Renormalized Volume of Hyperbolic 3-Manifolds Kirill Krasnov University of Nottingham Joint work with J. M. Schlenker (Toulouse) Review available as arxiv: 0907.2590
Fefferman-Graham expansion From M. Anderson hep-th/0403087 Let g be a complete, conformally compact Einstein metric on M n+1 i.e. ρ : ρ 2 g ρ =0, M extends to M dρ 0 M Then: M asymptotically hyperbolic i.e. R g +1 = O(ρ 2 ) can define boundary metric γ := ρ 2 g M with conformal class [γ] canonically defined
Foliate M (near M ) by ρ = const surfaces ds 2 = dρ2 ρ 2 + 1 ( ρ 2 γ + ρ 2 g (2) + ρ 4 g (4) +... { nodd ρ n 1 g (n 1) + ρ n g (n) + ρ n+1 g (n+1) + ) n even ρ n g (n) + ρ n log ρ h (n) +... +... ) n odd g (2k) k (n 1)/2 computable in terms of γ g (n) free apart from γ g (n) =0, Tr γ g (n) =0 n even g (2k) k (n 2)/2 h (n) } traceless, transverse computable in terms of g (n) free apart from γ g (n) = τ, Tr γ g (n) = δ γ
n {γ,g (n) } free data can integrate Einstein eqs in a finite neighborhood of M However, if MM connected and π 1 (M, M) =0 Then γ can be freely varied, while g (n) is determined by γ Hard to characterize explicitely
Interesting to compare with bounded metrics ds 2 = dρ 2 + γ + ρb +... no linear term - no second fundamental form in AH case!
Volume V (ɛ) = { ɛ dρ ρ n+1 det(γ + ρ 2 g (2) +...) = ɛ n v (0) + ɛ n+2 v (2) +... + ɛ 1 v (n 1) + V + O(ɛ) ɛ n v (0) + ɛ n+2 v (2) +... + ɛ 2 v (n 2) + log(ɛ) L + V + O(ɛ) n odd n even V L } canonical invariants of M Define Renormalized volume RV ol(m) := V depends on the foliation for n even
Example: n =3 Anderson 01 M 4 W 2 =8π 2 χ(m) RV ol(m) renormalized volume controls the norm of the Weyl curvature L 2
Very special case: n =2 Weyl curvature vanishes identically, AH=hyperbolic FG expansion stops Skenderis, Solodukhin hep-th/9910023 ds 2 = dρ2 ρ 2 + 1 ρ 2 (1+ ) ρ2 2 g (2)γ 1 γ (1+ ρ2 2 γ 1 g (2) ) g (2) free apart from γ g (2) =0, Tr γ g (2) = R γ There is no log term! Compare with the known fact that in 3D given the first and second fundamental forms can write down the equidistant foliation metric explicitly
Geometrization M is completely determined by [γ] (plus possibly topological information) E.g. Schottky contractible E.g. quasi-fuchsian [γ 1 ] [γ] [γ 2 ] g (2) is determined by [γ] of all boundary components Tracefree part of g (2) - holomorphic quadratic differential - projective structure
RV ol(m,ρ) depends on foliation ( ρ -coordinate) Lemma: Given γ on all components of M! foliation of M by equidistant surfaces S ρ such that 1 2 ( γρ + B ρ + γ 1 ρ B ρ γρ 1 ) 1 = ρ 2 γ here γ ρ,b ρ are fundamental forms of S ρ Can be described explicitly - Epstein surfaces in the covering space. Breaks down inside (typically) RV ol(m,ρ) =RV ol(γ)
Can be computed explicitly, but even without an explicit result variational formulas can be obtained de Haro, Skenderis, Solodukhin hep-th/0002230 δrv ol(γ) = 1 4 da γ g (2) Ric γ, δγ where g (2) = g (2) 1 2 R γγ tracefree subcase of a general n result, computation involving a regularization before the variation is taken Important formula!
