Polyakov conjecture for hyperbolic singularities

We propose the form of the Liouville action satisfying Polyakov conjecture on the accessory parameters for the hyperbolic singularities on the Riemann sphere.


Introduction
Let us consider the Fuchsian equation where In the context of the Liouville field theory T (z) plays the role of the zz-component of the energy-momentum tensor and the real positive numbers ∆ j are conformal weights. The complex numbers c j are called accessory parameters. The requirement that T (z) is regular at the infinity implies the relations The Polyakov conjecture concerns the following version of the Riemann-Hilbert problem [1][2][3]. For a given set of positive weights {∆ j } n j=1 one has to adjust the accessory parameters in such a way that the Fuchsian equation (1) admits a fundamental system of solutions with SU (1, 1) monodromies around all singularities.
The interest to this problem comes from its close relation to the Liouville equation on the punctured Riemann sphere X = ( C ∪ {∞} ) \ {z 1 , . . . , z n }. If χ 1 (z), χ 2 (z) are linearly independent solutions of (1) then the function ϕ(z,z) determined by the relation e ϕ(z,z) = 4 |w ′ | 2 (1 − |w| 2 ) 2 , w(z) = satisfies the Liouville equation for all that z for which w(z) is well defined. It is convenient to use normalized solutions with Wronskian satisfying: so that the relation (4) can be written in a simple form Note that ϕ(z,z) is real by construction. If in addition χ 1 (z), χ 2 (z) satisfy the SU (1, 1) monodromy condition then ϕ(z,z) is single-valued. Under some restrictions on conformal weights the relation above can be made more precise. The case of all parabolic singularities was analyzed by Poincaré in the context of the uniformization problem [7]. He showed that the Liouville equation (5) has a unique realvalued regular on X solution with the following behavior at the punctures: This solution defines a metric ds 2 = e ϕ |dz| 2 which is complete on X. It has constant negative curvature −1 and parabolic singularities at each z j . The energy-momentum tensor of the solution ϕ, is a holomorphic function on X of the form (2) with the conformal weights ∆ j = 1 2 , j = 1, . . . , n.
It follows from (8) was proved by Picard [8,9] (see also [10] for the modern proof). The solution can be interpreted as the conformal factor of the complete, hyperbolic metric on X with the conical singularities of the opening angles 0 < θ j < 2π at the punctures z j . The energy-momentum tensor takes the form (2) with , j = 1, . . . , n .
The Polyakov conjecture states that the (properly defined and normalized) Liouville action functional S[φ] evaluated on the classical solution ϕ(z,z) is a generating function for the accessory parameters of the monodromy problem described above : This formula was derived within path integral approach to the quantum Liouville theory as the quasi-classical limit of the conformal Ward identity [11][12][13]. In the case of the parabolic singularities on n-punctured Riemann sphere a rigorous proof based on the theory of quasiconformal mappings was given by Zograf and Takhtajan [5]. It was also pointed out that the same technique applies for the elliptic singularities of finite order. Note that only in these two cases the monodromy group of the Fuchsian equation is (up to conjugation in SL(2, C)) a discrete subgroup in SU (1, 1), and the map w(z) defined by (4) solves the uniformization problem [5]. An alternative method, working both in the case of parabolic and general elliptic singularities, was recently developed by Cantini, Menotti and Seminara [1][2][3]. Yet another proof, based on a direct calculation of the regularized Liouville action for parabolic and general elliptic singularities, was proposed by Takhtajan and Zograf in [6].
The aim of this Letter is to find the action which satisfies (9) for the singularities of the hyperbolic type. The SU (1, 1) monodromy problem is well posed in this case, but whether it has a solution for arbitrary conformal weights ∆ j > 1 2 and arbitrary locations of punctures z j is up to our knowledge an open problem. In the present Letter we assume that such solution exists.
As one can expect from the case of two punctures [14] the corresponding solution to the Liouville equation determined by the relation (4) develops concentric line singularities around each puncture. We assume that these line singularities do not intersect and that they are the only singularities of the classical solution.
The singular behavior around punctures can be described in terms of local conformal maps. This allows for the construction of an appropriately regularized Liouville action functional. One can then apply the method of Takhtajan and Zograf [6] to prove the Polyakov conjecture. The problem of the existence and the uniqueness of the solution to the Fuchsian equation with the properties stated above goes beyond the scope of the present letter. Let us only mention that in the case of three hyperbolic singularities an explicit solution in terms of the hypergeometric functions exists [4,15]. There is also a simple geometrical construction yielding a large class of solutions with an arbitrary number of hyperbolic singularities [15].
The choice of the Fuchsian equation (1) as a starting point for the construction of the Liouville field theory is a convenient way to impose the crucial constraint on admissible classical solutions -the holomorphic form (2) of their energy-momentum tensor T (z). We hope that the singular hyperbolic solutions and the corresponding Liouville action will be helpful in understanding the factorization problem in the geometrical approach to the Liouville theory developed by Takhtajan [12,13,16,17]. This was actually our main motivation for the present paper.
Finally let us note that the hyperbolic solutions provide multi black hole solutions of the 3-dim gravity [18][19][20].

