Abelian extensions via prequantization

arXiv:0910.3906v1 [math.DG] 20 Oct 2009

Abelian extensions via prequantization Cornelia Vizman West University of Timisoara, Department of Mathematics Bd. V.Parvan 4, 300223-Timisoara, Romania e-mail: [email protected]

Abstract We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2–form ω (ω–invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2–form ω up to a linear isomorphism (ω–equivariant diffeomorphisms). Every abelian extension of a simply connected Lie group can be obtained as the pull-back of such a prequantization abelian extension.

Keywords: prequantization, diffeomorphism group, flux 1–cocycle, abelian extension MSC: 22E65, 58B20

1

Introduction

On the universal cover of the identity component of the group Symp(M, ω) of symplectic diffeomorphisms of a symplectic manifold (M, ω), one defines the symplec^ tic flux [MS07]. It is the group homomorphism S˜ω : Symp(M, ω)0 → H 1 (M, R), R1 S˜ω ([ϕ]) = 0 [iηt ω]dt, where [ϕ] is the homotopy class of a path ϕt in Symp(M, ω) starting at the identity and ηt is the time dependent vector field dtd ϕt ◦ ϕ−1 t . The 1 flux subgroup Π = S˜ω (π1 (Symp(M, ω))) of H (M, R) is discrete [O06], hence S˜ω descends to a Lie group homomorphism Sω : Symp(M, ω)0 → H 1 (M, R)/Π, with kernel the group of hamiltonian diffeomorphisms [B78]. When ω has an integral cohomology class, then there exists a principal circle bundle q : P → M and a principal connection 1–form θ ∈ Ω1 (P ) (a contact form in the symplectic case) with curvature ω. Let h(ℓ) denote the holonomy around the loop ℓ in M. The identity component of the group Hol(M, ω) = {ϕ ∈ Diff(M) : ∀ℓ ∈ C ∞ (S 1 , M), h(ϕ ◦ ℓ) = h(ℓ)}

1

of holonomy preserving diffeomorphisms of M is the group Ham(M, ω) of hamiltonian diffeomorphisms. There are two prequantization central extensions integrating 0 → R → C ∞ (M) → Xham (M, ω) → 0, the natural central extension of the Lie algebra of hamiltonian vector fields, namely 1 → T → Quant(P, θ) → Hol(M, ω) → 1 1 → T → Quant(P, θ)0 → Ham(M, ω) → 1,

(1)

where T = R/Z and Quant(P, θ) denotes the group of quantomorphisms, i.e. the group of θ-preserving automorphisms of the principal bundle P [K70] [S70] [RS81]. The above mentioned results where generalized to vector valued closed 2–forms in [NV03]. Let M be a connected smoothly paracompact manifold, possibly infinite dimensional, and ω a closed 2–form on M with values in a Mackey complete locally convex space V . Let Diff(M, ω) be the group of ω–invariant diffeomorphisms and X(M, ω) the Lie algebra of ω–invariant vector fields. A flux homomorphism integrating the Lie algebra homomorphism flux : X(M, ω) → H 1 (M, V ),

flux(η) = [iη ω],

can be defined if ω has a discrete period group Γ ⊂ V . It is the group homomorphism Flux : Diff(M, ω)0 → H 1 (M, V )/H 1(M, Γ), defined similarly to the symplectic flux Sω . The identity component Diff ex (M, ω) of the kernel of Flux is called the group of exact ω–invariant diffeomorphisms. In the symplectic setting it coincides with the group of hamiltonian diffeomorphisms. Let A be the abelian Lie group V /Γ. Assuming M is smoothly paracompact, there exists a principal A-bundle q : P → M and a principal connection 1–form θ ∈ Ω1 (P, V ) with curvature ω ∈ Ω2 (M, V ). Again the group Hol(M, ω) of holonomy preserving diffeomorphisms is a subgroup of Diff(M, ω) and its identity component is Diff ex (M, ω). The prequantization central extensions (1) corresponding to a vector valued 2–form are 1 → A → Diff(P, θ)A → Hol(M, ω) → 1 1 → A → Diff(P, θ)A 0 → Diff ex (M, ω) → 1,

(2)

where Diff(P, θ)A is the group of θ-preserving automorphisms of the principal bundle P . Let Diff(P )A denote the group of automorphisms of P . Both central extensions (2) are contained in the abelian extension 1 → C ∞ (M, A) → Diff(P )A → Diff(M)[P ] → 1

(3)

of the group Diff(M)[P ] of diffeomorphisms preserving the isomorphism class [P ] of the principal bundle P . In the same context (a closed V –valued 2–form ω on M, curvature of a principal bundle P → M), we introduce the group of diffeomorphisms preserving the 2

holonomy up to a linear isomorphism of V , subgroup of the group of ω–equivariant diffeomorphisms. We show equivariant versions of (2), which we call prequantization abelian extensions, as well as a non-abelian extension generalizing (3). The group of ω–equivariant diffeomorphisms (diffeomorphisms preserving ω up to a linear isomorphism of V ): Diff eq (M, ω) = {(ϕ, u) ∈ Diff(M) × GL(V ) : ϕ∗ ω = u · ω} contains the group Diff(M, ω) of ω–invariant diffeomorphisms as a subgroup. The flux homomorphism Flux can be extended to a flux 1–cocycle Fluxeq : Diff eq (M, ω)0 → H 1 (P, V )/H 1(P, Γ) for the Diff eq (M, ω)–action induced by the natural GL(V )–action on V . It integrates the Lie algebra 1–cocycle fluxeq : Xeq (M, ω) → H 1 (P, V ),

fluxeq (η, γ) = [q ∗ iη ω − γ · θ]

defined on the Lie algebra of ω–equivariant vector fields Xeq (M, ω) := {(η, γ) ∈ X(M) × gl(V ) : Lη ω = γ · ω}. Its kernel is the Lie subalgebra of hamiltonian vector fields. All equi-hamiltonian functions h : P → V have to be almost A–invariant in the sense that for all a ∈ A, the function h − h ◦ ρ(a) is constant on P , where ρ denotes the principal A–action eq on P . The identity component Diff eq is called the ex (M, ω) of the kernel of Flux group of equi-hamiltonian diffeomorphisms. The group Holeq (M, ω) of diffeomorphisms of M preserving the holonomy up to a linear isomorphism of V is another subgroup of Diff eq (M, ω) with identity component Diff eq ex (M, ω). One can write prequantization abelian extensions containing the prequantization central extensions (2) 1 → A → Diff eq (P, θ)A → Holeq (M, ω) → 1 eq 1 → A → Diff eq (P, θ)A 0 → Diff ex (M, ω) → 1,

(4)

where Diff eq (P, θ)A is the group of projectable diffeomorphisms of P preserving θ up to a linear isomorphism of V , called equi-quantomorphisms. Unlike the quantomorphisms, which are A–equivariant diffeomorphisms of P , the equi-quantomorphisms are almost A–equivariant diffeomorphisms in the following sense: if ψ is an equi-quantomorphism with ψ ∗ θ = u · θ for u ∈ GL(V ) (uniquely determined by ψ), then ψ ◦ ρ(a) = ρ(¯ u(a)) ◦ ψ,

a ∈ A,

where u¯ is the unique group automorphism of A satisfying u¯ ◦ exp = exp ◦u. The group Diff eq (P )A of almost equivariant diffeomorphisms of P is a non-abelian extension (5) 1 → CA∞ (P, A) → Diff eq (P )A → Diff(M)[P ] → 1. 3

of Diff(M)[P ] by the group of almost A–invariant smooth maps from P to A CA∞ (P, A) = {f ∈ C ∞ (P, A) : ∀a ∈ A, (f ◦ ρ(a))f −1 constant on P } with group multiplication (f1 · f2 )(y) = f1 (ρ(y, f2 (y)))f2(y). The group of A– invariant smooth maps from P to A, identified with C ∞ (M, A), is an abelian subgroup of CA∞ (P, A). The non-abelian extension (5) contains the abelian extensions (4) and (3). Let ω be a closed V –valued 2–form on a possibly infinite dimensional manifold M. A Lie group G with a smooth hamiltonian action λ : G → Diff ex (M, ω) has a ˆ by A, obtained by pulling back the prequantization central Lie group extension G ˆ central extension (2) and the G–action λ lifts to a smooth G–action on P by quantomorphisms [NV03]. Examples include the Bott-Virasoro group and Ismagilov’s central extension of the group of exact volume preserving diffeomorphisms integrating Lichnerowicz 2–cocyles. A similar result holds for abelian extensions and is explained below. An equi-hamiltonian action of a Lie group G consists of a smooth action λ on M together with a linear action b on V , such that (λ, b) : G → Diff eq ex (M, ω) ⊂ ˆ Diff(M) × GL(V ). It determines an abelian Lie group extension G of G by A ˙ ˙ ))(x0 ) on g, where integrating the Lie algebra 2–cocycle (X, Y ) 7→ −ω(λ(X), λ(Y ˙ x0 ∈ M is fixed and λ : g → X(M) denotes the infinitesimal action. This extension is obtained by pulling back the prequantization abelian extension (4). ˆ The G–action lifts to a smooth G–action on P by equi-quantomorphisms. Examples include abelian extensions of the group of diffeomorphisms of S 1 or S 2 , and abelian extensions of the group of volume preserving diffeomorphisms of a compact manifold. ˆ → G → 1 of a simply connected Every abelian Lie group extension 1 → A → G Lie group G by a G–module A = V /Γ can be obtained in this way. Let gˆ = g⋉σ V be the corresponding abelian Lie algebra extension, which is defined by the V – valued 2–cocycle σ on g, and let p : ˆg → V denote the projection on the second ˆ is a principal A-bundle with principal connection θ = peq , the factor. Then G ˆ ˆ with identity value p, and with curvature G–equivariant V –valued 1–form on G eq ω = −σ , the closed G–equivariant V –valued 2–form on G with identity value −σ. Let λ denote the left translation on G and b the linear G–action on V induced by the G–module structure of A. Then the G–action (λ, b) is equi-hamiltonian and determines the given abelian Lie group extension of G. The author thanks Karl-Hermann Neeb for many very useful comments and suggestions.

2

Flux 1–cocycle

Let M be a connected smoothly paracompact manifold, possibly infinite dimensional, and ω a closed 2–form on M with values in a Mackey complete locally 4

convex space V . The Lie algebra of ω–equivariant vector fields on M, Xeq (M, ω) := {(η, γ) ∈ X(M) × gl(V ) : Lη ω = γ · ω}, with Lie bracket [(η1 , γ1), (η2 , γ2)] = (−[η1 , η2 ], γ1 γ2 − γ2 γ1 ),

(6)

is a Lie subalgebra of the direct product Lie algebra X(M) × gl(V ). We take the negative sign convention (as in the Lie algebra X(M) of the group of diffeomorphisms of M). The Lie algebra of ω–invariant vector fields with the opposite bracket sits in Xeq (M, ω) as a Lie subalgebra ι : X(M, ω) → Xeq (M, ω),

ι(η) = (η, 0).

