Abelian Extensions of the Group of Diffeomorphisms of a Torus.

Yuly Billig School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, K1S 5B6, Canada e-mail: billig@math.carleton.ca

Abstract. In this Letter we construct abelian extensions of the group of diffeomorphisms of a torus. We consider the jacobian map, which is a crossed homomorphism from the group of diffeomorphisms into a toroidal gauge group. A pull-back under this map of a central 2-cocycle on a gauge group turns out to be an abelian cocycle on the group of diffeomorphisms. We show that in the case of a circle, the Virasoro-Bott cocycle is a pullback of the Heisenberg cocycle. We also give an abelian generalization of the Virasoro-Bott cocycle to the case of a manifold with a volume form.

Mathematics Subject Classifications (2000). 22E65, 58B25, 81R10.

Key words. Group of diffeomorphisms, abelian extensions, Virasoro-Bott group.

0. Introduction.

The Lie group that corresponds to the Virasoro Lie algebra is the central extension of the group of diffeomorphisms of a circle with the Virasoro-Bott cocycle (2.1). It is wellknown that the Virasoro cocycle on the Lie algebra of vector fields on a circle does not admit a generalization as a central 2-cocycle to the case of higher dimensional manifolds. Nonetheless there exist abelian extensions of the Lie algebra of vector fields on a torus that generalize the Virasoro Lie algebra. These abelian extensions play an important role in the representation theory of toroidal Lie algebras ([EM], [L], [BB], [Bi]).

The goal of this Letter is to construct the corresponding abelian extensions for the group of diffeomorphisms of a torus.

We approach this problem by linking the abelian cocycles on the group of diffeomorphisms of a torus with the central cocycles on the toroidal gauge groups. The central extensions of the toroidal gauge groups were studied in [LMNS] (see also [PS], [FK]).

The connection between the group of diffeomorphisms of a torus and toroidal gauge groups is given by the jacobian map. Let us briefly outline this correspondence. The differential of a diffeomorphism of a torus is a mapping of the tangent bundle into itself. However, using the fact that torus has a trivial tangent bundle, we can globally identify all tangent spaces with ℝ^{N}. After this identification, the differential of a diffeomorphism of a torus becomes a mapping of a torus into GL_{N}(ℝ). In the standard coordinates, this mapping is given by the jacobian matrix. In this way we obtain the jacobian map from the group of diffeomorphisms of a torus into a toroidal gauge group:

J : Diff(𝕋^{N}) → Map(𝕋^{N}, GL_{N}(ℝ)).

In all constructions of this Letter the torus may be replaced with an arbitrary manifold with a trivial tangent bundle.

The jacobian mapping is not a group homomorphism, but what is called a crossed homomorphism (see definition in Section 3). In case N = 1, we get a crossed homomorphism from the group of diffeomorphisms of a circle into abelian loop group Map(S^{1}, ℝ^{∗}). A central extension of this loop group is an infinite dimensional Heisenberg group. We point out that the Virasoro-Bott cocycle may be interpreted as a pull-back of the Heisenberg cocycle under the jacobian map (Theorem 2.1). This observation motivates our constructions of 2-cocycles on the group of diffeomorphisms in higher dimensions.

First we describe our construction of 2-cocycles by pull-back in a very general set-up. We show (Lemma 3.1) that under a crossed homomorphism of two groups

j : D → M,

a pull-back of a D invariant central 2-cocycle on M is an abelian 2-cocycle on D.

We apply this lemma to construct an abelian extension of the group of diffeomorphisms of a manifold with a volume form (Theorem 4.1), generalizing the Virasoro-Bott group, for which the manifold is the circle. This extension (as well as the Virasoro-Bott extension) trivializes on the subgroup of volume preserving diffeomorphisms.

We also construct another extension of the group of diffeomorphisms of a torus (Theorem 3.2) that remains non-trivial when restricted to the subgroup of volume preserving diffeomorphisms.

We would like to mention here that Ovsienko and Roger described abelian extensions of the group of diffeomorphisms of a circle with the modules of tensor densities [OR].

