Werner's Measure on Self-Avoiding Loops and Welding

Werner's conformally invariant family of measures on self-avoiding loops on Riemann surfaces is determined by a single measure $\mu_0$ on self-avoiding loops in ${\mathbb C} \setminus\{0\}$ which surround $0$. Our first major objective is to show that the measure $\mu_0$ is infinitesimally invariant with respect to conformal vector fields (essentially the Virasoro algebra of conformal field theory). This makes essential use of classical variational formulas of Duren and Schiffer, which we recast in representation theoretic terms for efficient computation. We secondly show how these formulas can be used to calculate (in principle, and sometimes explicitly) quantities (such as moments for coefficients of univalent functions) associated to the conformal welding for a self-avoiding loop. This gives an alternate proof of the uniqueness of Werner's measure. We also attempt to use these variational formulas to derive a differential equation for the (Laplace transform of) the"diagonal distribution"for the conformal welding associated to a loop; this generalizes in a suggestive way to a deformation of Werner's measure conjectured to exist by Kontsevich and Suhov (a basic inspiration for this paper).


Introduction
Given a topological space S, let Comp(S) denote the set of all compact subsets of S with the Vietoris topology, and let Loop(S) := {γ ∈ Comp(S) : γ is homeomorphic to S 1 } with the induced topology (see Appendix C). Suppose that for each Riemann surface S, µ S is a positive Borel measure on Loop(S). Following Werner, this family of measures is said to satisfy conformal restriction if for each conformal embedding S 1 → S 2 , the restriction of µ S2 to Loop(S 1 ) equals µ S1 ; the family is nontrivial if the measure of the set {γ ∈ Loop({0 < |z| < A}) \ Loop({|z| < a}) : γ surrounds 0} is finite and positive, for some 0 < a < A. In [16] Werner proved the following remarkable result. Theorem 1.1. There exists a nontrivial family of measures {µ S } on self-avoiding loops on Riemann surfaces which satisfies conformal restriction. This family is unique up to multiplication by an overall positive constant.
Below we will introduce a normalization which will uniquely determine this family of measures (see (1.5)).
Essentially because any self-avoiding loop on a Riemann surface is contained in an embedded annulus, the family {µ S } is (in principle) uniquely determined by where φ : (U, 0) → (V, 0) is a conformal isomorphism. 1 We will refer to c W as Werner's constant, which depends on the normalization (1.5) below. At the present time we can only say that c W ≥ 1.
Our purpose is to explore other possible explicit formulas for µ 0 , especially in terms of welding. To put this in perspective, it is convenient to slightly digress and recall the "Fundamental Theorem of Welding", and some associated terminology. Theorem 1.3. Suppose that σ ∈ QS(S 1 ), the group of quasisymmetric homeomorphisms of S 1 . Then is a univalent holomorphic function in the open unit disk ∆, with quasiconformal extension to C ∪ {∞}, m ∈ S 1 is a rotation, 0 < a ≤ 1 is a dilation, the mapping inverse to l, is a univalent holomorphic function on the open unit disk about infinity, ∆ * , with quasiconformal extension to C ∪ {∞}, and the compatibility condition holds. This factorization is unique.
Definition. A homeomorphism σ of S 1 has a triangular factorization (or admits a conformal welding) if σ = l • ma • u where is a holomorphic function in ∆ with a continuous extension to a homeomorphism on D := closure(∆), m ∈ S 1 is a rotation, 0 < a is a dilation, the mapping inverse to l, is a holomorphic function on ∆ * with a continuous extension to a homeomorphism on D * = closure(∆ * ), and the compatibility condition mau(S 1 ) = L(S 1 ) holds.
Quasisymmetric homeomorphisms have unique weldings. For less regular homeomorphisms, there are additional sufficient conditions for the existence of weldings (see [2] and references), but there are many examples of homeomorphisms which do not admit weldings, and weldings which are not unique (see [3]).
Remarks. (a) To clarify (1.4), the σ image of W is by definition the set of homeomorphisms which admit a triangular factorization with rotation m = 1.
(b) The map W is not 1 − 1 because triangular factorization fails (in a dramatic way) to be unique (The source of nonuniqueness: there exist homeomorphisms of the 2-sphere which are conformal off of a Jordan curve, and which are not linear fractional transformations).
The ideal goal of this paper is to calculate, in some explicit way, the image measure W * µ 0 , and to show that we can recover µ 0 from this image. As we will see in Section 2, conformal invariance implies that where ν 0 is an inversion invariant finite measure, which we normalize to have unit mass. This reduces the task of computing W * µ 0 to computing the inversion invariant probability measure ν 0 . A lofty goal is to compute the joint distribution for the coefficients of u (A sufficiently explicit calculation would yield a probabilistic proof of the Bieberbach Conjecture/de Branges Theorem). Since these coefficients are bounded, it in principle suffices to compute their joint moments. We will show that there is an algorithm for doing this, although it is rather unwieldly. We can explicitly calculate some low order moments, for example: For various reasons it is of special interest to calculate the "diagonal distribution" for the welding homeomorphism.
Conjecture. If ν 0 is normalized to be a probability measure, then . This conjecture is closely related to Proposition 18 in [16]. This is discussed in Section 7. There is a generalization of this conjecture to a natural deformation of Werner's measure which is conjectured to exist in [10]; in fact our hope is that this extended conjecture might be useful in proving existence of this deformation.
1.1. Outline of the Paper. In Section 2 we prove some basic facts about the welding map W . The idea of the proof of the diagonal distribution conjecture is to show that the Laplace transform of the diagonal distribution for ν 0 satisfies a differential equation, using the infinitesimal conformal invariance of µ 0 . For this purpose in Section 3 we recall some classical variational formulas of Duren and Schiffer. In Section 4 we discuss the infinitesimal action from a representation theoretic point of view, and we recast the Duren-Schiffer formulas in terms of generating functions, using a stress-energy tensor formulation common in conformal field theory. In Section 5 we establish the version of infinitesimal conformal invariance of µ 0 needed for our purposes. In Section 6 we apply this to compute moments of the coefficients of u. In Section 7 we discuss the relation between the diagonal distribution conjecture and Proposition 18 of [16], and outline a strategy for a proof; we also briefly indicate how the conjecture generalizes to the deformation which is conjectured by Kontsevich and Suhov to exist in [10].

