Berezin Quantization, Conformal Welding and the Bott–Virasoro Group

Following Nag–Sullivan, we study the representation of the group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Diff}^+(S^1)$$\end{document}Diff+(S1) of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift provides quantization of conformal welding, gives a new representative of the Bott–Virasoso cocycle class, and leads to a surprising identity for the Takhtajan–Teo energy functional on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Diff}^+(S^1)$$\end{document}Diff+(S1).


Introduction
Coadjoint orbits of the canonical central extension 1 → S 1 → Diff + (S 1 ) → Diff + (S 1 ) → 1 of the group G = Diff + (S 1 ) of orientation preserving diffeomorphisms of the circle (also called Virasoro coadjoint orbits) attracted attention both in mathematics and physics literature since long time, see e.g.[1,5,7,12,15].The coadjoint action on the hyperplane corresponding to the coordinate c (dual to Lie(S 1 ) ∼ = R) is defined on the space of quadratic differentials on the circle T (x)dx 2 , and it is given by formula where Sch(χ) is the Schwarzian derivative For c = 0, one of the Virasoro coadjoint orbits is of special importance.It corresponds to T (x) = c 24 dx 2 , and it is the unique orbit with the stabiliser isomorphic to the group PSL(2, R).This orbit (also called the Teichmüller orbit) naturally embeds in the universal Teichmüller space T (1) 1 : O Teich ∼ = Diff + (S 1 )/PSL(2, R) ⊂ QS(S 1 )/PSL(2, R), 1 Teichmüller spaces for curves of all finite genera naturally embed in T (1).where QS(S 1 ) is the group of quasi-conformal mappings of the circle.
Consider the space Hyp(D) of geodesically complete hyperbolic metrics on the unit disk D ⊂ C. A typical example in this class is the standard Poincaré metric.The group of orientation preserving diffeomorphisms Diff + (D) acts transitively on Hyp(D).Now consider the group Diff + (D, ∂D) ⊂ Diff + (D) which fixes the boundary of the disk ∂D ∼ = S 1 .It was argued in the physics literature (see [11]) that O Teich is symplectomorphic to the following moduli space: O Teich ∼ = Hyp(D)/Diff + (D, ∂D).
Formal Duistermaat-Heckman integrals over this space were defined and studied in [13,2].
Recall that for χ ∈ QS(S 1 ) there exist two univalent holomorphic functions for x ∈ R.Here D * is the unit disk centered at infinity.The functions f + (z) and f − (z) are called components of conformal welding of χ.The Kähler potential of the Weil-Petersson metric on O Teich is given by the Takhtajan-Teo (TT) energy functional (see [14] 2 ): In this paper, we focus our attention on the subgroup Diff + hol (S 1 ) ⊂ Diff + (S 1 ) which is characterized by the property that the map z = e ix → e iχ(x) extends to a holomorphic function on an annulus A r,R = {z ∈ C; r < |z| < R} for some r < 1 < R. For this subgroup, following Nag-Sullivan [10] we define a group homomorphism to the group of restricted symplectic transformations acting on the Hilbert space H = H + ⊕ H − of holomorphic functions (modulo constants): Diff + hol (S 1 ) → Sp res (H + ⊕ H − ).Here H + is spanned by z n for n ≥ 1, and H − by z n for n ≤ −1, and ||z n || = |n|.We then use the metaplectic representation of Sp res (H + ⊕ H − ) defined by Berezin formalism of normal symbols (see [4]) to construct operators (1) N χ = N f −1 + * N f − .Here * is the product of operators acting on the Fock space F defined by the polarization H = H + ⊕ H − .In this sense, equation (1) defines a quantization of conformal welding.
Our first main result is as follows: Theorem 1.1.For χ, φ ∈ Diff + hol (S 1 ), we have where C(χ, φ) ∈ C * is a multiplicative group 2-cocylce with the property that and f ± define a conformal welding of χ, and g ± define a conformal welding of φ.Furthermore, for χ Equation ( 2) is surprizing since the cocycle C(χ, φ) depends only on the components f − and g + !For a more precise statement of Theorem 1.1, see Theorem 5.4 and Theorem 5.10.
Our second main result is the following theorem (see also Theorem 5.12): Theorem 1.2.For χ ∈ Diff + hol (S 1 ), the operator U χ = e S(χ)/48π N χ is unitary, and the cocycle C(χ, φ) satisfies the equality The left hand side and the right hand side of (3) have rather different analytic forms.Equation (3) follows from Berezin formalism, but at this point we are not aware of its direct proof.