Remark 1. Let V0 be the closure of the image of ω : T M ×M T M → V . The restriction of γ to V0 is uniquely determined by η. When V0 = V , then γ is determined by η and one can identify the Lie algebra Xeq (M, ω) with its image on the first factor: {η ∈ X(M) : ∃γ ∈ gl(V ) s.t. Lη ω = γ · ω}. When V0 6= V , an ω–invariant vector field η can determine other ω–equivariant vector fields beside (η, 0), namely (η, γ) with γ|V0 = 0. Remark 2. The period group Γ of the closed V -valued 2–form ω is the image of the homomorphism H2 (M, R) → V determined by integrating ω. It is a subgroup of V0 and for any (η, γ) R∈ Xeq (M, R ω) the restriction of γ to Γ is trivial. Indeed, for any 2–cycle σ in M, γ( σ ω) = σ Lη ω = 0. In particular γ = 0 if V is generated by Γ, so ι(X(M, ω)) = Xeq (M, ω) in this case. For a closed R–valued 2–form ω, the Lie algebra of ω–equivariant vector fields is strictly bigger than the Lie algebra of ω–invariant vector fields if and only if Γ = 0, i.e. ω is exact. The projection on the second factor, (η, γ) ∈ Xeq (M, ω) 7→ γ ∈ gl(V ),

(7)

is a Lie algebra homomorphism, so V becomes a Xeq (M, ω)–module in a natural way. Restricted to the image ι(X(M, ω)), the action is trivial. If the period group Γ ⊂ V of ω is discrete, then A = V /Γ is an abelian Lie group with abelian Lie algebra V , and there exists a principal A–bundle q : P → M with connection form θ ∈ Ω1 (P, V ) and curvature form ω ∈ Ω2 (M, V ). Let exp : V → A be the canonical projection, ρ the principal A–action and ρ˙ : V → X(P ) its infinitesimal action. In particular dθ = q ∗ ω and iρ(v) ˙ θ = v for all v ∈ V . Proposition 3. The linear map fluxeq : (η, γ) ∈ Xeq (M, ω) 7→ [q ∗ iη ω − γ · θ] ∈ H 1 (P, V ) 5

(8)

is a Lie algebra 1–cocycle for the natural Xeq (M, ω)–module structure on H 1 (P, V ) induced by the Xeq (M, ω)–action (7) on V . Its cohomology class [fluxeq ] ∈ H 1 (Xeq (M, ω), H 1(P, V )) is independent of the choice of the connection θ. Proof. The 1–form q ∗ iη ω − γ · θ on P is closed for any ω–equivariant vector field (η, γ) because Lη ω = γ · ω and q ∗ ω = dθ. For (η1 , γ1 ), (η2 , γ2 ) ∈ Xeq (M, ω), fluxeq ([(η1 , γ1 ),(η2 , γ2)]) = −[q ∗ i[η1 ,η2 ] ω + (γ1 γ2 − γ2 γ1 ) · θ] = [d(q ∗ ω(η1 , η2 )) + q ∗ iη2 Lη1 ω − q ∗ iη1 Lη2 ω − γ1 γ2 · θ + γ2 γ1 · θ] = [q ∗ iη2 (γ1 · ω) − γ1 γ2 · θ] − [q ∗ iη1 (γ2 · ω) − γ2 γ1 · θ] = γ1 · fluxeq (η2 , γ2 ) − γ2 · fluxeq (η1 , γ1 ), so the 1–cocycle condition for fluxeq is satisfied for the natural Xeq (M, ω)–action. Two connection 1–forms on P differ by the pull-back q ∗ α of a closed V –valued 1–form α on M. Then the corresponding flux 1–cocycles differ by the linear map (η, γ) 7→ γ · [q ∗ α], which is a 1–coboundary on the Lie algebra Xeq (M, ω). The 1–cocycle fluxeq is called the infinitesimal flux 1–cocycle. Its kernel, deeq noted by Xeq ex (M, ω), is a Lie subalgebra of X (M, ω) and is called the Lie algebra of equi-hamiltonian vector fields on (M, ω). We say that h ∈ C ∞ (P, V ) is an equi-hamiltonian function for the equi-hamiltonian vector field (η, γ) ∈ Xeq ex (M, ω) if q ∗ iη ω − γ · θ = dh. (9) Remark 4. The infinitesimal flux homomorphism is defined on the Lie algebra of ω–invariant vector fields X(M, ω) by flux : η ∈ X(M, ω) 7→ [iη ω] ∈ H 1 (M, V ). The kernel Xex (M, ω) of flux is the ideal of exact ω–invariant vector fields (hamiltonian vector fields when ω is symplectic). Because fluxeq ◦ι = q ∗ ◦ flux, the inclusion ι descends to an inclusion ι : Xex (M, ω) → Xeq ex (M, ω). Not every smooth V –valued function on P can play the role of an equi∞ hamiltonian function. We denote by Cadm (P, V ) the space of all possible equihamiltonian functions, also called admissible functions. ∞ Proposition 5. If h ∈ Cadm (P, V ) is an equi-hamiltonian function for the equihamiltonian vector field (η, γ), then there exists a group homomorphism γ¯ : A → V with γ¯ ◦ exp = γ, such that for all a ∈ A and v ∈ V :

1. Lρ(v) ˙ h = −γ(v). 2. h − h ◦ ρ(a) = γ¯ (a).

6

Proof. Since h is an equi-hamiltonian function for the equi-hamiltonian vector field (η, γ), we have that dh = q ∗ iη ω − γ · θ. Then from Lρ(v) ˙ h = iρ(v) ˙ dh = −γ(v) we get the identity 1. From q ◦ ρ(a) = q and ρ(a)∗ θ = θ follows that ρ(a)∗ dh = dh, so h − h ◦ ρ(a) is a constant function on the connected manifold P . This assures the existence of a group homomorphism γ¯ : A → V satisfying the identity 2. From that we easily get that γ¯ ◦exp ∈ gl(V ). To show that γ¯ ◦exp = γ, we differentiate at t = 0 the identity h − h ◦ ρ(exp tv) = γ¯ (exp tv) for v ∈ V , and we obtain that Lρ(v) ¯ (v). ˙ h = −T1 γ Together with identity 1, this gives γ = T1 γ¯ = T0 (¯ γ ◦ exp) = γ¯ ◦ exp. Defining the space of almost A–invariant functions as CA∞ (P, V ) = {h ∈ C ∞ (P, V ) : ∀a ∈ A, h − h ◦ ρ(a) = constant on P },

(10)

∞ the proposition above says that Cadm (P, V ) ⊂ CA∞ (P, V ). It follows that there exists γh ∈ gl(V ) uniquely determined by the almost A–invariant function h such that Lρ(v) ˙ h = −γh (v) for all v ∈ V . The group of ω–equivariant diffeomorphisms

Diff eq (M, ω) := {(ϕ, u) ∈ Diff(M) × GL(V ) : ϕ∗ ω = u · ω} is a subgroup of the direct product group Diff(M)×GL(V ). The second projection (ϕ, u) ∈ Diff eq (M, ω) 7→ u ∈ GL(V ) is a group homomorphism, so V becomes a natural Diff eq (M, ω)–module. The group of ω–equivariant diffeomorphisms contains the group of ω–invariant diffeomorphisms as a subgroup via the injective homomorphism i : Diff(M, ω) → Diff eq (M, ω),

i(ϕ) = (ϕ, 1V ).

The restriction of the second component u ∈ GL(V ) of the ω–equivariant diffeomorphism (ϕ, u) to V0 ⊆ V (defined in Remark 1) is determined by its first component ϕ ∈ Diff(M). When V0 = V , then one identifies Diff eq (M, ω) with its projection on the first factor, the group {ϕ ∈ Diff(M) : ∃u ∈ GL(V ) s.t. ϕ∗ ω = u · ω}. This is the case for a closed R2 –valued 2–form ω = (ω1 , ω2 ) on a compact manifold M with ω1 6= 0 and ω2 6= 0 (so V0 = V = R2 ). E.g. the flow of the ω–equivariant vector field η satisfying Lη ω1 = −ω2 and Lη ω2 = ω1 , if it exists, is a 1–parameter subgroup ϕt of ω–equivariant diffeomorphisms of M satisfying ϕ∗t ω1 = (cos t)ω1 − (sin t)ω2 and ϕ∗t ω2 = (sin t)ω1 + (cos t)ω2 . Remark 6. There is a natural (Diff(M) × GL(V ))–action on the vector space Ω2 (M, V ) of V –valued 2–forms on M: (ϕ, u) · ω = u · ((ϕ−1 )∗ ω)

7

with infinitesimal action of the Lie algebra X(M) × gl(V ) with Lie bracket (6) given by (η, γ) · ω = −Lη ω + γ · ω. The isotropy group of a closed 2–form ω ∈ Ω2 (M, V ) coincides with the group of ω–equivariant diffeomorphisms Diff eq (M, ω), and its isotropy Lie algebra coincides with the Lie algebra of ω–equivariant vector fields Xeq (M, ω). A curve ϕ in Diff(M) is called a smooth curve if the corresponding map (t, x) 7→ (ϕ(t)(x), ϕ(t)−1 (x)) in M × M is smooth. Similarly a curve u in GL(V ) is smooth if the map (t, v) 7→ (u(t)(v), u(t)−1(v)) in V × V is smooth. Let Diff eq (M, ω)0 be the normal subgroup of those elements in Diff eq (M, ω) which can be connected to the identity by a smooth curve in Diff eq (M, ω) ⊂ Diff(M) × GL(V ). Remark 7. The abelian group A = V /Γ is a natural Diff eq (M, ω)0 –module. Indeed, eq for any 2–cycle σ in M andR for any R(ϕ, u) ∈ Diff (M, ω)0 , the 2–cycles σ and R ∗ ϕ(σ) are homologous, so u( σ ω) = σ ϕ ω = σ ω and u fixes the elements of the period group Γ. In particular u descends to a group automorphism u¯ of A, and the Diff eq (M)–action on V descends to an action on the abelian group A. If V is generated by Γ, then u = 1V and ϕ is ω–invariant for all ω–equivariant diffeomorphisms (ϕ, u). Proposition 8. The following equivalences hold for smooth paths ϕt in Diff(M) and ut in GL(V ) starting at the identity: (ϕt , ut ) ∈ Diff eq (M, ω)⇔ (δ l ϕt , δ l ut ) ∈ Xeq (M, ω) ⇔(δ r ϕt , δ r ut ) ∈ Xeq (M, ω). This follows from Remark 31 in the Appendix, where also the left and right derivative δ l and δ r are defined. In particular if the flow of an ω–equivariant vector field (η, γ) exists, then it consists of ω–equivariant diffeomorphisms. Lemma 9. For any loop ℓ in P and any smooth path of ω–equivariant diffeomorphisms (ϕt , ut ) starting at the identity, we define the 2–chain σ in M by σ(t, s) = ϕt (q(ℓ(s))), t, s ∈ [0, 1], swept out by the loop q ◦ ℓ in M under the isotopy ϕt . Then Z Z 1 Z Z Z ∗ l ut · (q iδl ϕt ω − δ ut · θ)dt = ω − u · θ + θ. ℓ