Notation and Conventions
. Given a complex number z, we often write z * for the complex conjugate, especially when z is represented by a complicated expression.
Throughout this paper, we view vector fields on a manifold as the Lie algebra of diffeomorphisms of the manifold; the induced bracket is the negative of the usual bracket obtained by viewing vector fields as derivations of functions on the manifold.

The Welding Map
In this section we consider the welding map (1.4).
The distributions for ρ 0 and ρ ∞ are invariant with respect to dilation, i.e. equivalent to Haar measure for R + .
where ν 0 is a finite measure (which we will normalize to have unit mass).
(c) The measure dν 0 (σ) is inversion invariant and invariant with respect to conjugation by C : z → z * .
where u and L are written as in Definition 1.
(f ) The welding map is equivariant with respect to rotations in the sense that Proof. We first claim that The inequality r < ρ ∞ (γ) implies that γ cannot be contained in {|z| < r}. In Thus γ is in the ball of radius 4R. This proves the claim.
By conformal invariance and the nontriviality assumption of Werner, the set of loops (surrounding zero) with r < ρ ∞ < R has µ 0 finite measure, for any r < R. This implies that there is a essentially unique disintegration of µ 0 of the form where the fiber measures are probability measures.
The invariance of µ 0 with respect to dilation, γ → ργ, implies that the ρ ∞ distribution ω is also dilation invariant, i.e. it is a Haar measure for R + . The invariance of µ 0 with respect to z → 1 z implies that the same is true for ρ 0 . This proves (a).
Since µ 0 is determined up to multiplication by a constant, we can suppose that dω(ρ ∞ ) = dρ ∞ ρ ∞ The action by dilation transports one fiber to another. Hence dilation invariance also implies that all the fiber measures are the same. This implies that W * µ 0 is a product measure, as claimed in part (b).
For part (c), we first use the invariance of µ 0 with respect to z → 1 z * , which maps γ to 1 γ * : This implies the invariance of ν 0 with respect to inversion.
The measure µ 0 is also invariant with respect to C : z → z * . In this case This implies that ν 0 is invariant with respect to conjugation by C. This proves (c).
Part (d) is obvious. For part (e) (essentially the well-known area theorem from the theory of univalent functions), the main point is that and By continuity of measure, these formulas hold for all γ. This implies part (e). Part (f) follows from Remarks. (a) In connection with part (c), in general, if a homeomorphism σ has a triangular factorization lmau, then σ −1 has a triangular factorization with In particular inversion stabilizes the set of σ having triangular factorization with m = 1.
(b) In connection with part (f ), equivariance with respect to rotations, see Subsection 2.2 below.

Unresolved Foundational Issues. Theorem 1.3 implies that W induces a bijection
where a quasicircle is a Jordan curve which admits a parameterization by the restriction to S 1 of a quasiconformal homeomorphism of C ∪ {∞}.
(b) W is 1 − 1 on a set of full µ 0 measure. (c) Almost surely with respect to ν 0 , σ has a unique triangular factorization (with m = 1). This is suggested by [2].

