Holography principle and arithmetic of algebraic curves Yuri Manin and Matilde Marcolli Adv.Theor.Math.Phys. Vol.5 N.3 (200) 67650 Baltimore January 2003
Arithmetic surfaces Projective algebraic curve X defined over Q; equations w/ Zcoefficients scheme X Z ; closed fiber of X Z at p Spec(Z) reduction X Z mod p infinitesimal neighborhoods: reductions of X Z mod p n. Limit n : p adic completion of X Z arithmetic infinity: embedding Q C (absolute value vs. padic valuations) Arakelov: Hermitian geometry of X C analog of p adic completions of X Z : Green functions provide intersection indices of arithmetic curves over the fiber at infinity.
Green function compact Riemann surface X C, Green function g µ,a : divisor A = m (), positive real analytic 2form dµ Laplace equation: g A satisfies g A = πi (deg(a) dµ δ A ), with δ A the δ current ϕ m ϕ(). Singularities: z = loc coord in neighb of g A m log z loc real analytic. Normalization: g A satisfies X g Adµ = 0. B = y n y (y) divisor, A B =, g µ (A, B) := y n y g µ,a (y) symmetric biadditive g µ depends on µ. In case of degree zero divisors, deg A = deg B = 0, g µ (A, B) = g(a, B) conformal invariant. 2
If A = Div(w A ), w A meromorphic function g(a, B) = log y B w A (y) n y P (C): in terms of crossratio a, b, c, d P (C): a, b, c, d = (a b)(c d) (a d)(c b) log a, b, c, d General case: A, B degree zero div on X C, ω A differential of third kind w/ purely imaginary periods and residues m at : Green function g(a, B) = Re γ B ω A, γ B = chain with boundary B basis of differentials of third kind on X C 3
Space time Anti de Sitter: AdS d+ space time satisfying Einstein s eq w/constant curvature R < 0 (empty space with negative cosmological constant) In general relativity: AdS 3+ = S R 3, metrically hyperboloid u 2 v 2 + 2 + y 2 + z 2 = in R 5, ds 2 = du 2 dv 2 + d 2 + dy 2 + dz 2. To avoid time like closed geodesics univ cover topologically R 4. Boundary at infinity of AdSd+ AdS3+ is a compactification of d dimensional Minkowsky space time Euclidean signature AdS d+ H d+ real hyperbolic space In quantum gravity AdS 2+ and H 3 4
The Holography principle Bulk space (asymptotically AdS) and Boundary (conformal boundary at infinity): Gravity on bulk space field theory on the boundary 2 + : Geodesic propagator on bulk bosonic propagator on the boundary (for Riemann surfaces: Boson/Fermion equivalence) Genus zero case Euclidean signature: P (C): H 3 Boundary propagator (i.e. Green function) = geodesic propagator on the bulk space g((a) (b), (c) (d)) = ordist (a {c, d}, b {c, d}) d b*{c,d} a y b a*{c,d} u c The propagators: a c, b d, logarithmic divergence: intrinsic, no choice of cutoff functions (cf.balasubramanian Ross Phys.Rev.D 3 6 (2000) 4) 5
Bañados Teitelboim Zanelli black hole Genus one case: H 3 /(q Z ) X q (C) = C /(q Z ) (Jacobi uniformization) q : (z, y) (qz, q y) ( ) 2π(i r r q = ep + ) l ( Ml ± ) M 2 l 2 + J r± 2 = 2 mass and angular momentum of black hole, /l 2 = cosmological constant. Arakelov Green function and BTZ black hole Operator product epansion of path integral on the elliptic curve X q (C) (AlvarezGaume,Moore,Vafa Comm.Math.Phys. 06 (986)) g(z, ) = log ( q B 2(log z / log q )/2 z n= q n z q n z in terms of geodesics (gravity on bulk space): = 2 l(γ 0) B 2 ( lγ0 ( z, ) l(γ 0 ) ) + n 0 l γ ( 0, z n ) + n l γ ( 0, z n ). ) 6
oo H 3 X q 0 oo Η 3 0 δ 0 X q n γ 0 0 q n 0 2 = {0, }; z n = q n z {, }, z n = q n z {, } 7
Higher genus case Bosonic field propagator on algebraic curve X C via ω (a) (b) := ν (a) (b) l X l (a, b)ω gl, differentials of the third kind with purely imaginary periods (FerrariSobczyk, J.Math.Phys. 4 9 (2000)) All correlation functions: G(z,..., z m ; w,..., w l ) = m l q i φ(z i, z i )φ(w j, w j ) q j, j= i= q i = system of charges at positions z i interacting with q j = charges at positions w j obtained from basic correlator G µ (a b, z) (in terms of ω (a) (b) (z)) 8
Schottky uniformization PSL(2, C) = orientation preserving isometries of H 3 = C R +, 3dim real hyperbolic space. Schottky group: Γ PSL(2, C) Γ is discrete, free group of rank g The action of Γ on H 3 etends to an action on P (C) by fractional linear transformations Γ is purely loodromic Kleinian group i.e. generator γ Γ {z ± (γ)} P (C) fied points P (C) Λ Γ = limit set of Γ: accumulation pts. of Γorbits: Γinvariant, totally disconnected (Cantor set for g 2) closed subset of P (C) Ω Γ := P (C) Λ Γ connected, nonsimply connected Γinvariant domain of discontinuity of Γ X C = Ω Γ /Γ 9
Basis of differentials of the third kind as averages over the Schottky group: (base point z 0 Ω Γ, domain of discont) diff third kind ν (a) (b) := d log a, b, γz, γz 0 γ Γ and diff first kind ({z + (γ), z (γ)} fied points) ω γ = h C( γ) d log hz + (γ), hz (γ), z, z 0 (solve for X l (a, b) for purely imaginary periods) Obtain eplicit epression for the Green function: g((a) (b), (c) (d)) = log a, b, hc, hd h Γ g l= X l (a, b) h S(g l ) log z + (h), z (h), c, d C( γ) = Γ/γ Z ; S(γ) = conjug.class 0
Green function and Krasnov black holes Global quotients of AdS 2+ by a Schottky group Γ PSL(2, C); Euclidean signature: H 3 /Γ hyperbolic handlebody, conformal boundary X C Eplicit bulk/boundary correspondence: each term in the Bosonic field propagator for X C in terms of geodesics in Euclidean Krasnov black hole H 3 /Γ. h Γ ordist(a {hc, hd}, b {hc, hd}) g + X l (a, b) l= h S(g l ) ordist(z + (h) {c, d}, z (h) {c, d}). Coefficients X l (a, b) also in terms of geodesic propagators