Proof. Using Z

0

σ

ℓ

ℓ

ϕ∗t ω

= ut · ω, we compute Z 1 Z Z 1 1 ∗ ˙ ut · q iδl ϕt ω dt = ut · dt (q ∗ iδl ϕt ω)(ℓ(s))ds 0 ℓ 0 0 Z 1Z 1 ˙ = ut · ω(δ l ϕt (q(ℓ(s))), T q.ℓ(s))dsdt Z0 1 Z0 1 Z ˙ = ω(ϕ˙ t (q(ℓ(s))), T ϕt.T q.ℓ(s))dsdt = ω 0

0

σ

and the result follows. 8

The quotient space H 1 (P, V )/H 1(P, Γ) is a natural Diff eq (M, ω)0 –module because, as we have seen in Remark 7, Diff eq (M, ω)0 acts trivially on Γ ⊂ V , hence it acts trivially on H 1 (P, Γ) ⊂ H 1(P, V ). The map Fluxeq : Diff eq (M, ω)0 → H 1 (P, V )/H 1(P, Γ) Z 1 eq Flux (ϕ, u) = ut · fluxeq (δ l ϕt , δ l ut )dt mod H 1 (P, Γ) 0 hZ 1 i = ut · (q ∗ iδl ϕt ω − δ l ut · θ)dt mod H 1 (P, Γ),

(11)

0

for any piecewise smooth path of ω–equivariant diffeomorphisms (ϕt , ut) from the identity to (ϕ, u), is a well defined group 1–cocycle, called the flux 1–cocycle associated to the closed vector valued form ω with discrete period group Γ. The map Fluxeq is well defined because Lemma 9 implies that for a loop (ϕt , ut ) of ω–equivariant diffeomorphisms based at the identity, the integral over a loop ℓ R1 ∗ in P of the 1–form 0 ut · (q iδl ϕt ω − δ l ut · θ)dt is the integral of ω over a 2–cycle σ, hence it belongs to the group Γ of periods of ω. The 1–cocycle condition for Fluxeq is verified as in Proposition 34 from the Appendix. Remark 10. The flux homomorphism associated to a closed vector valued 2–form ω is defined by Flux : Diff(M, ω)0 → H 1 (M, V )/H 1 (M, Γ) Z 1 Z 1 1 Flux(ϕ) = [iδr ϕt ω]dt mod H (M, Γ) = [iδl ϕt ω]dt mod H 1 (M, Γ) (12) 0

0

for any smooth curve ϕt in Diff(M, ω) connecting the identity and ϕ. The group Diff ex (M, ω) = (Ker Flux)0 is called the group of exact ω–invariant diffeomorphisms. For a symplectic manifold (M, ω), the symplectic flux homomorphism Sω is obtained by the factorization of a smaller subgroup, Π ⊆ H 1 (M, Γ), called the flux subgroup, so Sω : Diff(M, ω)0 → H 1 (M, R)/Π. In this case the group of hamiltonian diffeomorphisms Ham(M, ω) coincides with Diff ex (M, ω), because Ker Sω = (Ker Flux)0 [NV03]. eq The group Diff eq ex (M, ω) = (Ker Flux )0 is called the group of equi-hamiltonian diffeomorphisms. The flux 1–cocycle Fluxeq and the flux homomorphism Flux are related by Fluxeq ◦i = q ∗ ◦ Flux, hence i descends to an injective homomorphism Diff ex (M, ω)) → Diff eq ex (M, ω). The next proposition follows from Corollary 37 in the Appendix.

Proposition 11. For any piecewise smooth path of ω–equivariant diffeomorl l eq phisms (ϕt , ut ), we have (ϕt , ut ) ∈ Diff eq ex (M, ω)⇔ (δ ϕt , δ ut ) ∈ Xex (M, ω)⇔ (δ r ϕt , δ r ut ) ∈ Xeq ex (M, ω). 9

Remark 12. In the special case when ω = dα for an α ∈ Ω1 (M, V ) (in particular the period group Γ = 0), the flux homomorphism is given by Flux(ϕ) = [ϕ∗ α−α] ∈ q H 1 (M, V ). To compute the flux 1–cocycle Fluxeq in this case, let P = M × V → M be the trivial V -bundle with principal connection 1–form θ = q ∗ α + θV and curvature ω, where θV = δ l (1V ) ∈ Ω1 (V, V ) stands for the Maurer-Cartan form on V . We get Fluxeq : Diff eq (M, ω) → H 1(P, V ),

Fluxeq (ϕ, u) = q ∗ [ϕ∗ α − u · α].

Indeed, let (ϕt , ut ) be a path of ω–equivariant diffeomorphisms joining the identity and (ϕ, u). From dtd [ϕ∗t α] = [ut · iδl ϕt ω] we obtain dtd [q ∗ ϕ∗t α − ut · θ] = ut · [q ∗ iδl ϕt ω − δ l ut · θ]. Integrating this cohomology class from 0 to 1 gives the expression of the flux cocycle Fluxeq (ϕ, u) = q ∗ [ϕ∗ α − u · α], because θ − q ∗ α = θV is an exact 1–form on P . In this case the group of equi-hamiltonian diffeomorphisms is ∗ Diff eq ex (M, ω) = {(ϕ, u) ∈ Diff(M) × GL(V ) : ϕ α − u · α exact}.

3

Infinitesimal equi-quantomorphisms and prequantization

Appropriate prequantization procedures have been developed for symplectic, presymplectic, Poisson and also Dirac manifolds [WZ05]. In this section we suggest a prequantization procedure for a closed vector valued 2–form in the equivariant setting. We consider again a principal A-bundle q : P → M for the abelian Lie group A = V /Γ, with principal A–action ρ, infinitesimal action ρ˙ : V → X(P ), principal connection θ ∈ Ω1 (P, V ) and curvature ω ∈ Ω2 (M, V ). By definition, the horizontal lift of a vector field η ∈ X(M) is the unique vector field η hor , q-related to η, satisfying iηhor θ = 0.

Infinitesimal quantomorphisms A vector field ξ ∈ X(P ) is called projectable if it is q-related to a vector field η ∈ X(M), and we denote η = q∗ ξ. Projectable vector fields can be characterized by T q ◦ ξ ◦ ρ(a) = T q ◦ ξ for all a ∈ A. The Lie algebra of vertical vector fields on P is a Lie subalgebra of the Lie algebra of projectable vector fields. To every function h ∈ C ∞ (P, V ) one associates the vertical vector field ρ(h) ˙ on ∞ P by ρ(h)(y) ˙ = ρ(h(y))(y). ˙ We endow C (P, V ) with a Lie bracket such that the ∞ injective mapping ρ˙ : C (P, V ) → X(P ) becomes a Lie algebra homomorphism for the opposite Lie bracket on X(P ). This leads to [h1 , h2 ] = Lρ(h ˙ 2 ) h1 − Lρ(h ˙ 1 ) h2 ,

10

(13)

because i[ρ(h ˙ 1 ),ρ(h ˙ 2 )] θ = Lρ(h ˙ 1 ) h2 − Lρ(h ˙ 2 ) h1 . The Lie algebra of projectable vector fields Xproj (P ) is a non-abelian Lie algebra extension of X(M) ρ˙

q∗

0 → C ∞ (P, V ) → Xproj (P ) → X(M) → 0.

(14)

The bracket (13) on pull-back functions q ∗ f for f ∈ C ∞ (M, V ) vanishes, so C (M, V ) is an abelian Lie subalgebra of C ∞ (P, V ). The Lie algebra X(P )A of A–invariant vector fields on P (infinitesimal automorphisms of P ) consists of vector fields ξ such that ρ(a)∗ ξ = ξ for all a ∈ A, or equivalently Lρ(v) ˙ ξ = 0 for all v ∈ V . Restricting (14) to the Lie algebra X(P )A we obtain an abelian Lie algebra extension ∞

ρ˙

q∗

0 → C ∞ (M, V ) → X(P )A → X(M) → 0

(15)

defined by the curvature form ω on M viewed as a Lie algebra 2–cocycle on X(M) with values in the X(M)–module C ∞ (M, V ). The Lie algebra of infinitesimal quantomorphisms is X(P, θ)A = {ξ ∈ X(P )A : Lξ θ = 0} = {ξ ∈ Xproj (P ) : Lξ θ = 0}. We check the non-trivial inclusion: if q∗ ξ = η and Lξ θ = 0, then [ξ, ρ(v)] ˙ = 0 for all v ∈ V because it is a vertical vector field (q-related to [η, 0] = 0) and i[ξ,ρ(v)] θ = Lξ v = 0, so ξ ∈ X(P )A . ˙ Restricting (15) further to X(P, θ)A , we get the central extension ρ˙

q∗

0 → V → X(P, θ)A → Xex (M, ω) → 0,

(16)

the vector valued analog of the prequantization Lie algebra central extension. Indeed, let ξ ∈ X(P, θ)A . Both ξ and θ ∈ Ω1 (P, V ) being A–invariant, the function iξ θ ∈ C ∞ (P, V ) is A–invariant too, hence it descends to a function q∗ iξ θ on M. Now Lξ θ = 0 and q∗ ξ = η imply iη ω = d(−q∗ iξ θ), so ξ is q-related to the hamiltonian vector field η. On the other hand ρ(h) ˙ ∈ X(P, θ)A implies 0 = Lρ(h) θ = dh, so ˙ the only vertical infinitesimal quantomorphisms are of the form ρ(v), ˙ v ∈ V . The Lie algebra cohomology class describing this extension is the class of the 2–cocycle on Xex (M, ω) given by (η1 , η2 ) 7→ −ω(η1 , η2 )(x0 ), x0 ∈ M. A function f ∈ C ∞ (M, V ) is a hamiltonian function for the vector field ηf if iηf ω = df , so the hamiltonian functions on M have to be constant along the leaves ∞ of Ker ω ⊂ T M. They form the subspace of admissible functions Cadm (M, V ). A hamiltonian vector field associated to such an admissible function f can be determined only up to a section in Γ(Ker ω). Remark 13. The linear map ∞ (M, V ) ξ ∈ X(P, θ)A 7→ −q∗ iξ θ ∈ Cadm

11

(17)

is surjective. Indeed, given an admissible function f , there exists a hamiltonian vector field ηf , and the vector field ξf = ηfhor − ρ(q ˙ ∗ f ) is an infinitesimal quantomorphism with iξf θ = −q ∗ f . The kernel of (17) is Γ(Ker ω)hor . ∞ In the symplectic case (Ker ω = 0 and A = T) we have Cadm (M) = C ∞ (M) and the hamiltonian vector field ηf is uniquely determined by its hamiltonian function f . The linear map (17) is a bijection with inverse f ∈ C ∞ (M) 7→ ξf := ηfhor − (q ∗ f )E ∈ X(P, θ)A ,

(18)

E = ρ(1) ˙ denoting the infinitesimal generator of the circle action on P . This is the symplectic prequantization, in the construction due to Souriau, which associates to each function f ∈ C ∞ (M) the infinitesimal quantomorphism ξf on P [S70].

