On the coadjoint Virasoro action

The set of coadjoint orbits of the Virasoro algebra at level 1 is in bijection with the set of conjugacy classes in a certain open subset $\widetilde{\rm SL}(2,\mathbb{R})_+$ of the universal cover of ${\rm SL}(2,\mathbb{R})$. We strengthen this bijection to a Morita equivalence of quasi-symplectic groupoids, integrating the Poisson structure on $\mathfrak{vir}^*_\mathsf{1}(S^1)$ and the Cartan-Dirac structure on $\widetilde{\rm SL}(2,\mathbb{R})_+$, respectively.


Introduction
Let C be an unparametrized circle: a compact, connected, oriented 1-manifold.The Virasoro Lie algebra vir(C) is the canonical central extension of the Lie algebra Vect(C) of vector fields.
Its smooth dual at level 1, denoted vir * 1 (C), has a geometric interpretation as the space of projective structures on C, and also as a space of Hill operators on C. The group Diff + (C) of orientation preserving diffeomorphisms acts on vir * 1 (C) by affine-linear transformations; we shall refer to this action as the coadjoint Virasoro action.
A classification of the coadjoint Virasoro orbits was achieved in the work of Kirillov [17], Lazutkin-Pankratova [19], Goldman [13], Segal [27], and Witten [31].In the approach of [13,27], this classification is described in terms of a diagram (1) D(C) Here D(C) is the space of developing maps for projective structures on C. A developing map is an orientation-preserving immersion γ : C → RP(1) of the simply connected cover of C, such that γ is quasi-periodic, with monodromy given by the action of an element of PSL(2, R).The right arrow takes γ to its lifted monodromy in the universal cover, and has as its image a certain open subset indicated by a subscript '+'.
The left arrow takes γ to the corresponding projective structure.The space D(C) has natural commuting actions of PSL(2, R) and of the universal cover of Diff + (C), and the two arrows are the quotient maps for these two actions.Consequently, the diagram sets up a bijection between the coadjoint orbits in vir * 1 (C) and conjugacy classes in SL(2, R) + .In this article, we show that this correspondence of group actions extends canonically to a correspondence between the Poisson structure on vir * 1 (C) and the Cartan-Dirac structure on SL(2, R) + .More precisely, we construct a Morita equivalence, in the sense of Xu [32]: (2) (G 1 , ω 1 ) (D(C), ̟ D ) x x q q q q q q q q q q q ' ' In this diagram, (G 1 , ω 1 ) is a symplectic groupoid integrating the Poisson structure on vir * 1 (C), and (G 2 , ω 2 ) is a quasi-symplectic groupoid integrating the Cartan-Dirac structure on SL(2, R) + .The groupoid G 2 is the action groupoid for the conjugation action of PSL(2, R).On the other hand, G 1 is not simply an action groupoid -rather, it is obtained as a quotient of an action groupoid.The Morita equivalence between these quasi-symplectic groupoids has the space of developing maps as its Hilsum-Skandalis bimodule, and comes with a distinguished bi-invariant 2-form ̟ D ∈ Ω2 (D(C)).For C = S1 , writing quasi-periodic paths as γ = (sin φ : cos φ) with φ : R → R, this 2-form is given by an expression (3) S(φ) ∧ dφ φ ′ + boundary terms involving the Schwarzian derivative S(φ); the boundary terms are needed to make the expression invariant under diffeomorphisms.The Morita equivalence (2) of quasi-symplectic groupoids gives rise to a 1-1 correspondence of the associated Hamiltonian spaces.It hence allows us to associate (certain) Hamiltonian Virasoro spaces to finite-dimensional PSL(2, R)-spaces with SL(2, R) + -valued moment maps and vice versa.One such example, motivated by the recent physics literature on Jackiw-Teitelboim gravity (e.g., [26,29]), is the Teichmüller moduli space of conformally compact hyperbolic metrics on surfaces with boundary.Details of these and similar examples will be discussed in forthcoming work.
The Morita equivalence for the Virasoro Lie algebra is related to a similar Morita equivalence for loop groups, via Drinfeld-Sokolov reduction.For any connected Lie group G with an invariant metric on its Lie algebra g, the loop algebra Lg has a central extension Lg.Taking G = PSL(2, R), the Drinfeld-Sokolov procedure [10] realizes vir * 1 (S 1 ) as a Marsden-Weinstein reduced space of Lg between Hamiltonian loop group spaces and finite-dimensional q-Hamiltonian spaces.See [32] for its interpretation as a Morita equivalence.
Throughout this article, we will treat infinite-dimensional manifolds as Fréchet manifolds [14], but without entering any technical discussion.Recent references giving a detailed treatment of the relevant techniques in a related context are [9,25].Alternatively, one could work with diffeologies [15].Recall that diffeology is particularly well-suited for dealing with differential forms, which actually suggests diffeological symplectic and quasi-symplectic groupoids as convenient stand-ins for infinite-dimensional Poisson and Dirac manifolds.
The organization of the paper is as follows.Section 2 discusses Hill operators and the Virasoro algebra.The approach will be coordinate-free; the coordinate expressions are spelled out towards the end of the section.Section 3 studies the geometry of the space of developing maps.In particular, we determine the stabilizers for the actions of diffeomorphisms; this leads us to the construction of the groupoid G 1 .Section 4 describes the 2-form ̟ D on the space of developing maps, giving the Morita equivalence of quasi-symplectic groupoids and resulting in a correspondence of their Hamiltonian spaces.Section 5 gives the construction of ̟ D by Drinfeld-Sokolov reduction, which accounts for its basic properties.The appendix gives a general review of Morita equivalence of quasi-symplectic Lie groupoids, as defined by Xu [32].

Virasoro Lie algebra
We begin by reviewing some background on the Virasoro Lie algebra, and its relationship with Hill operators.We shall follow the coordinate free approach from [11,27]; further information may be found in [16,17,24].
2.1.Hill operators.Let C be a 1-dimensional oriented manifold.For r ∈ R, we denote by |Λ| r C → C the r-density bundle on C; its space of sections is denoted by |Ω| r C = Γ(|Λ| r C ).The orientation on C identifies |Λ| r C with the cotangent bundle for r = 1, and with the tangent bundle for r = −1.The principal symbol of a k-th order differential operator ; the principal symbol of a composition of two such operators is the product of the principal symbols.The formal adjoint of D is the k-th order differential operator given by C (Du)v = C u(D * v) whenever u has compact support; its principal symbol is σ k (D * ) = (−1) k σ k (D).