We believe that our findings admit extensions to other representations of the group of diffeomorphisms of the circle defined in terms of free fields.In particular, this applies to representations of affine Kac-Moody algebras on Wakimoto modules.We also believe that our results may find applications in Theoretical Physics.Our original motivation comes from the work [8] which introduced conformal welding in the study of Fermions in a gravitational field in 2 dimensions (see also the analysis of the gravitational Wess-Zumino functionals in [3]).
The structure of the paper is as follows: in Section 2, we recall the definition of the Bott-Virasoro 2-cocycle on Diff + (S 1 ), and we extend it to the groupoid of conformal maps.In Section 3, we explain how holomorphic maps define symplectic transformations on the space H, and we define their Grunsky coefficients.In Section 4, we set up the Berezin formalism for normal and unitary symbols of operators on the Fock space F .Finally, in Section 5 we describe quantization of conformal welding, the cocycles for normal and unitary symbols and their relation to the TT functional.award of the Simons Foundation to the Hamilton Mathematics Institute of the Trinity College Dublin under the program "Targeted Grants to Institutes".

Group cocycles
In this Section, we recall the notion of a group 2-cocycle.We then focus our attention on the Bott-Virasoro cocycle on the group of orientation preserving diffeomorphisms of the circle Diff + (S 1 ) and on its extension to holomorphic maps.
2.1.Group 2-cocycles: definition and basic properties.Let G be a group and K be the basic field (in this article, R or C) viewed as a trivial G-module.
for all f, g, h ∈ G.The definition implies c(e, g) = c(e, e) = c(g, e) for all g ∈ G.
Also, for all k ∈ K the assignment c(f, g) = k is a 2-cocycle.
For every map b : Note that for b(f which has the property c(e, e) = 0. We will use the following cyclic property of 2-cocycles: Proposition 2.1.Assume that a 2-cocycle c has the property c(f, f −1 ) = 0 for all f ∈ G.Then, where f gh = e.
Assume that the group G possesses a Lie algebra g = Lie(G), denote by exp : g → G the exponential map, and assume that finite products of the type exp(u 1 ) . . .exp(u m ) cover G.We define a map β : g × G → K by formula (6) β(u, g) = d dt c(exp(tu), g)| t=0 .
The condition c(g, e) = c(e, e) implies that β(u, e) = 0 for all u ∈ g.
Proof.Put f = exp(tu) in equation ( 4) and differentiate in t at t = 0. We obtain Hence, if β(u, g) = 0 then the normalized cocycle c(f, g) vanishes, as required. Define Recall that a ∈ ∧ 2 g * is a Lie algebra 2-cocycle, and that it satisfies the equation ( 7) Note that there is no analog of Proposition 2.2 which would allow to reconstruct maps β and c starting from the map a (or the map α).Indeed, adding a trivial cocycle δb such that d/dt b(exp(tu))| t=0 = 0 does not affect the maps α and a, but it changes β and c, in general.Let ρ : G → End(V ) be a projective representation, and assume that ρ(f )ρ(g) = e ic(f,g) ρ(f g), where c : G × G → C is a complex valued function.Then, c verifies the identity (4) modulo 2πZ.This is a direct consequence of associativity of the product in End(V ).If the group G is connected, and the function c is a continuous function, then it is actually a 2-cocycle.Indeed, in this case the defect in equation ( 4) is also a continuous function of f, g, h which vanishes for f = g = h = e.Hence, it vanishes for all f, g, h ∈ G. Furthermore, assume that V is a Hilbert space and that ρ : This group fits into a short exact sequence and defines a central extension of G.If the cocycle c is real valued, this central extension is by the circle S 1 (instead of C * ).
2.2.The Bott-Virasoro cocycle.Consider the group G = Diff + (S 1 ) of orientation preserving diffeomorphisms of the circle.We recall the following basic fact: is a normalized real valued group 2-cocycle.Furthermore, it satisfies the cyclic property (5).
Proof.For convenience of the reader, we give a proof of this statement.The left hand side of equation ( 4) is as follows: and the right hand has the following form: Note that the second lines in the two expressions coincide term by term, and the third line of the right hand side of ( 4) is obtained from the third line of the left hand side by the change of variable x → ψ(x).
It is instructive to compute the map β BV : Here we made a change of variables y = φ(x) in the integral.The map α BV is given by formula, and it is skew-symmetric in u and v.

Extension to holomorphic maps.