Variational Formulas
Since the measure µ 0 has a local form of conformal invariance, it is natural to suspect that there are senses in which the measure is infinitesimally conformally invariant. For this reason we need to consider how φ ± vary when the curve γ is varied by a local deformation z → z + ǫv(z), where v(z) is holomorphic in C \ {0}. This deformation corresponds to a real vector field W * consists of antiholomorphic vector fields. It is spanned by {L n : n ∈ Z}.
The precise relationship between −→ W and W is that there is a real embedding Loosely speaking, −→ W is the Witt algebra considered as a real Lie algebra (see page 115 of [4]). The reason for maintaining a distinction is that the variational formulas below will naturally define a real representation of the real Lie algebra −→ W . However it is convenient to express this representation in terms of an associated complex representation of W. We will write the map (3.1) as and similarly for −→ iL n . The corresponding actions on a function of γ are given by where in the last line we have implicitly chosen a parameterization for γ, and similarly for −→ iL n .
When n ≥ −1, L n = −z n+1 ∂ ∂z is regular at z = 0. In this case it is very easy to find the variations of φ + with respect to Proof.
→ L 0 is infinitesimal dilation. In this case the formulas in part (a) are obvious, because σ(γ) is unchanged when γ is dilated.
For n ≥ −1 and ǫ sufficiently small, a uniformization for the region inside γ + ǫγ n+1 is the composition φ + + ǫ(φ + ) n+1 This uniformization has to be composed with a linear fractional transformation to obtain the correct normalization. Consequently where λ (having unit norm) and ω are determined by the conditions that this uniformization vanishes at z = 0 and has positive derivative at z = 0. Suppose n ≥ 0. In this case the linear fractional transformation is the identity for all ǫ. This implies part (a).
Part (b), when n + 1 = 0, is slightly more involved. In this case To calculate the derivatives at zero, we use the normalizations for the mapping (3.3). Because 0 is mapped to zero, Secondly the derivative of the map (3.3) at z = 0 must be positive. Thus and (because λ has unit norm) Plugging these derivatives into (3.4) yields This implies part (b). Parts (c) and (d) are similar. For part (c), when n = 0, note that To calculate the derivatives at zero, we use the normalizations for the mapping (3.8). Because 0 is mapped to zero, Secondly the derivative of the map (3.8) at z = 0 must be positive. Thus This implies part (d).

(Lack of ) Equivariance for W .
We have already observed that the welding map is equivariant with respect to the actions of rotation of loops and conjugation of homeomorphisms; see (f) of Lemma 1. Given |w| < 1, define φ 1 (w) ∈ P SU (1, 1) (viewed as the group of automorphisms of the Riemann sphere which stabilize the circle) by Remark. The formula in (c) illustrates how the welding map is trying, with limited success, to intertwine the action of P SU (1, 1) on loops with its action by conjugation on the welding homeomorphism.

Proof. A uniformization for the region inside γ(ǫ) is the composition
This uniformization has to be precomposed with a linear fractional transformation to obtain the correct normalization. Consequently 1 + ω(ǫ)λ(ǫ)z and λ (having unit norm) and ω are determined by the conditions that this uniformization vanishes at z = 0 and has positive derivative at z = 0.
The first condition implies This implies the formula in (a).
In a similar way and one precedes as before. This leads to (b) and (c).