Hyperbolic singularities
Let us assume that {χ 1 , χ 2 } is a normalized solution to the SU (1, 1) monodromy problem for hyperbolic weights Then the fundamental system defined by is also normalized and has SL(2, R) monodromy around all punctures. In terms of {ψ 1 , ψ 2 } the formula (7) reads The advantage of using solutions with SL(2, R) monodromy is that any hyperbolic element Thus for each singularity z j there exists an element B j ∈ SL(2, R) such that the system has a diagonal monodromy at z j . It follows that ψ j ± have the canonical form where ϑ j ∈ R, and u j ± (z) are analytic functions Expanding the energy-momentum tensor one gets The Fuchsian equation (1) then implies It is a well known property of the Schwarz derivative and the Fuchsian equation (1) that the ratio satisfies the relation For each hyperbolic singularity we define It follows from (11,12,13) that ρ j (z) is an analytic function on D j and: Using (14) and the properties of the Schwarz derivative one gets where Let us consider the Fuchsian equation on the complex ρ plane and a normalized fundamental system of solutions with the diagonal monodromy at ρ = 0 of the following form The corresponding solution of the Liouville equation reads [14] ϕ j (ρ,ρ) = log λ 2 j |ρ| 2 sin 2 (λ j log |ρ|) .
The metric e ϕ j d 2 ρ has infinitely many closed geodesics: and infinitely many singular lines: Using the transformation rule (17) and the expansion (16) one can show that on D j ⊂ X the metric e ϕ d 2 z coincides with the pull-back of the metric e ϕ j d 2 ρ by the map ρ j (z). As is an open neighborhood of 0 there are infinitely many geodesics G l and singular lines S l contained in ρ j (D j ). Their inverse images G l = ρ −1 j ( G l ), S l = ρ −1 j ( S l ), are closed singular lines and closed geodesics of the metric e ϕ d 2 z on X. This provides a detailed description of the singular hyperbolic geometry in a sufficiently small neighborhood of the hyperbolic singularity: an alternating sequence of the concentric closed geodesics and closed singular lines. Let us stress that all these geodesic have the same length, uniquely determined by the conformal weight: The question arises what happens to this geometry when one goes away from the singularity. We assume that there exists a set {Γ j } n j=1 of closed geodesics with the following properties: • Γ j separates z j from all other geodesics Γ i (i = j); • the map ρ j extends to a conformal invertable map on the hole H j around z j defined as the part ofĈ containing z j and bounded by Γ j ; • the metric e ϕ d 2 z is regular on the surface M ≡Ĉ \ n j=1 H j .
The assumption is well justified by the properties of the general 3-puncture solution and by the geometric construction of the n-puncture solutions [15]. In particular, it implies that each Γ j is an inverse image by ρ j of one of the standard closed geodesics G l in the ρ-plane. It can be parameterized as for some l ∈ Z. The orientation of the j-th boundary component ∂M j ≡ Γ j corresponds to the parameter t decreasing from 2π to 0. Using (16) one gets for ρ ∈ ρ j (H j )

Liouville action
The standard Liouville action on a surface M ⊂ C with regular boundary components reads where d 2 z = i 2 dz ∧dz and κ z is a geodesic curvature of ∂M (computed in the flat metric on the complex plane). It yields the boundary conditions n a ∂ a φ + 2κ z = 0 (24) and the equation of motion (5). The classical solution ϕ(z,z) defines on M a hyperbolic metric e φ d 2 z with geodesic boundaries. If M is unbounded one has to impose an appropriate asymptotic conditions on admissible solutions. It can be done by means of a modified action The presence of the additional boundary terms forces φ(z,z) to behave asymptotically as This implies that T (z) is regular at infinity and the limit (25) exists. Let ϕ(z,z) denote a solution of the Liouville equation (5)  On each hole H j there exists a unique flat metric with the only singularity at z j such that the boundary ∂H j = Γ j is geodesic and its length is 2πλ j . It can be constructed as the pull-back by ρ j (z) of the metric λ 2 j |ρ| 2 d 2 ρ which yields the following formula for its conformal factor Using the expansion (16) one gets Let us note that ϕ j (z,z) satisfies C 1 sewing relations along the boundary We define on H j the regularized classical action where H ǫ j denotes H j with a disc of radii ǫ around z j cut out. With this notation our proposal for the Liouville action in the case of hyperbolic singularities can be written in the following form: It should be stressed that the Polyakov conjecture determines the classical Liouville action S L [ϕ] only up to an arbitrary function of conformal weights. This freedom is tacitly assumed in the formula above. Using the sewing relations (29) one can rewrite the classical action The modifications of the action related to the asymptotic behavior at infinity are independent of the locations of singularities and are irrelevant for our derivation of the Polyakov conjecture. For the sake of brevity they are suppressed in the formula above.

Polyakov conjecture
Using the equations of motion ∂∂ϕ = 1 2 e ϕ , ∂∂ϕ j = 0, and the sewing relations (29) one gets The first term on the r.h.s. of (34) results from the change of the shape and the position of the boundary components ∂M k induced by the change of z j . The second one is due to the change of the position of the circle |z − z j | = ǫ (by construction, all the remaining "small holes" preserve their positions; their radii, equal to ǫ, are fixed From (22) and (21) one also has: The terms in the first line of (35) denoted by dots contain non-zero, integer powers of e it and vanish upon integration. The expansion (28) implies Hence, up to the terms that vanish in the limit ǫ → 0 :