The Group of Quasisymmetric Homeomorphisms of the Circle and Quantization of the Universal Teichm\"uller Space

In the first part of the paper we describe the complex geometry of the universal Teichm\"uller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo M\"obius transformations may be treated as a smooth part of $\mathcal T$. In the second part we consider the quantization of universal Teichm\"uller space $\mathcal T$. We explain first how to quantize the smooth part $\mathcal S$ by embedding it into a Hilbert-Schmidt Siegel disc. This quantization method, however, does not apply to the whole universal Teichm\"uller space $\mathcal T$, for its quantization we use an approach, due to Connes.


Introduction
The universal Teichmüller space T , introduced by Ahlfors and Bers, plays a key role in the theory of quasiconformal maps and Riemann surfaces. It can be defined as the space of quasisymmetric homeomorphisms of the unit circle S 1 (i.e. homeomorphisms of S 1 , extending to quasiconformal maps of the unit disc ∆) modulo Möbius transformations. The space T has a natural complex structure, induced by its realization as an open subset in the complex Banach space B 2 (∆) of holomorphic quadratic differentials in the unit disc ∆. The space T contains all classical Teichmüller spaces T (G), where G is a Fuchsian group, as complex submanifolds. The space S := Diff + (S 1 )/Möb(S 1 ) of normalized diffeomorphisms of the circle may be considered as a "smooth" part of T .
Our motivation to study T comes from the string theory. Physicists have noticed (cf. [15,3]) that the space Ω d := C ∞ 0 (S 1 , R d ) of smooth loops in the d-dimensional vector space R d may be identified with the phase space of bosonic closed string theory. By looking at a natural symplectic form ω on Ω d , induced by the standard symplectic form (of type "dp ∧ dq") on the phase space, one sees that this form can be, in fact, extended to the Sobolev completion of Ω d , coinciding with the space V d := H 1/2 0 (S 1 , R d ) of half-differentiable vector-functions. Moreover, the latter space is the largest in the scale of Sobolev spaces H s 0 (S 1 , R d ), on which ω is correctly defined. So the form ω itself chooses the "right" space to be defined on. From that point of view, it seems more natural to consider V d as the phase space of bosonic string theory, rather than Ω d . In this paper we set d = 1 to simplify the formulas and study the space V := V 1 = H 1/2 0 (S 1 , R). ⋆ This paper is a contribution to the Special Issue on Kac-Moody Algebras and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/Kac-Moody algebras.html According to Nag-Sullivan [12], there is a natural group, attached to the space V =H 1/2 0 (S 1 ,R), and this is precisely the group QS(S 1 ) of quasisymmetric homeomorphisms of the circle. Again one can say that the space V itself chooses the "right" group to be acted on. The group QS(S 1 ) acts on V by reparametrization of loops and this action is symplectic with respect to the form ω. The universal Teichmüller space T = QS(S 1 )/Möb(S 1 ) can be identified by this action with a space of complex structures on V , compatible with ω.
The second half of the paper is devoted to the quantization of the universal Teichmüller space T . We start from the Dirac quantization of the smooth part S = Diff + (S 1 )/Möb(S 1 ). This is achieved by embedding of S into the Hilbert-Schmidt Siegel disc D HS . Under this embedding the diffeomorphism group Diff + (S 1 ) is realized as a subgroup of the Hilbert-Schmidt symplectic group Sp HS (V ), acting on the Siegel disc by operator fractional-linear transformations. There is a holomorphic Fock bundle F over D HS , provided with a projective action of Sp HS (V ), covering its action on D HS . The infinitesimal version of this action is a projective representation of the Hilbert-Schmidt symplectic Lie algebra sp HS (V ) in a fibre F 0 of the Fock bundle F. This defines the Dirac quantization of the Siegel disc D HS . Its restriction to S gives a projective representation of the Lie algebra Vect(S 1 ) of the group Diff + (S 1 ) in the Fock space F 0 , which defines the Dirac quantization of the space S.
However, the described quantization procedure does not apply to the whole universal Teichmüller space T . By this reason we choose another approach to this problem, based on Connes quantization. (We are grateful to Alain Connes for drawing our attention to this approach, presented in [5].) Briefly, the idea is the following. The QS(S 1 )-action on T , mentioned above, cannot be differentiated in classical sense (in particular, there is no Lie algebra, associated to QS(S 1 )). However, one can define a quantized infinitesimal version of this action by associating with any quasisymmetric homeomorphism f ∈ QS(S 1 ) a quantum differential d q f , being an integral operator on V with kernel, given essentially by the finite-difference derivative of f . In these terms the quantization of T is given by a representation of the algebra of derivations of V , generated by quantum differentials d q f , in the Fock space F 0 .
I. Universal Teichmüller space 2 Group of quasisymmetric homeomorphisms of S 1 2.1 Definition of quasisymmetric homeomorphisms Definition 1. A homeomorphism h : S 1 → S 1 is called quasisymmetric if it can be extended to a quasiconformal homeomorphism w of the unit disc ∆.
Recall that a homeomorphism w : ∆ → w(∆), having locally L 1 -integrable derivatives (in generalized sense), is called quasiconformal if there exists a measurable complex-valued function µ ∈ L ∞ (∆) with µ ∞ := ess sup z∈∆ |µ(z)| =: k < 1 such that the following Beltrami equation holds for almost all z ∈ ∆. The function µ is called a Beltrami differential or Beltrami potential of w and the constant k is often indicated in the name of the k-quasiconformal maps.
In the case when k = 0 the homeomorphism w, satisfying (1), coincides with a conformal map from D onto w(D). For a diffeomorphism w its quasiconformality means that w transforms infinitesimal circles into infinitesimal ellipses, whose eccentricities (the ratio of the large axis to the small one) are bounded by a common constant K < ∞, related to the above constant k = µ ∞ by the formula Universal Teichmüller Space 3 The least possible constant K is called the maximal dilatation of w and is also sometimes indicated in the name of K-quasiconformal maps. The inverse of a quasiconformal map is again quasiconformal and the same is true for the composition of quasiconformal maps. This implies that orientation-preserving quasisymmetric homeomorphisms of S 1 form a group of quasisymmetric homeomorphisms of the circle QS(S 1 ) with respect to composition.
Any orientation-preserving diffeomorphism h ∈ Diff + (S 1 ) extends to a diffeomorphism of the closed unit disc ∆, which is evidently quasiconformal, according to the above criterion. So Diff + (S 1 ) ⊂ QS(S 1 ), and we have the following chain of embeddings Here, Möb(S 1 ) denotes the Möbius group of fractional-linear automorphisms of the unit disc ∆, restricted to S 1 .