Variational Formulas, II.
It is far more difficult to calculate → L −n φ + for n > 1. In this case z −n+1 is regular at z = ∞. This is the situation considered in [5], with slight modifications. In the following statement, let U denote the mapping inverse to u.
If degree(u j ) = j, then P n is a homogeneous polynomial of degree n.
Remarks. (a) It is natural to restate the relationship between the P n and U in terms of quadratic differentials In Section 7 it will be convenient to rewrite this as and to set P n (φ + ) = ρ −n 0 P n (u). Hopefully this will not cause any confusion. (b) Similarly the residue formula for B m is naturally understood as the integral over γ of the natural pairing of the holomorphic vector field −v(t) d dt and the holomorphic quadratic differential U (t) m (∂ln(U (t))) 2 .
For later reference we note some elementary properties of the polynomials P n .
Proposition 4. (a) P n (u) is a homogeneous polynomial in u 1 , .., u n of degree n, where degree(u j ) = j, with integer coefficients.
(b) P n (u) = −2nu n + terms involving u 1 , .., u n−1 (c) u n is a homogeneous polynomial in P 1 , .., P n of degree n, where degree(P j ) = j, with rational coefficients.
At this point we have formulas for the action of the real Witt algebra on the coefficients of φ + . If we write where w = 1 z is the standard coordinate at infinity, then we can also write down formulas for the action of the real Witt algebra on the coefficients of φ − . We will postpone this until the next section. . This representation is real, in the sense that the set of real functions is stable, or equivalently that the action commutes with complex conjugation of functions.
To be precise, fix λ ∈ C. The Duren-Schiffer formulas imply that there is a real representation of the real Lie algebra −→ W by real derivations on the spaces of complex-valued functions W by π 0 . By abstract nonsense there is an associated complex representation of W by complex derivations of the algebra of complexvalued functions of self-avoiding loops, defined by There is also a representation This is a complex representation of W = W * by complex derivations. In turn, in terms of the real embedding (3.1) The point of this translation is that the complex representations π and π are easier to analyze. In fact (on proper domains) they can be expressed in terms of highest weight representations, and this allows us to access well-known results from the theory of highest weight representations of the Virasoro algebra (at the moment the central charge c = 0, so that we are only considering the Witt algebra).
If degree(u j ) = j, then P n is a homogeneous polynomial of degree n.
(e) For k ≥ 1 Using the Lemma below, this can be restated in the following way.
Remark. This second statement seems more clean than the first. However, as we will see when we introduce the energy-momentum tensor, the first statement has the advantage of being stated in terms of the inverse of φ + . Which is the better point of view remains to be sorted out.
To avoid cumbersome notation, we will often identify L n with its corresponding operator, π(L n ). Suppose that we write u 0 = 1 and a k = ρ 0 u k , so that If n > 0 and k ≥ 1, then according to (e) and Res(( This can be restated more cleanly in the following way.
Using the Lemma we can write The mild surprise is that this expression leads to a formula which is valid for all n.
Proof. We just need to check that this formula agrees with our previous calculations when n ≥ 0. This is straightforward.
Now we want to add things up as in the preceding section. As before we write φ + (z) = a k z k+1 , where a k = ρ 0 u k and it is understood that u 0 = 1. By part (e) By the change of variable lemma of the preceding subsection ) 1 * where the notation (..) k denotes the kth Fourier coefficient. This equals As in the preceding subsection, we obtain the following uniform formula.
Proof. We just need to check that this formula agrees with the formulas in Proposition 7. This is again straightforward.

4.4.
Formulas for π 0 , Revisited. We can use Theorems 4.1 and 4.2 to recast the Duren-Schiffer variational formulas in the following form.
It is obviously desirable to find a direct proof of these formulas which reflects their structure.

4.5.
Calculations with φ − . On the one hand, in the standard w coordinate at The l n coordinates for φ − are analogous to the u n coordinates for φ + , and variational formulas for φ − essentially arise from substituting l j 's for u j 's in our earlier formulas. On the other hand, in the standard z coordinate, and it is occasionally useful to employ the b m coordinates. The relation between the two sets of coordinates is standard. Proof.
The φ − analog of Theorem 4.1 and Theorem 4.2 is the following theorem. In the statement, for a Laurent expansion convergent in an annulus R < |z| < ∞, we use the notation Res( g m z m , z = ∞) = −g −1 (This is actually the residue of the differential g(z)dz at z = ∞ in the Riemann sphere).

4.6.
Representation-Theoretic Consequences. The formulas of the preceding section imply that π is a complex representation of the Witt algebra W by derivations of the algebra Ω 0 (ρ 0 ) ⊗ C[u 1 , u 2 , ..], where Ω 0 (ρ 0 ) denotes any algebra of smooth functions of ρ 0 . Consider the action of W on the vector space where λ is a fixed complex number. For n > 0 the operators L n kill ρ λ 0 , and the spectrum of L 0 on the W-module generated by ρ λ 0 is {λ/2 + n : n = 0, 1, ..}. We will refer to this as a lowest weight module (admittedly there are conflicting conventions). The following proposition follows from well-known facts about such representations (see [7]).

