Periods of meromorphic quadratic differentials and Goldman bracket

Transcription

1 Periods of meromorphic quadratic differentials and Goldman bracket Dmitry Korotkin Concordia University, Montreal Geometric and Algebraic aspects of integrability, August 05, 2016

2 References D.Korotkin, Periods of meromorphic quadratic differentials and Goldman bracket, to appear M.Bertola, D.Korotkin, C.Norton, Symplectic geometry of the moduli space of projective structures in homological coordinates, arxiv: D.Korotkin, P.Zograf, Prym class and tau-functions, Contemporary Math. (2013) A.Kokotov, D.Korotkin, Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula, J.Diff.Geom., 82, (2009)

3 Main equation C g - Riemann surface of genus g. "Schrödinger equation" on C g : ϕ uϕ = 0 where ϕ is a ( 1/2)-differential (locally), and 2u - meromorphic projective connection on C g with n simple poles. Parametrization of space of all "potentials": ϕ + ( 1 2 S 0 + Q)ϕ = 0 S 0 - "base" projective connection, Q - meromorphic quadratic differential with n simple poles.

4 Canonical symplectic structure on T M g Moduli space of pairs (C g, Q) is Q g,n = T M g,n ; dimq g,n = 6g 6 + 2n {q i } 3g 3+n - any complex coordinates on M g,n (say, 3g 3 entries of period matrix and (v 1 /v 2 )(y k )); {v i } - normalized abelian differentials. {d q i } 3g 3+n - basis in cotangent space; {p i } 3g 3+n - coordinates of cotangent vector in this basis. Symplectic structure and symplectic potential: ω can = 3g 3+n dp i dq i θ can = 3g 3+n p i dq i

5 Homological Darboux coordinates Let all zeros of Q be simple: {x i } 4g 4+n. Canonical cover (spectral, Hitchin, Seiberg-Witten...) Ĉ: v 2 = Q in T C g ; 4g 4 + 2n branch points at {x i, y i }; genus 4g 3 + n; involution µ : Ĉ Ĉ Decomposition of H 1 (Ĉ, Z) into even and odd parts: H 1 (Ĉ, Z) = H H + where dim H + = 2g, dim H = 6g 6 + 2n. Generators of H : {a i, b i } 3g 3+n ; intersection a i b j = δ ij. Homological coordinates A i = a i v, B i = b i v.

6 Canonical cover Figure: Canonical basis of cycles on the canonical cover Ĉ

7 Homological and canonical symplectic structures Homological symplectic structure on T M g : Theorem 1. ω hom = 3g 3+n ω hom = ω can da i db i Thus (A i, B i ) are Darboux coordinates for ω can on the main stratum of T M g (all zeros of Q are simple).

8 Symplectic structure on the space of projective connections Space S g : pairs (C g, S), S holomorphic projective connection on C g. Affine bundle over M g. Given the "base" projective connection S 0 on C g which holomorphically depends on moduli of C g, write any S as S = S 0 + 2Q, for some holomorphic quadratic diff. Q. The map F S 0 : Q g S g is used to induce symplectic structure on S g from ω can. Equivalence: S 0 S 1 if corresponding symplectic structures on S g coincide. Generating function G 01 : δ µ G = µ(s 1 S 0 ) C g

9 Equivalent projective connections Schottky projective connection S Sch ( ) = {w, }, where w is the Schottky uniformization coordinate; {, }- Schwarzian derivative. main example: Bergman projective connection S B. Canonical bimeromorphic differential B(x, y) on C g : a α B(, y) = 0, ( B(x, y) = 1 (ξ(x) ξ(y)) S B(ξ(x)) +... ) dξ(x)dξ(y) B depends on Torelli marking (choice of canonical basis in homologies on C) Generating function from S Sch to S B : Zograf s F-function F = Z B (1); Z B- Bowen s zeta-function of Schottky group. Generating function corresponding to change of Torelli marking defining S B is given by det(cω + D) (cocycle of determinant of Hodge vector bundle).

