 2 if 1 < { x } + { y } < 2 .   { x } if 0 � { x } < 1 / 2  6. (( x + 1 / 2)) = 0 if { x } = 1 / 2   { x } − 1 if 1 / 2 < { x } < 1 .

17 The reverse sawtooth function µ ( x ) I. ◮ Definition The reverse sawtooth function is µ : R → ( − 1 , 1] ; x �→ µ ( x ) = 1 − 2 { x } ◮ • 1 − 1 0 1 2 3 • • • • • • − 1

18 The reverse sawtooth function µ ( x ) II. ◮ Proposition The reverse sawtooth function has the following jumping and nonadditivity properties: � − 2(( x )) if x ∈ R \ Z 1. µ ( x ) = 1 if x ∈ Z . 2. µ is continuous at x ∈ R \ Z , with a jump 2 at each x ∈ Z . � +1 if 0 � { x } + { y } < 1 3. µ ( x ) + µ ( y ) − µ ( x + y ) = − 1 if 1 � { x } + { y } < 2. 4. µ (0) = 1, µ (1 / 2) = 0. 5. µ ( x + 1) = µ ( x ) for x ∈ R . � 0 if x ∈ R \ Z 6. µ ( x ) + µ ( − x ) = 2 if x ∈ Z . � +1 if 0 � { x } < 1 / 2 7. µ ( x ) − µ ( x + 1 / 2) = 2 µ ( x ) − µ (2 x ) = − 1 if 1 / 2 � { x } < 1.

19 The function E ( x ) ◮ Definition (Barge and Ghys, Cocycles d’Euler et de Maslov, 1992) The E -function is � [ x ] + 1 / 2 if x ∈ R \ Z E : R → R ; x �→ x − (( x )) = ([ x ] − [ − x ]) / 2 = x if x ∈ Z ◮ Proposition For any x , y ∈ R E ( x + y ) − E ( x ) − E ( y ) = (( x )) + (( y )) − (( x + y ))   0 if x ∈ Z or y ∈ Z or x + y ∈ Z  = − 1 / 2 if 0 < { x } + { y } < 1   1 / 2 if 1 < { x } + { y } < 2 .