In this Section, we extend the Bott-Virasoro cocycle to a certain class of holomorphic maps.In more detail, let S be a connected open subset of the complex plane with π 1 (S) ∼ = Z.By the Uniformization Theorem, such a domain is holomorphically isomorphic to an annulus We will consider triples (S, f, T ), where S and T are two such domains, and f : S → T is a holomorphic isomorphism between S and T .Sometimes it is convenient to label the domain and the range of f by S f and T f , respectively.Note that the domain S f contains a closed curve C f which represents the generator of By the analytic continuation principle, the holomorphic function f is uniquely determined by its restriction to C f .Triples (S f , f, T f ) form a groupoid with composition law Two triples are composable if S f = T g .The curves g(C g ) and C f are homotopic to each other in S f = T g .We define a subset Diff + hol (S 1 ) ⊂ Diff + (S 1 ) by the following property: extends to a univalent holomorphic map f on an annulus A r,R with r < 1 < R. It is easy to see that Diff + hol (S 1 ) is a subgroup of Diff + (S 1 ).Consider two diffeomorphisms χ, φ ∈ Diff + hol (S 1 ) and the corresponding univalent holomorphic functions f and g such that f (e ix ) = e iχ(x) , g(e ix ) = e iφ(x) .
By making the annulus S g smaller if needed, one can always achieve T g ⊂ S f .By restricting f to T g , one obtains a pair of composable holomorphic maps, and Since the analytic function f • g is uniquely determined by its values on the unit circle, we conclude that it corresponds to the diffeomorphism χ • φ.
) and f, g the corresponding composable holomorphic functions.Then, Proof.The proof is by a direct calculation.In particular, for z = e ix , g(z) = e iφ(x) we have zg ′ (z)/g(z) = φ ′ (x), and g(z For a pair of composable univalent holomorphic maps f and g, one can use the right hand side of equation ( 8) as a definition of a functional of a pair (f, g): Here the integration is over the curve C g on which both homorphic functions g and f • g are well defined.Note that C BV is complex valued, in general.This is in contrast to the cocycle c BV which takes values in R.
Proposition 2.5.The map C BV is a groupoid 2-cocylce.That is, for all composable triples (f, g, h), it satisfies the equation Furthermore, it satisfies the cyclic property (5).
Proof.The proof of the cocycle condition is analogous to the one of Theorem 2.3.The only non-trivial step in the proof is as follows: one needs to check that ′ Two integrals are related by the change of variable w = h(z).After this change of variables, the integration contour on the left hand side is C h , and on the right hand side it is h −1 (C g ).Both these curves represent the generator of π 1 (S h ), and therefore they are homotopic to each other.For the cyclic property, put The proof of Proposition 2.1 applies verbatim to the case of groupoids.This completes the proof.
Note that the expression ( 9) can be re-written using the change of variables z = g −1 (w).We get Holomorphic functions f : S f → T f and h : T g → S g are actually defined on the same domain S f = T g which contains the curve C f .We compute the expression β for C BV .By putting f (w) = w + tu(w) + O(t 2 ), we obtain Remark 2.6.Yet another groupoid cocycle which has the cyclic property is given by formula The proof is similar to those of Theorem 2.3 and of Proposition 2.5.

Symplectic transformations
In this Section, we recall the notion of symplectic transformations associated to holomorphic maps.
3.1.Symplectic transformations in finite dimensions.Recall the following standard setup: let U be a complex vector space, and ω ∈ ∧ 2 U be a non-degenerate (symplectic) 2-form.A linear map A ∈ End(U) is called symplectic if it preserves ω: The following set of examples is of special interest for us: let V be a finite dimensional complex vector space.Then, one can equip the direct sum U = V ⊕V * with a natural symplectic form Here This transformation is symplectic if and only if the following conditions are verified: Note that if α and δ are invertible, the following operators are symmetric: (α −1 β), (βδ −1 ), (γα −1 ), (δ −1 γ).In this case, one can also express α and δ in terms of three other operators: 3.2.Symplectic transformations and holomorphic maps.We now pass to the infinite dimensional context and apply the theory of symplectic transformations to holomorphic functions.As in the previous Section, let S ⊂ C be a connected domain with π 1 (S) = Z, and let C ⊂ S be a closed oriented curve which realizes the generator of π 1 (S).We consider the space H S of holomorphic functions on S and define the following 2-form Here δ is the de Rham differential on H S and ∂ z is the z-derivative.It is clear that the definition of ω S is independent of the choice of the curve C. Let f : S → T be a holomorphic isomorphism.It induces an isomorphism In turn, the map f * induces a pull-back map of differential forms that we denote by (f * ) * .