Beurling-Ahlfors criterion
There is an intrinsic description of quasisymmetric homeomorphisms of S 1 in terms of cross ratios. Recall that the cross ratio of four different points z 1 , z 2 , z 3 , z 4 on the complex plane is given by the quantity The equality of two cross ratios ρ(z 1 , z 2 , z 3 , z 4 ) = ρ(ζ 1 , ζ 2 , ζ 3 , ζ 4 ) is a necessary and sufficient condition for the existence of a fractional-linear map of the complex plane, transforming the quadruple z 1 , z 2 , z 3 , z 4 into the quadruple ζ 1 , ζ 2 , ζ 3 , ζ 4 . In the case of quasiconformal maps the cross ratios of quadruples may change but in a controlled way. This property, reformulated in the right way for orientation-preserving homeomorphisms of S 1 , yields a criterion of quasisymmetricity, due to Ahlfors and Beurling. The required property reads as follows: for an orientation-preserving homeomorphism h : S 1 → S 1 it should exist a constant 0 < ǫ < 1 such that the following inequality holds for any quadruple z 1 , z 2 , z 3 , z 4 ∈ S 1 with cross ratio ρ(z 1 , z 2 , z 3 , z 4 ) = 1 2 . Theorem 1 (Beurling-Ahlfors, cf. [1,9]). Suppose that h : S 1 → S 1 is an orientation-preserving homeomorphism of S 1 . Then it can be extended to a quasiconformal homeomorphism w : ∆ → ∆ if and only if it satisfies condition (2). Douady and Earle (cf. [6]) have found an explicit extension operator E, assigning to a quasisymmetric homeomorphism h its extension to a quasiconformal homeomorphism w of ∆, which is conformally invariant in the sense that g(w • h) = w • g(h) for any fractional-linear automorphism of ∆.
Though quasisymmetric homeomorphisms of S 1 , in general, are not smooth, they enjoy certain Hölder continuity, provided by the following Theorem 2 (Mori, cf. [1]). Let w : ∆ → ∆ be a K-quasiconformal homeomorphism of the unit disc onto itself, normalized by the condition: w(0) = 0. Then the following sharp estimate |w(z 1 ) − w(z 2 )| < 16|z 1 − z 2 | 1/K holds for any z 1 = z 2 ∈ ∆. In other words, the homeomorphism w satisfies the Hölder condition of order 1/K in the disc ∆. is called the universal Teichmüller space. It can be identified with the space of normalized quasisymmetric homeomorphisms of S 1 , fixing the points ±1 and −i.
As we have pointed out earlier, there is an inclusion We consider the homogeneous space as a "smooth" part of T .
The space T can be provided with the Teichmüller distance function, defined by for any quasisymmetric homeomorphisms g, h ∈ T , extended to quasiconformal homeomorphisms of the disc ∆. Here, K(h • g −1 ) denotes the maximal dilatation of the quasiconformal map h • g −1 . This definition does not depend on the extensions of g, h to ∆ and defines a metric on T . The universal Teichmüller space is a complete connected contractible metric space with respect to the introduced distance function (cf. [9]). Unfortunately, this metric is not compatible with the group structure on T , given by composition of quasisymmetric homeomorphisms (cf. [9, Theorem 3.3]). The term "universal" in the name of the universal Teichmüller space is due to the fact that T contains, as complex submanifolds, all classical Teichmüller spaces T (G), where G is a Fuchsian group (cf. [10]). If a Riemann surface X is uniformized by the unit disc ∆, so that X = ∆/G, then the corresponding Techmüller space T (G) may be identified with the quotient where QS(S 1 ) G is the subset of G-invariant quasisymmetric homeomorphisms in QS(S 1 ). The universal Teichmüller space T itself corresponds to the Fuchsian group G = {1}.
Since quasisymmetric homeomorphisms of S 1 are defined in terms of quasiconformal maps of ∆, i.e. in terms of solutions of Beltrami equation in ∆, one can expect that there is a definition of T directly in terms of Beltrami differentials. Denote by B(∆) the set of Beltrami differentials in the unit disc ∆. It follows from above that it can be identified (as a set) with the unit ball in the complex Banach space L ∞ (∆). Given a Beltrami differential µ ∈ B(∆), we can extend it to a Beltrami differentialμ on the extended complex plane C by settingμ equal to zero outside the unit disc ∆. Then, applying the existence theorem for quasiconformal maps on the extended complex plane C (cf. [1]), we get a normalized quasiconformal homeomorphism w µ , satisfying Beltrami equation (1) on C with potentialμ. This homeomorphism is conformal on the exterior ∆ − of the closed unit disc ∆ on C and fixes the points ±1, −i. The image ∆ µ := w µ (∆) of ∆ under the quasiconformal map w µ is called a quasidisc. We associate with Beltrami differential µ ∈ B(∆) the normalized quasidisc ∆ µ . Introduce an equivalence relation between Beltrami differentials in ∆ by saying that two Beltrami differentials µ and ν are equivalent if w µ | ∆ − ≡ w ν | ∆ − . Then the universal Teichmüller space T will coincide with the quotient T = B(∆)/∼ of the space B(∆) of Beltrami differentials modulo introduced equivalence relation. In other words, it coincides with the space of normalized quasidiscs in C.