Infinitesimal equi-quantomorphisms For the equivariant setting we observe that the space CA∞ (P, V ) of almost A– invariant functions defined in (10) endowed with the Lie bracket [h1 , h2 ] = γh2 ◦ h1 − γh1 ◦ h2 ,

(19)

is a Lie subalgebra of C ∞ (P, V ) with Lie bracket (13), because Lρ(h ˙ 1 ) h2 = −γh2 ◦h1 ∞ for h1 , h2 ∈ CA (P, V ). On the other hand Xeq (P )A = {ξ ∈ X(P ) : ∃γ ∈ gl(V ) s.t. ∀v ∈ V, Lρ(v) ˙ ˙ ξ = ρ(γ(v))}

(20)

is the Lie algebra of almost A–invariant vector fields. Its characterization as Xeq (P )A = {ξ ∈ X(P ) : ∃¯ γ : A → V s.t. ∀a ∈ A, ρ(a)∗ ξ − ξ = ρ(¯ ˙ γ (a))}

(21)

can be deduced from the identity d (ρ(exp tv)∗ ξ dt

− ξ − ρ(¯ ˙ γ (exp tv))) = ρ(exp tv)∗ (Lρ(v) ˙ ˙ ξ − ρ(γ(v))),

γ¯ being a group homomorphism with γ¯ ◦ exp = γ. Every almost A–invariant vector field is projectable, since applying T q to the characterizing relation in (21) we get T q ◦ ξ ◦ ρ(a) = T q ◦ ξ for all a ∈ A. We restrict the extension (14) to the Lie algebra Xeq (P )A of almost A–invariant vector fields, obtaining a new Lie algebra extension ρ˙

q∗

0 → CA∞ (P, V ) → Xeq (P )A → X(M) → 0,

(22)

with CA∞ (P, V ) the space of almost A–invariant functions defined in (10). This can be seen as follows: for an arbitrary function h ∈ C ∞ (P, V ), the necessary and sufficient condition for the vertical vector field ρ(h) ˙ to be almost A–invariant is Lρ(v) ˙ = ρ(γ(v)); ˙ but we know from (13) that Lρ(v) ˙ = −ρ(L ˙ ρ(v) ˙ ρ(h) ˙ ρ(h) ˙ h), so the ∞ condition above becomes Lρ(v) ˙ h = −γ(v), which means h ∈ CA (P, V ). 12

To pass to an abelian extension by V , we have to consider the Lie algebra Xeq (P, θ)A = {ξ ∈ Xproj (P ) : ∃γξ ∈ gl(V ) s.t. Lξ θ = γξ · θ} of infinitesimal equi-quantomorphisms. The linear map γξ ∈ gl(V ) is determined eq A by ξ because γξ (v) = γξ (iρ(v) ˙ θ) = iρ(v) ˙ Lξ θ. The Lie algebra X (P, θ) contains as A a Lie subalgebra the Lie algebra X(P, θ) of infinitesimal quantomorphisms. The infinitesimal equi-quantomorphism ξ determines an ω–equivariant vector field (q∗ ξ, γξ ) on M. Moreover (q∗ ξ, γξ ) is an equi-hamiltonian vector field for the ∞ equi-hamiltonian function h = −iξ θ ∈ Cadm (P, V ), because dh = −diξ θ = iξ dθ − Lξ θ = q ∗ iq∗ ξ ω − γξ · θ. An equivariant version of Remark 13 holds. Proposition 14. The linear map ∞ ξ ∈ Xeq (P, θ)A 7→ −iξ θ ∈ Cadm (P, V ).

(23)

is surjective with kernel Γ(Ker ω)hor . ∞ Proof. Given h ∈ Cadm (P, V ), let (ηh , γh ) be an equi-hamiltonian vector field with equi-hamiltonian function h and let ξh := ηhhor − ρ(h). ˙ Then ξh ∈ Xeq (P, θ)A ∗ because Lξh θ = Lηhhor θ − Lρ(h) θ = q iηh ω − dh = γh · θ. The linear correspondence ˙ (23) is surjective since −iξh θ = h. Let ξ ∈ Xeq (P, θ)A be an element in the kernel of (23). Then ξ is a horizontal lift: there exists η ∈ X(M) such that ξ = η hor . But ξ is an infinitesimal quantomorphism, so γξ · θ = Lηhor θ = iηhor dθ = q ∗ iη ω. We get γξ = 0, so iη ω = 0, which means η ∈ Γ(Ker ω).

The next proposition shows the inclusion Xeq (P, θ)A ⊂ Xeq (P )A . Proposition 15. Any infinitesimal equi-quantomorphism with Lξ θ = γξ · θ satisfies Lρ(v) ˙ ξ (v)) for all v ∈ V . ˙ ξ = ρ(γ Proof. The infinitesimal equi-quantomorphism ξ is projectable and ρ(v) ˙ is vertical, so Lρ(v) ˙ ξ] is also vertical. A short computation using Lξ θ = γξ · θ gives ˙ ξ = [ρ(v), i[ρ(v),ξ] θ = γ (v). These two facts imply Lρ(v) ˙ ξ (v)). ˙ ξ ˙ ξ = ρ(γ Theorem 16. The Lie algebra of infinitesimal equi-quantomorphisms is an abelian extension of the Lie algebra Xeq ex (M, ω) of equi-hamiltonian vector fields by eq the natural Xex (M, ω)–module V . An abelian Lie algebra 2–cocycle on Xeq ex (M, ω) defining this abelian extension is ((η1 , γ1 ), (η2 , γ2 )) 7→ −ω(η1 , η2 )(x0 ), for any fixed element x0 ∈ M.

13

Proof. We show that the following sequence of Lie algebras is exact: ρ˙

p

0 → V → Xeq (P, θ)A → Xeq ex (M, ω) → 0,

(24)

where p(ξ) = (q∗ ξ, γξ ). The injectivity of ρ˙ is clear. The surjectivity of p follows from the surjectivity of (23). The inclusion ρ(V ˙ ) ⊆ Ker p follows from p ◦ ρ˙ = 0. For the reversed inclusion let ξ ∈ Ker p ⊂ Xeq (P, θ)A . Then Lξ θ = 0 and ξ = ρ(h) ˙ ∞ for some h ∈ C (P, V ). From Lρ(h) θ = dh and from the connectedness of P ˙ follows that ξ ∈ ρ(V ˙ ). The induced action of Xeq ex (M, ω) on V is the natural one because from Proposition 15 we get [ρ(v), ˙ ξ] = ρ(γ ˙ ξ (v)) for all v ∈ V and eq A ξ ∈ X (P, θ) . We determine the 2–cocycle defined with the linear section s of (24) given by s(η, γ) = η hor − ρ(h), ˙ where h is the unique equi-hamiltonian function of the equi-hamiltonian vector field (η, γ) vanishing at a fixed point y0 ∈ q −1 (x0 ). First we observe that given the equi-hamiltonian vector fields (η1 , γ1 ) and (η2 , γ2 ) with equi-hamiltonian functions h1 and h2 vanishing at y0 , the equi-hamiltonian function vanishing at y0 for the bracket (−[η1 , η2 ], γ1 γ2 − γ2 γ1 ) is γ1 ◦ h2 − γ2 ◦ h1 + q ∗ ω(η1 , η2 ) − ω(η1 , η2 )(x0 ). Indeed, d(γ1 ◦ h2 − γ2 ◦ h1 + q ∗ ω(η1 , η2 )) = γ1 · (q ∗ iη2 ω − γ2 · θ) − γ2 · (q ∗ iη1 ω − γ1 · θ) + q ∗ diη2 iη1 ω = q ∗ (iη2 Lη1 ω − iη1 Lη2 ω + diη2 iη1 ω) − (γ1 ◦ γ2 − γ2 ◦ γ1 ) · θ = −q ∗ i[η1 ,η2 ] ω − (γ1 ◦ γ2 − γ2 ◦ γ1 ) · θ. Since ω is the curvature of the principal bundle P , the identity [η1hor , η2hor ] − [η1 , η2 ]hor = −ρ(q ˙ ∗ ω(η1 , η2 )) holds by Proposition 5. The equi-hamiltonian functions h1 and h2 belong to CA∞ (P, V ), so by (19) we have [ρ(h ˙ 1 ), ρ(h ˙ 2 )] = ρ(γ ˙ 2 ◦h1 − hor ∗ γ1 ◦ h2 ). Using also the fact that [ρ(h ˙ 1 ), η2 ] = −ρ(L ˙ η2hor h1 ) = −ρ(q ˙ ω(η1 , η2 )), we compute [s(η1 , γ1 ), s(η2 , γ2 )] − s([(η1 , γ1 ), (η2 , γ2 )]) = −[η1hor − ρ(h ˙ 1 ), η2hor − ρ(h ˙ 2 )] + [η1 , η2 ]hor − ρ(γ ˙ 2 ◦ h1 − γ1 ◦ h2 − q ∗ ω(η1 , η2 ) + ω(η1 , η2 )(x0 )) = [ρ(h ˙ 1 ), η2hor ] − [ρ(h ˙ 2 ), η1hor ] − [ρ(h ˙ 1 ), ρ(h ˙ 2 )] ∗ − ρ(γ ˙ 2 ◦ h1 ) + ρ(γ ˙ 1 ◦ h2 ) + 2ρ(q ˙ ω(η1 , η2 )) − ρ(ω(η ˙ 1 , η2 )(x0 )) = −ρ(ω(η ˙ 1 , η2 )(x0 )), thus obtaining a Lie algebra 2–cocycle for the abelian extension (24). Remark 17. Under the assumption V = V0 , the closure of the image of ω : T M ×M T M → V , the Lie algebra of equi-hamiltonian vector fields can be identified with its projection on the first factor: ∗ Xeq ex (M, ω) = {η ∈ X(M) : ∃γ ∈ gl(V ) s.t. q iη ω − γ · θ exact}.