C
given by multiplication, with σ 0 (M u ) = u.The formal adjoint is A Hill operator is a second order differential operator

C
with L * = L and with principal symbol σ 2 (L) = 1.(The choices r 1 = − 1 2 , r 2 = 3 2 are the unique ones for which these conditions make sense.)We denote by Hill(C) the affine space of all Hill operators; the underlying linear space is the space |Ω| 2 C of quadratic differentials.It is often convenient to choose a connection on the bundle |Λ| C , and the Hill operators on C are of the form C is a quadratic differential.Using (4) we may define the Wronskian of two − 1 2 -densities as (5) W (u 1 , u 2 ) = u 1 ∂u 2 − u 2 ∂u 1 ∈ |Ω| 0 C ; this does not depend on the choice of ∂.
The group Diff + (C) of orientation preserving diffeomorphisms acts on the spaces of rdensities by push-forward.This induces an affine action on Hill(C) by ( 6) with underlying linear action the push-forward of quadratic differentials.Given a Hill operator L and any v ∈ Vect(C), the Lie derivative ] is a differential operator of order 0, and hence is multiplication by a quadratic differential.Denoting the latter by −D L v, this defines a linear map (7) describes the infinitesimal action on the space of Hill operators.From the coordinate description below, we will see below that D L is a third order differential operator, with Finally, we note that the Hill operator determines for all x 0 ∈ C, a symmetric bilinear form ('bilinear concomitant') on the 2-jet bundle J 2 (T C) At x 0 ∈ C, this bilinear form is given by where I ⊆ C is an open interval around x 0 , I + = {x ∈ I : x ≥ x 0 } is its positive side with respect to the orientation of C and v 1 , v 2 are representatives of the given 2-jets with compact support in I. From now on, we will write ).If x 0 , x 1 ∈ C are the end points of a positively oriented arc in C, we obtain (10) x 1 x 0 2.2.Virasoro Lie algebra.The construction of vir(C) (following [11]) uses the following well-known principle.Let K be a Lie group, with an affine K-action on an affine space E such that the underlying linear action is the coadjoint action of K on k * .Then one obtains a central extension 0 of the Lie algebra.As a vector space, k is the space of affine-linear maps ξ : E → R; the associated linear map ξ : k * → R is an element of k.The Lie bracket is given by where ξ Suppose C is compact, connected, and oriented.Using the integration pairing between |Ω| 2

C
and |Ω| −1 C ∼ = Vect(C), the action of Diff + (C) on quadratic differentials is indeed the coadjoint action on the smooth dual of Vect(C).As remarked above, this is the linear action underlying the affine action of Diff + (C) on the space of Hill operators.The resulting central extension (11) 0 is the Virasoro Lie algebra.As a vector space, vir(C) consists of affine-linear functionals v : Hill(C) → R for which the underlying linear functional |Ω| 2 C → R is given by pairing with an element v ∈ Vect(C).Using the description (8) of the infinitesimal action, we see that the Lie bracket on vir(C) is given by (12) [ From now on, we refer to the action of Diff + (C) on the affine space (13) vir * 1 (C) = Hill(C) as the coadjoint Virasoro action. where Here we used (14) for F regarded as an R/Z-valued function; for example F ′′′ is a 3-density.
The infinitesimal generators of the action are , which verifies that D * L = −D L , and gives the formula x 0 .
for the bilinear form (9). Taking the Hill operator L 0 with the zero Hill potential (i.e., L 0 u = u ′′ ) as the base point for the affine space Hill(S 1 ), we see that vir(S 1 ) is the central extension of Vect(S 1 ) defined by the Gelfand-Fuchs cocycle (20) c

The space of developing maps
For the rest of this paper, we take C to be compact, connected, and oriented (i.e., diffeomorphic to an oriented circle).We let C be a simply connected covering space, so that C = C/Z.The diffeomorphism of C corresponding to translation by 1 ∈ Z will be denoted κ.

C
a fundamental system of solutions (defined on the universal cover), with normalized Wronskian The normalization determines the fundamental system uniquely up to the action of SL(2, R).Since u 1 , u 2 have no common zeroes, their ratio is a well-defined local diffeomorphism.Under the action of κ, this map transforms according to γ(κ • x) = h • γ(x) (with the standard action of PSL(2, R) on RP(1)), for some h ∈ PSL(2, R).Definition 3.1.[30] A developing map is an orientation preserving local diffeomorphism γ : C → RP(1) which is quasi-periodic, in the sense that for some monodromy h ∈ PSL(2, R).We denote by the space of developing maps.Thus, every normalized fundamental system for a Hill operator defines a developing map; a different choice of fundamental system changes γ by the natural PSL(2, R)-action, Conversely, every developing map γ arises in this way from a unique Hill operator L.
Every γ admits a normalized lift, unique up to an overall sign.With ∂ as in (4), there is a unique Hill operator having the normalized lift as its fundamental system of solutions.Explicitly, (22) L (The right hand side defines an operator over C; one checks that it descends to C.) This defines a surjective map p : D(C) → Hill(C) = vir * 1 (C), which is the quotient map for the principal PSL(2, R)-action (21).