The structure of the Letter is the following. In Section 1 we present a geometric approach to the abelian extensions of the Lie algebra of vector fields on a torus and introduce the jacobian map from the group of diffeomorphisms of a torus into a toroidal gauge group. In Section 2 we give two constructions of the infinite dimensional Heisenberg group and exhibit the link between the Virasoro-Bott cocycle and the Heisenberg cocycle. In Section 3 we describe the general construction of a pull-back of a cocycle under a crossed homomorphism and use this approach to get an abelian extension of the group of diffeomorphisms of a torus. Finally in Section 4 we obtain another, inequivalent abelian cocycle that generalizes the Virasoro-Bott cocycle to the case of a manifold with a volume form.

1. Abelian extensions of the Lie algebra of vector fields of a torus.

It is well-known that the Lie algebra W_{N} of vector fields on an N-dimensional torus 𝕋^{N} (Witt algebra) has a non-trivial central extension only when N = 1. In that case there exists a unique non-trivial central extension of W_{1}, which is the Virasoro Lie algebra. The Virasoro cocycle can not be generalized as a central 2-cocycle to higher dimensions. However when N > 1, there exist abelian extensions of W_{N} that generalize the Virasoro cocycle. These abelian extensions appeared in the representation theory [EM], [L], where it was discovered that it is easier to construct representations for certain abelian extensions of W_{N} rather than for the Witt algebra itself.

Let us describe the construction of these abelian extensions of W_{N}.

Let F be the algebra of C^{∞} real-valued functions on a torus 𝕋^{N} = ℝ^{N}/ℤ^{N}. Note that the algebra of complex-valued functions on a torus, ℂ ⊗ F has a convenient topological basis

{exp (2πi ∑_{j=1}^{N} r_{j}x_{j}) | r_{j} ∈ ℤ} .

Often this basis is used in algebraic setting (considering Fourier polynomials instead of Fourier series). However, the group of diffeomorphisms of a torus can not be realized using Fourier polynomials, so we will be working with C^{∞} functions instead. We define the group Diff(𝕋^{N}) as the group of C^{∞} diffeomorphisms of a torus. An element F ∈ Diff(𝕋^{N}) can be viewed as an N-tuple of functions F = (F_{1}(x), …, F_{N}(x)), such that the map

x = (x_{1}, …, x_{N}) ↦ (F_{1}(x), …, F_{N}(x))

is a bijection 𝕋^{N} → 𝕋^{N}.

The Lie algebra W_{N} consists of the vector fields v = ∑_{j=1}^{N} v_{j}(x)\frac{∂}{∂x_{j}} with v_{j}(x) ∈ F.

Consider the space of differential k-forms on 𝕋^{N}:

Ω^{k} = { ∑_{1≤j_{1}<...<j_{k}≤N} a_{j_{1}...j_{k}}(x) dx_{j_{1}} ∧ … ∧ dx_{j_{k}} | a_{j_{1}...j_{k}}(x) ∈ F} .

There is a differential map d : Ω^{k} → Ω^{k+1}:

d (a(x)dx_{j_{1}} ∧ … ∧ dx_{j_{k}}) = ∑_{j=1}^{k} \frac{∂a}{∂x_{j}} dx_{j} ∧ dx_{j_{1}} ∧ … ∧ dx_{j_{k}}.

The Lie algebra of vector fields acts in an obvious way on F: v · a(x) = ∑_{j=1}^{N} v_{j}(x)\frac{∂a}{∂x_{j}}.

The algebra of functions F is also a right module for the group Diff(𝕋^{N})

a(x)(F) = a(F (x)), a ∈ F, F ∈ Diff(𝕋^{N}),

and so is the spaces of differential forms:

(a(x)dx_{j_{1}} ∧ … ∧ dx_{j_{k}}) (F) = a(F (x))dF_{j_{1}}(x) ∧ … ∧ dF_{j_{k}}(x).  (1.1)

It is easy to see that the differential map is a homomorphism of Diff(𝕋^{N}) modules.

The action of the group Diff(𝕋^{N}) on k-forms gives rise to the action of its Lie algebra W_{N}:

v · (a(x)dx_{j_{1}} ∧ … ∧ dx_{j_{k}}) =

(v · a(x))dx_{j_{1}} ∧ … ∧ dx_{j_{k}} + ∑_{r=1}^{k} a(x)dx_{j_{1}}(x) ∧ … ∧ dv_{j_{r}}(x) ∧ … ∧ dx_{j_{k}}.

