Discrete Laplace-Darboux sequences, Menelaus theorem and the - PowerPoint PPT Presentation

Discrete Laplace-Darboux sequences, Menelaus’ theorem and the pentagram map by W.K. Schief Technische Universit¨ at Berlin ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Australia

1. Discrete Laplace-Darboux transformations (Doliwa 1997) Conjugate lattice: � 2 → ✁ 3 Φ : ✂ 2 ∼ � 2 = { ( n 1 , n 2 ) ∈ ✂ 2 : n 1 + n 2 odd } = with planar faces. Laplace-Darboux transformations: + Φ − Φ Φ L + : 2 1 ] �→ Φ + [Φ ¯ 2 , Φ 1 , Φ 2 , Φ ¯ Φ _ L − : 1 ] �→ Φ − 1 [Φ ¯ 2 , Φ 1 , Φ 2 , Φ ¯ Φ 1 Φ _ 2

2. Laplace-Darboux sequences Facts: (1) Φ + and Φ − likewise constitute conjugate lattices 4 3 1 2 3 4 2 1 (2) ” L + ◦ L − = L − ◦ L + = id ” (3) There exist invariants h ( n ) associated with the conjugate lattices Φ ( n ) = ( L + ) n (Φ) . These obey a gauge-invariant version of the discrete 2-dimensional Toda equation, i.e. a discretisation of (ln h ( n ) ) xy = − h ( n − 1) + 2 h ( n ) − h ( n +1)

3. The combinatorics of Laplace-Darboux sequences Combinatorial picture: Interpretation: Laplace-Darboux sequences generate three-dimensional lattices of face- centred cubic (fcc) combinatorics: � 3 → ✁ 3 Φ : � 3 = { ( n 1 , n 2 , n 3 ) ∈ ✂ 3 : n 1 + n 2 + n 3 odd } Φ Φ 3 _ 3 with the properties Φ 2 Φ 3 = L + (Φ ¯ 2 , Φ 1 , Φ 2 , Φ ¯ 1 ) Φ _ 1 Φ 3 = L − (Φ ¯ 1 Φ ¯ 2 , Φ 1 , Φ 2 , Φ ¯ 1 ) Φ _ 2

4. Laplace-Darboux lattices Observation: Laplace-Darboux lattices are ‘symmetric’ in n 1 , n 2 , n 3 , that is the two- dimensional sublattices Φ( n 1 = const , n 2 , n 3 ) and Φ( n 1 , n 2 = const , n 3 ) may also be regarded as conjugate lattices which are related by Laplace-Darboux transfor- mations! Interpretation: � 3 = set of vertices of a collection of octahedra (1) 3 _ 2 1 1 _ 2 _ 3

..... (2) Bipartite structure of octahedra 4 1 Φ 4 3 3 1 2 2 Definition. A Laplace-Darboux lattice is a map � 3 → ✁ 3 Φ : (1) which maps the four black faces and six vertices of any octahedron to a (planar) configuration of four lines and six points of intersection.

5. Theorem of Menelaus (100 AD; Euclid ?) Q 12 Theorem of Menelaus. Three points Q 12 , Q 23 , Q 31 Q 3 Q 31 on the (extended) edges of a triangle with vertices Q 1 , Q 2 , Q 3 are collinear if and only if Q 1 Q 23 Q 1 Q 12 Q 2 Q 23 Q 3 Q 31 = − 1 . Q 2 Q 12 Q 2 Q 23 Q 3 Q 31 Q 1 Conclusion: Laplace-Darboux lattices � 3 → ✁ 3 Φ : are characterized by the multi-ratio condition Φ ¯ 2 Φ 1 Φ ¯ 3 Φ 2 Φ ¯ 1 Φ 3 = − 1 Φ 1 Φ ¯ Φ 2 Φ ¯ Φ 3 Φ ¯ 3 1 2 which holds on each octahedron. Convention: The above figure is termed Menelaus configuration.