3.2.
The quotient map q.The PSL(2, R)-action on RP(1) lifts to a SL(2, R)-action on the universal cover, RP(1) = R.Hence, writing γ ∈ D(C) in the form γ = (sin φ : cos φ) as above, we have with the lifted monodromy h ∈ SL(2, R).The map φ is only determined up to multiples of π, but the lifted monodromy does not depend on the choice.This defines a map taking γ to h.Let SL(2, R) + be the range of this map, giving the diagram ( 25) Proposition 3.4.The image of the map q is the subset since φ is increasing.This proves the inclusion ⊆.For the opposite inclusion, suppose that h ∈ SL(2, R) and φ 0 ∈ R with h • φ 0 > φ 0 are given.Pick a base point x 0 ∈ C, and choose an increasing function φ ∈ C ∞ ( C, R) with φ(x 0 ) = φ 0 and with the property φ(κ(x)) = h • φ(x) for all x.Then γ = (sin(φ) : cos(φ)) is a developing map with lifted monodromy h.
For a more concrete description, recall the classification of conjugacy classes in SL(2, R).Let (26) ) is conjugate to exactly one of the following types: (i) elliptic/central: r α for α ∈ R is the homotopy class of the path t → exp(tαJ 1 ).In particular, r πn , n ∈ Z are the central elements of SL(2, R).
The horizontal line in this picture depicts the elliptic/central classes, with the central elements r πn as the nodes n.The vertical lines represent the hyperbolic classes, and the fat dots represent the parabolic classes.The space SL(2, R) + / PSL(2, R) is the subset The article of Balog-Feher-Palla [3] (see also [7]) contains a detailed discussion of the developing maps and Hill potentials for the representatives (27).The simplest examples are given by 'exponential paths': is a developing map (The condition γ ′ > 0 holds since it is a small perturbation of the path for A = 0).This may be used to realize all h in some open neighborhood of the central element r πn .In particular, the conjugacy classes of p ± n , n ∈ N may be realized in this way, as well as h β,n for β small.As developing maps realizing h β,n with n ∈ N for arbitrary β > 0 one may take [3, Section 3.5.2] 3.3.Action of diffeomorphisms.For the following discussion, we shall use the Poincaré translation number [12].The translation number is a quasi-homomorphism τ : SL(2, R) → R, defined in terms of the action of SL(2, R) on R = RP(1) by ( 30) for arbitrary φ 0 ∈ R.This is well-defined and independent of the choice of φ 0 ; it is furthermore conjugation invariant and continuous, and it satisfies the quasi-homomorphism property |τ ( g One may think of the τ ( g) as the 'asymptotic slope' of the action of g on R. We have In terms of the picture of conjugacy classes, τ is simply the horizontal coordinate.
Recall that the action of h ∈ PSL(2, R) on RP(1) has two fixed points if h is hyperbolic, and a single fixed point if h is parabolic.Lemma 3.5.Let γ ∈ D(C) be a developing map with lifted monodromy h = q(γ).
(a) If τ ( h) > 0, then γ : R → RP(1) is surjective (and hence is a covering).(b) If τ ( h) = 0, and h is hyperbolic, then the range of γ is an open arc whose end points are the two fixed points of h.(c) If τ ( h) = 0, and h is parabolic, then the range of γ is the complement of the unique fixed point of h.
Proof.Write γ = (sin φ : cos φ) ∈ D(C).We shall use the property this property shows that φ is surjective, and hence is a global diffeomorphism.Suppose τ ( h) = 0.In particular, h must be hyperbolic or parabolic, and so its action on RP(1) has (one or two) fixed points.Their pre-images are fixed points for the action of h (since τ ( h) = 0).Since φ is increasing, equation (31) shows that the range of φ cannot contain any such fixed points, but that for any given x ∈ C the limits lim N →±∞ φ(κ N (x)) are fixed points.Hence, the range of φ must be the interval between two successive fixed points.
Let Diff Z ( C) be the Z-equivariant diffeomorphisms of C. Any such diffeomorphism descends to an orientation preserving diffeomorphism of C, resulting in an identification with the universal covering group of Diff + (C).
Proposition 3.6.The map q : D(C) → SL(2, R) + is the quotient map for the action of Diff Z ( C) on the space of developing maps.
Proof.Let γ i = (sin φ i : cos φ i ), i = 1, 2 be two developing maps with q(γ 1 ) = q(γ 2 ) = h.Since the φ i are increasing functions, they are diffeomorphisms onto their range.By Lemma 3.5, the maps φ 1 , φ 2 have the same range, after possibly changing φ 2 by a multiple of π.But then To summarize, the maps p and q in the diagram (25) are the quotient maps for commuting actions of PSL(2, R) and Diff Z ( C), respectively.The projection p : D(C) → vir * 1 (C) intertwines the Diff Z ( C)-action with the action of Diff + (C) on Hill(C).We therefore obtain an identification of orbit spaces This recovers the classification of coadjoint Virasoro orbits in the form due to Goldman and Segal [13,27].
Nonetheless, D(C) does not give a Morita equivalence of action groupoids for the two actions, since it does not identify the stabilizer groups.This may be traced back to the fact that the Diff Z ( C)-action on the space of developing maps is not free: Example 3.7.Let C = S 1 = R/Z, and let γ = (sin(αx) : cos(αx)), α > 0 be the developing map realizing the elliptic/central element r α .The stabilizer of γ is acting by translations.
The pre-image of κ ∈ Diff Z ( C) L under this map is (κ, h).Hence, where Z is the subgroup generated by (κ, h).On the other hand, the projection (F, g) → g induces a surjection With these preparations, we obtain the following description of the stabilizer groups: Proposition 3.8.Let γ ∈ D(C) be as above, written as γ = (sin φ : cos φ).
• If τ ( h) > 0, there is an isomorphism of stabilizer groups with the unique The stabilizer group Diff Z ( C) γ is a cyclic group generated by the unique diffeomorphism where F g is defined as φ•F g = g •φ, with g the unique lift of translation number τ ( g) = 0.

It induces an isomorphism
The stabilizer group Diff Z ( C) γ is trivial.

Proof.
• Case 1: Conversely, given g ∈ SL(2, R) h , use (37) to define F. This is well-defined since φ is a diffeomorphism, and F is periodic by the calculation We hence obtain an isomorphism Diff This isomorphism takes κ to h.Using (35), this gives the description of Diff + (C) L .• Case 2: τ ( h) = 0, so φ is a diffeomorphism onto an open interval between two fixed points of the action of h on R, of length π/2 if h is hyperbolic and π if h is parabolic.If (F, g) stabilizes γ, there is, as before, a unique lift g such that (37) holds.But this equation means, in particular, that g preserves the range of φ, and so it has translation number zero.Conversely, given g ∈ PSL(2, R) h , let g be the unique lift of translation number 0. The equation (37) uniquely determines F. The same calculation as in Case 1 shows that F is periodic.Then (F, g) stabilizes γ.
The description of Diff Z ( C) γ follows from the above, since the stabilizer consists of all F such that (F, e) ∈ (Diff Taking the quotient by the subgroup generated by h, we see that the stabilizer Diff Taking the quotient by the subgroup generated by h, we see that the stabilizer Diff + (C) L is R/αZ ∼ = SO (2).In a similar way, one obtains explicit descriptions of the stabilizers for hyperbolic and elliptic elements, matching the calculations of [3].
We have seen that Diff Z ( C) γ is a cyclic subgroup if τ ( h) > 0, and trivial if τ ( h) = 0.For a better understanding of how these stabilizer groups fit together, take C = S 1 = R/Z, and let be the Poincaré translation number [12] of F ∈ Diff Z (R).This is independent of x 0 , and we shall choose x 0 = 0. Consider γ = (sin φ : cos φ) ∈ D(C), with lifted monodromy h = q(γ) satisfying τ ( h) > 0. Let F ∈ Diff Z (R) γ be the generator of the stabilizer of γ, so that φ(F(x)) = φ(x) + π.
To see this, observe first that τ ( h) may be computed as πx N where x N is any sequence with lim N →∞ x N = ∞.(The last equality follows since φ is increasing.)In particular, we may take x N = F N (0), which gives as claimed.This is consistent with Example 3.7.
Example 3.10.It may be puzzling that the Diff Z ( C)-action on D(C) results in a smooth orbit space SL(2, R) + , despite having non-constant stabilizers.This is possible since the action is not proper.For a finite-dimensional example, let N = R 2 / ∼ be the quotient under the equivalence relation generated by (x, y) ∼ (x, y + 1 x ) for x = 0.By the standard criterion for quotients of manifolds, N is a smooth Hausdorff manifold; one may picture it as a union of circles R/ 1 x Z, which for x = 0 degenerates to R. The R-action t • (x, y) = (x, y − t) on R 2 descends to a locally free R-action on N , with orbit space N/R = R.The stabilizer of [(x, y)] ∈ N under this action is trivial when x = 0, and equal to 1 x Z when x = 0.This space N will appear again in Appendix A, as an example of a symplectic groupoid.

3.5.
Local sections.To complete our discussion of the geometry of the space of developing maps, let us show that the quotient map q : D(C) → SL(2, R) + is a submersion, in the sense that it admits local sections.We may assume C = S 1 = R/Z.Local sections can be obtained by the following observations: (a) For fixed The first condition shows that U is contained in SL(2, R) + (see Proposition 3.4), while the second condition ensures that h is the homotopy class of a unique exponential path t → exp(tA), for A ∈ g.Hence, the map taking h ∈ U to the path γ(x) = exp(xA) • γ 0 , defines a section of q over U .The U 's of this form (for choices of φ 0 ) cover the subset of SL(2, R) + given by 0 ≤ τ ( h) <

Morita equivalence for the coadjoint Virasoro action
We shall now exhibit D(C) as a bimodule for a Morita equivalence of quasi-symplectic groupoids.Background on Morita equivalence of groupoids, and the extension to quasisymplectic groupoids, is provided in Appendix A.