This action is called the Lie derivative action.

Next we need to discuss the jacobian of a diffeomorphism of a torus, and the jacobian of a vector field.

Let F ∈ Diff(𝕋^{N}), F : 𝕋^{N} → 𝕋^{N}. The differential of F is a map of tangent bundles:

dF : T (𝕋^{N}) → T (𝕋^{N}),

with linear maps

T_{x}(𝕋^{N}) → T_{F (x)}(𝕋^{N}).

However torus is a flat manifold and its tangent bundle is trivial: T (𝕋^{N}) = 𝕋^{N} × ℝ^{N}, so all tangent spaces can be globally identified with ℝ^{N}, and we define the jacobian of F as a map

F^{J} : 𝕋^{N} → Hom(ℝ^{N}, ℝ^{N}).

Since F is an invertible transformation of the torus, F^{J} is an invertible linear homomorphism, so

F^{J} : 𝕋^{N} → GL_{N}(ℝ).

In coordinates (x_{1}, … x_{N}), the matrix of the jacobian of a diffeomorphism F is

F^{J} = (\frac{∂F_{i}}{∂x_{k}} )_{i,j=1,...,N} .

The jacobian map J : F ↦ F^{J} associates with every element of Diff(𝕋^{N}) an element of a gauge group Map(𝕋^{N}, GL_{N}). However the resulting map

J : Diff(𝕋^{N}) → Map(𝕋^{N}, GL_{N})  (1.2)

is not a group homomorphism, but what is called a crossed homomorphism [K]:

(F G)^{J} = F^{J}(G) · G^{J} (chain rule).  (1.3)

In the above formula we used the action of Diff(𝕋^{N}) on the toroidal gauge group Map(𝕋^{N}, GL_{N}) by automorphisms:

m(x) ↦ m(F (x)), where m ∈ Map(𝕋^{N}, GL_{N}), F ∈ Diff(𝕋^{N}), x ∈ 𝕋^{N}.  (1.4)

In a similar way, the jacobian of a vector field v is defined as a matrix

v^{J} = (\frac{∂v_{i}}{∂x_{k}} )_{i,j=1,...,N} ,

thus giving a linear map from W_{N} to the toroidal analog Map(𝕋^{N}, gl_{N}) ≅ gl_{N} ⊗ F of the loop algebra.

Now we can use the jacobians to give a simple descriptions of the abelian extensions of W_{N}. We will actually consider two possible extension spaces. In the first case, the 2-cocycles will have values in the factor module K = Ω^{1}/dΩ^{0} . This module is of special interest since it is the space of the universal central extension for the toroidal Lie algebras. Two such K valued 2-cocycles have appeared in the representation theory ([EM], [L], [BB]):

τ_{1}(v, w) = Tr (v^{J}dw^{J})

and

τ_{2}(v, w) = Tr (v^{J}) Tr (dw^{J}) .

Viewed as Ω^{1} valued expressions these are not 2-cocycles, but modulo dΩ^{0} these become skew-symmetric and satisfy the Jacobi identity.

There is a closely related pair of 2-cocycles on W_{N}, obtained from τ_{1}, τ_{2} by applying the differential map d : Ω^{1} → Ω^{2}. Since d^{2} = 0, this map factors through the factor module Ω^{1}/dΩ^{0}. The differentials dτ_{1}, dτ_{2} are thus Ω^{2} valued 2-cocycles on W_{N}:

dτ_{1}(v, w) = Tr (dv^{J} ∧ dw^{J})

and

dτ_{2}(v, w) = Tr (dv^{J}) ∧ Tr (dw^{J}) .

Dzhumadildaev proved that for N > 1 the space of Ω^{2} valued 2-cocycles H^{2}(W_{N}, Ω^{2}) is two-dimensional and is spanned by dτ_{1}, dτ_{2} [D].

These cocycles give 2-parametric abelian extensions of the Lie algebra of vector fields on a torus with extension spaces K or Ω^{2} (see e.g. Section 2 of [BB]).

The main goal of this Letter is to construct the corresponding abelian extensions for the group of diffeomorphisms of a torus.