Complex structure of the universal Teichmüller space
We introduce a complex structure on the universal Teichmüller space T , using its embedding into the space of quadratic differentials.
Given an arbitrary point [µ] of T , represented by a normalized quasidisc w µ (∆), consider a map assigning to a Beltrami differential µ ∈ [µ] the Schwarz derivative of the conformal map w µ on ∆. Due to the invariance of Schwarzian under Möbius transformations, the image of µ under the above map depends only on the class [µ] of µ in T . Moreover, it is a holomorphic quadratic differentials in ∆ − . The latter fact follows from the transformation properties of Beltrami differentials, prescribed by Beltrami equation (according to (1), Beltrami differential behaves as a (−1, 1)-differential with respect to conformal changes of variable). Composing the above map with a fractional-linear biholomorphism of ∆ − onto the unit disc ∆, we obtain a map associating a holomorphic quadratic differential ψ(µ) in ∆ with a point [µ] of the universal Teichmüller space T . The space B 2 (∆) of holomorphic quadratic differentials in ∆ is a complex Banach space, provided with a natural hyperbolic norm, given by for a quadratic differential ψ. It can be proved (cf. [9]) that ψ[µ] 2 ≤ 6 for any Beltrami differential µ ∈ B(∆).
The constructed map Ψ : T → B 2 (∆), called a Bers embedding, is a homeomorphism of T onto an open bounded connected contractible subset in B 2 (∆), containing the ball of radius 1/2, centered at the origin (cf. [9]).
Using the constructed embedding, we can introduce a complex structure on the universal Teichmüller space T by pulling it back from the complex Banach space B 2 (∆). It provides T with the structure of a complex Banach manifold. (Note that the topology on T , induced by the map Ψ, is equivalent to the one, determined by the Teichmüller distance function.) Moreover, the composition of the natural projection with the constructed map Ψ yields a holomorphic map with respect to the natural complex structure on B(∆) (cf. [10]).
II. QS-action on the Sobolev space of half-differentiable functions 4 Sobolev space of half-differentiable functions on S 1