Groupoids. By applying Example
The groupoid G 1 , presented here as a quotient by PSL(2, R) of the submersion groupoid, may also be seen as a quotient of the action groupoid There is an exact sequence of groupoids over vir * 1 (C), (41) where K is the family of discrete groups Recall again (Proposition 3.8) that ( 42) is trivial when τ (q(γ)) = 0, and is a cyclic subgroup otherwise.
Proof.By definition, G 1 consists of pairs of developing maps having the same lifted monodromy, up to simultaneous transformation by PSL(2, R).Since Diff Z ( C) acts transitively on the fibers of q, this defines a surjective groupoid morphism G 1 → G 1 taking (F, p(γ)) to the PSL(2, R)-orbit of (F • γ, γ).The kernel of this groupoid morphism is K.
4.2.The 2-form ̟ D .In this section, we will promote (39) to a Morita equivalence of quasisymplectic groupoids.This involves a 2-form ̟ D on D(C), which is 'natural' in the sense that its definition does not involve choices, and in particular does not depend on a parametrization of C. Throughout, we will work with the metric X, Y = 2 tr(XY ) on g.

Theorem 4.3.
There is a natural 2-form ̟ D ∈ Ω 2 (D(C)), with the following properties: (a) The exterior differential is R)-invariant, and its contractions with generating vector fields for , and its contractions with generating vector fields for v ∈ Vect(S 1 ) are The proof of this result is contained in the subsequent sections.Proposition 4.13 presents a coordinate-free formula for ̟ D , which in particular gives its invariance properties.Theorem 5.6 exhibits ̟ D as the pullback of a similar 2-form ̟ P for loop groups; the formulas for contractions, and the exterior differential, are immediate consequences of the properties of ̟ P .In Section 5.4 we will give a conceptual explanation of this description, as a Drinfeld-Sokolov reduction.Proof.As recalled in Appendix A, a closed multiplicative 2-form ω on a Lie groupoid G ⇒ M determines a map ρ : A = Lie(G) → T * M , given on the level of sections by ρ(σ) = i * M ι(σ L )ω where σ L is the left-invariant vector field corresponding to σ, and i M is the inclusion of units.If G is source-connected, ω is uniquely determined by ρ.
The Lie algebroid of G 1 is the action algebroid A = Vect(C) ⋉ vir * 1 (C), and the map ̺ is given on constant sections by ( 44) (See Example A.12.) The groupoid G 1 has the same Lie algebroid, A. By the above, there can be at most one closed multiplicative form ω 1 ∈ Ω 2 (G 1 ) having the same map (44), and its pullback to G 1 must coincide with ω can .In particular, ω 1 must be symplectic.
To construct ω 1 we use the description (40) of the groupoid.The 2-form pr (where pr 1 , pr 2 are the projections from the fiber product to the two factors) is multiplicative, and + pr * 2 q * η = 0 since q • pr 1 = q • pr 2 by definition of the fiber product.This 2-form is invariant under the diagonal PSL(2, R)-action, and in fact is PSL ( It hence descends to a closed multiplicative 2-form ω 1 ∈ Ω 2 (G 1 ).The left invariant vector field on (45) corresponding to v ∈ Vect(C) is given by (0, v D ), and descends to the corresponding vector field on G 1 .By part (c) of Theorem 4.3 we have This confirms that the map ̺ for ω 1 is given by (44).
These data define a Hamiltonian space for G 1 = G 1 /K (see ( 42)) if and only if the subgroupoid K acts trivially.In other words, the stabilizers of points u ∈ M must satisfy The condition that the associated space M = (M × vir * 1 (C) D(C))/ Diff Z ( C) be finite-dimensional amounts to an assumption that T u Φ M has finite dimensional kernel everywhere; as explained in the appendix (see (77)) this kernel is identified with ker(T m Φ M ), where m is a point corresponding to u. Remark 4.6.dim(ker T u Φ M ) < ∞ if and only if M is 'transversally finite-dimensional', in the sense that the Diff Z ( C)-orbits have finite codimension.Indeed, the correspondence identifies the normal spaces to the orbits through u and m.
To give a concrete formula, choose a parametrization C ∼ = S 1 = R/Z.The evaluation map will be denoted simply by F(x); thus d F(x) is an ordinary 1-form on Diff(S 1 Proof.Let v = f (x) ∂ ∂x be a given vector field on S 1 , and denote by t → exp(tv) ∈ Diff + (S 1 ) its flow.The resulting left-invariant vector field on the diffeomorphism group acts by right translations, F → F • exp(−tv).Hence Remark 4.9.We may also directly verify that ( 46) is left-invariant, and satisfies the Maurer-Cartan equation Using the Maurer-Cartan form, we may write down the 2-form of the symplectic groupoid ). Recall that the choice of parametrization defines a splitting of the central extension, describing the latter in terms of the Gelfand-Fuchs cocycle (20).The splitting also identifies vir * 1 (S 1 ) with the space of quadratic differentials (Hill potentials).We write T ∈ Ω 0 (vir * 1 (S 1 ), |Ω| 2 S 1 ) for the map taking an element of vir * 1 (S 1 ) to the corresponding Hill potential, thus dT is a 1-form on vir * 1 (S 1 ) with values in quadratic differentials.As a special case of (83), we obtain (47) (In (51), we are treating u i as a |Ω| -valued functions on D(C).Strictly, the normalized lifts are only defined up to an overall sign; thus u i are defined on a double cover D(C).) Proof.To verify (50), choose a local coordinate t on C, and let V = f (t) ∂ ∂φ be a vector field along γ.The corresponding vector field on C is The alternative expression (51) reduces to (50) by using the formula (23) for the normalized lifts.
Note the similarity of (50) with the expression for the left-invariant Maurer-Cartan form on Diff + (S 1 ).For the following result, let with J 1 as in (26) and with Up to a factor, this is is the momentum map for the SL(2, R)-action on the symplectic vector space (R 2 , ds 1 ∧ ds 2 ).In terms of the SL(2, R)-invariant 1-form , we may form substitute for s, defining a vector field See Remark 4.12 below for the relevance of these vector fields.(7)) is given by C ) is the tautological 1-form of the affine space vir * 1 (C) = Hill(C).In particular, D L Θ is Z-invariant.

Proof.
(a) Equivariance under the action of Diff Z ( C) is clear from the coordinate-free definition.The PSL(2, R)-invariance follows from (51), since the 1-form s 1 ds 2 − s 2 ds 1 ∈ Ω 1 (R 2 ) is SL(2, R)-invariant.(It is the contraction of the symplectic form ds 1 ∧ ds 2 by the Euler vector field.)(b) We may assume .This function is SL(2, R)-equivariant, using the standard action on R 2 .Using the formula (51) for Θ, the expression for ι(X D )Θ is a consequence of the formula (52) for ι(X R 2 )(s 1 ds 2 − s 2 ds 1 ).