Remark 1. We can see that linear combinations of τ_{1}, τ_{2} are in 1-1 correspondence with the symmetric invariant bilinear forms on gl_{N}(ℝ). Since for N > 1 the Lie algebra gl_{N}(ℝ) is not simple, but only reductive, gl_{N}(ℝ) = sl_{N}(ℝ) ⊕ ℝI, the space of symmetric invariant forms on gl_{N}(ℝ) is two-dimensional and is spanned by the forms Tr(AB) and Tr(A)Tr(B). Thus a linear combination of τ_{1} and τ_{2} can be written as

τ(v, w) = (v^{J}|dw^{J}),

where (·|·) is a symmetric invariant bilinear form of gl_{N}(ℝ).

Remark 2. Cocylcles τ_{2} and dτ_{2} trivialize on the subalgebra of divergence zero vector fields, because the divergence of a vector field is equal to the trace of the jacobian: div(v) = Tr(v^{J}).

Remark 3. In coordinates, cocycles τ_{1}, τ_{2} are written as follows (cf. [BB]):

τ_{1} (v(x)\frac{∂}{∂x_{i}} , w(x)\frac{∂}{∂x_{j}} ) = \frac{∂v}{∂x_{j}} ∑_{k=1}^{N} \frac{∂^{2}w}{∂x_{i}∂x_{k}} dx_{k},

τ_{2} (v(x)\frac{∂}{∂x_{i}} , w(x)\frac{∂}{∂x_{j}} ) = \frac{∂v}{∂x_{i}} ∑_{k=1}^{N} \frac{∂^{2}w}{∂x_{j}∂x_{k}} dx_{k}.

Remark 4. When N = 1, cocycles τ_{1} and τ_{2} coincide (the forms Tr(AB) and Tr(A)Tr(B) are equal in gl_{1}), and become the Virasoro cocycle on the Witt algebra W_{1}. Thus for N > 1, these cocycles are two distinct abelian generalizations of the Virasoro cocycle.

2. Realization of the Virasoro-Bott cocycle via the Heisenberg group.

The Virasoro-Bott cocycle on the group of diffeomorphisms of S^{1} has a rather mysterious form [Bo]:

(F, α)(G, β) = (F G, α + β + B(F, G)) ,

where F, G ∈ Diff(S^{1}), α, β ∈ ℝ and

B(F, G) = ∫_{0}^{1} ln |F′(G(t))| d ln |G′(t)|.  (2.1)

The cocycle B(F, G) seems to have more analytic flavour rather than algebraic, and looks quite different from other cocycles on the infinite dimensional groups, e.g., affine groups. We are now going to elucidate the situation with the unusual form of this cocycle. We will link the Virasoro-Bott cocycle with the Heisenberg cocycle, which is defined in more algebraic terms. The infinite dimensional Heisenberg group is the affine group which corresponds to GL_{1}, thus the Heisenberg cocycle may be viewed as the simplest case of the affine group cocycle.

The link between Diff(S^{1}) and the loop group of GL_{1} has been seen in fact in (1.2) in the previous section, where the jacobian J maps the former group into the latter. We will recover the Virasoro-Bott cocycle by evaluating the Heisenberg cocycle on the jacobians of the diffeomorphisms.

Let us recall the construction of the infinite dimensional Heisenberg group. We will in fact give two versions of the Heisenberg group, one algebraic, and another analytic.

In our first construction, we will directly exponentiate the infinite dimensional Heisenberg Lie algebra. This Lie algebra has the basis {H_{j}, c|j ∈ ℤ} and the Lie bracket

[H_{n}, H_{m}] = nδ_{n,−m}c,

and c is central.

By the direct exponentiation we will get the Heisenberg group:

{exp(αc) exp(∑_{j≤0} a_{j}H_{j}) exp(∑_{j>0} a_{j}H_{j})} ,  (2.2)

where a_{k} ∈ ℝ with only finitely many non-zero a_{k}. Using the formula

exp(A) exp(B) = exp([A, B]) exp(B) exp(A),

which holds when [A, B] commutes with A and B, we get the multiplication law in the Heisenberg group:

exp(αc) exp(∑_{j≤0} a_{j}H_{j}) exp(∑_{j>0} a_{j}H_{j}) × exp(βc) exp(∑_{j≤0} b_{j}H_{j}) exp(∑_{j>0} b_{j}H_{j})

= exp((α + β + ∑_{j>0} ja_{j}b_{−j})c) exp(∑_{j≤0}(α_{j} + β_{j})H_{j}) exp(∑_{j>0}(α_{j} + β_{j})H_{j}).