Definition
The Sobolev space of half-differentiable functions on S 1 is a Hilbert space , consisting of functions f ∈ L 2 (S 1 , R) with zero average over the circle, having generalized derivatives of order 1/2 again in L 2 (S 1 , R). In terms of Fourier series, a function f ∈ L 2 (S 1 , R) with Fourier series if and only if it has a finite Sobolev norm of order 1/2: The space H 1/2 0 (S 1 , R) is well known and widely used in classical function theory (cf. [18]). However, our motivation to employ this space comes from its relation to string theory (cf. below).

Kähler structure
Because of (3), this form is correctly defined on V . Moreover, H 1/2 0 (S 1 , R) is the largest Hilbert space in the scale of Sobolev spaces H s 0 (S 1 , R), s ∈ R, on which this form is defined. It should be also underlined that the form ω is the only natural symplectic form on V (we shall make this point clear in Section 5.1).
We return to our motivation for studying the space V . It is well known to physicists (cf., e.g., [15,3]) that the space Ω d = C ∞ 0 (S 1 , R d ) of smooth loops in the d-dimensional vector space R d can be identified with the phase space of bosonic closed string theory. The space Ω d has a natural symplectic form, which coincides with the image of the standard symplectic form (of type "dp ∧ dq") on the phase space of closed string theory under the above identification. This form, computed in terms of Fourier decompositions, coincides precisely with the form ω, given by (4). As we have remarked, the latter form may be extended to the Sobolev space V d := H 1/2 0 (S 1 , R d ) and this space is the largest in the scale H s 0 (S 1 , R d ) of Sobolev spaces, on which ω is correctly defined. One can say that symplectic form ω "chooses" the Sobolev space V d . This is in contrast to Ω d , which was taken for the phase space of string theory simply because it's easier to work with smooth loops. By this reason, we find it more natural to consider V d as the phase space of string theory, which motivates the study of V d in more detail. In our analysis we set d = 1 for simplicity.
Apart from symplectic form, the Sobolev space V has a complex structure J 0 , which can be given in terms of Fourier decompositions by the formula This complex structure is compatible with symplectic form ω and, in particular, defines a Kähler metric g 0 on V by g 0 (ξ, η) := ω(ξ, J 0 η) or, in terms of Fourier decompositions, In other words, V has the structure of a Kähler Hilbert space.
The complexification Hilbert space and the Kähler metric g 0 on V extends to a Hermitian inner product on V C , given by We extend the symplectic form ω and complex structure operator J 0 complex linearly to V C .
The space V C is decomposed into the direct sum of the form where W ± is the (∓i)-eigenspace of the operator J 0 ∈ End V C . In other words, The subspaces W ± are isotropic with respect to symplectic form ω and the splitting V C = W + ⊕ W − is an orthogonal direct sum with respect to the Hermitian inner product ·, · , given by (5).

QS-action on the Sobolev space
Note that any homeomorphism h of S 1 , preserving the orientation, acts on L 2 0 (S 1 , R) by change of variable. In other words, there is an operator T h : This operator has the following remarkable property.
Proposition 2 (Nag-Sullivan [12]). For any h ∈ QS(S 1 ) we have for all ξ, η ∈ V . Moreover, the complex-linear extension of QS-action to the complexification V C preserves the holomorphic subspace W + if and only if h ∈ Möb(S 1 ). In the latter case, T h acts as a unitary operator on W + .
We have pointed out in Section 4.2 that the Sobolev space V is "chosen" by the symplectic form ω. In the same way, one can say that the space V chooses the reparametrization group QS(S 1 ). Indeed, this is the biggest reparametrization group, leaving V invariant, according to Proposition 1. On the other hand, it is a group of "canonical transformations", preserving the symplectic form ω, according to Proposition 2. So we have a natural phase space (V, ω) together with a natural group QS(S 1 ) of its canonical transformations.
Here is an assertion, making clear in what sense ω is a unique natural symplectic form on V .