In this case the abelian extension (24) can be seen as a restriction of (22). 14

4

Prequantization abelian group extension

Given f ∈ C ∞ (P, A) we denote by ρ(f ) the fiber preserving smooth map y ∈ P 7→ ρ(f )(y) = ρ(y, f (y)) ∈ P. ∞ The space Cfiber (P, P ) of fiber preserving smooth maps has a monoid structure with respect to the composition of maps. So there is a unique monoid structure on ∞ C ∞ (P, A) such that the bijective mapping ρ : C ∞ (P, A) → Cfiber (P, P ) becomes an isomorphism, namely

(f1 · f2 )(y) = f1 (ρ(y, f2 (y)))f2(y).

(25)

The image by ρ of the group of invertible elements C ∞ (P, A)inver in C ∞ (P, A) is ∞ the group of invertible elements in Cfiber (P, P ), i.e. the group of fiber preserving diffeomorphisms of P . The group Diff proj (P ) of projectable diffeomorphisms is the group of diffeomorphisms of P which map fibers to fibers, i.e. those ψ ∈ Diff(P ) such that q ◦ ψ = ϕ ◦ q for some ϕ ∈ Diff(M). Projectable diffeomorphisms of P can be characterized by q ◦ ψ ◦ ρ(a) = q ◦ ψ for all a ∈ A. We write ϕ = q∗ ψ and the diffeomorphism ϕ belongs to Diff(M)[P ] , the group of diffeomorphisms preserving the isomorphism class [P ] of the principal bundle P . The exact sequence of groups ρ

q∗

1 → C ∞ (P, A)inver → Diff proj (P ) → Diff(M)[P ] → 1

(26)

is a non-abelian group extension with infinitesimal version the non-abelian Lie algebra extension (14).

Invariant setting Let Diff(P )A be the group of A–equivariant diffeomorphisms of P , i.e. automorphisms of the principal bundle P . An abelian extension is obtained by restricting the previous exact sequence to the subgroup Diff(P )A ⊂ Diff proj (P ): ρ

q∗

1 → C ∞ (M, A) → Diff(P )A → Diff(M)[P ] → 1, with infinitesimal version the abelian Lie algebra extension (15). Indeed, ρ(f ) is A–equivariant if and only if f is A–invariant, so f = f¯◦ q for some f¯ ∈ C ∞ (M, A). A central extension can be obtained by a further restriction to the group of quantomorphisms Diff(P, θ)A = {ψ ∈ Diff(P )A : ψ ∗ θ = θ} = {ψ ∈ Diff proj (P ) : ψ ∗ θ = θ}. The quantomorphisms of P descend to holonomy preserving diffeomorphisms on M. Denoting by h(ℓ) ∈ A the holonomy around a loop ℓ in M, let Hol(M, ω) = {ϕ ∈ Diff(M) : ∀ℓ ∈ C ∞ (S 1 , M), h(ϕ ◦ ℓ) = h(ℓ)} 15

be the group of holonomy preserving diffeomorphisms. It is a subgroup of the group Diff(M, ω) of ω-preserving diffeomorphisms. Similar to the prequantization central extension (1) due to [K70] [S70] [RS81], one has a prequantization central extension for a vector valued 2–form [NV03]: ρ

q∗

1 → A → Diff(P, θ)A → Hol(M, ω) → 1.

(27)

For this we use Proposition 33 in the Appendix. Given f ∈ C ∞ (P, A), ρ(f )∗ θ = θ if and only if δ l (f ) = 0, so f is a constant ∈ A. Passing to connected components of the identity we get the other prequantization central extension ρ

q∗

1 → A → Diff(P, θ)A 0 → Diff ex (M, ω) → 1.

(28)

Equivariant setting The group of almost A–equivariant diffeomorphisms of P is Diff eq (P )A = {ψ ∈ Diff(P ) : ∃¯ uψ ∈ Aut(A) s.t. ψ ◦ ρ(a) = ρ(¯ uψ (a)) ◦ ψ, ∀a ∈ A}, where Aut(A) denotes the group of group automorphisms of A. One can describe Diff eq (P )A as the group of those diffeomorphisms ψ of P such that the vertical vector fields ρ(v) ˙ and ρ(u ˙ ψ (v)) are ψ-related. It is a subgroup of Diff proj (P ) because q ◦ ψ ◦ ρ(a) = q ◦ ρ(¯ uψ (a)) ◦ ψ = q ◦ ψ for all a ∈ A. The fiber preserving diffeomorphism ρ(f ) for f ∈ C ∞ (P, A) is almost A–equivariant if and only if ρ(f ) ◦ ρ(a) = ρ(¯ uρ(f ) (a)) ◦ ρ(f ), which can be written as −1 −1 f (y) f (ρ(y, a)) = a u¯ρ(f ) (a) for all y ∈ P . We define the set of almost A– invariant maps CA∞ (P, A) = {f ∈ C ∞ (P, A) : ∀a ∈ A, f −1 (f ◦ ρ(a)) constant on P }. For an almost A–invariant function h ∈ CA∞ (P, V ), exp ◦h is an almost A–invariant map with u¯f = exp ◦¯ γh . The existence of a unique u¯f ∈ Aut(A) such that f (y)−1f (ρ(y, a)) = a−1 u¯f (a) for all a ∈ A and y ∈ P follows easily. We observe that ρ(f ) ∈ Diff eq (P )A if and only if f is an almost A–invariant map with u¯f = u¯ρ(f ) . A group multiplication on CA∞ (P, A) is (f1 · f2 )(y) = f1 (y)¯ uf1 (f2 (y)), and it becomes a subgroup of C ∞ (P, A)inver with multiplication (25). Indeed, for f1 , f2 ∈ CA∞ (P, A), f1 (ρ(y, f2 (y)))f2(y) = f1 (y)¯ uf1 (f2 (y)). The abelian group ∞ C (M, A), identified with the group of A–invariant maps f : P → A, is a subgroup of CA∞ (P, A) (in this case u¯f = 1A ). Restricting (26) we obtain a non-abelian group extension with infinitesimal version (22): ρ

q∗

1 → CA∞ (P, A) → Diff eq (P )A → Diff(M)[P ] → 1. 16

(29)

The rest of this section is devoted to “integrate” the abelian Lie algebra extension (24) to a prequantization abelian extension. We define the group of equiquantomorphisms as Diff eq (P, θ)A = {ψ ∈ Diff proj (P ) : ∃uψ ∈ GL(V ) s.t. ψ ∗ θ = uψ · θ}. It contains the group Diff(P, θ)A of quantomorphisms as a subgroup. The linear isomorphism uψ is uniquely determined by ψ. When ϕ ∈ Diff(M) with q◦ψ = ϕ◦q, by differentiating the relation ψ ∗ θ = uψ · θ we get ϕ∗ ω = uψ · ω, hence (ϕ, uψ ) is an ω–equivariant diffeomorphism. The proof of the next proposition follows from Remark 31 in the Appendix. Proposition 18. For a smooth curve ψt in Diff(P ) starting at the identity we have ψt ∈ Diff eq (P, θ)A ⇔ δ l ψt ∈ Xeq (P, θ)A ⇔ δ r ψt ∈ Xeq (P, θ)A . In particular if the flow of an infinitesimal equi-quantomorphism exists, then it consists of equi-quantomorphisms. Proposition 19. The group Diff eq (P )A of almost A–equivariant diffeomorphisms contains the group Diff eq (P, θ)A of equi-quantomorphisms as a subgroup. More precisely, the deviation from A–equivariance of an equi-quantomorphism ψ is measured by the isomorphism uψ ∈ GL(V ). Proof. Let ψ be an equi-quantomorphism with ψ ∗ θ = uψ · θ. Then ψ −1 is an equi-quantomorphism too, with (ψ −1 )∗ θ = (uψ )−1 · θ. For any a ∈ A, the diffeomorphism ψ ◦ ρ(a) ◦ ψ −1 is fiber preserving and θ–invariant, in particular it is of the form ρ(f ) with f ∈ C ∞ (P, A)inver depending on a. From Proposition 33 in the Appendix, 0 = ρ(f )∗ θ − θ = δ l (f ), so that f is a constant denoted u¯(a) ∈ A. We obtain that ψ ◦ ρ(a) = ρ(¯ u(a)) ◦ ψ, so u¯ ∈ Aut(A). The infinitesimal version of this identity is T ψ ◦ ρ(v) ˙ = ρ(u(v)) ˙ ◦ ψ, where u ∈ GL(V ) with u¯ ◦ exp = exp ◦u. It remains to be shown that u = uψ . This follows from the above mentioned ∗ ∗ fact that ρ(v) ˙ and ρ(u(v)) ˙ are ψ-related: uψ (v) = iρ(v) θ) = u(v) ˙ (ψ θ) = ψ (iρ(u(v)) ˙ for all v ∈ V . The group Holeq (M, ω) of diffeomorphisms preserving the holonomy up to a group automorphism of A, called the group of almost holonomy preserving diffeomorphisms is Holeq (M, ω) = {(ϕ, u¯) ∈ Diff(M) ×Aut(A) : ∀ℓ ∈ C ∞ (S 1 , M), h(ϕ ◦ ℓ) = u¯(h(ℓ))}. The group Holeq (M, ω) acts in a natural way on the abelian group A. Adapting the idea of the proof of Theorem 2.7 in [NV03] to the equivariant setting, we will show that ρ p 1 → A → Diff eq (P, θ)A → Holeq (M, ω) → 1 (30) is an exact sequence of groups integrating the abelian Lie algebra extension (24). 17

Lemma 20. Given (ϕ, u ¯) ∈ Holeq (M, ω) and, for a fixed x0 ∈ M, a bijection ψx0 : q −1 (x0 ) → q −1 (ϕ(x0 )) satisfying ψx0 ◦ ρ(a) = ρ(¯ u(a)) ◦ ψx0 for all a ∈ A, there exists a unique equi-quantomorphism ψ of P extending ψx0 and descending to the diffeomorphism ϕ of M. Proof. Let Pt(c) : q −1 (x0 ) → q −1 (x) denote the parallel tansport map along a curve c from x0 to x in M. It defines a map ψx = Pt(ϕ ◦ c) ◦ ψx0 ◦ Pt(c)−1 : q −1 (x) → q −1 (ϕ(x))