This version is an algebraic version of the Heisenberg group since only finite products are allowed in (2.2). The group of diffeomorphisms on the other hand is of analytic nature, so we would need to consider a certain completion of the algebraic version of the Heisenberg group. The construction of the analytic version of the Heisenberg group will follow the general pattern of the affine extensions of the loop groups [PS].

Consider the group Map(S^{1}, ℝ^{∗}) of C^{∞} loops in ℝ^{∗} = GL_{1}(ℝ), and its connected component of identity Map_{0}(S^{1}, ℝ^{∗}) = Map(S^{1}, ℝ_{+}^{∗}). The group structure in a loop group is given by pointwise multiplication. In order to define a central extension of Map(S^{1}, ℝ_{+}^{∗}), we need to consider contractions of elements f ∈ Map(S^{1}, ℝ_{+}^{∗}) to identity. The general scheme of affine extensions simplifies in our case since there exists a canonical contraction for each f. This is due to the fact that the exponential map from ℝ (the Lie algebra of ℝ_{+}^{∗}) to ℝ_{+}^{∗} is bijective.

Indeed let x = ln(f), x ∈ Map(S^{1}, ℝ). Then f̃(τ) = exp(τx), τ ∈ [0, 1] is a homotopy between identity and f. This canonical contraction respects multiplication: f̃g = f̃g̃ because ℝ^{∗} is commutative, and the usual properties of the logarithm and the exponential functions hold. In the general scheme of affine extensions one has to consider all possible contractions of elements of a loop group to identity. In our case this is not necessary because of the existence of a canonical contraction. The affine cocycle is given by the formula [PS]:

C(f, g) = ∫_{S^{1}} ∫_{[0,1]} f̃^{−1}df̃∧ dg̃g^{−1}.

Here f̃, g̃ are viewed as functions of two variables: t – the parameter along S^{1} and τ – the parameter of the homotopy. Let x(t) = ln(f(t)), y(t) = ln(g(t)). Then we can calculate C(f, g) explicitly in the following way:

C(f, g) = ∫_{S^{1}} ∫_{[0,1]} \frac{1}{f̃ ; g̃} (\frac{∂f̃}{∂τ}\frac{∂g̃}{∂t} − \frac{∂f̃}{∂t}\frac{∂g̃}{∂τ}) dτ ∧ dt

= ∫_{S^{1}} ∫_{[0,1]} τ (x(t)\frac{dy}{dt} − \frac{dx}{dt}y(t)) dτ ∧ dt.

Integrating in τ, and then integrating the second term by parts with respect to t we get

C(f, g) = ∫_{S^{1}} x(t)dy(t).

We see that the formula for the Heisenberg cocycle becomes very similar to the formula for the Virasoro-Bott cocycle:

C(f, g) = ∫_{S^{1}} ln(f)d ln(g).  (2.3)

Now x(t) and y(t) are periodic C^{∞} functions. These can be represented by Fourier series:

x(t) = \frac{a_{0}}{2} + ∑_{j>0} a_{j} cos(2πjt) + ∑_{j>0} a_{−j} sin(2πjt),

y(t) = \frac{b_{0}}{2} + ∑_{j>0} b_{j} cos(2πjt) + ∑_{j>0} b_{−j} sin(2πjt).

The functions x(t), y(t) are of class C^{∞} if and only if the Fourier coefficients satisfy the condition

lim_{j→±∞} j^{k}a_{j} = lim_{j→±∞} j^{k}b_{j} = 0 for all k ∈ ℕ.  (2.4)

Now by direct integration of (2.3) we can get the formula for the cocycle in terms of the Fourier coefficients:

C(f, g) = π ∑_{j∈ℤ} ja_{j}b_{−j}.

The analytic version of the Heisenberg group is the set of pairs (f, α), where f ∈ Map(S^{1}, ℝ_{+}^{∗}), α ∈ ℝ, and the group law is

(f, α)(g, β) = (fg, α + β + C(f, g)).