On the other hand, using Θ
Remark 4.12.For γ ∈ D(C), the map is a Lie algebra action on C: it is bracket preserving as a consequence of the Maurer-Cartan equation for Θ.From the definition of Θ, one finds that ̺ γ (X) is simply the pullback of the generating vector field X RP(1) under the local diffeomorphism γ : C → RP(1).Under the Z-action, where we used that Υ is SL(2, R)-equivariant.In particular, the vector field ̺ γ (X) is periodic (and so descends to C) if and only if X is fixed under the adjoint action of the monodromy.These vector fields are exactly the elements of Vect(C) p(γ) , the Lie algebra of the stabilizer Diff + (C) p(γ) .See [17] and [24, Section 1.3].
4.6.The 2-form.We now have all the ingredients to describe the 2-form on the space of developing maps.The formula involves an integral of the density valued 2-form over the segment from x 0 to κ(x 0 ) for some choice of x 0 ∈ C. The result becomes independent of x 0 , by adding a boundary term involving the bilinear concomitant B L,x 0 , see (9).
does not depend on the choice of x 0 ∈ C.
Proof.Let x 1 = x 0 be another choice.Without loss of generality, assume x 1 > x 0 with respect to the orientation.We have where we used that D L Θ is Z-invariant (Proposition 4.11).Using the integration by parts formula (10) with Finally, note that B L,x i (Θ, Θ) = 0 by symmetry.
Using the explicit expression for the boundary terms, as given in (19), we can write down the formula for the case C = S 1 = R/Z, as follows: Here T is the Hill potential corresponding to L = p(γ), and T x , Θ x , Θ ′ x , Θ ′′ x are the evaluations at x ∈ R. Writing everything in terms of T = (φ ′ ) 2 + 1 2 φ ′′′ /φ ′ − 3 4 (φ ′′ /φ ′ ) 2 and Θ = −dφ/φ ′ , this becomes a highly complicated expression in terms of φ and its first three derivatives.
The properties of ̟ D , as described in Theorem 4.3 may now be proved directly, using the properties of the 1-form Θ.On the other hand, they will also be immediate consequences of the construction by Drinfeld-Sokolov reduction, described in the next Section.

Construction via Drinfeld-Sokolov method
We shall now give a construction of the 2-form ̟ D , using the Drinfeld-Sokolov approach [10].Throughout, we take C to be a compact, connected, oriented 1-manifold, presented as C = C/Z for a universal cover C. The quotient map will be denoted π : C → C.

5.1.
Principal bundles over C. Throughout this subsection, G will denote an arbitrary connected Lie group; later we will specialize to G = PSL(2, R).Let Q → C be a principal G-bundle, and let Gau(Q) be its group of gauge transformations, with Lie algebra the space of sections of the adjoint bundle.Let A(Q) be the affine space of principal connections on Q; the underlying linear space is Ω 1 (C, Q × G g).The group Gau(Q) acts on A(Q) by gauge transformations of connections, with generating vector fields A → −∂ A ξ where is the covariant derivative.
The action groupoid for the gauge action is Morita equivalent to the transformation groupoid for the conjugation action of G on itself (55) Here the equivalence bimodule is the space of quasi-periodic sections of the pullback bundle π * Q → C. Furthermore, q(τ ) = h is the monodromy, and p(τ ) is the unique principal connection such that τ is horizontal (for the pullback connection).Since we are dealing with action groupoids, the Morita equivalence just means that the two actions commute, and that they are both principal actions, with p and q as their respective quotient maps.In particular, (55) gives a 1-1 correspondence between Gau(Q)-orbits in A(Q) and conjugacy classes in G.
(c) One may also allow for disconnected G.Here Q is possibly non-trivial, and the image of the map q is a union of components of G, depending on the topological type of Q.This generalization is relevant for the setting of twisted loop groups.
The gauge action of Gau(Q) on the space of connections extends to the larger group Aut(Q) of all principal bundle automorphisms.Let Aut + (Q) be the subgroup of automorphisms such that the induced action on the base C preserves orientation.It fits into an exact sequence where Aut Z (π * Q) are the Z-equivariant automorphisms of π * Q.Note that the action of this group on P(Q) commutes with the action of G.
We shall need a certain 'tautological' 1-form Proof.Properties (a)-(c) are simple consequences of the definition.Consider the transformation property under κ.At any given x ∈ C, this says that By definition, elements of P(Q) satisfy τ (κ(x)) = q(τ ) • τ (x).That is, ev κ(x) is given by ev x followed by the gauge action of q(τ ) on Q π(x) .Choose a trivialization Q π(x) ∼ = G, so that θ x = θ L , and the gauge action is multiplication from the right by q(τ ) −1 .Equation (61) follows from the property of Maurer-Cartan forms.
5.2.The 2-form ̟ P .Suppose now that g = Lie(G) comes equipped with a non-degenerate invariant symmetric bilinear form •, • : g × g → R, henceforth referred to as a metric.The metric, together with integration, defines a non-degenerate Aut + (Q)-invariant pairing between Ω 1 (C, Q × G g) and gau(Q).We may hence regard Ω 1 (C, Q × G g) as the smooth dual of gau(Q), with the Gau(Q)-action as the coadjoint action.It is the linear action underlying the affine Gau(Q)-action on the affine space A(Q).By the same mechanism as explained at the beginning of section 2.2, this defines a central extension As a vector space, gau(Q) consists of all affine-linear functionals ξ : A(Q) → R for which the underlying linear functional on Ω 1 (C, Q × G g) is given by an element ξ ∈ gau(Q); the bracket reads as From now on, we consider the affine action of Gau(Q) on A(Q) = gau(Q) * 1 as the coadjoint gauge action.
The following result is due to [2] for the case Q = S 1 × G; see also [1,20].
Theorem 5.3.There is a natural 2-form with the following properties: (a) The exterior differential of ̟ P is and its contractions with generating vector fields for the G-action are (c) ̟ P is Gau(Q)-invariant, and its contractions with generating vector fields, for ξ ∈ gau In fact, For a given choice of base point x 0 ∈ C, we have Here naturality means that ̟ P does not depend on any additional choices.In particular; it does not depend on the choice of x 0 .
Proof.The G-invariance is clear from (63) and the invariance of the metric on g.The independence of (63) of the choice of x 0 is verified by an argument parallel to that for Θ: Replacing x 0 with x 1 changes the first term by 1 2 x 0 , which exactly compensates the change of the boundary term.This also shows that their lifts, the 2-forms ̟ P(Q 1 ) and ̟ P(Q 2 ) are related by pullback.In particular, ̟ P is Aut + (Q)-invariant.The remaining claims are proved in [2] (see also [20, Appendix C]) for the case Q = S 1 × G, hence we just need to make the relevant identifications for this case.We have Gau(Q) = C ∞ (S 1 , G) = LG, while P(Q) can be identified with paths γ : R → G satisfying γ(x + 1) = q(γ)γ(x), by the map taking γ to the section τ and putting x 0 = 0, we have which is the expression used in [2] (except for a different sign convention).