Embedding of the universal Teichmüller space into an infinite-dimensional Siegel disc
The Propositions 1 and 2 imply that quasisymmetric homeomorphisms act on the Hilbert space V by bounded symplectic operators. Hence, we have a map Here, Sp(V ) is the symplectic group of V , consisting of linear bounded symplectic operators on V , and U(W + ) is its subgroup, consisting of unitary operators (i.e. the operators, whose complex-linear extensions to V C preserve the subspace W + ).
In terms of the decomposition Such an operator belongs to symplectic group Sp(V ), if it has the form with components, satisfying the relations The space standing on the right hand side of (6), can be regarded as an infinite-dimensional analogue of the Siegel disc, since it may be identified with the space of complex structures on V , compatible with ω. Indeed, any such structure J determines a decomposition of V C into the direct sum of subspaces, isotropic with respect to ω. This decomposition is orthogonal with respect to the Kähler metric g J on V C , determined by J and ω. The subspaces W and W are identified with the (−i)-and (+i)-eigenspaces of the operator J on V C respectively. Conversely, any decomposition (7) of the space V C into the direct sum of isotropic subspaces determines a complex structure J on V C , which is equal to −iI on W and +iI on W and is compatible with ω. This argument shows that symplectic group Sp(V ) acts transitively on the space J (V ) of complex structures J on V , compatible with ω. Moreover, a complex structure J, obtained from a reference complex structure J 0 by the action of an element A of Sp(V ), is equivalent to J 0 if and only if A ∈ U(W + ). Hence, The space on the right can be, in its turn, identified with the Siegel disc D, defined as the set We identify, as above, the right hand side with a subspace J HS (V ) of the space J (V ) of compatible complex structures on V . We call complex structures J ∈ J HS (V ) Hilbert-Schmidt.
As before, the space J HS (V ) of Hilbert-Schmidt complex structures on V can be realized as a Hilbert-Schmidt Siegel disc D HS = {Z : W + → W − is a symmetric Hilbert-Schmidt operator withZZ < I}.

III. Quantization of S
6 Statement of the problem

Dirac quantization
We start by recalling a general definition of quantization of finite-dimensional classical systems, due to Dirac. A classical system is given by a pair (M, A), where M is the phase space and A is the algebra of observables. The phase space M is a smooth symplectic manifold of even dimension 2n, provided with a symplectic 2-form ω. Locally, it is equivalent to the standard model, given by symplectic vector space M 0 := R 2n together with standard symplectic form ω 0 , given in canonical coordinates (p i , q i ), i = 1, . . . , n, on R 2n by The algebra of observables A is a Lie subalgebra of the Lie algebra C ∞ (M, R) of smooth real-valued functions on the phase space M , provided with the Poisson bracket, determined by symplectic 2-form ω. In particular, in the case of standard model M 0 = (R 2n , ω 0 ) one can take for A the Heisenberg algebra heis(R 2n ), which is the Lie algebra, generated by coordinate functions p i , q i , i = 1, . . . , n, and 1, satisfying the commutation relations for any f, g ∈ A. We also assume the following normalization condition: r(1) = I.
For complexified algebras of observables A C or, more generally, complex involutive Lie algebras of observables (i.e. Lie algebras with conjugation) their Dirac quantization is given by an irreducible Lie-algebra representation satisfying the normalization condition and the conjugation law: r(f ) = r(f ) * for any f ∈ A.
We are going to apply this definition of quantization to infinite-dimensional classical systems, in which both the phase space and algebra of observables are infinite-dimensional. For infinitedimensional algebras of observables it is more natural to look for their projective Lie-algebra representations. The above definition of quantization will apply also to this case if one replaces the original algebra of observables with its suitable central extension.

Statement of the problem
We start from the Dirac quantization of an infinite-dimensional system (V, A) with the phase space, given by the Sobolev space of half-differentiable functions V := H The choice of the introduced Lie algebra A for the algebra of observables is motivated by the following physical considerations. As we have pointed put, the space V d is a natural Sobolev completion of the space Ω d := C ∞ 0 (S 1 , R d ) of smooth loops in R d . In the same way, the Lie algebra A = heis(V ) ⋊ sp HS (V ) is a natural extension of the Lie algebra heis(Ω d ) ⋊ Vect(S 1 ), where Vect(S 1 ) is the Lie algebra of the diffeomorphism group Diff + (S 1 ). The algebra heis(Ω d ) can be identified with the Lie algebra of coordinate functions on Ω d , while the algebra Vect(S 1 ) is generated by certain quadratic functions on Ω d (cf. [3]). One can say that the Lie algebra heis(Ω d ) ⋊ Vect(S 1 ) is an infinite-dimensional analogue of the Poincarè algebra of the d-

Heisenberg representation
In this Section we recall the well known Heisenberg representation of the first component heis(V ) of algebra of observables A. A detailed exposition of this subject may be found in [13,8,2].