(31)

which does not depend on the choice of c, because for every loop ℓ at x0 Pt(ϕ ◦ ℓ) ◦ ψx0 ◦ Pt(ℓ)−1 = ρ(¯ u(h(ℓ))) ◦ ψx0 ◦ ρ(h(ℓ))−1 = ψx0 . The maps ψx , x ∈ M, glue to a diffeomorphism ψ of P which satisfies ψ ◦ ρ(a) = ρ(¯ u(a))◦ψ for all a ∈ A. Its infinitesimal version is: ρ(v) ˙ and ρ(u(v)) ˙ are ψ-related, where u ∈ GL(V ) is given by exp ◦u = u¯ ◦ exp. The tangent map T ψ : T P → T P maps horizontal vectors to horizontal vectors because for any horizontal lift chor of the curve c, the curve ψ◦chor is the horizontal lift starting at ψ(chor (0)) of the curve ϕ ◦ c. Indeed, Pt(c)−1 (chor (t)) = chor (0), so by (31) we obtain (ψ ◦ chor )(t) = Pt(ϕ ◦ c|[0,t] )(ψ(chor (0)). Now one can show that ψ ∗ θ = u · θ: (ψ ∗ θ)(η hor + ρ(v)) ˙ = θ(T ψ.η hor ) + θ(ρ(u(v))) ˙ = u(v) = (u · θ)(η hor + ρ(v)) ˙ for all η ∈ X(M) and v ∈ V . Theorem 21. The group Diff eq (P, θ)A of equi-quantomorphisms is an abelian extension of the group Holeq (M, ω) of almost holonomy preserving diffeomorphisms by the natural Holeq (M, ω)–module A, i.e. (30) is an exact sequence of groups. Proof. For ψ ∈ Diff eq (P, θ)A with ϕ ◦ q = q ◦ ψ and ψ ∗ θ = uψ · θ, we define p(ψ) = (q∗ ψ, u ¯ψ ), where u¯ψ ∈ Aut(A) with u¯ψ ◦ exp = exp ◦uψ . In particular ψ is almost A–equivariant: ψ ◦ ρ(a) = ρ(¯ uψ (a)) ◦ ψ by Proposition 19. We verify that h(ϕ ◦ ℓ) = u¯ψ (h(ℓ)) for any ℓ ∈ C ∞ (S 1 , M), showing that ϕ is an almost holonomy preserving diffeomorphism. First we observe that if ℓhor is a horizontal lift of the loop ℓ, then ψ ◦ ℓhor is a horizontal lift of the loop ϕ ◦ ℓ: θ((ψ ◦ ℓhor )′ (t)) = θ(T ψ.(ℓhor )′ (t)) = (ψ ∗ θ)((ℓhor )′ (t)) = uψ (θ((ℓhor )′ (t)) = 0. Then the desired identity follows from the computation ρ((ψ ◦ ℓhor )(0), h(ϕ ◦ ℓ)) = (ψ ◦ ℓhor )(1) = ψ(ρ(ℓhor (0), h(ℓ))) = ρ(ψ(ℓhor (0)), u¯ψ (h(ℓ))). Thus p : Diff eq (P, θ)A → Holeq (M, ω) is well defined. By the previous lemma it is also surjective, and (30) is an exact sequence of groups. 18

Adapting the proof of Corollary 2.8 in [NV03] to the equivariant setting, one shows that the identity component Diff eq (P, θ)A 0 of the group of equi-quantomorphisms is an abelian extension of the group Diff eq ex (M, ω) of equi-hamiltonian eq diffeomorphisms by the natural Diff ex (M, ω)–module A: eq 1 → A → Diff eq (P, θ)A 0 → Diff ex (M, ω) → 1.

(32)

We call this the prequantization abelian extension. Remark 22. Like in Remark 17, if V = V0 the closure of the image of ω : T M ×M T M → V , the group of equi-hamiltonian diffeomorphisms can be identified with its projection on the first factor. In this case (30) can be seen as a restriction of (29), and the same is true for the prequantization abelian extension.

Group 2–cocycle In the special case when ω = dα for some α ∈ Ω1 (M, V ), the principal bundle with q curvature ω is P = M × V → M with principal connection 1–form θ = q ∗ α + θV , where θV denotes the Maurer-Cartan form on V . Then the prequantization central extension (28) for (M, dα) is defined by V –valued group 2–cocycles [ILM06]: c(ϕ1 , ϕ2 ) = f (ϕ2 )(x) − f (ϕ1 ϕ2 )(x) + f (ϕ1 )(ϕ2 (x)) on Diff ex (M, ω), where f : Diff ex (M, ω) → C ∞ (M, V ) is a map satisfying f (1M ) = 0 and d(f (ϕ)) = α − ϕ∗ α. Here x ∈ M is arbitrary, because one observes that c(ϕ1 , ϕ2 ) does not depend on x. We show that the prequantization abelian extension (32) is described by a similar V -valued group 2–cocycle on Diff eq ex (M, ω). Theorem 23. Given ω = dα for α ∈ Ω1 (M, V ), the identity component of the group of equi-quantomorphisms Diff eq (P, θ)A 0 is the abelian extension of the group eq of equi-hamiltonian diffeomorphisms Diff ex (M, ω) by V with cohomology class defined by the V –valued group 2–cocycle c on Diff eq ex (M, ω), c((ϕ1 , u1), (ϕ2 , u2 )) = u1 (f (ϕ2 , u2 )(x)) − f (ϕ1 ϕ2 , u1 u2 )(x) + f (ϕ1 , u1 )(ϕ2 (x)), ∞ where f : Diff eq ex (M, ω) → C (M, V ) is a map satisfying f (1M , 1V ) = 0 and d(f (ϕ, u)) = u · α − ϕ∗ α. Different choices for f define cohomologous cocycles.

Proof. The existence of the map f follows from Remark 12. Let ψ be an equi-quantomorphism of (P = M × V, θ = q ∗ α + θV ). Then it is of the form ψ(x, v) = (ϕ(x), mψ (x, v)), where ϕ = q∗ ψ and mψ : M × V → V . The condition ψ ∗ θ = u · θ becomes dmψ = d(f (ϕ, u) + u) because ψ ∗ (θV ) = dmψ and ψ ∗ q ∗ α = q ∗ ϕ∗ α = q ∗ (u · α − d(f (ϕ, u))). Hence the map mψ is of the form mψ (x, v) = f (ϕ, u)(x) + u(v) + a for some a ∈ V so ψ(x, v) = (ϕ(x), f (ϕ, u)(x) + u(v) + a). 19

In this way we define a bijection ψ 7→ ((ϕ, u), a) between Diff eq (P, θ)A and the cartezian product Diff eq ex (M, ω) × V ⊂ Diff(M) × GL(V ) × V . The following computation shows that the group Diff eq (P, θ)A 0 is isomorphic to the abelian extension defined by the given V –valued group 2–cocycle c on Diff eq ex (M, ω): (ϕ1 , u1 , a1 ) ◦ (ϕ2 , u2 , a2 )(x, v) = (ϕ1 , u1 , a1 )(ϕ2 (x), f (ϕ2 , u2)(x) + u2 (v) + a2 ) = (ϕ1 ϕ2 (x), f (ϕ1 , u1 )(ϕ2 (x)) + u1 f (ϕ2 , u2)(x) + u1 u2 (v) + u1 (a2 ) + a1 ) = (ϕ1 ϕ2 (x), f (ϕ1 ϕ2 , u1u2 )(x) + u1 u2 (v) + a1 + u1 (a2 ) + c((ϕ1 , u1 ), (ϕ2 , u2))) = (ϕ1 ϕ2 , u1 u2 , a1 + u1 (a2 ) + c((ϕ1 , u1 ), (ϕ2 , u2 )))(x, v) for all (x, v) ∈ P , which means ((ϕ1 , u1 ), a1 )◦((ϕ2 , u2 ), a2 ) = ((ϕ1 , u1)(ϕ2 , u2), a1 + (ϕ1 , u1 ) · a2 + c((ϕ1 , u1), (ϕ2 , u2 ))).

5

Group extensions via prequantization

Let G be a connected Lie group, λ a smooth G-action on M with infinitesimal ˙ and ω ∈ Ω2 (M, V ) a G–invariant closed 2–form. The G–action λ is action λ, called a hamiltonian action if iλ(X) ω ∈ Ω1 (M, V ) is exact for all X ∈ g. In this ˙ case λ is a group homomorphism : G → Diff ex (M, ω). Assuming the period group Γ of ω is discrete and denoting A = V /Γ, we consider again a principal A-bundle q : P → M with connection θ ∈ Ω1 (P, V ) and curvature ω. In Theorem 3.4 from [NV03] is proven that, given a hamiltonian action λ of a ˆ of G connected Lie group G on (M, ω), there exists a central Lie group extension G ˆ by A and a smooth G–action on (P, θ) by quantomorphisms, lifting the G–action. The central extension of G is a pull-back of the prequantization central extension ˙ ˙ ))(x0 ), (28) and a corresponding Lie algebra cocycle is (X, Y ) 7→ −ω(λ(X), λ(Y ˆ X, Y ∈ g, where x0 ∈ M is fixed. The manifold structure on G is obtained from the pull-back bundle of P by an orbit map of G on M. These results are generalized in this section to obtain abelian Lie group extensions of G associated to a G–equivariant 2–form ω. We consider a smooth action λ on M and a linear action b on V such that ω is G–equivariant, i.e. λ∗g ω = bg · ω. Let λ˙ : g → X(M) and b˙ : g → gl(V ) denote the infinitesimal g-actions. The pair ˙ (λ, b) is called an equi-hamiltonian G–action if the 1–form q ∗ iλ(X) ω − b(X) · θ is ˙ exact for all X ∈ g. In this case (λg , bg ) ∈ Diff eq ex (M, ω) for each g ∈ G, by Proposition 11. Then the pull-back of the prequantization abelian extension (32) by (λ, b): ˆ = {(g, ψ) ∈ G × Diff(P ) : q∗ ψ = λg , ψ ∗ θ = bg · θ}, G

(33)

i ˆ p is an abelian group extension 1 → A → G → G → 1 with i(a) = (e, ρ(a)) and p(g, ψ) = g. The induced G–module structure on A comes from the linear action b on V .

20

ˆ by equi-quantomorphisms ˆ is a Lie group with a smooth action λ To show that G of (P, θ), lifting the action λ, we use Lemma 3.2 and Lemma 3.3 from [NV03] and we adapt the proof of Theorem 2.7 to our equivariant setting. Theorem 24. Given ω ∈ Ω2 (M, V ) a closed 2–form with discrete period group Γ and an equi-hamiltonian G–action (λ, b), there is an abelian Lie group extension ˆ of G by the G–module A = V /Γ, integrating the V –valued Lie algebra 2–cocycle G ˙ ˙ ))(x0 ), (X, Y ) 7→ −ω(λ(X), λ(Y

X, Y ∈ g,

(34)

whose cohomology class does not depend on the choice of the point x0 ∈ M. There ˆ on P lifting the G–action and such that θ is G–equivariant, ˆ ˆ is also a G–action λ ∗ ˆ ˆ i.e. λgˆθ = bp(ˆg) · θ for any gˆ ∈ G. Proof. Let λx0 : g ∈ G 7→ λg (x0 ) ∈ M be the orbit map, and let λ∗x0 P → G be the pull-back of the A-bundle P → M. From Lemma 3.2 follows that an element ˆ 7→ (g, ψ(y0)) ∈ λ∗x P , and y0 ∈ P with q(y0 ) = x0 defines a bijection (g, ψ) ∈ G 0 ˆ by this bijection does not depend the smooth manifold structure transported on G on the choice of x0 . ˆ on P by ˆ We define the G–action λ ˆ : ((g, ψ), y) ∈ G ˆ × P 7→ ψ(y) ∈ P. λ ˆ It lifts the G–action λ : G × M → M and θ is G–equivariant because ψ is an ˆ equi-quantomorphism. Its restriction to A is λa = ρ(a) for a ∈ A. ˆ is smooth, we put product coordinates on P and G. ˆ In order to show that λ Any smooth local section sM : U ⊂ M → P of q, with sM (x0 ) = y0 , defines a local ˆ of p, for g sufficiently close to the identity smooth section sG (g) = (g, ψg ) ∈ G of G in order to have λg (x0 ) ∈ U. Here ψg is uniquely defined by the condition ψg (y0) = sM (λg (x0 )). We fix the open neighbourhood UG of e and the convex neighbourhood UM of x such that UG · UM ⊂ U. Then we define a function f : UG × UM → A with f (g, x0 ) = 1 for all g ∈ UG by the relation ψg (sM (x)) = ρ(sM (λg (x)), f (g, x)).