6. The dSKP equation Introduction of shape factors α, β, γ, δ according to Φ ¯ 2 − Φ 1 = α (Φ 1 − Φ ¯ 3 ) Φ ¯ 3 − Φ 2 = β (Φ 2 − Φ ¯ 1 ) Φ ¯ 1 − Φ 3 = γ (Φ 3 − Φ ¯ 2 ) Φ 1 − Φ 2 = δ (Φ 2 − Φ 3 ) αβγ = − 1 !! ⇔ Theorem. Laplace-Darboux lattices are governed by the coupled system αβγ = − 1 , α 23 β 13 γ 12 = − 1 , ( α 23 γ 12 − 1)( γ +1) = ( αγ − 1)( γ 12 +1) or, equivalently, by the discrete Schwarzian KP (dSKP) equation ( φ ¯ 2 − φ 1 )( φ ¯ 3 − φ 2 )( φ ¯ 1 − φ 3 ) 2 ) = − 1 ( φ 1 − φ ¯ 3 )( φ 2 − φ ¯ 1 )( φ 3 − φ ¯ � 3 → for a scalar function φ : ✁ which parametrises the shape factors according to α = φ ¯ 2 − φ 1 β = φ ¯ 3 − φ 2 γ = φ ¯ 1 − φ 3 , , . φ 1 − φ ¯ φ 2 − φ ¯ φ 3 − φ ¯ 3 1 2

7. Parametrisations Alternative parametrisation: α = − ψ ¯ β = − ψ ¯ γ = − ψ ¯ 3 1 2 , , , ψ ¯ ψ ¯ ψ ¯ 2 3 1 leading to the discrete modified KP (dmKP) equation ψ ¯ 2 − ψ ¯ + ψ ¯ 3 − ψ ¯ + ψ ¯ 1 − ψ ¯ 3 1 2 = 0 . ψ 1 ψ 2 ψ 3 Introduction of a τ -function according to ψ ¯ 2 − ψ ¯ τ ¯ 3 τ 1 ψ ¯ 3 − ψ ¯ τ ¯ 3 τ 2 ψ ¯ 1 − ψ ¯ τ ¯ 3 τ 3 1¯ 2¯ 1¯ 2¯ 1¯ 2¯ 3 1 2 = κ [1] , = κ [2] , = κ [3] , ψ 1 τ ¯ 2 τ ¯ ψ 2 τ ¯ 1 τ ¯ ψ 3 τ ¯ 1 τ ¯ 3 3 2 leading to the discrete Toda or Hirota-Miwa equation κ [1] τ ¯ 1 τ 1 + κ [2] τ ¯ 2 τ 2 + κ [3] τ ¯ 3 τ 3 = 0 .

8. Periodic reductions Motivation: Analogue of classical classification scheme of Laplace-Darboux sequences Periodic reduction of the dSKP equation: φ ( n 1 , n 2 , n 3 ) = φ ( n 1 , n 2 , n 3 + p ) , p even Classical analogue: Periodic 2-dim Toda lattice: (ln h ( n ) ) xy = − h ( n − 1) + 2 h ( n ) − h ( n +1) , h ( n + p ) = h ( n ) Consistent ‘quasi-periodicity’ assumption: Φ( n 1 , n 2 , n 3 ) = λ Φ( n 1 , n 2 , n 3 + p ) , ( λ = spectral parameter!) (i.e. periodicity in the setting of projective geometry.)

9. Period 2 2 1 3 In the simplest case p = 2 , we obtain for φ = φ | n 3 =0 , _ ¯ φ = φ | n 3 =1 : 3 _ _ 2 1 φ ¯ 3 = φ 3 2 − φ 1 )(¯ 1 − ¯ ( φ ¯ φ − φ 2 )( φ ¯ φ ) 2 ) = − 1 ( φ 1 − ¯ 1 )(¯ φ )( φ 2 − φ ¯ φ − φ ¯ (¯ 2 − ¯ φ 1 )( φ − ¯ φ 2 )(¯ φ ¯ φ ¯ 1 − φ ) 2 ) = − 1 (¯ φ 1 − φ )(¯ φ 2 − ¯ 1 )( φ − ¯ φ ¯ φ ¯ or, equivalently, (ˆ 2 − ˆ φ 1 )(ˆ φ − ˆ φ 2 )(ˆ 1 − ˆ φ ¯ φ ¯ φ ) 2 ) = − 1 (ˆ φ 1 − ˆ φ )(ˆ φ 2 − ˆ 1 )(ˆ φ − ˆ φ ¯ φ ¯ for { ˆ φ } = { φ } ∪ { ¯ φ } . This is a discrete Schwarzian Liouville equation (?!?) known in the theory of discrete holomorphic functions (Schramm circle patterns).