5.3.
The Drinfeld-Sokolov embedding.We shall now specialize to the structure group G = PSL(2, R), with the metric on g = sl(2, R) given by X, Y = 2 tr(XY ).We shall construct a principal G-bundle Q → C and an inclusion D(C) ֒→ P(Q) under which the 2-form ̟ D is the pullback of the corresponding form ̟ P on P(Q).
The following constructions are inspired by Segal's paper [28].Since solutions of a given Hill operator are uniquely determined by their 1-jet at any given point, one considers the jet bundle ).This is a rank 2 vector bundle, with a reduction of the structure group to SL(2, R), using the trivialization of det(E x ) = ∧ 2 E x by j x (u 1 ) ∧ j x (u 2 ), for − 1 2 -densities u 1 , u 2 with Wronskian W (u 1 , u 2 ) = −1.We take Q → C to be the associated principal PSL(2, R)-bundle, obtained as the quotient of the SL(2, R)-frame bundle by the action of the center.As before, we denote by π * Q → C its pullback.
For any developing map γ ∈ D(C) we have the normalized lift (Definition 3.2) u 1 , u 2 , defined up to an overall sign.At any point x ∈ C, this defines an SL(2, R)-frame for E x defined up to an overall sign, i.e, an element of π * Q x .The quasi-periodicity of the normalized lift means that the resulting section of π * Q is quasi-periodic.This defines an inclusion (65) ι : D(C) → P(Q).
The map (65) is G-equivariant, and descends to the Drinfeld-Sokolov embedding (66) ι : vir * 1 (C) = Hill(C) ֒→ A(Q).Remark 5.5.The Drinfeld-Sokolov embedding may also be seen as follows.By the existence and uniqueness theorem for second order ODE's, every solution to a Hill operator L is uniquely determined by its 1-jet.Hence, L determines a linear connection ∇ on E, with the property that for all local sections u of |Λ| In turn, ∇ defines an element of the space A(Q) of projective connections.
The main result of this subsection relates the 2-forms on the space of developing maps and on the space of quasi-periodic sections.
Theorem 5.6.The 2-forms on the space D(C) of developing maps and on the space P(Q) of quasi-periodic sections are related by pullback: In particular, the properties of the 2-form ̟ D , as listed in Theorem 4.3, follow from the corresponding properties of ̟ P (Theorem 5.3).
Proof.We shall show separately that the integrands match, i.e. (67) and the boundary terms (for given choice of x 0 ) match as well.For the calculation, we may assume C = S 1 = R/Z.The r-densities |∂x| r define a trivialization of the r-density bundles |Λ| r C , |Λ| r

C
, which we will use to identify densities with functions.In particular, we obtain a trivialization of E, given at x ∈ S 1 by Using square brackets to denote the image of an SL(2, R)-matrix in PSL(2, R), the inclusion ι is given by Recall that Ξ is pointwise the pullback of the left-invariant Maurer Cartan form θ L .Thus, ι * Ξ = τ −1 dτ .As it turns out, it will be more convenient to work with given in terms of the parametrization as ι * Ξ = (dτ )τ −1 .The integral term in the formula for ̟ P (Q) can be computed as 1 2 The matrix entries may be expressed in terms of as required.Consider next the boundary terms.Again, we find it more convenient to express these in terms of Ξ: ).Here we used T 1 = T 0 by periodicity of the Hill potential.
Remark 5.7.The proof showed in particular that the linear connection ∇ = δ+A corresponding to the Hill operator L = d 2 dx 2 + T (x) is given by (68).The inclusion ι is Diff + (C)-equivariant.Indeed, the action of Diff + (C) on jets determines an action on E by linear bundle automorphisms, preserving orientation, and hence an action on Q by principal bundle automorphisms.This defines a group homomorphism Diff + (C) ֒→ Aut + (Q) which splits the exact sequence (57).It lifts to a group homomorphism splitting (58).From the coordinate-free construction, it is clear that the embedding (65) is equivariant with respect to (69).Let us also give the coordinate expression.
Proposition 5.8.For C = S 1 = R/Z, and Q = S 1 × G, the canonical inclusion is given by where .