Here we made a change of variables z = f −1 (w), and then used the fact that f (C S ) is homotopic of C T in T .
For an annulus A r,R = {z ∈ C; r < |z| < R} with r < 1 < R, one can choose C S to be the unit circle.We will consider the space H = H A /C of holomorphic functions modulo constants.Using the Fourier transform, we obtain a formula for ω A : This form is symplectic on H.In what follows, it will be more convenient to work with functions which do not contain the superfluous constant a 0 .One can view Fourier components {a n , a * n } as coordinates on the infinite dimensional symplectic space of holomorphic functions.
The space H admits a polarization where H + is spanned by monomials z n with n ≥ 1 and H − by monomials z n with n ≤ −1.Holomorphic maps induce symplectic transformations In more detail, This equation implies ( 13) where α m,n , β m,n , γ m,n , δ m,n are infinite dimensional matrices representing operators α, β, γ, δ.
The map from holomorphic maps to symplectic transformations is a group antihomomorphism: Proposition 3.2.Let f, g be two composable holomorphic maps and A f , A g be the corresponding symplectic transformations.Then, Proof.The proof is by a direct computation.
3.3.Grunsky coefficients and symplectic transformations.We will need the following simple properties of holomorphic functions.
Let f (z) be a univalent holomorphic function on neighborhood of zero with F m,n z m w n is regular in z and w.Here F m,n are the Grunsky coefficients of f (z) (see [9] for details).
In a similar fashion, let f (z) be a univalent holomorphic function on a neighborhood of infinity with The following proposition will be important for the rest of the paper: 0 be a univalent holomorphic map on a neighborhood of zero.Then, the corresponding symplectic transformation is upper-triangular, the operator γ vanishes, and the symmetric operator (α −1 β) is of the following form: n=−∞ f n z n with f 1 = 0 be a univalent holomorphic map on a neighborhood of infinity.Then, the corresponding symplectic transformation is lower triangular, the operator β vanishes, and the symmetric operator (γα −1 ) is of the following form: 15) is regular at zero and its integral over C vanishes.Hence, the operator γ vanishes.Furthermore, for n > m the function f (z) n−1 f ′ (z)/z m in equation ( 13) is also regular at zero which implies α m,n = 0. Therefore, α is an upper-triangular (infinite) matrix.A similar argument shows that δ is also upper-triangular.
Let h be the inverse function of f .Equation ( 13) implies where C ′ is some (possibly different from C) circle around zero.Combining with equation ( 14), we obtain Summing up a geometric series and making a substitution z = h(u) yields Finally, integration by parts over u gives rise to Hence, it is given by the Taylor series (19).Proof of equation ( 20) is similar.

Metaplectic representation and Berezin formalism
In this Section, we recall the metaplectic representation of the symplectic group and Berezin formalism in finite and infinite dimensions.Choose a Hermitian scalar product (•, •) on V .Then, the symmetric algebra SV * also carries a Hermitian product, and it can be completed to a Fock space The Fock space F carries a natural action (by unbounded operators) of the Heisenberg algebra, where â • 1 = 0 for all a ∈ V and where ∂ a is a constant vector field acting on SV * .Introduce an orthonormal basis {a i } of V and the dual basis {a * i } of V * .In this basis, ω takes the canonical form Then, operators âi , â * i on F are conjugate to each other under the Hermitian structure on F .To multi-indices I = (i 1 , . . ., i m ), J = (j i , . . ., j n ) we associate monomials a I = a i 1 . . .a im , a J = a * j 1 . . .a * jn .To a power series in formal variables a i , a * i N q (a, a * ) = I,J q I,J a I a * J one associates an operator q = I,J q I,J â * J âI .
If the sum is finite, this operator is well defined, and N q (a, a * ) is called its normal symbol.Sometimes, q is well defined even for infinite series N q (a, a * ).The operator product q • r is represented by a formal Gaussian integral in terms of normal symbols: Note that this formal integral in defined modulo sign since in general it involves a square root of the determinant (see below for a more detailed discussion).

Berezin formalism.