(35)

In other words, for g ∈ UG the expression of ψg in product coordinates is ψg : (x, a) ∈ UM × A 7→ (λg (x), f (g, x)¯bg (a)) ∈ U × A, where ¯bg ∈ Aut(A) with ¯bg ◦ exp = exp ◦bg . This can be deduced from (ψg ◦ ρ(a))(sM (x)) = (ρ(¯bg (a)) ◦ ψg )(sM (x)) = ρ(f (g, x)¯bg (a))(sM (λg (x))). l The connection 1–form in product coordinates U × A is θ = q ∗ α + qA∗ θA , with −1 ∗ 1 l l qA : q (U) → A the second projection, α = sM θ ∈ Ω (U, V ) and θA = δ (1A ) ∈

21

Ω1 (A, V ) the Maurer-Cartan form on A. In particular dα = ω on U ⊂ M. The condition ψg∗ θ = bg · θ implies δ l (fg ) = λ∗g α − bg · α. Indeed, l l 0 = ψg∗ (q ∗ α + qA∗ θA ) − bg · (q ∗ α + qA∗ θA ) = q ∗ (λ∗g α − bg · α) + δ l (m ◦ (¯bg ◦ qA , fg ◦ q)) − q ∗ δ l (¯bg ) = q ∗ (λ∗ α − bg · α − δ l (fg )) A

g

because qA ◦ ψg = m ◦ (¯bg ◦ qA , fg ◦ q) for m the group multiplication map of A and δ l (m ◦ (h1 , h2 )) = δ l h1 + δ l h2 for h1 , h2 : P → A. Using the Poincar´e Lemma applied to the closed 1–form λ∗g α − bg · α on the convex set UM , we obtain that f is a smooth function. Then (ρ(a) ◦ ψg )(ρ(sM (x), a′ )) = ρ(sM (λg (x)), f (g, x)a¯bg (a′ )) ˆ in product coordinates: assures the following expression of λ ˆ λ((g, a), (x, a′)) = (λg (x), f (g, x)a¯bg (a′ )), ˆ thus showing the smoothness of λ. The A–valued local group 2–cocycle on G corresponding to the section sG is c(g1 , g2 ) = f (g1 , λg2 (x0 )),

(36)

because (35) ψg1 ψg2 (y0 ) = ψg1 (sM (λg2 (x0 ))) = ρ(sM (λg1 g2 (x0 )), f (g1, λg2 (x0 ))) = ρ(ψg1 g2 (y0 ), f (g1, λg2 (x0 ))).

This shows the smoothness of multiplication and inversion in an identity neighˆ That the left multiplications are smooth, follows from the fact that borhood in G. ˆ acts by smooth maps on the bundle P . Now Lemma 3.3 in [NV03] implies that G ˆ is a Lie group. G The corresponding Lie algebra extension 0 → V → ˆg → g → 0 is the pull˙ b) ˙ : g → Xeq (M, ω) of the abelian extension (24), hence a defining Lie back by (λ, ex ˙ b) ˙ of the 2–cocycle from Theorem algebra 2–cocycle is (34), the pull-back by (λ, 16. Remark 25. Given an exact 2–form ω = dα ∈ Ω2 (M, V ) and an equi-hamiltonian ˆ of G by G–action (λ, b), Theorem 24 provides an abelian Lie group extension G the G–module V . It can be defined also by a group 2–cocycle, the pull-back of the cocycle c from Theorem 23 by the group homomorphism (λ, b) : G → Diff eq ex (M, ω): c(g1 , g2 ) = bg1 (f (g2)(x)) − f (g1 g2 )(x) + f (g1 )(λg2 (x)), where f (g) ∈ C ∞ (M, V ) with f (e) = 0 and d(f (g)) = bg · α − λ∗g α, and x ∈ M arbitrary. If f (g)(x0) = 0 for all g ∈ G, one obtains (36) like in the proof of the previous theorem. ˆ ˆ The G–action on P = M × V lifting the G–action and such that θ is G– ′ ′ equivariant is (g, v) · (x, v ) = (λg (x), f (g)(x) + bg (v ) + v). 22

q ˆ → Proposition 26. Every abelian Lie group extension 1 → A = V /Γ → G G → 1 of a simply connected Lie group G can be obtained as a pull-back of the prequantization abelian extension (32).

ˆ = g ⋉σ V be the corresponding abelian Lie algebra extension, which Proof. Let g is defined by a V -valued 2–cocycle σ on g, and let p : gˆ → V denote the projection ˆ is a principal A-bundle with principal connection on the second factor. Then G eq ˆ ˆ with identity value θe = p, and curvature θ = p , the G-equivariant 1–form on G eq ω = −σ , the closed G-equivariant 2–form on G with identity value ωe = −σ. Indeed, the Chevalley-Eilenberg differential of the 1–cochain p on gˆ being the opposite of the pullback of the 2–cocycle σ on g, we have dpeq = −q ∗ σ eq . The G-action (λ, b) is equi-hamiltonian, where λ denotes the left translation on G and b the linear G-action on V induced by the G-module structure of A. ˆ can be expressed with the help of This we check now. The coadjoint action in G ˆ → Lin(g, V ) as a group 1–cocycle κ : G Ad(ˆ g )(X, v) = (Ad(g)X, bg (v) − κ(ˆ g )(Ad(g)X)). The expression of the Lie bracket in gˆ involving σ assures that the Lie algebra 1– ′ ˙ cocycle corresponding to κ is α : gˆ → Lin(g, V ), α(X ′, v ′ )(X) = b(X)v +σ(X, X ′ ), ˙ ◦ p − σˇ. which means α = bˇ From Proposition 6.4 in [N04] follows that for any X ∈ g the differential of the eq ˙ ˆ V ) defined by κ is dκX = b(X)·p map κX ∈ C ∞ (G, +q ∗ iX r σ eq , where X r denotes the right invariant vector field on G with identity value X r (e) = X. This follows ˙ p)eq = b(X) ˙ also from Proposition 36 from the Appendix, observing that (bˇ◦ · peq and evX ◦(σˇ)eq = −iX r σ eq . ˙ Since λ(X) = X r , we obtain that (λ, b) is an equi-hamiltonian action: ˙ ˙ ˙ ω − b(X) · θ] = −[dκX ] = 0. fluxeq (λ(X), b(X)) = [q ∗ iλ(X) ˙ The pull-back of the prequantization abelian extension eq eq ˆ peq )A 1 → A → Diff eq (G, 0 → Diff ex (G, −σ ) → 1

by the equi-hamiltonian action (λ, b) is an abelian Lie group extension of G by A, integrating the Lie algebra 2–cocycle σ because ˙ ˙ ))(e) = σ eq (X r , Y r )(e) = σ(X, Y ). −ω(λ(X), λ(Y ˆ since they have the same Lie algebras and It coincides with the given extension G G is simply connected.

6

Examples

ˆ by an n-torus Tn is a For a connected Lie group G, any abelian extension G central extension. This happens because the automorphism group GLn (Z) of Tn 23

is discrete. This is the reason why the examples of abelian extensions presented in this section are infinite dimensional. The obstructions for the integration of a general Lie algebra 2–cocycle [N02] [N04] are presented in the Appendix. Theorem 24 provides a geometric construction of several abelian Lie group extensions of diffeomorphism groups. In the examples below, the manifold M will always be a homogeneous manifold G/H, with H a connected Lie subgroup of G and V a G-module. The G–equivariant closed 2–form ω on G/H is uniquely defined by a V -valued Lie algebra 2–cocycle σ on g satisfying two properties: 1. The kernel of σ contains the Lie algebra h of H, so that σ descends to a skew-symmetric bilinear form on g/h. 2. σ is H–equivariant, i.e. σ(Ad(g)X, Ad(g)Y ) = b(g) · σ(X, Y ) for all X, Y ∈ g and g ∈ H. The subgroup H being connected, the last condition is ˙ equivalent to σ([Z, X], Y ) + σ(X, [Z, Y ]) = b(Z) · σ(X, Y ) for all X, Y ∈ g and Z ∈ h. Example 27. We consider the group Diff + (S 1 ) of orientation preserving diffeomorphisms of the circle and its modules Fλ of λ-densities on the circle: bλ (ϕ)f = (ϕ′ )λ (f ◦ ϕ) for ϕ ∈ Diff + (S 1 ) and f ∈ C ∞ (S 1 ). The X(S 1 )–module structure on Fλ is given by b˙ λ (X)f = Xf ′ + λX ′ f for X ∈ X(S 1 ). The abelian extensions of X(S 1) defined with the Fλ –valued cocycles Z 1 σ0 (X, Y ) = (X ′ Y ′′ − X ′′ Y ′ )dx ∈ R ⊂ F0 0