It is easy to see that this cocycle is equivalent to the cocycle from the first version of the Heisenberg group. Indeed, the isomorphism between the latter analytic version and the completion of the algebraic version can be given as follows:

φ (exp (\frac{a_{0}}{2} + ∑_{j>0} a_{j} cos(2πjt) + ∑_{j>0} a_{−j} sin(2πjt)) , α) =

exp((\frac{α}{2π} + \frac{1}{2} ∑_{j>0} ja_{j}a_{−j})c) exp(∑_{j≤0} a_{j}H_{j}) exp(∑_{j>0} a_{j}H_{j}).

The condition on coefficients {a_{j}} for the completion of the first version of the Heisenberg group should be taken the same as the C^{∞} condition (2.4).

Remark 5. We can lift the cocycle (2.3) to the group Map(S^{1}, ℝ^{∗}) = Map(S^{1}, ℝ_{+}^{∗}) × ℤ_{2} by taking absolute values of the loops:

C(f, g) = ∫_{S^{1}} ln |f|d ln |g|.  (2.5)

Now we can give interpretation of the Virasoro-Bott cocycle by means of the cocycle of the Heisenberg group.

Theorem 2.1. Let J be the jacobian map J : Diff(S^{1}) → Map(S^{1}, ℝ^{∗}). The Virasoro-Bott cocycle on Diff(S^{1}) can be obtained by evaluating the Heisenberg cocycle (2.5) on F^{J}(G) and G^{J}:

B(F, G) = C(F^{J}(G), G^{J}).

Proof. This follows immediately from comparing the formulas (2.1) and (2.5).

Remark 6. Using the jacobian map we also get a similar relation between the Virasoro and Heisenberg cocycles at the level of Lie algebras. The Heisenberg cocycle on the abelian Lie algebra of functions on a circle is

h(u_{1}(x), u_{2}(x)) = ∫_{S^{1}} u_{1}(x)du_{2}(x).

The Virasoro cocycle on W_{1} is obtained by evaluating the Heisenberg cocycle on the jacobians (i.e. first derivatives) of vector fields:

(v(x)\frac{∂}{∂x}, w(x)\frac{∂}{∂x}) = h(v^{J}, w^{J}) = ∫_{S^{1}} v′(x)dw′(x).

Remark 7. Integration along S^{1} in the formulas (2.3) and (2.5) realizes the isomorphism between Ω^{1}/dΩ^{0} and ℝ. We thus can also write the Heisenberg cocycle as a Ω^{1}/dΩ^{0} valued cocycle

C(f, g) = ln |f|d ln |g|.  (2.6)

Proposition 2.2. Let X be a C^{∞} real manifold. Then the formula (2.6) gives an Ω^{1}(X)/dΩ^{0}(X) valued central 2-cocycle on the group Map(X, ℝ^{∗}).

Proof. We first verify the claim for the subgroup Map(X, ℝ_{+}^{∗}). Applying the logarithm function, we get that the multiplicative group Map(X, ℝ_{+}^{∗}) is isomorphic to the additive group of C^{∞} functions on X. Under this isomorphism (2.5) transforms into

C_{1}(u, v) = u(x)dv(x),  (2.7)

where u = ln(f), v = ln(g). To see that (2.7) satisfies the cocycle condition, we need to check that

C_{1}(u, v) + C_{1}(u + v, w) = C_{1}(v, w) + C_{1}(u, v + w),

which follows immediately from the linearity of (2.7).

Finally, the group Map(X, ℝ^{∗}) is a direct product Map(X, ℝ_{+}^{∗}) × ℤ_{2}^{n}, where n is the number of connected components of X, thus a central cocycle on the subgroup Map(X, ℝ_{+}^{∗}) lifts to a cocycle on Map(X, ℝ^{∗}).

Even though the above theorems are just very simple observations, they allow us to construct interesting generalizations of the Virasoro-Bott cocycle that will be explored in the next two sections.

3. Abelian extension of the group of diffeomorphisms of a torus.

The Virasoro cocycle can not be generalized as a central cocycle to the case of a Lie algebra of vector fields in dimension greater than one. However in the case of a torus, it is known that there are abelian generalizations of the Virasoro cocycle. We thus may expect that we can also construct abelian extensions of the group of diffeomorphisms of a torus.

In this section we will construct an abelian extension that corresponds to Lie algebra cocycle dτ_{1}. The space of the extension will be Ω^{2}(𝕋^{N}). This cocycle will be non-vanishing on the subgroup of volume-preserving diffeomorphisms.