10. Period 2 + ‘tangential’ shifts discrete (Schwarzian) sinh-Gordon equation (Hirota)! discrete (Schwarzian) Korteweg-de Vries equation! discrete (Schwarzian) Boussinesq equation!

11. The continuum limit In general, consider the reduction T = T µ 1 T ν µ + ν = even τ ¯ 3 = Tτ 3 , 2 , Then, the discrete Toda equation assumes the form ( σ = τ 3 ) 2 τ 2 = τ ¯ 1 τ 1 − τ ¯ − ǫ [1] ǫ [2] σTσ − ǫ [1] ǫ [2] τT − 1 τ. σ ¯ 1 σ 1 − σ ¯ 2 σ 2 = Continuum limit: (ln τ ) xy = − σ 2 (ln σ ) xy = − τ 2 τ 2 , σ 2 so that ( σ 2 /τ 2 = exp ω ) ω xy = 4 sinh ω, Hence, continuum limit = sinh-Gordon equation for any T (cf. classical theory)!

12. The pentagram map Evolution of polygons on the plane (Schwartz 1992, Ovsienko, Schwartz & Tabachnikov 2009): ✂ → ✁ � 2 in fact ) ✁ 2 Polygon: C : ( Discrete time step: C �→ C ∗ C n+2 C n−2 * C n Cross ratios: x n = q ( D n , A n , C n − 2 , C n − 1 ) C n+1 C n−1 y n = q ( D n , B n , C n +2 , C n +1 ) A n B n C n 1 − x n − 1 y n − 1 Dynamical system: x ∗ n = x n 1 − x n +1 y n +1 D n 1 − x n +2 y n +2 y ∗ n = y n +1 1 − x n y n Results: (a) Integrable if the polygon is closed (modulo a projective transformation) (b) Boussinesq equation in the continuum limit

13. The Menelaus connection 2 _ _ _ _ 1 11 1 1 1 1 1 _ 2 Observation: The ‘pentagram lattice’ is nothing but a Laplace-Darboux sequence con- strained by Φ ¯ 3 = Φ 111 ⇔ Φ 3 = Φ ¯ 1¯ 1¯ 1 and therefore governed by ( φ ¯ 2 − φ 1 )( φ 111 − φ 2 )( φ ¯ 1 ) 1 − φ ¯ 1¯ 1¯ 2 ) = − 1 . ( φ 1 − φ 111 )( φ 2 − φ ¯ 1 )( φ ¯ 1 − φ ¯ 1¯ 1¯

14. The Schwarzian Boussinesq equation D Lemma: q ( A, B, D, C ) = − M ( E, G, C, F, H, B ) F H C Hence: B G E − ( φ ∗ − φ ∗ )( φ ∗ − φ ∗ )( φ ∗ − φ ∗ ) x n = ( φ ∗ − φ ∗ )( φ ∗ − φ ∗ )( φ ∗ − φ ∗ ) A − ( φ ∗ − φ ∗ )( φ ∗ − φ ∗ )( φ ∗ − φ ∗ ) y n = ( φ ∗ − φ ∗ )( φ ∗ − φ ∗ )( φ ∗ − φ ∗ ) Note: A is not a lattice point! and the evolution equations for x n and y n reduce to the above reduction of the dSKP equation! Continuum limit: φ 1 = φ + ǫφ u + O ( ǫ 2 ) , φ 2 = φ + ǫ 2 φ v + O ( ǫ 3 ) φ vv − φ uu v + 3 φ 2 4 { φ ; u } u φ u = 0 φ 2 u Schwarzian Boussinesq equation Note: The above discrete SBQ equation is non-standard!