Consider variations not changing the conformal structure: δγ =2uγ (with fixed total area) F = RV ol(γ)+ λ 2 da γ variation δf = 1 4 da γ Ric γ + λγ, δγ = Ric γ = λγ critical points (local maxima) - constant curvature metrics
Define: RV ol(m) =RV ol(γ) computed on γ : R γ = 1 renormalized volume maximized among all volumes for the same conformal structure and area of the boundary canonically defined for M otherwise up to a constant (with the normalization chosen)
Consider variations of the conformal class δrv ol(m) = 1 da γ g (2), δγ 4 Well-known that γ g (2) =0, Tr γ g (2) =0 where Q = g (2) = Re(Q) holomorphic quadratic differential on M (with respect to complex structure of γ )
Can characterize Q explicitly - encodes the projective structure of M φ i : i M H 2 holomorphic w.r.t. [γ i ] φ i - uniformization map for the i M component Q i = S(φ i ) Schwarzian derivative difference of two projective structures on i M Q i - holomorphic cotangent vector on T g = Q i = 4 i RV ol(m)
Can now be shown that i Q i = ω WP Schottky - Takhtajan-Zograf 88 = Weil-Petersson symplectic form Schottky - KK 00 Renormalized Volume is a Kaehler potential for WP form on the appropriate moduli space - Schottky, Teichmueller for Teichmueller uses quasi-fuchsian manifolds; the quantity 1 Q 1 does not depend on the complex structure of the other component - use Fuchsian McMullen 00 can be established by complex-analytic methods in both cases needs a technical result on reciprocity in quasi-fuchsian case
Remark: The variational formula is reminiscent of that of Schlaefli If V is a volume of a hyperbolic polyhedron dv = 1 2 L e dθ e e and so if defines dual volume V = V 1 L e θ e 2 then e dv = 1 θ e dl e 2 e more than an analogy - can be derived from a version of the Schlaefli formula KK & J. M. Schlenker math/0607081
Historical remarks: The Renormalized Volume can be computed explicitly Schottky manifolds KK hep-th/0005106 Quasi-Fuchsian, Kleinian manifolds Takhtajan-Teo math/0204318 RV ol(γ) =S Liouv (γ) Liouville action as defined by Takhtajan and Zograf 88 (Takhtajan and Teo 02) The Kaehler potential property follows from this relation to Liouville and results of TZ-TT Kaehler potential on T g is an essentially 3D quantity!
Application 1: McMullen s QF reciprocity A stronger statement can be formulated as: Consider φ 1 : T g T g T T g T T g Simultaneous uniformization, corresponding projective structure Theorem: φ 1 (T g T g ) - Lagrangian submanifold in T T g T T g A version of L : p i (x) = V x i = dp i L 2 V dx i = dx j dx i =0 x i i i x j
Application 2: Schottky manifolds Consider φ 2 : S g T S g Schottky space, corresponding projective structure Theorem: φ 2 (S g ) Lagrangian submanifold in T S g
Application 3: Grafting map is symplectic Consider φ 3 : T g ML T T g = CP projective structure obtained by grafting a conformal metric as specified by a measured geodesic lamination Consider the corresponding hyperbolic end; the inner boundary is a surface pleated along ML ; a version of renormalized volume proves Theorem: φ 3 is symplectic; alternatively, the image in T g ML CP corresponding to hyperbolic ends is a Lagrangian submanifold
Why is this interesting to a physicist? Geometric quantization in real polarization: Foliation of P by Lagrangian submanifolds; states as integral leaves E.g. Harmonic Oscillator H = p 2 + q 2 p H = E q Integral leaves H = E n n +1/2
This program is realized in the context of flat (or hyperbolic) polyhedra! S Regge = e θ e (L)L e δs Regge = e θ e (L)δL e L θ Phase space P tet = {L e, θ e } Flat tetrahedron gives a Lagrangian submanifold θ e = θ e (L) Ψ(L e ) e (i/ ) P e θ e(l)l e Ponzano-Regge SU(2) irreps (6j)-symbol of Wigner Can also be done for hyperbolic tets - (6j)-symbol of the quantum group SL q (2, C) Freidel, Roche
Back to hyperbolic manifolds: Λ < 0 3D gravity can be formulated as a Hamiltonian system phase space - space of complex projective structures P = {γ, g (2) } = T T g = CP Quantization of P H = L 2 (T g ) Can show that the gravitational symplectic structure coincides with that in T T g According to axioms of TQFT, 2-surfaces - Hilbert spaces, 3-manifolds - states E.g. Schottky What is the corresponding state? Same question can be asked in the context of Chern- Simons theory on a handlebody, compare Weitsman 91
Do we have a Lagrangian foliation of T T g? Can one use level surfaces of traces of the 3g-3 holonomies of the flat PSL(2, C) connection? reminiscent to the Lagrangian foliation in Hitchin 88 The self-duality equations on a Riemann surface Possible to construct Schottky states in L 2 (T g )? Interesting functions on T g if expanded in powers of Semi-classically Ψ e i RV ol(c)/ Quantization of hyperbolic 3-manifolds
Conclusions: 3D Renormalized Volume as a Kaehler potential for the 2D Teichmueller space (Hyperbolic) 3-Manifolds correspond to Lagrangian submanifolds in the boundary phase space - natural setup for quantization