Fock space
Fix an admissible complex structure J ∈ J (V ). It defines a polarization of V , i.e. a decomposition of V C into the direct sum where W (resp. W ) is the (−i)-eigenspace (resp. (+i)-eigenspace) of the complex structure operator J. The splitting (8) is the orthogonal direct sum with respect to the Hermitian inner product z, w J := ω(z, Jw), determined by J and sympletic form ω. The Fock space F (V C , J) is the completion of the algebra of symmetric polynomials on W with respect to a natural norm, generated by ·, · J . In more detail, denote by S(W ) the algebra of symmetric polynomials in variables z ∈ W and introduce an inner product on S(W ), defined in the following way. It is given on monomials of the same degree by the formula where the summation is taken over all permutations {i 1 , . . . , i n } of the set {1, . . . , n} (the inner product of monomials of different degrees is set to zero), and extended to the whole algebra S(W ) by linearity. The completion S(W ) of S(W ) with respect to the introduced norm is called the Fock space of V C with respect to complex structure J: If {w n }, n = 1, 2, . . . , is an orthonormal basis of W , then an orthonormal basis of F J can be given by the family of polynomials where K = (k 1 , . . . , k n , 0, . . . ), k i ∈ N ∪ 0, and k! = k 1 ! · · · · · k n !.

Heisenberg representation
There is an irreducible representation of the Heisenberg algebra heis(V ) in the Fock space where ∂ v is the derivative in direction of v ∈ V . Extending r J to the complexified algebra heis C (V ), we obtain for v =z ∈ W and for z ∈ W . We set also r J (c) := λ · I for the central element c ∈ heis(V ), where λ is an arbitrary fixed non-zero constant. Introduce the creation and annihilation operators on In particular, for z ∈ W a * J (z)f (w) = w, z J f (w), For an orthonormal basis {w n } of W , we define the operators a * n := a * (w n ), a n := a(w n ), n = 1, 2, . . . , if a n f J = 0 for n = 1, 2, . . . . In other words, it is a non-zero vector, annihilated by operators a n . It is uniquely defined by r J (up to a multiplicative constant) and in the case of the initial Fock space F 0 = F (V, J 0 ) we set f 0 ≡ 1. Acting on vacuum f J by creation operators a * n , we can define the action of representation r J on any polynomial, which implies the irreducubility of r J .
So we have the following Proposition 6 (cf. [13,8,2]). There is an irredicible Lie algebra representation We shall see in the next Section that this representation is essentially unique.
8 Symplectic group action on the Fock bundle

Shale theorem
To construct an irreducible representation of the second component sp HS (V ) of the algebra of observables A, we study an action of the Hilbert-Schmidt symplectic group Sp HS (V ) on the Fock spaces F J . This action is provided by the following theorem.
This theorem was proved by Shale [17] in 1962, an independent proof was given in Berezin's book [2], published in Russian in 1965 (Berezin obtained also an explicit formula for the intertwining operator U J ).
The following Proposition gives a description of U J in terms of the Hilbert-Schmidt Siegel disc D HS , based on the identification of J HS (V ) with D HS .
Proposition 7 (Segal [16]). There is a projective unitary action of the group Sp HS (V ) on Fock spaces, defined by the unitary operator U J , given by the formula (11) below.
Here is an idea of Segal's construction, details may be found in [16]. Given an admissible complex structure J ∈ J HS (V ), we identify it with a point Z in the Siegel disc D HS . Regarding Z as an element of the symmetric square S 2 (W ), we can associate with it an element e Z/2 of S(W ). The inner product of two such elements has a simple expression The normalized elements play the role of coherent states (cf., e.g., [2]). In terms of these states the action of the group Sp HS (V ) on Fock spaces, defined by is given by the formula where µ : C * → S 1 is the radial projection.