10 Main tool: Variational formulas For any s i H define s i H (s i s j = δ ij ); P i = s i v. Then B(x, y) = 1 P i 2 t si where z(x) and z(y) are kept constant. v j (x) = 1 P i 2 t si Ω jk = 1 P i 2 B(x, t)(b(t, y) v(t) v j (t)(b(t, x) v(t) t s i v j v k v

11 Poisson bracket for potential u(z) Let S 0 = S B ; ψ = φ v; z(x) = x x 0 v - "flat" coordinate on C g and Ĉ. Main equation: ψ zz u(z)ψ = 0 where and S v ( ) = { x v, } u(z) = 1 1 S B S v 2 Q Invariant matrix form (on Ĉ): dψ = ( 0 v uv 0 Define h(x, y) = B2 (x,y) Q(x)Q(y) ) Ψ

12 Poisson bracket for potential u(z) (continued) where 4πi 3 {u(z), u(ζ)} = L zh (ζ) (z) L ζ h (z) (ζ) L z = z 2u(z) z u z (z) is known as "Lenard" operator in KdV theory; h (y) (x) = x x 1 h(y, )v( ) The first example of holomorphic Poisson bracket on a Riemann surface (get as a Dirac bracket from Atiyah-Bott symplectic structure??).

13 Monodromy representation and Goldman bracket Fundamental group: π 1 (C g \ {y i } n, x 0) with generators (γ i, α j, β j ) and relation g j=1 α jβ j α 1 j β 1 n j γ i = id Monodromy matrices: M αi, M βi, M yi with relation n M yi g j=1 M 1 β j Mα 1 j M βj M αj = I Goldman s bracket on character variety V g,n : {trm γ, trm γ } = 1 2 (trm γp γ trm γp γ 1) p γ γ

14 Relation to results of S.Kawai, Math Ann (1996) Kawai: "canonical symplectic structure on T M g implies Goldman bracket if S 0 = S Bers ". Together with our results: S B and S Bers are in the same equivalence class; implies existence of generating function. Conjecture: G = 6πi log Z [Γ C0,η](1) det(ω Ω 0 ) where Z is the Selberg zeta-function Z(s) = γ m=0 (1 qs+m γ ) corresponding to quasi-fuchsian group Γ C0,η; Ω and Ω 0 are period matrices of C 0 and C.

15 Tau-function of Scrödinger equation Motivation: Jimbo-Miwa tau-function for Schlesinger system: dψ dx = N A i x x i Ψ: log τ JM = 1 x i 2 res tr(dψψ 1 ) 2 x i dx A straightforward analog of this definition in the case of Schrödinger equation (no isomonodromy!) log τ := 1 P si 4πi si ( tr(dψψ 1 ) 2 v ) + 2v where P si = s i v; this gives rise to Bergman tau-function log τ = 1 P si 4πi si S B S v v τ σ = det 6 (CΩ + D) τ τ(ɛq) = ɛ 1/6(5g 5+n) τ(q)

16 Open: "Yang-Yang" function (ϕ i, l i ) - complexified Fenchel-Nielsen Darboux coordinates on character variety V g,n. ω can = i d l i dϕ i = i dp i dq i dg YY = i l i dϕ i i p i dq i (Nekrasov-Rosly-Shatashvili); G YY - "Yang-Yang" function (depends on pants decomposition on the Character variety side; transforms with dilogarithms; depends also on Torelli marking; transforms as a section of Hodge line bundle).

17 Simplest example: genus 0 with 4 simple poles Poles: 0, 1, t, ; B(x, y) = Poisson structure: Equation (Heun): Q = ϕ + Homological coordinates: µ dx dt (x t) 2, S B = 0; µ x(x 1)(x t) (dx)2 {µ, t} = t(1 t) 4πi µ x(x 1)(x t) ϕ = 0 a,b dx x(x 1)(x t)