Proposition 8. For any λ ∈ C,
(a) the representation generated by the π action of W on ρ λ 0 is a realization of the unique irreducible lowest weight representation of the Virasoro algebra with central charge c = 0 and h = 1 2 λ.
where u j has degree j. Otherwise there is a proper containment.
(b) Similarly, the representation generated by the π action of W on ρ −λ ∞ is a realization of the highest weight representation of the Virasoro algebra with central charge c = 0 and h = − λ 2 .
where l j has degree j. Otherwise there is a proper containment.
Remark. The realization of the lowest weight representation in part (a) is related in a relatively simple way to the realization, using geometric quantization techniques, due to Kirillov and Yuriev in [9]. In [9] W acts on a space of sections of a line bundle (parameterized by c = 0 and h = λ/2) over (a somewhat imprecisely defined) space of Schlicht functions u ∈ S (normalized univalent functions on the disk, viewed as a homogeneous space for Dif f (S 1 )). In coordinates (by trivializing the line bundle) this vector space is identified with C[u 1 , u 2 , ..], polynomials in the coefficients of the univalent function u, and the formulas for the action appear in (8) of [9] (with c = 0, and one takes the negative of the operators, because we consider the opposite of the bracket in [9]). The intertwining operator from Kirillov and Yuriev's realization to our realization in (a) is given by the map where U = t(1 + n>0 U n t n ) is the inverse to the univalent function u = z(1 + n>0 u n z n ). An advantage of our realization is that the operators are derivations of an algebra, which makes them more amenable to calculations. This will appear in the first author's dissertation. 4.7. Stress-Energy Formulation. Consider the standard holomorphic coordinate z = x + iy. In real coordinates the symmetric stress tensor has the form dz dz Conformal invariance is implied by the trace condition trace(T ) = T 11 + T 22 = 0 (see pages 101 and 103 of [4]). In complex coordinates this implies that T is diagonal.
In a conformal field theory with central charge c = 0 is a holomorphic quadratic differential (See page 155 of [4]. Note: for c = 0, the stress energy "tensor" is actually a holomorphic projective connection; see page 136 of [4] or page 532 of [14]). We are seeking a completely natural formulation for the action of the Witt algebra Proof. By definition By part (a) of Proposition 6, this equals This proves the first statement. The proof of the second statement is similar.

Corollary 2.
In the sense of hyperfunctions Proof. From a formal power series point of view, this follows immediately from the proposition. From the point of view of analysis, this equality has to be interpreted in a hyperfunction sense, because the first term is holomorphic in U + and the second term is holomorphic in U − .