Dirac quantization of V and S
We can unite Fock spaces F J into a Fock bundle over D HS , having the following properties. The proof of holomorphicity of the Fock bundle F → D HS is analogous to the proof of holomorphicity of the determinant bundle over the Hilbert-Schmidt Grassmannian, given in [13]. Note that the Fock bundle is trivial, since the Siegel disc D HS is contractible (even convex), so the statement follows from the Hilbert space version of the Oka principle (cf. [4]). An explicit trivialization of F → D HS is provided by the action (11). This action defines a projective unitary action of the group Sp HS (V ) on F, covering the Sp HS (V )-action on Siegel disc D HS .
The infinitesimal version of this action yields a projective representation of the symplectic algebra sp HS (V ) in the Fock space F 0 . We present an explicit description of this representation, due to Segal.
Recall that symplectic algebra sp HS (V ) is the Lie algebra of symplectic Hilbert-Schmidt group Sp HS (V ), which consists of linear operators A in V C , having the following block representations Here, α is a bounded skew-Hermitian operator and β is a symmetric Hilbert-Schmidt operator on F 0 . The complexified Lie algebra sp HS (V ) C consists of operators of the form where α is a bounded operator, while β andγ are symmetric Hilbert-Schmidt operators on F 0 . The projective representation of complexified symplectic algebra sp HS (V ) C is given by the formula Here, D α is the derivation of F 0 in α-direction, defined by The operator M β is the multiplication operator on F 0 , defined by and the operator M * γ is the adjoint of M γ : M * γ f (w) = γ∂ w , ∂w f (w). This is a projective representation with cocycle intertwined with the Heisenberg representation r 0 of heis(V ) in F 0 . Thus we have the following Proposition 9 (Segal [16]). There is a projective unitary representation given by formula (12) with cocycle (13). This representation intertwines with the Heisenberg representation r 0 of heis(V ) in F 0 .
The Heisenberg representation r 0 in the Fock space F 0 , described in Proposition 6, and symplectic representation ρ, constructed in Proposition 9, define together Dirac quantization of the system (V, A), where A is the central extension of A, determined by (13).
The constructed Lie-algebra representation of sp HS (V ) in the Fock space F 0 may be also considered as Dirac quantization of a classical system, consisting of the phase space D HS = Sp HS (V )/U(W + ) and the algebra of observables, given by the central extension of Lie algebra sp HS (V ).
The restriction of this construction to the smooth part S = Diff + (S 1 )/Möb(S 1 ) of the universal Teichmüller space T = QS(S 1 )/Möb(S 1 ) yields the Dirac quantization of S. Namely, we have the following The Diff + (S 1 )-action on the Fock bundle, mentioned in Proposition, was explicitly constructed in [7]. The infinitesimal version of this action yields a unitary projective representation of the Lie algebra Vect(S 1 ) in the Fock space F 0 . We can consider this construction as Dirac quantization of the phase space S, provided with the algebra of observables, given by the central extension of the Lie algebra Vect(S 1 ), called the Virasoro algebra.

IV. Quantization of T 9 Dirac versus Connes quantization
Unfortunately, the method, used in previous Chapter for the quantization of S, does not apply to the whole space T . Though we still can embed T into the Siegel disc D, we are not able to construct a projective action of symplectic group Sp(V ) on Fock spaces. According to theorem of Shale, it is possible only for the Hilbert-Schmidt subgroup Sp HS (V ) of Sp(V ). So one should look for another way of quantizing the universal Teichmüller space T . We are going to use for that the "quantized calculus" of Connes and Sullivan, presented in Chapter IV of the Connes' book [5] and [12].
Recall that in Dirac's approach we quantize a classical system (M, A), consisting of the phase space M and the algebra of observables A, which is a Lie algebra, consisting of smooth functions on M . The quantization of this system is given by a representation r of A in a Hilbert space H, sending the Poisson bracket {f, g} of functions f, g ∈ A into the commutator 1 i [r(f ), r(g)] of the corresponding operators. In Connes' approach the algebra of observables A is an associative involutive algebra, provided with an exterior differential d. Its quantization is, by definition, a representation π of A in a Hilbert space H, sending the differential df of a function f ∈ A into the commutator [S, π(f )] of the operator π(f ) with a self-adjoint symmetry operator S with S 2 = I. The differential here is understood in the sense of non-commutative geometry, i.e. as a linear map d : A → Ω 1 (A), satisfying the Leibnitz rule (cf. [5]).
In the following Reformulating the notion of Connes quantization of algebra of observables A, one can say that it is a representation of the algebra Der(A) of derivations of A in the Lie algebra End H. Recall that a derivation of an algebra A is a linear map: A → A, satisfying the Leibnitz rule. Clearly, derivations of an algebra A form a Lie algebra, since the commutator of two derivations is again a derivation.
If all observables are smooth real-valued functions on M , the two approaches are equivalent to each other. Indeed, the differential df of a smooth function f is symplectically dual to the Hamiltonian vector field X f and this establishes a relation between the associative algebra In the case when the algebra of observables A contains non-smooth functions, its Dirac quantization is not defined in the classical sense. In Connes approach the differential df of a non-smooth observable f ∈ A is also not defined classically, but its quantum counterpart d q f , given by may still be defined, as it is demonstrated by the following example, borrowed from [5].
Suppose that A is the algebra L ∞ (S 1 , C) of bounded functions on the circle S 1 . Any function f ∈ A defines a bounded multiplication operator in the Hilbert space H = L 2 (S 1 , C): The operator S is given by the Hilbert transform S : L 2 (S 1 , C) → L 2 (S 1 , C): where the integral is taken in the principal value sense and K(ϕ, ψ) is the Hilbert kernel The differential df of a general observable f ∈ A is not defined in the classical sense, but its quantum analogue is correctly defined as an operator in H for functions f ∈ V . Namely, we have the following This inequality is equivalent to the condition f ∈ V (cf. [12]). The quantum differential d q f = [S, M f ] of a function f ∈ V is an integral operator on V , given by with the kernel, given by where K(ϕ, ψ) is defined by (14). Note that the quasiclassical limit of this operator, defined by taking the value of the kernel on the diagonal (i.e. by taking the limit for s → t), coincides (up to a constant) with the multiplication operator h → f ′ h, so the quantization means in this case essentially the replacement of the derivative by its finite-difference analogue. This finite-difference analogue is an integral operator, given by The correspondence between functions f ∈ A and operators M f on H has the following remarkable properties (cf. [14]): 1. The differential d q f is a finite rank operator if and only if f is a rational function. This list may be supplemented by further function-theoretic properties of elements of A, having curious operator-theoretic characterizations (cf. [5]).