σ1 (X, Y ) = X ′ Y ′′ − X ′′ Y ′ ∈ F1 σ2 (X, Y ) = X ′ Y ′′′ − X ′′′ Y ′ ∈ F2 integrate to abelian extensions of Diff + (S 1 ). Corresponding group cocycles are presented in [OR98]. These abelian extensions can also be obtained geometrically by the Theorem 24, taking M to be the contractible homogeneous space Diff + (S 1 )/S 1 , where S 1 is identified with the subgroup of rotations of S 1 . The existence of the Diff + (S 1 )–equivariant Fλ –valued closed 2–form ωλ on M defined by the 2–cocycle σλ for λ = 0, 1, 2 is ensured by the S 1 –equivariance of σλ and the fact that the constant vector fields belong to the kernel of σλ . The abelian extensions of X(S 1) by Fλ defined with the 2–cocycles σ¯0 (X, Y ) = XY ′ − X ′ Y ∈ F0 σ ¯1 (X, Y ) = XY ′′ − X ′′ Y ∈ F1 σ ¯2 (X, Y ) = XY ′′′ − X ′′′ Y ∈ F2 g + (S 1 ) [N04]. integrate to abelian extensions of the universal covering group Diff For the geometric construction of these abelian extensions, in Theorem 24 we take g + (S 1 ) acting by left translations on M = Diff g + (S 1 ), which is contractible. G = Diff 24

g + (S 1 )–equivariant Fλ –valued closed 2–form ω The Diff ¯ λ is uniquely defined by its value σ ¯λ at the identity, for λ = 0, 1, 2. Example 28. Given a volume form µ on M, a non-trivial Ω1 (M)/dΩ0 (M)–valued Lie algebra 2–cocycle on X(M) is σ(X, Y ) = (div X)d(div Y ). A group 2–cocycle on Diff(M) integrating σ is constructed in [B03]. For the geometric construction of an abelian Lie group extension of Diff(M) by its module Ω1 (M)/dΩ0 (M), we remark that the 2–cocycle σ is Diff(M, µ)– equivariant and the Lie algebra X(M, µ) of divergence free vector fields is contained in the kernel of σ, hence there is a closed Diff(M)–equivariant 2–form ω on the homogeneous space Diff(M)/ Diff(M, µ). By a result of Moser, this space can be identified with the contractible space of all volume forms of total mass 1. Now the Theorem 24 can be applied to ω. Example 29. Let θ be a connection 1–form on the principal GL(n, R)-bundle of frames π : P (M) → M. Gelfand’s cocycle presented in [R06] is the Ω2 (M)–valued ˜ Y˜ ∈ 2–cocycle σ on X(M) defined by π ∗ σ(X, Y ) = tr(LX˜ θ ∧ LY˜ θ), where X, X(P (M)) are canonical lifts of X, Y ∈ X(M). In the special case M = Tn the n-torus, a group 2–cocycle on Diff(Tn ) integrating σ is constructed in [B03]. For the 2–sphere S 2 , Theorem 24 provides a gemetric construction of an abelian Lie group extension of Diff(S 2 ) by Ω2 (S 2 ) integrating Gelfand’s cocycle. We choose the connection θ on the principal bundle of frames of S 2 induced by the canonical Riemannian metric on S 2 . The connected component of the isometry group of S 2 is SO(3) ⊂ Diff(S 2 ), so LX˜ θ = 0 for X ∈ so(3) ⊂ X(S 2 ). The 2–cocycle σ is SO(3)–equivariant and the Lie algebra so(3) is contained in the kernel of σ, hence there exists a closed Diff(S 2 )–equivariant 2–form ω on the homogeneous space Diff(S 2 )/ SO(3), given at the identity by σ. By a result of Smale [S59], this homogeneous space is contractible, and Theorem 24 can be applied to ω.

7

Appendix

Logarithmic derivative In this subsection we collect some properties of the logarithmic derivative [KM97]. Given M a manifold and G a Lie group with Lie algebra g, the right logarithmic derivative of a function h ∈ C ∞ (M, G) is δ r h ∈ Ω1 (M, g): (δ r h)(Xx ) := (Tx h.Xx )h(x)−1 , ∀Xx ∈ Tx M. Remark 30. A left logarithmic derivative δ l is defined similarly. There is a relation between left and right logarithmic derivatives δ r h = Ad(h)δ l h = −δ l (h−1 ). 25

When h : R → (R+ , ·), then δ r h(∂t ) = δ l h(∂t ) = (log h)′ (t) is the derivative of the logarithm of h. Given a curve h in G we will identify the 1–form δ r h with the curve i∂t δ r h in g. When ϕt ∈ Diff(M) is a diffeotopy of the manifold M, then δ r ϕt is the associated time dependent vector field on M, and δ l ϕt = ϕ∗t δ r ϕt . For ω a differential form, we have dtd ϕ∗t ω = ϕ∗t Lδr ϕt ω. This is a particular case of the identity dtd (gt · v) = δ r gt · (gt · v) = gt · (δ l gt · v), for a smooth G–module V and a curve gt in G. Remark 31. Given a closed form ω ∈ Ωp (M, V ), a path ϕt of diffeomorphisms of M and a path ut of linear isomorphisms of V , both starting at the identity, the following equivalences hold: ϕ∗t ω = ut · ω ⇔ Lδr ϕt ω = δ r ut · ω ⇔ Lδl ϕt ω = δ l ut · ω. We show that Lδr ϕt ω = δ r ut · ω implies ϕ∗t ω = ut · ω. The computation dtd (ϕ∗t ω − ut · ω) = ϕ∗t Lδr ϕt ω − δ r ut · (ut · ω) = δ r ut · (ϕ∗t ω − ut · ω) shows that the curve ωt = ϕ∗t ω − ut · ω is the unique solution of the differential equation dtd ωt = δ r ut · ωt with initial condition ω0 = 0, hence ωt = 0. Remark 32. The right logarithmic derivative satisfies the Leibniz rule δ r (h1 h2 ) = δ r h1 + Ad(h1 )δ r h2 for h1 , h2 ∈ C ∞ (M, G). In other words δ r is a group 1–cocycle on C ∞ (M, G) with values in the C ∞ (M, G)-module Ω1 (M, g). The logarithmic derivative of a G-valued function h on a connected manifold M vanishes if and only if h is constant. The left Maurer-Cartan form on a Lie group G is the 1–form l θG = δ l (1G ) ∈ Ω1 (G, g). l l It satisfies θG (Xg ) = g −1 ·Xg for Xg ∈ Tg G, and δ l h = h∗ θG for any h ∈ C ∞ (M, G). l l Both θG and δ h satisfy the right Maurer-Cartan equation

1 l l l dθG + θG ∧ θG = 0, 2

1 dδ l h + δ l h ∧ δ l h = 0. 2

Given α ∈ Ω1 (M, g) which satisfies the right Maurer-Cartan equation dα+ 21 α∧α = 0 and U ⊆ M simply connected, there exists a smooth function h : U → G with δ l h = α on U, called the Cartan developing of α. In the special case of a domain in R2 , when h(t, s) ∈ G is a smooth two parameter family, we get d d η − ξ = [ξ, η], dt dt d where ξ(t, s) = ( dtd h(t, s))h(t, s)−1 and η(t, s) = ( ds h(t, s))h(t, s)−1 .

26

(37)

Proposition 33. Let θ ∈ Ω1 (P, V ) be a principal connection of the principal A-bundle q : P → M with A = V /Γ. Then for any f ∈ C ∞ (P, A), ρ(f )∗ θ = θ + δ l (f ),

(38)

where ρ denotes the principal A-action. Proof. Let q −1 (U) ∼ = U × A be a principal bundle chart with q and qA the two projections on U and A. The principal connection 1–form in this chart is l θ = q ∗ α + qA∗ θA , l where θA = δ l (1A ) ∈ Ω1 (A, V ) is the Maurer-Cartan form on A, and α ∈ Ω1 (U, V ) closed. The diffeomorphism ρ(f ) written in this chart is ρ(f ) = (q, m ◦ (qA , f )), with m denoting the multiplication map in A. Now

ρ(f )∗ θ = q ∗ α + (m ◦ (qA , f ))∗ δ l (1A ) = q ∗ α + δ l (qA ) + δ l (f ) = θ + δ l (f ), because δ l (m ◦ (h1 , h2 )) = δ l h1 + δ l h2 for h1 , h2 : P → A.

1–Cocycles We list in this subsection some properties of 1–cocycles on Lie algebras and Lie groups [N04]. Let G be a Lie group with Lie algebra g and let V be a smooth Gmodule. Then V is a g-module and the pull-back action by the universal covering ˜ → G makes V a G-module. ˜ homomorphism G A V -valued group 1–cocycle is a locally smooth map a : G → V with a(gg ′) = a(g) + g · a(g ′ ),

(39)

and a V -valued Lie algebra 1–cocycle is a linear map α : g → V with α([X, X ′ ]) = X · α(X ′ ) − X ′ · α(X). There is a natural map a 7→ de a from locally smooth group 1–cocycles on the Lie group G to Lie algebra 1–cocycles on its Lie algebra g. Proposition 34. There exists a unique group 1–cocycle on the universal covering ˜ integrating the Lie algebra 1–cocycle α : g → V group G Z 1 ˜ a ˜ : G → V, a ˜([g]) = gt · α(δ l gt )dt, 0

˜ is the homotopy class of the piecewise smooth path gt in G starting where [g] ∈ G at the identity. 27

Proof. Using the identity (37), one shows that the map a ˜ is well defined. The 1–cocycle condition (39) for a ˜ is easily verified noticing that the smooth path gt gt′ , t ∈ [0, 1] and the piecewise smooth path ht , defined by ht = g2t if t ≤ 21 and ′ ht = g1 g2t−1 if t ≥ 12 , are homotopic. Remark 35. If the group Πα of periods of the closed 1–form αeq is discrete, the for any discrete subgroup Π of V containing Πα , the 1–cocycle a ˜ descends to a 1–cocycle a : G → V /Π. Proposition 36. If a : G → V is a group 1–cocycle integrating the Lie algebra 1–cocycle α : g → V , then the following identities hold: 1. da = αeq ∈ Ω1 (G, V ). 2.

d (a(gt )) dt

= gt · α(δ l gt ) = α(δ r gt ) + δ r gt · a(gt ), for gt a path in G starting at the identity.

From Proposition 36 and Remark 30 we deduce the following: Corollary 37. Let a : G → V /Π be a group 1–cocycle integrating the Lie algebra 1–cocycle α : g → V . Then Ker a is a subgroup of G and Ker α is a Lie subalgebra of g such that, for any smooth curve g in G starting at the identity, the following are equivalent: gt ∈ Ker a⇔δ l gt ∈ Ker α⇔δ r gt ∈ Ker α.

Lie group extensions According to the general theory developed in [N02] and [N04], there are two obstructions for the integration of a Lie algebra cocycle σ on g with values in a G-module V to a Lie group extension of G by a quotient group of V : the period map and the flux homomorphism. The period map is the group homomorphism Z G perσ : π2 (G) → V , perσ ([c]) = c∗ σ eq for c ∈ C ∞ (S 2 , G). S2

Its image Πσ is called the period group of σ. The flux homomorphism Fσ : π1 (G) → H 1 (g, V ), [γ] 7→ [Iγσ ], assigns to each piecewise smooth loop γ in G based at the identity, the cohomology class of the 1–cocycle Z Iγσ : g → V,

iX r σ eq .

Iγσ (X) = −

γ

Theorem 38. For a Lie algebra V -valued 2–cocycle σ on g with discrete period group Πσ and vanishing flux homomorphism Fσ , the Lie algebra extension ˆg = g ⋉σ V integrates to an abelian Lie group extension ˆ → G → 1. 1 → V /Πσ → G 28

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