Infinitesimal Invariance
Suppose that γ ∈ Loop 1 (C \ {0}). In terms of the standard coordinate z, In terms of the coordinate w = 1 z , The variational formulas of the preceding section imply that the vector space of functions of the form p(u 1 , .., u n , u 1 , .., u n )f (ρ 0 ) where p is a polynomial of any number of variables, and f has compact support in R + , is stable with respect to the action of the Witt algebra (this applies both to the real action and the complexified actions). Since the Witt algebra is stable with respect to z → w = 1 z , the vector space of functions of the form p(l 1 , .., l n , l 1 , .., l n )f (ρ ∞ ) where p is a polynomial of any number of variables, and f has compact support in R + , is also stable with respect to the action of the Witt algebra. Consequently the vector space of "test functions" spanned by functions of the form where p is a polynomial and f has compact support in R + × R + , is stable with respect to the Witt algebra (for the real or complexified actions). In reference to F , since u n and l n are bounded (by constants depending only on n), p is bounded. The compact support condition on f implies that F is supported on Loop 1 of a fixed finite type annulus. Since µ 0 has finite measure on loops in a finite type annulus, F is integrable. Kontsevich and Suhov have conjectured that there is a converse of this result which holds generally for their conjectural family of measures µ c deforming µ 0 (see section 2.5.2 of [10]).
For the purposes of this paper, we need to be able to apply integration by parts to functions which involve the bounded function a λ (λ > 0), rather than a function having compact support in ρ 0 , ρ ∞ . One complication is that for L ∈ W, is not necessarily bounded.
where p is a polynomial and f has compact support in R + . Then for any L ∈ W×W, for Re(λ) sufficiently large, The same conclusion applies if we replace f (ρ ∞ ) by f (ρ 0 ).
Proof. Fix a smooth positive function g(ρ 0 ) having compact support for ρ 0 ∈ R + and identically 1 in a neighborhood of ρ = 1. By Proposition 10, for each δ > 0, Since g is fixed, the first term goes to zero as δ → 0. We can apply dominated convergence to the second term, for sufficiently large λ (so that the part of the integrand not involving g is bounded, and hence the integral is well-defined). This implies the Lemma. Proof. Suppose that L = L n orL n with n < 0. Fix a smooth family of functions g δ (ρ ∞ ) which converges to the δ function at ρ ∞ = 1. Using L n (ρ ∞ ) = 0 and Lemma 5, Since L(F ) is bounded for sufficiently large Re(λ), the left hand side of the last equality converges to L(F )dν 0 as δ → 0. This implies part (a). If L = L n orL n with n > 0, the same argument applies with g δ (ρ 0 ) in place of g δ (ρ ∞ ).
If L = → L 0 , then L(F ) = 0. We have previously observed that if L = −→ iL 0 , then L exponentiates to rotational symmetry of C \ {0}, and this corresponds to invariance of ν 0 with respect to the conjugation action of rotations on homeomorphisms.
In the sections below, we will repeatedly apply a variation of the preceding proof in the following way.
Suppose that n > 0 and L = L n or L =L n . Then as in the proof We can take the limit as δ → 0, because the support of g δ remains bounded, and ρ −n 0 will be bounded in this support region. This implies which can be written heuristically as where δ 1 denotes the Dirac delta function at 1. There are similar integral formulas involving L −n , but then we must use an approximation to δ 1 (ρ ∞ ).
We will use the following notation throughout this section. . We will refer to elements in the vector space C[u,ū] (n,n) as being of level n.
The rationale for the notation is the following. The outer tensor product, W ×W, acts on the tensor product C[u] ⊗ C[ū]. The product of the corresponding rotation groups acts, and induces a bigrading. In (b) we are considering the 0-eigenspace for the real embedded rotation group.
If x ∈ C[u,ū] (n,n ′ ) , then one may verify using the rotational invariance of Werner's measure. Therefore, we restrict ourselves to computing integrals of elements at levels n = 1, 2, . . . (i.e., n = n ′ ). Suppose 1 ≤ m ≤ n. In general, we can obtain integral identities by computing and applying infinitesimal invariance. As we will see in the following sections, we are particularly interested in the cases m = 1, 2.
Remark. In (6.2) it is necessary to restrict consideration to L −m for m ≥ 1, because we actually need this derivative to fix ρ ∞ . Otherwise we cannot apply integration by parts to obtain integrals.
We will now give an example where we compute the integrals for all elements of level 2. The single equationL −2 (ρ 2 0 u 2 1 ) = −6u 1ū1 + 14u 2 6.2. Expressions for L −1 . Consider (6.2) in the case m = 1. The first expression we derive for this operator is purely algebraic.
Lemma 6. Suppose that kp k = n and kq k = n−1 and let u pūq := k u p k kū q k k . Then The first sum of terms are of level n − 1, and the other terms are of level n.
Proof. We calculate This simplifies to the expression in the statement of the lemma.
The second expression is in terms of divergence-type differential operators. We also note that the homogeneity condition on the domains can be expressed in terms of divergence-type operators.
is of the form N 1 ⊗ 1, where Proof. We will prove (b'): If n = 1, then R 1 : C → C[ū] (1) is injective by dimension considerations. If n ≥ 2, then consider the representationπ of W on C[ρ 0 , u]. For the lowest-weight representation generated by ρ n−1 0 , we have c = 0 and h = −( n−1 2 ) (see Section 4.6). This is a reducible Verma module if and only if When the Verma module is irreducible, the creation operator L −1 is injective at each level, i.e. R 1 is injective. Notice that the same thing would be true for L −k for any k > 0.
where we are denoting a partial inverse to R 1 by R −1 1 . Unfortunately this does not make any sense for most Q.
Definition. For a single complex variable z, we define 1 √ k! z k to be an orthonormal basis for C[z]. For a tensor product such as we take the tensor product Hilbert space structure, meaning that 1 √ p! u p is an orthonormal basis, where p! := p 1 !p 2 !..

Proposition 14. (a)
The adjoint of where i.e. the cokernel of R 1 (or the orthogonal complement of the image of R 1 ). Then which has dimension (p(n) − p(n − 1)) 2 > 0 for n > 1.
We will now give a slight generalization of Theorem 1.4 using the algebraic expression for L −1 .
Corollary 3. Suppose that weight(p) = n. Then Proof. The formula in Lemma 6 implies Thus we obtain a recursion relation The terms on the right hand side of the same form with weight = n − 1. Since j≥1 jp j = n, induction implies the right hand side equals 1. This implies the corollary.
6.3. Expressions for L −2 . We now consider the operator (6.2) in the case m = 2, which is substantially more complicated than in the m = 1 case. Recall that p (−1) k denotes the Laurent coefficient of z u(z) and P 2 = 7u 2 1 − 4u 2 .
Proposition 16. Fix n ≥ 2 and let K m = kernel(R t m : Therefore, in principle, we can determine all moments by using only L −1 and L −2 .
Proof. Consider the cylic π-representation generated by ρ n 0 : which is an irreducible Verma module. Therefore, the n-th graded component, , has a basis consisting of elements of the form L −ij · · · L −i1 (ρ n 0 ), where 0 < i 1 ≤ · · · ≤ i j and i 1 +· · ·+i j = n. The claim follows since U( k≥1 CL −k ) is generated by L −1 and L −2 .
6.4. The Recursion Relation. Consider u P ⊗ū Q ∈ C[u] (n) ⊗C[ū] (n) . In principle, we can writeū for some polynomialsf j ∈ C[ū] (n−j) . We can then compute The question now becomes how to divideū Q into two pieces. In theory, this can be done using the orthogonal decomposition This gives a recursion relation for moments. The drawback is that we have to find all of the moments at a given level (indexed by n, which involves u 1 , .., u n ) to proceed. Because p(n) grows very rapidly, it has proven to difficult to numerically calculate enough moments to identify the distribution of, say, u 1 .