Quantization of the universal Teichmüller space
We apply these ideas to the universal Teichmüller space T . In Section 5.1 we have defined a natural action of quasisymmetric homeomorphisms on V . As we have remarked, this action does not admit the differentiation, so classically there is no Lie algebra, associated with QS(S 1 ) or, in other words, there is no classical algebra of observables, associated to T . (The situation is similar to that in the example above.) We would like to define a quantum algebra of observables, associated to T . First of all, extend the QS(S 1 )-action on V to symmetry operators by setting for h ∈ QS(S 1 ). This action agrees with a natural action of QS(S 1 ) on the universal Teichmüller space T = QS(S 1 )/Möb(S 1 ), considered as a space of compatible complex structures on V . The quantized infinitesimal version of the action (16) is given by the integral operator d q h : V → V , equal to d q h = [S, δh] with δh given by (15). Let us recall the steps of the Dirac quantization of Sobolev space V : 1) we start from Sp HS (V )-action on V ; 2) extend it to Sp HS (V )-action on complex structure operators J; 3) this action generates a projective unitary action of Sp HS (V ) on Fock spaces F (V, J); 4) the infinitesimal version of this action yields a projective unitary representation of the Lie algebra sp HS (V ) in Fock space F 0 , described in Section 8.2.
In the case of T we have: 1) QS(S 1 )-action on V ; 2) this action extends to QS(S 1 )-action on symmetry operators S, given by h → S h .
However, compared to Dirac quantization of V , the next step in the quantization scheme is absent. Because of the Shale theorem, we cannot extend the QS(S 1 )-action on symmetry operators S to Fock spaces F (V, S). Also we cannot differentiate the QS(S 1 )-action on V . But we have a quantized infinitesimal version of h : S → S h , given by quantum differential d q h = [S, δh].
We extend this operator to F 0 by defining it first on the basis elements (9) of the Fock space F 0 with the help of Leibnitz rule, and then by linearity to all finite elements of F 0 . The completion of the obtained operator yields an operator d q h on F 0 . These extended operators d q h with h ∈ QS(S 1 ) generate a quantum derivation algebra Der q (QS), associated to T . This algebra should be considered as a quantum Lie algebra of observables, associated to T . So, instead of steps (3), (4) in the Dirac quantization of V , we construct directly a quantum Lie algebra of observables Der q (QS), corresponding to the non-existing classical Lie algebra of observables on T . Moreover, we can use the quantum Lie algebra Der q (QS) as a substitution of a classical Lie algebra of QS(S 1 ).
Conclusion. The Connes quantization of the universal Teichmüller space T consists of two stages: 1. The first stage ("first quantization") is a construction of quantized infinitesimal version of QS(S 1 )-action on V , given by quantum differentials d q h = [S, δh] with h ∈ QS(S 1 ).
2. The second step ("second quantization") is an extension of quantum differentials d q h to the Fock space F 0 . The extended operators d q h with h ∈ QS(S 1 ) generate the quantum algebra of observables Der q (QS), associated to T .
We note also that the correspondence principle for the constructed Connes quantization of T means that this quantization reduces to the Dirac quantization while restricted to S.