The description of Diff
Replacing x with x+ t, and taking the first order Taylor approximation, the coefficient function is replaced by Proposition 5.9.The symplectic groupoid G 1 ⇒ vir * 1 (C) is the restriction (as a groupoid) of Proof.By the general property (85) of Morita equivalences, the groupoid Gau To summarize, the Morita equivalence (39) for the coadjoint Virasoro action is obtained from the Morita equivalence (56) by restriction.This extends to the Hamiltonian spaces: If (M, ω M ) is a Hamiltonian Gau 0 (Q)-space, then the pre-image of vir * 1 (C) ⊆ A(Q) under the momentum map Φ M : M → A(Q) is a Hamiltonian Virasoro space.5.4.Drinfeld-Sokolov reduction.The result ̟ D = ι * ̟ P has a conceptual explanation.We shall follow the coordinate-free description of Drinfeld-Sokolov reduction [10] given in Segal's paper [28]; see also [11].Let P → C = C/Z be an oriented rank 1 projective bundle, with a distinguished section σ ∈ Γ(P).Let Q → C be the associated principal G = PSL(2, R)-bundle.The gauge group G = Gau(Q) acts by projective transformations on P. The section σ determines subgroups where B are the gauge transformations fixing σ, and N are those which furthermore act trivially on σ * V P, where V P ⊆ T P is the vertical bundle.The corresponding Lie algebras act as infinitesimal gauge transformations on P, and so are realized as spaces of vertical vector fields on P. The metric on g = sl(2, R) induces a bundle metric on Q × G g, and defines a non-degenerate C ∞ (C)-valued bilinear form on gau(Q).Using local trivializations, we see that Lie(G)/ Lie(B) = Γ(σ * V P) is non-singularly paired with Lie(N ).Combined with integration over C, this identifies (the smooth dual).The space A(Q) of principal connections may be regarded as the space of projective connections on P. Given such a connection, the composition of T σ : T C → T P followed by vertical projection defines a bundle map T C → σ * V P, which we may also think of as an element of (72).This gives a B-equivariant map (73) Ψ : A(Q) → Lie(N ) * .
By construction, Ψ(A) = 0 if and only if σ is an A-horizontal section.
Proposition 5.10.The restriction of the central extension gau(Q) to Lie(N ) ⊆ gau(Q) is canonically trivial.
Proof.By definition, gau(Q) consists of affine-linear functionals ξ : A(Q) → R whose underlying linear functional Ω 1 (C, Q × G g) → R is given by an element ξ ∈ gau(Q) (via pairing and integration).For ξ ∈ Lie(N ), the map is a distinguished lift.This defines a splitting.
The inclusion A(Q) = gau(Q) * 1 ֒→ gau(Q) * is the momentum map for the gauge action.Hence, we may regard (73) as a momentum map (equivariant in the usual sense) for the action of the subgroup N .
Let us now return to the case that P is the projectivization of E = J 1 (|Λ| ).The kernel of the natural map to |Λ| − 1 2 C defines a rank 1 subbundle ℓ ⊆ E, which in turn determines a section σ of P. From the jet sequence , we have This gives a canonical identification (See, e.g., [16].)We may assume C = S 1 = R/Z.Following the discussion from the proof of Theorem 5.6, and using the trivialization E ∼ = C × R 2 constructed there, the quotient map E → E/ℓ is projection to the second component of R 2 .Hence the subbundle ℓ is the subbundle of C × (R ⊕ 0) ⊆ C × R 2 , and σ ∈ Γ(P) is the constant section (1 : 0) of S 1 × RP (1).
Let N ⊆ B ⊆ G be the connected subgroups whose Lie algebras are the subalgebras of g = sl(2, R), consisting of strictly upper triangular and lower triangular matrices, respectively.
Using the metric on g to identify g ∼ = g * , the space Lie(N ) * is identified with C ∞ (S 1 , n − ), where n − are the strictly lower triangular matrices.The map Ψ takes a connection 1-form A ∈ Ω 1 (S 1 , g) ∼ = C ∞ (S 1 , g) to its lower left corner, A 21 .The gauge action of N reads as We see that on the subset where A 21 = f has no zeroes, the action is free.Furthermore, for any fixed f , the subset where the diagonal entries are zero serves as a global slice.In particular, this is true for f = −1, proving that matrices of the form (68) are a global slice for the N -action on Ψ −1 (−1).
Similarly, the symplectic groupoid G 1 ⇒ vir * 1 (C) may be regarded as the symplectic reduction of Gau 0 (Q) ⋉ A(Q) ⇒ A(Q) under the action of N × N .This follows by the argument from the proof of Proposition 5.9, since the two groupoids are described as (P(Q) (quotient under equivalence relation (qg 2 , q ′ ) ∼ (q, g 2 q ′ )) is a Morita equivalence from G 1 to G 3 .(b) A Morita equivalence identifies the 'transverse geometry' of the groupoids.In particular, it induces a homeomorphism of orbit spaces for x i = J i (q), i = 1, 2, and isomorphisms of their representations on the normal spaces to orbits, Example A.2. Let K be a Lie group, and π : P → M a left principal K-bundle.Suppose q : P → N is an equivariant submersion onto another K-manifold N .Let P × N P ⇒ P be the submersion groupoid (a subgroupoid of the pair groupoid), and (P × N P )/K ⇒ M its quotient under the diagonal action of K. We obtain an Morita equivalence with the action groupoid K ⋉ N ⇒ N : Taking N = P , we obtain a Morita equivalence between the action groupoid K ⋉ P ⇒ P and the trivial groupoid M ⇒ M ; for N = pt we obtain a Morita equivalence between the gauge groupoid G(P ) = (P × P )/K ⇒ M and the group K ⇒ pt.
Example A.3 (Morita equivalence of group actions).Let Q be a manifold with commuting actions of Lie groups K 1 and K 2 , both of which are principal actions.Let M 1 = Q/K 2 with the induced action of K 1 and M 2 = Q/K 1 with the induced action of K 2 .Then Q defines a Morita equivalence of the action groupoids Example A.4.The Morita equivalence Q gives rise to a 1-1 correspondence between G 1 -spaces and G 2 -spaces, as follows.Given a G 2 -action on a manifold P 2 , along a map Φ 2 : P 2 → M 2 , one obtains a G 1 -action on the manifold along the induced map Φ 1 : P 1 → M 1 .One recovers P 1 = Q − ⋄ P 2 .In fact, one may understand this correspondence as a Morita equivalence of the two action groupoids There is a natural map Ψ : R → Q inducing the maps Φ i .As consequences, if p i ∈ P i correspond to each other in the sense that p i = f i (r), then the normal spaces to the orbits T p i P i /T p i (G i • p i ), the stabilizer groups (G i ) p i , as well as their representations on the normal spaces, are identified.Furthermore, one has canonical identifications (77) ker(T p 1 Φ 1 ) ∼ = ker(T p 2 Φ 2 ), since these are identified with ker(T r Ψ), and similarly for the cokernel.
For further details and examples, we refer to [8].
→ G is the groupoid multiplication and pr 1 , pr 2 : G (2) → G are the two projections (here G (2) denotes the space of composable arrows).This property implies that ω pulls back to 0 on M (see [5,32]).It therefore defines a bundle map The pair (G, ω) is called a symplectic groupoid [6] if ω is a multiplicative symplectic 2-form.In this case, the manifold of units acquires a Poisson structure, and the symplectic groupoid is called its integration.
Example A.5.The main example of a symplectic groupoid is the cotangent groupoid T * G ⇒ g * of a Lie group.Using left trivialization T * G = G × g * , the groupoid structure is that as an action groupoid for the coadjoint action; in particular, t(k, µ) = k • µ, s(k, µ) = µ.The 2-form is given by the canonical symplectic form of the cotangent bundle: Quasi-symplectic groupoids [5,18,32] (called twisted pre-symplectic groupoids in [5]) are a generalization of symplectic groupoids.Here ω need not be closed, but instead should satisfy (82) dω = s * η − t * η for a given closed 3-form η ∈ Ω 3 (M ).As a non-degeneracy condition, one requires that the map (a, ρ) : Lie(G) → T M ⊕ T * M given by the anchor a the map (79), is an inclusion as a Lagrangian (i.e., maximal isotropic) subbundle.In this case, the range of (a, ρ) defines an η-twisted Dirac structure on M , with (G, ω) its integration in the sense of [5].
Remark A.6.By [5, Corollary 3.4], a multiplicative 2-form ω on a source-connected Lie groupoid G ⇒ M , satisfying (82) for a given closed 3-form η, is uniquely determined by its values along the units.(This is not true for source-disconnected groupoids, in general.)Since ω pulls back to zero on M , it follows that ω is uniquely determined by the associated map ρ.
Example A.9.For M = g * with the coadjoint action, the choice η = 0, α(X) = dµ, X reproduces the example A.5 of a symplectic groupoid.In this case, Example A.10.Let G be a Lie group with an invariant nondegenerate symmetric bilinear form •, • on its Lie algebra g.Then Example A.14. Recall that T * R = R 2 , with the standard symplectic form ω = dx ∧ dy, and groupoid structure given by fiberwise addition, is the source 1-connected symplectic groupoid integrating R with the zero Poisson structure.The other source connected symplectic groupoids integrating R with the zero Poisson structure are obtained quotients of T * R by closed subgroupoids K ⇒ R with discrete fibers.Thus, each K x ⊆ R is either trivial, or is a cyclic subgroup.In order for the quotient to be a manifold, it is necessary and sufficient that there is some open neighborhood of the zero section in T * R does not contain non-trivial elements of K.The resulting G ⇒ R is a family of Lie groups G x = R/K x .The symplectic form dx ∧ dy descends to G, making the latter into a s-connected symplectic groupoid.It is an action groupoid if and only if the family of subgroups K x is constant.
A.4. Hamiltonian spaces.A (left) Hamiltonian space (P, ω P ) for a quasi-symplectic groupoid G ⇒ M , with 2-form ω ∈ Ω 2 (G) and 3-form η ∈ Ω 3 (M ), is given by a manifold P with a 2-form ω P , equipped with a (left) G-action along a map Φ : P → M , such that (a) A * ω P = pr * G ω + pr * P ω P (b) dω P = −Φ * η (c) ker(ω P ) ∩ ker(T Φ) = 0.Here pr G , pr P are the projections from G × M P to G and P , respectively, and A : G × M P → P is the action map.We remark that (a) encodes both an invariance property of ω P and a moment map property for Φ (which enters the definition of the fiber product G × M P ).For the case of symplectic groupoids, conditions (b) and (c) are equivalent to ω P being symplectic, and one recovers the notion of Hamiltonian actions of symplectic groupoids due to Mikami-Weinstein [22].The generalization to quasi-symplectic groupoids is due to Xu [32].
Remark A.15.The conditions imply that µ ∈ M is a regular value of Φ if and only if the isotropy group G µ acts locally freely on the level set Φ −1 (µ).The symplectic form descends to the orbifold P µ = Φ −1 (µ)/G µ ('symplectic quotient').See [32,Theorem 3.18] Remark A.16.One may similarly define a right Hamiltonian space (P, ω P ), as a manifold P with a right-action of G ⇒ M along a map Φ : P → M satisfying the same conditions as above, but with (b) replaced by dω P = Φ * η.Using inversion, a right Hamiltonian space for G may be regarded as a left Hamiltonian space for the quasi-symplectic groupoid (G, −ω).
Example A.17. (P, ω P ) = (G, ω) a left/right Hamiltonian space for G ⇒ M , using the left/right action of G on itself.
Example A.18. Consider the case of an action groupoid A.8. Using the identification G × M P = G × P , the action map A represents an ordinary G-action A : G × P → P .Property (a) is equivalent to the condition that ω P is G-invariant, with ι(X P )ω P = −Φ * α(X).This shows, in particular, that X P | p with α(X)| p = 0 are contained in the kernel of ω P ; Property (c) amounts to the condition that this is the entire kernel: ker(ω P )| p = {X P | p : α(X) = 0}.
Example A. 19.Let us describe the Hamiltonian spaces for the symplectic groupoid G = T * R/K from example A.14.Using the quotient map T * R → G, these are in particular Hamiltonian spaces for T * R: Thus, P is a symplectic manifold with a Hamiltonian Φ ∈ C ∞ (P, R) generating an R-action.For such a space to be a G-space, the subgroupoid K must act trivially.This gives the condition on the stabilizer groups, K Φ(p) ⊆ G p for all p ∈ P .Conversely, if this condition is satisfied, then there is a well-defined G-action given by [(x, y)] • p = y • p, so that (P, ω P , Φ) is a Hamiltonian G-space.