The diagonal distribution
To determine the joint distribution for (ρ 0 , ρ ∞ ), Lemma 1 implies that it suffices to determine the distribution for H = −log(a) ≥ 0, which is a kind of height function for (7.1) {σ ∈ Homeo(S 1 ) : ∃ unique welding σ = lau} Conjecture. For some β 0 < 5π 2 4 , the ν 0 distribution for a is given by Equivalently the Laplace transform for λ > 0, where K 1 is a modified Bessel function.
We will first explain how this conjecture is related to a remarkable calculation of Werner in Section 7 of [16]. We will then discuss some ideas which are hopefully (interesting and) relevant to a proof. Finally we will briefly indicate how the conjecture naturally generalizes to the deformation of Werner's measure considered in [10]. 7.1. A Formula of Werner. As in Section 7 of [16], consider the function where A is a finite type annulus with modulus ρ = ρ(A), i.e. ρ > 0 is the unique number such that A is conformally equivalent to {1 < |z| < e ρ } As we will explain below in more detail and as a consequence Werner shows that F (ρ) is asymptotic to constant · exp(− β ρ ) as ρ → 0, where β = 5π 2 4 ; see Proposition 18 of [16]. This leads to the constraint on β 0 in the diagonal distribution conjecture.
Proof. The Cauchy integral formula implies, for sufficiently smooth γ, Since γ is outside the unit disk and φ −1 This implies the first inequality. The last inequality also follows from this. We noted previously that the equality (c) There is an asymptotic expansion Proof. (a) Using the factorization dµ 0 = dρ∞ ρ∞ × dν 0 , By making the change of variables ρ ∞ = e y , we obtain the expression in part (a). (b) There is a Laurent expansion Therefore there is an expansion where the divergence of the logarithm and the Laurent expansion at x = 0 perfectly cancel, allowing us to figure out c 0 .
Proof. This follows from (b) of the first Lemma and (a) of the second Lemma.
Here is another approach, although not quite as sharp: Werner's formula for the measure of the latter set is c W ρ, where c W is Werner's constant (see below).

