Periods of meromorphic quadratic differentials and Goldman bracket Dmitry Korotkin Concordia University, Montreal Geometric and Algebraic aspects of integrability, August 05, 2016
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:1506.07918 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, 35-100 (2009)
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.
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
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.
Canonical cover Figure: Canonical basis of cycles on the canonical cover Ĉ
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).
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
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)) 2 + 1 6 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).
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
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
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 = 1 2 3 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??).
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
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.
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)
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).
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)