In the finite dimensional context, the group of symplectic transformations Sp(V ⊕ V * ) has a double cover called the metaplectic group.We will also need the associated central extension of Sp(V ⊕ V * ) by C * : It comes with a natural representation on the Fock space F which can also be viewed as a projective representation of Sp(V ⊕ V * ).For a given symplectic transformation A ∈ Sp(V ⊗V * ) one says that an invertible operator Â on F implements it if it represents a lift of A in Sp(V ⊕ V * ).In more detail, it means that We will call a symplectic transformation A ∈ Sp(V ⊕ V * ) admissible if its components α : V → V and δ : V * → V * are invertible.The following theorem summarizes a result of Berezin (see [4]): Theorem 4.1.Let A ∈ Sp(V ⊕ V * ) be an admissible symplectic transformation.Then, A is implemented by a unique operators Â with a normal symbol N A whose constant term is equal to 1.This normal symbol is given by formula We will call a pair of admissible symplectic transformations A 1 , A 2 composable if A 1 A 2 is also an admissible transformation.We use a similar terminology for triples.The following proposition gives a product rule in terms of normal symbols: Proof.The proof is by a direct calculation of the Gaussian integral (21).
Note that the product rule for normal symbols (24) is not quite well defined because of the square root of the determinant.In fact, the subset of admissible elements in the metaplectic group admits the following description: .
Hence, we can re-write the product rule of normal symbols in terms of the metaplectic group as follows: where A = A 2 A 1 .One can summarize the properties of the product rule as follows: Proposition 4.3.The expression is a multiplicative 2-cocylce.That is, for all composable triples A 1 , A 2 , A 3 we have Proof.The statement follows from the fact that This fact reflects associativity of the operator product * .
For operators with normal symbols G (A,z) = z −1 N A , we obtain a group antihomomorphism: The group Sp(V ⊕V * ) contains a subgroup USp(V ⊕V * ) which has the following property: ãi is the conjugate of ã * i for all i.This condition imposes an extra requirement on the components α, β, γ, δ of A: Here α, β are complex conjugate of α and β, respectively.In particular, we obtain This implies that α is invertible, and that so is δ = α.Furthermore, we have the following useful identity: and This implies that all transformations A ∈ USp(V ⊕ V * ) are admissible, all pairs A 1 , A 2 are composable, and Theorem 4.1 and Proposition 4.2 apply without further assumptions.Also, the map (A, z) → G (A,z) defines a group anti-homomorphism from the corresponding subgroup of the metaplectic group MUSp(V ⊕ V * ) to unitary operators on the Fock space F .

4.3.
The infinite dimensional case.Most of the facts reviewed in the previous Section generalize to the infinite dimensional setup.Let V be a Hilbert space.This allows to identify In what follows, we list special features which distinguish the infinite dimensional situation from the finite dimensional one.Instead of the symplectic group Sp(V ⊕ V * ), one considers the restricted symplectic group Sp res (V ⊕ V * ) = {A ∈ Sp(V ⊕ V * ); α, δ are Fredholm, β, γ are Hilbert − Schmidt}.
For admissible elements of this subgroup, Theorem 4.1 and Proposition 4.2 hold true verbatim.In particular, the determinant is invertible, the operators (α −1 2 β 2 ) and (γ 1 α −1 1 ) are Hilbert-Schmidt, and hence the operator ) is of trace class.The product formula (24) still makes sense on the metaplectic double cover.
However, the determinant det(α) is not well defined, in general.Therefore, the admissible part of the metaplectic group does not allow for a simple description using the equation z 2 = det(α), and the cocycle It is instructive to write the corresponding Lie algebra cocycle a N (x 1 , x 2 ) on a pair of elements of the symplectic Lie algebra: where a i and d i are bounded operators, and b i and c i are Hilbert-Schmidt operators.An easy calculation shows that The right hand side is well defined because both terms b 1 c 2 and c 1 b 2 are of trace class.
The restricted group USp res (V ⊕ V * ) is defined as before: Again, all elements A ∈ USp res (V ⊕ V * ) are admissible and all pairs A 1 , A 2 are composable.
Another important result of [4] is as follows: Then, the normal symbol defines a unitary operator on F .
Here we have used equation ( 27).Note that the resulting Fredholm determinant is well defined since the operator (α −1 β)(α −1 β) * is of trace class.Operators U A satisfy the product rule where the cocycle C U (A 1 , A 2 ) is given by formula .
It is defined on the metaplectic double cover, and it takes values in S 1 ∼ = {z ∈ C; |z| = 1} instead of C * .This cocycle is non trivial, in general.The corresponding Lie algebra cocycle is the same as for C N (A 1 , A 2 ) (up to second order, the normalization factor is symmetric in A 1 , A 2 ):

Quantization of conformal welding
In this Section, we apply Berezin quantization to triangular decomposition of symplectic transformations induced by holomorphic maps, and in particular to conformal welding.5.1.Triangular decomposition and conformal welding.In this Section, we discuss an analogue of triangular decomposition for holomorphic maps.