As we have seen in Section 1, the two abelian cocycles on W_{N}, τ_{1} and τ_{2}, correspond to the decomposition of gl_{N} into sl_{N} ⊕ ℝI. The cocycle that will be constructed in this section corresponds to the sl_{N} component. In Section 4 we will construct a cocycle that corresponds to the component of scalar matrices.

Motivated by Theorem 2.1, our idea is to use the crossed homomorphism (1.2) and pull back a central 2-cocycle on Map(𝕋^{N}, GL_{N}) to an abelian 2-cocycle on Diff(𝕋^{N}) .

We will first describe our construction in a very general set-up. Let a group D acts on a group M on the right by automorphisms:

m ↦ m(d), for m ∈ M, d ∈ D.

Definition. A crossed homomorphism is a map

j : D → M,

such that

j(d_{1}d_{2}) = j(d_{1})(d_{2}) · j(d_{2})  (3.1)

for all d_{1}, d_{2} ∈ D.

Suppose we indeed have a crossed homomorphism j : D → M.

Let K be an abelian (additive) group and let

c : M × M → K

be a central 2-cocycle on M. Suppose that D also acts on K on the right by automorphisms. Lemma 3.1. If the K valued 2-cocycle c on M is D invariant, i.e.,

c(m_{1}, m_{2})(d) = c(m_{1}(d), m_{2}(d)), for all m_{1}, m_{2} ∈ M, d ∈ D,

then

b(d_{1}, d_{2}) = c(j(d_{1})(d_{2}), j(d_{2}))  (3.2)

defines an abelian K valued 2-cocycle on D.

Proof. Using the fact that c satisfies the central cocycle condition

c(m_{1}, m_{2}) + c(m_{1}m_{2}, m_{3}) = c(m_{1}, m_{2}m_{3}) + c(m_{2}, m_{3}),  (3.3)

we need to show that b is an abelian cocycle:

b(d_{1}, d_{2})(d_{3}) + b(d_{1}d_{2}, d_{3}) = b(d_{1}, d_{2}d_{3}) + b(d_{2}, d_{3}).  (3.4)

From the definition (3.2), we get that the left hand side is

c(j(d_{1})(d_{2}), j(d_{2}))(d_{3}) + c(j(d_{1}d_{2})(d_{3}), j(d_{3})),

which using the invariance of cocycle c and (3.1), becomes

c(j(d_{1})(d_{2}d_{3}), j(d_{2})(d_{3})) + c(j(d_{1})(d_{2}d_{3})j(d_{2})(d_{3}), j(d_{3})).

By (3.3) this equals

c(j(d_{1})(d_{2}d_{3}), j(d_{2})(d_{3})j(d_{3})) + c(j(d_{2})(d_{3}), j(d_{3}))
= c(j(d_{1})(d_{2}d_{3}), j(d_{2}d_{3})) + c(j(d_{2})(d_{3}), j(d_{3}))
= b(d_{1}, d_{2}d_{3}) + b(d_{2}, d_{3}),

which is the left hand side of (3.4). The Lemma is proved.

We will apply this Lemma to the groups D = Diff(𝕋^{N}) and M = Map(𝕋^{N}, GL_{N}). The extension group K will be Ω^{2}(𝕋^{N}) and the crossed homomorphism between Diff(𝕋^{N}) and Map(𝕋^{N}, GL_{N}) is the jacobian map (1.2). The action of Diff(𝕋^{N}) on Map(𝕋^{N}, GL_{N}) and Ω^{2} is given by (1.4) and (1.1).

The central 2-cocycle on Map(𝕋^{N}, GL_{N}) that we will use is (cf. [PS], [LNMS]):

C(f(x), g(x)) = Tr (f^{−1}df ∧ dg g^{−1}) .  (3.5)

Let us verify the cocycle condition

C(f, g) + C(fg, h) = C(f, gh) + C(g, h).  (3.6)

The left hand side here is

Tr (f^{−1}df ∧ dg g^{−1}) + Tr ((fg)^{−1}d(fg) ∧ dh h^{−1})
= Tr (f^{−1}df ∧ dg g^{−1}) + Tr (g^{−1}f^{−1}d(f)g ∧ dh h^{−1}) + Tr (g^{−1}dg ∧ dh h^{−1}) .