Werner's Constant.
As in the rest of these notes, we assume ν 0 is a probability measure. We let c W denote the constant such that if γ is a loop which surrounds ∆, Proposition 17. c W ≥ 1.
Proof. On the one hand Therefore by Werner's formula for the measure of the latter set, On the other hand where the last inclusion uses Koebe's Quarter Theorem. Therefore Because ν 0 (e −x ≤ a ≤ 1) ↑ 1 as x ↑ 1 it follows that F (ρ) behaves like a linear function with slope one for ρ >> 1. This behavior is compatible with the estimate above using Werner's formula if and only if c W ≥ 1. This implies the proposition. 7.3. Some Ideas. The conjectural Laplace transform (7.2) satisfies the ODE Thus we need to show that (λlog(a) 2 − β 0 )a λ dν 0 (σ) = 0, λ > 0 for some constant β 0 . Roughly speaking, we are trying to calculate the second moment for the distribution of H = −log(a). To calculate the second moment for a standard normal complex variable, one can apply ∂∂ to exp(−|z| 2 /2) and use infinitesimal invariance of the background Lebesgue measure; our strategy is to do the same with the stress tensor T (t) in place of ∂, a λ in place of the Gaussian, and Werner's measure in place of Lebesgue measure.
(b) This follows in a similar way, using the fact that L m kills P n (u 1 , .., u n ).
(c) and (d) are proven in a similar way, and will not be used.
Recall that (this is a holomorphic quadratic differential which is well-defined in U + ) and (this is a holomorphic quadratic differential which is well-defined in U − ; note that The fact that these two quadratic differentials do not have a common domain, or at the very best, are possibly defined on the rough loop γ, is a crucial point.
Proof. This follows from the Lemma and infinitesimal conformal invariance.
The basic question now is whether there is a constant β 0 such that λlog(a) 2 − β 0 is a limit, in an appropriate measure theoretic sense relative to ν 0 , of linear combinations of the functions λP n (φ + )P n (φ − ) − 2n, as n varies.
Question. Do there exist constants c n such that N n=1 c n P n (u)P n (l)a −n → log(a) 2 as N → ∞ in some measure-theoretic sense relative to ν 0 ? This is definitely false for all σ. To see this, suppose that Thus for this particular u P n (u) = (m + 1)w m N = −P n (l) * , n = mN and zero otherwise. Also If we actually have an identity, then for each N = 1, 2, ..
If we set x = |w N | 2 , then this is equivalent to This is clearly impossible: we cannot consistently solve for the constants. Furthermore the radius of convergence for the LHS is 1, and the radius of convergence for the RHS is 1 2 . A more promising approach seems to be to use the stress-energy tensor. Here is one heuristic calculation: There is a canonical trivialization of the determinant line bundle in genus zero, so that the conjectured KS measure can be viewed as a scalar measure which is invariant with respect to global conformal transformations; see Section 2.5 of [10]. We denote this measure restricted to Loop 1 (C \ {0}) by µ c ; properly normalized, this is the Werner measure when c = 0. d(W * µ)(σ, ρ ∞ ) = dν c (σ) × dρ ∞ ρ ∞ (c) The measure dν c (σ) is inversion invariant and invariant with respect to conjugation by C : z → z * .
(d) The measure dν c (σ) is supported on σ having triangular factorization σ = lau, i.e. m = 1. This is a rigorous lemma (contingent on the existence of µ c ), because the various statements use only global conformal invariance of µ c .
There is a natural conjecture for the diagonal distribution.
Conjecture. The ν c distribution for H = −log(a) is the inverse gamma distribution with parameters α = 1 − c and some β c > 0 (possibly proportional to h + (c), the larger value of two values of the conformal anomaly h corresponding to c < 1).
In other words we are conjecturing that a λ dν c (σ) = 2(β c λ) where K α is a modified Bessel function. This function of λ satisfies the differential equation λf ′′ (λ) + cf ′ (λ) − β c f (λ) = 0 This differential equation obviously makes sense for values of the parameters which are not necessarily positive. But for example if c = 1, i.e. α = 0, then the particular solution we are considering, K 0 , is not finite at λ = 0, so that the probabilistic interpretation is lost (This is obvious by noting that the pdf is not integrable at ∞ when α = 0). In terms of our conjecture this means that when c = 1, the σ distribution for the conjectured Kontsevich-Suhov measure is not finite, according to us.
To motivate this, in a heuristic way, we imagine that µ c is absolutely continuous with respect to Werner's measure µ 0 : µ c = δ c dµ 0 . We then apply infinitesimal invariance in the following way. Suppose that n > 0. Then L n L −n (a λ )δ c δ(ρ 0 = 1)dµ 0 = ( L n L −n (a λ ) δ c + L −n (a λ )L n (δ c ))δ(ρ 0 = 1)dµ 0 = ( λ 2 P n (u)P n (l)a −n − 2nλ + λP n (l)ρ n ∞ cQ n (u, l))δ(ρ 0 = 1)a λ δ c dµ c where we have tentatively written L n (δ c ) = cQ n (u, l)δ c (This should rigorously be expressed in terms of divergences, as proposed in section 2.5.2 of [10]). From this, by dividing by λ, we can deduce that (λP n (u)P n (l)a −n + cP n (l)Q n (u, l)ρ n ∞ − 2n)a λ dν c = 0 Now we would have to take linear combinations and limits, to log(a) 2 from the first term, log(a) from the second term (involving c), and a constant β c from the third term.

Appendix. The Vietoris Topology
Suppose that S is a topological space. The Vietoris topology on Comp(S) has a base consisting of sets of the form {K ∈ Comp(S) : K ⊂ U, K ∩ U i = φ, i = 1, .., n} where U, U 1 , .., U n are open subsets of S. Given K 0 ∈ Comp(S), suppose we tightly cover K 0 with open sets U i , 1 ≤ i ≤ n, and let U = ∪ i U i . Then "K is close to K 0 " means that (i) K ⊂ U , so every point in K is close to a point in K 0 , and (ii) for each point x 0 ∈ K 0 , x 0 ∈ U i , for some i, hence K ∩ U i = φ implies x 0 is close to some point in K. If S is metrizable, with metric d, then the Vietoris topology is compatible with the associated Hausdorff metric topology on Comp(S), where the Hausdorff metric is δ(K 1 , K 2 ) = max{ sup p1∈K1 (d(p 1 , K 2 )), sup p2∈K2 (d(K 1 , p 2 ))} For most topological properties τ , "S is τ " if and only if "Comp(S) is τ " (see section 4 of [12]). In particular if S is second countable and locally compact, then Comp(S) is second countable and locally compact.
Suppose that S is a Riemann surface with a fixed compatible complete metric. The associated Hausdorff metric on Loop(S) is obviously not complete, since for example a small circle can pinch down to a point. Does there exist a complete separable metric on Loop(S) compatible with the Vietoris topology?