3. 1 .
Developing maps.Let L ∈ Hill(C) be a Hill operator, and u 1 , u 2 ∈ |Ω| − 1 2 R) since both are identified as the quotients of D(C) under the action of Diff Z ( C) × PSL(2, R).

Remark 4 . 2 .
The picture simplifies when we restrict to the open subset Hyp 0 ⊆ SL(2, R) + consisting of hyperbolic elements of translation number τ = 0. Letting R 0 ⊆ vir * 1 (C) be the corresponding invariant subset, we obtain a Morita equivalence between the action groupoids Diff

Remark 4 . 7 .
Following Remark 4.2, the situation simplifies when G 2 is restricted to the open subset Hyp 0 ⊆ SL(2, R) of hyperbolic elements of translation number 0, and G 1 to the corresponding subset R 0 ⊆ vir * 1 (C).The Hamiltonian G 2 | Hyp 0 -spaces are just the Hamiltonian G 2 -spaces with Φ M (M ) ⊆ Hyp 0 .Since the quotient map SL(2, R) → PSL(2, R) restricts to a diffeomorphism from Hyp 0 to the hyperbolic elements of PSL(2, R), these can be further viewed as q-Hamiltonian PSL(2, R)-spaces with PSL(2, R)-valued moment map taking values in hyperbolic elements.On the other hand, since K is trivial over R 0 , the Hamiltonian G 1 | R 0 -spaces are identified with Hamiltonian Virasoro spaces whose moment map take values in R 0 , with no condition on stabilizers (but still assuming that T u Φ has finite dimensional kernel everywhere).Many examples of this type may be constructed from PSL(2, R)-representation varieties.4.4.The Maurer-Cartan form on Diff + (C).The infinite-dimensional group Diff + (C) has Vect(C) = |Ω| −1 C as its Lie algebra, and hence carries a left-invariant Maurer-Cartan form θ L Diff + (C) ∈ Ω 1 Diff + (C), |Ω| −1 C .By definition, this is characterized by left invariance together with the property ι

( a )
For g ∈ PSL(2, R) and F ∈ Diff Z ( C), g * Θ = Θ, F * Θ = Ad F Θ. (b) Θ satisfies the Maurer-Cartan equation The contractions with the generating vector fields for the Diff Z ( C)-action on D(C) are given by ι(v D )Θ = v for v ∈ Vect(C).(d) The contractions with the generating vector fields for the PSL(2, R)-action on D(C) are given by ι(X D )Θ = 2 tr(XΥ(u)) for X ∈ g.(e) Under the Z-action on |Ω| −1 C (c) This follows since γ * (v) = −v D | γ , as vector fields along γ.(The minus sign arises since the Diff Z ( C)-action on D(C) comes from the action on the source of γ : C → RP(1).)(d) We regard u = (u 1 u 2 ) ⊤ as a function on D(C) (more precisely, its double cover D(C)) with values in R 2 ⊗ |Ω| −1/2 C

Remark 5. 1 .
(a) A choice of parametrization C ∼ = S 1 and trivialization Q ∼ = S 1 ×G identifies Gau(Q) = LG, with the affine action on A(Q) = Ω 1 (S 1 , g).(b) If G is not simply connected, then the group Gau(Q) ∼ =LG will be disconnected, and one may prefer to work with its identity component Gau 0 (Q) ∼ = L G (where G is the universal cover of G).The quotient map under the action of Gau 0 (Q) defines a lifted monodromy map P(Q) → G, and gives a Morita equivalence (56)

with the symplectic form induced by pr * 1 ̟
P −pr * 2 ̟ P .The restriction of this groupoid to vir * 1 (C) is given by arrows for which both source and target are in vir * 1 (C).But this is exactly G 1 = (D(C)× G + D(C))/G, and the pullback of the symplectic form is induced by pr * 1 ̟ D −pr * 2 ̟ D .
). Letting x vary, this defines dF ∈ Ω 1 (Diff + (S 1 ), |Ω| 0 S 1 ).Similarly, x → ∂F ∂x ∂x is a positive density.Similar to our conventions in 2.3, we denote this density by F ′ ∈ Ω 0 (Diff + (S 1 ), |Ω| 1 S 1 ).Lemma 4.8.The left-invariant Maurer-Cartan form on Diff + (S 1 ) is given by respectively.(a)Morita equivalence is an equivalence relation.It is reflexive: Q = G gives a Morita equivalence from G to itself.It is symmetric: If Q gives a Morita equivalence from G 1 to G 2 , then the opposite bimodule Q op (equal to Q as a manifold, with the roles of J 1 , J 2 interchanged, and the new bi-action A op (g 2 , p, g 1