Let f : A r,R → C be a univalent holomorphic map defined on the annulus A r,R such that its range is contained in another annulus: f (A r,R ) ⊂ A r ′ ,R ′ .We say that f admits a triangular decomposition if there exist univalent holomorphic maps Here |z| > r} ∪ {∞} are the discs centered at 0 and ∞, respectively.Holomorphic functions f + and f − admit Taylor expansions at 0 and ∞: A special case of triangular decomposition is given by conformal welding of diffeomorphisms of the circle.Let f be a holomorphic function which corresponds to χ ∈ Diff + hol (S 1 ).That is, f (e ix ) = e iχ(x) .
In this case, one can choose the domain of f to be an annulus A r,R with r < 1 < R.
The following theorem follows from results of [6] on conformal welding for elements of Diff + (S 1 ).We will only be interested in the subgroup Diff + hol (S 1 ) ⊂ Diff + (S 1 ).Theorem 5.1.Let f : A r,R → C be a univalent holomorphic map which corresponds to a diffeomorphism of the circle χ.Then, it admits a unique triangular decomposition f = f −1 + • f − with (f + ) ′ (0) = 1.Remark 5.2.One says that the univalent holomorphic functions f ± provide a conformal welding of the diffeomorphism χ.Note that the normalization . This is achieved by dividing both f + and f − by f ′ − (∞).Equation (28) can also be re-written in the form the Gauss decomposition of matrices.It turns out that it induces a Gauss decomposition on the corresponding elements of the infinite dimensional symplectic group.Indeed, by Proposition 3.3, symplectic transformations A f + and A f − are upper-and lower-triangular, respectively.And by Proposition 3.2, we have a Gauss type decomposition (recall that the map f → A f is a group anti-homomorphism): A for components of the symplectic transformation A f .Then, we have Assuming that α f − and α −1 f + are invertible, this implies

5.2.
Normal symbols of holomorphic maps.In this Section, we apply Berezin theory of normal symbols to holomorphic maps and the corresponding symplectic transformations.
In order to do that, we equip the space of holomorphic functions (modulo constants) H with a structure of a Hilbert space (following [10]) by declaring ||z n || = |n| for all n = 0.Then, symplectic transformations induced by holomorphic maps admit a metaplectic projective representation on the corresponding Fock space.
In more detail, let f be a holomorphic map, and assume that the corresponding symplectic transformation A f is admissible.Then, it is convenient to denote by N f (instead of N A f ) its normal symbol.Proposition 5.3.Let f and g be a pair of composable holomorphic maps, and assume that the corresponding symplectic transformations A f and A g are admissible.Then, Proof.The first statement follows from Proposition 4.2.For the second statement, note that if f = ∞ n=1 f n z n , then by Proposition 3.3 γ f = 0. Similarly, if g = Theorem 5.4.Let f and g be a pair of composable holomorphic maps, and assume that they admit triangular decompositions.Then, .
The corresponding map β N is given by formula (33) Proof.For the first statement, we use Proposition 5.3, and we observe that by formula (30) In order to prove the formula for β N , put f (t) = exp(tu), where u = u(z) ∂ ∂z is a vector field with where exp(tu − )(z) is the image of z under the holomorphic map exp(tu − ).The derivative in t at t = 0 yields (after integrating over z and w) Next, we convert the trace in m, n into a double contour integral.Since the factors √ mn and 1/ √ mn cancel out, we obtain the desired result.
As an example, consider the case where C u = C v = D is the unit disk.In this case, we denote E + (v) = E(D, v).Let v be a Möbius transformation: as required.
By Theorem 5.7, the first term on the right hand side is invariant under the involution χ → χ −1 while the second term is anti-invariant.
5.4.Unitary symbols and quantization of welding.In this Section, we study properties of unitary symbols U f corresponding to diffeomorphisms of the circle using conformal welding.Let χ ∈ Diff + hol (S 1 ) and f the corrsponding holomorphic map.Recall that the symplectic transformation A f ∈ USp res (H + ⊕ H − ) belongs to the unitary symplectic group.This implies that α f is invertible, γ f = βf , and is Hilbert-Schmidt, the operator Proof.On the one hand, equation (39) implies: