Proof of Polyakov conjecture for general elliptic singularities

A proof is given of Polyakov conjecture about the accessory parameters of the SU(1,1) Riemann-Hilbert problem for general elliptic singularities on the Riemann sphere. Its relevance to 2+1 dimensional gravity is stressed.


Introduction
Polyakov made the following conjecture [1] on the accessory parameters β n which appear in the solution of the SU(1, 1) Riemann-Hilbert problem where S P is the regularized Liouville action [2], S P = lim ǫ→0 S ǫ with 2 (2) and X ǫ is the disk of radius 1/ǫ in the complex plane from which disks of radius ǫ around all singularities have been removed; γ n are the boundaries of the small disks and γ ∞ is the boundary of the large disk.
In eq.(1) S P has to be computed on the solution of the inhomogeneous Liouville equation which arises from the minimization of the action i.e.
4∂ z ∂zφ = e φ + 4π n g n δ 2 (z − z n ) with asymptotic behavior at infinity φ = −g ∞ ln zz + O(1). Such a conjecture plays an important role in the quantum Liouville theory [3] and in the ADM formulation of 2 + 1 dimensional gravity [4,5]. The conjecture is interesting in itself as it gives a new meaning to the rather elusive accessory parameters [6,7] of the Riemann-Hilbert problem. In particular it implies that the form ω = n β n dz n + c.c. is exact.
Zograf and Takhtajan [8] provided a proof of eq.(1) for parabolic singularities using the technique of mapping the quotient of the upper half-plane by a fuchsian group to the Riemann surface and exploiting certain properties of the harmonic Beltrami differentials. In addition they remark that the same technique can be applied when some of the singularities are elliptic of finite order. The case of only parabolic singularities is of importance in the quantum Liouville theory [3] as such singularities provide the sources from which to compute the correlation functions. On the other hand in 2 + 1 gravity one is faced with general elliptic singularities and here the mapping technique cannot be directly applied. (In the case of elliptic singularities with rational g n some progress in the mapping technique that are relevant to this problem were made in [14]). As a matter of fact we shall see that the case of elliptic singularities is more closely related to the theory of elliptic non linear differential equations (potential theory) than to the theory of fuchsian groups.
In a series of papers at the turn of the past century Picard [9] proved that eq.(3) for real φ with asymptotic behavior at infinity and −1 < g n , 1 < g ∞ (which excludes the case of punctures) and n g n + g ∞ < 0 admits one and only one solution (see also [10]). Picard [9] achieved the solution of (3) through an iteration process exploiting Schwarz alternating procedure. The same problem has been considered recently with modern variational techniques by Troyanov [11], obtaining results which include Picard's findings. The interest of such results is that they solve the following variant of the Riemann-Hilbert problem: at z 1 , . . . z n we are given not with the monodromies but with the class, characterized by g j , of the elliptic monodromies with the further request that all such monodromies belong to the group SU(1, 1). The last requirement is imposed by the fact that the solution of eq.(3) has to be single valued. Eq. (3) is the type of equation one encounters in the ADM treatment [4,5] of 2 + 1 gravity coupled with point particles in the maximally slicing gauge [12]. In this case z varies on the Riemann sphere, N of the z j are the particle singularities with residue g j = −1 + µ j and N − 2 of them are the so called apparent singularities z B with residues g B = 1. The inequalities on the values of g m are satisfied in 2 + 1 dimensional gravity due to the restriction on the masses of the particles 0 < µ n < 1 (in rationalized Planck units) and to the fact that the total energy µ must satisfy the bound n µ n < µ < 1. For this reason in this paper we shall confine ourselves to the Riemann sphere. After solving all the constraints, the hamiltonian nature of the particle equations of motion is a consequence of Polyakov conjecture; actually is the consequence of a somewhat weaker form of it [5] i.e. of the relation one obtains by taking the derivative of Polyakov conjecture with respect to the total energy.
From eq.(3) one can easily prove [10,13] that the function Q(z) defined by is analytic i.e. as pointed out in [13] Q(z) is given by the analytic component of the energy momentum tensor of a Liouville theory. Q(z) is meromorphic with poles up to the second order [6] i.e. of the form All solutions of eq.(3) can be put in the form being y 1 , y 2 two properly chosen, linearly independent solutions of the fuchsian equation w 12 is the constant wronskian. In fact following [10,13] as e −φ/2 solves the fuchsian equation (8) it can be put in the form with ψ j (z) solutions of eq.(8) with wronskian 1 and χ j (z) also solutions of eq.(8) with wronskian 1. The solution of eq.(3) (φ = real) with the stated behavior at infinity is unique [9,11]. Exploiting the reality of e φ it is possible by an SL(2C) transformation to reduce eq.(9) to the form eq.(7). In fact, being χ j linear combinations of the ψ j , the reality of e φ imposes with the 2×2 matrix H jk hermitean and det H = −1. By means of a unitary transformation, which belongs to SL(2C) we can reduce H to diagonal form diag(−λ, λ −1 ) and with a subsequent SL(2C) transformation we can reduce it to the form diag(−1, 1) i.e. to the form (7). Through eq.(5) φ contains the full information about the accessory parameters β n defined in eq.(6). It is important to notice that being all of our monodromies elliptic, we can by means of an SU(1, 1) transformation, choose around a given singularity z m (not around all singularities simultaneously) y 1 and y 2 with the following canonical behavior proof of the real analytic dependence of the accessory parameters on the z n in the case of rational g n has been given by Kra [14].
Starting from the singularity in z 1 we can consider the canonical pair of solutions around z 1 i.e. those solutions which behave as a single fractional power multiplied by an analytic function with coefficient one as given in eq. (11). We shall call such pair of solutions (y 1 1 , y 1 2 ) and let (y 1 , y 2 ) the solution which realize SU(1, 1) around all singularities. Obviously all conjugations with any element of SU(1, 1) is still an equivalent solution in the sense that they provide the same conformal factor φ. The canonical pairs around different singularities are linearly related i.e. (y 1 1 , y 1 2 ) = (y 2 1 , y 2 2 )C 21 . We fix the conjugation class by setting which however has to be absent in our case (no logarithm condition [7]) in order to have a single valued φ. In this case the monodromy matrix is simply the identity or minus the identity. The monodromy around z 1 thus belongs to SU(1, 1) for any choice of K. If D n denote the diagonal monodromy matrices around z n , we have that the monodromy around z 1 is D 1 and the one around z 2 is where with C 12 we have denoted the inverse of the 2 × 2 matrix C 21 .
In the case of three singularities (one of them at infinity) the counting of the degrees of freedom is the following: by using the freedom on K we can reduce As the matrices M n = K −1 C 1n D n C n1 K satisfy identically det M n = 1 and TrM n = 2 cos πg n we need to impose generically on M 2 only two real conditions e.g. Re b 2 = Re c 2 and Im b 2 = −Im c 2 . The same for M 3 . Thus is appears that we need to satisfy four real relations when we can vary only three real parameters. The reason why we need only three and not four is that for any solution of the fuchsian problem the following relation among the monodromy matrices is identically satisfied Rigorously the conditions for realizing SU(1, 1) monodromies are Satisfying the eight above equations is a sufficient (and necessary) condition to realize the SU(1, 1) monodromies. The fact that given a z 0 n in a neighborhood of such a point there exists one and only one solution to the eight equation (15) means that at least three of them are not identically satisfied in such a neighborhood and that the remaining are satisfied as a consequence of them. We shall denote such equations as The matrices A n = C n1 K which give the solution of the problem in terms of the canonical solutions around the singularities are completely determined by the two equations (y 1 , y 2 ) = (y due to the non vanishing of the wronskian of y In order to understand the dependence of β 3 andβ 3 on z n ,z n we apply around a solution (which due to Picard we know to exist) of the three equations, Weierstrass preparation theorem [15]. It states that in a neighborhood of a solution z 0 n k 0 β 0 3 , ∆ (i) can be written as where P (k) is a polynomial in k with coefficients analytic functions of z n ,z n , β 3 ,β 3 , while u(k, z n ,z n , β 3 ,β 3 ) is a "unit" i.e. an analytic function of the arguments, which does not vanish in a neighborhood of z 0 n , k 0 , β 0 . Thus our problem is reduced to the search of the real common zeros of the three polynomial P (i) (k). By algebraic elimination of the variable k we reach a system of two equations which depend analytically on the variables z n ,z n , β 3 ,β 3 and reasoning as above by elimination ofβ 3 we reach as condition to be satisfied by Picard's solution V (β 3 |c k (z n ,z n )) = 0 where V is a polynomial in β 3 with coefficients analytic functions of z n andz n . The derivative of β 3 with respect to z n (and similarly with respect toz n ) is then given by If V ′ vanishes identically on the β 3 (z n ) provided by Picard's solution we can adopt V ′ as determining such a function. The procedure can be repeated until V ′ does not vanish identically on Picard's solution and thus in a neighborhood of z 0 n the derivative ∂β 3 ∂z n exists except for a finite number of points. Actually β 3 is an analytic functions of z n andz n for all points of such a neighborhood in which V ′ does not vanish [15]. The extension to five or more singularities proceeds along the same line.
As we already mentioned if some of the g m is an integer we have the so called apparent singularities which have monodromy I if g m is even and monodromy −I if g m is odd. In this case we have to impose the so called no-logarithm conditions (see e.g. [7]) which result in a linear combination of the β n with n = m to be equal to the square of β m . Thus we can eliminate one β n in favor of β m and we have the same matching in the degrees of freedom.

Proof of Polyakov conjecture
As already stated we shall limit ourselves to the case of the Riemann sphere with a finite number of conical singularities, one of them at infinity, subject to the restrictions given by Picard and described in sect.1. The technique to prove Polyakov conjecture will be to express the original action in terms of a field φ M which is less singular than the original conformal field φ. This procedure will give rise to an action S for the field φ M which does not involve the ǫ → 0 process. Despite that, computing the derivative of the new action S is not completely trivial because one cannot take directly the derivative operation under the integral sign. In fact such unwarranted procedure would give rise to an integrand which is not absolutely summable. In the global coordinate system z on C one writes where φ B is a background conformal factor which is regular and behaves at infinity like with behavior at infinity φ 0 = (2 − g ∞ ) ln(zz) + O(1). Such a behavior fixes the value of α Then we have for φ M φ M is a continuous function on the Riemann sphere. The action which generates the above equation is where the splitting between the measure and the integrand has been introduced for later convenience. Due to the behavior of φ M and φ 0 at the singularities and at infinity the integral in eq.(25) converges absolutely. It is straightforward to prove that the action S computed on the solution of eq.(24) is related to the original Polyakov action S P also computed on the solution of eq.(24) by The behavior of φ M around the singularities z m can be deduced from eqs. (7,11). Thus in a finite neighborhood of z m we can write where the finite sum extends to the terms such that 2L(g n + 1) + M + N ≤ 3 and ρ(|ζ|) is  [16].
The procedure to compute the derivative will be to prove that where F is given in eq.(25) and X ǫ has been defined after eq.(2). This is achieved by writing F = (F − f ) + f where F − f is sufficiently regular, i.e.
continuous and absolutely integrable with ∂(F −f ) ∂zm continuous and | ∂(F −f ) ∂zm | < M for any z while z m varies in a finite interval, so that proving at the same time that Then by summing eq.(29) and eq.(30) we obtain eq.(28).
Using the expansions eq.(27) we shall choose the function f as where the finite sum extend to all singularities of F and M, N ≥ 0 and L ≥ 1 such that 2L(g m + 1) + M + N ≤ 3. We notice that the point being that each integral in the sum does not depend on z n due to translational invariance and thus we have to take only the derivative of the coefficients b nLM N . Moreover The last term is either zero due to the phase of the integrand or goes to zero for ǫ → 0 It is easily checked that the only contribution which survives in the limit ǫ → 0 is which can be computed by using eq.(24) and Using φ M − (α − 1)φ B = φ − n g n ln |z − z n | 2 and the expansion of A = 1 + c 1 ζ + · · · and B = 1 + c 2 ζ + · · · which are obtained by substituting into the differential equation (8)