Similarly, the left hand side of (3.6) becomes

Tr (f^{−1}df ∧ d(gh)(gh)^{−1}) + Tr (g^{−1}dg ∧ dh h^{−1})
= Tr (f^{−1}df ∧ dg g^{−1}) + Tr (f^{−1}d(f) ∧ gdh h^{−1}g^{−1}) + Tr (g^{−1}dg ∧ dh h^{−1}) .

Both sides are equal because

Tr (g^{−1}f^{−1}d(f)g ∧ dh h^{−1}) = Tr (f^{−1}d(f) ∧ g dh h^{−1}g^{−1})

by the commutativity of the trace: Tr(AB) = Tr(BA).

Finally we note that the cocycle (3.5) on Map(𝕋^{N}, GL_{N}) is Diff(𝕋^{N}) invariant, since the action of diffeomorphisms on differential forms (1.1) agrees with the action on functions (1.4):

Tr (f^{−1}df ∧ dgg^{−1}) (H) = Tr (f^{−1}(H)df(H) ∧ dg(H)g^{−1}(H)) .

We thus obtain the following result:

Theorem 3.2. The set Diff(𝕋^{N}) × Ω^{2}(𝕋^{N}) becomes an extension of the group of diffeomorphisms with the abelian group Ω^{2}(𝕋^{N}) with multiplication defined as follows:

(F, α)(G, β) = (F G, α(G) + β + Tr (f^{−1}df ∧ dgg^{−1})) ,

where f = F^{J}(G), g = G^{J}.

Remark 8. It is easy to compute that the corresponding Lie algebra extension of W_{N} is given by cocycle dτ_{1}.

Problem. Construct an abelian extension of Diff(𝕋^{N}) with abelian subgroup Ω^{1}/dΩ^{0} which corresponds to the Lie algebra 2-cocycle τ_{1}.

4. Abelian extension of the group of diffeomorphisms of a manifold with a volume form.

In this section we introduce an abelian extension of the group of diffeomorphisms of a manifold with a volume form that generalizes the Virasoro-Bott central extension.

Let X be a real C^{∞} manifold with the volume form ω. We note that for an arbitrary diffeomorphism F ∈ Diff(X), the quotient \frac{ω(F)}{ω} is a well-defined non-vanishing function. Moreover, the map

δ : Diff(X) → Map(X, ℝ^{∗}),

given by

δ(F) = \frac{ω(F)}{ω}

is a crossed homomorphism. Indeed,

δ(F G) = \frac{ω(F G)}{ω} = \frac{ω(F G)}{ω(G)} \frac{ω(G)}{ω} = δ(F)(G)δ(G).

As an immediate consequence of this observation, Lemma 3.1 and Proposition 2.2, we get the following:

Theorem 4.1. Let X be a C^{∞} real manifold with a volume form ω and let C(f, g) be the Heisenberg cocycle (2.6) on Map(X, ℝ^{∗}). Then

B(F, G) = C (\frac{ω(F G)}{ω(G)}, \frac{ω(G)}{ω})

= ln |\frac{ω(F G)}{ω(G)}| d ln |\frac{ω(G)}{ω}|  (4.1)

is an abelian Ω^{1}(X)/dΩ^{0}(X) valued 2-cocycle on the group of diffeomorphisms Diff(X).

Corollary 4.2. The 2-cocycle on the Lie algebra of vector fields on a manifold with a volume form ω that corresponds to the group cocycle (4.1) is

c(v, w) = div(v) d div(w),

where the divergence of a vector field v is given by div(v) = \frac{v·ω}{ω}.

Example 4.3. If X is a torus 𝕋^{N} with the volume form ω = dx_{1} ∧ … ∧ dx_{N}, then \frac{ω(F)}{ω} = det(F^{J}), and the cocycle (3.1) can be written as

B(F, G) = ln |det(F^{J}(G))|d ln |det(G^{J})|.

One can easily calculate that the corresponding Lie algebra cocycle on W_{N} is τ_{2}.

Acknowledgments. I am grateful to l’Institut des Mathématiques de Jussieu (Paris) and Mathematical Sciences Research Institute (Berkeley) for their hospitality. I also thank Dmitri Orlov for stimulating discussions.

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