Journal of Lie Theory Volume 11 (2001) 173–184 ◯^{C}2001 Heldermann Verlag

Explicit formulae for cocycles of holomorphic vector fields with values in λ densities

Friedrich Wagemann

Communicated by K.-H. Neeb

Abstract. In this article, we give explicit formulae for the generators of H^{2}(Hol(Σ_{r}), F_{λ}(Σ_{r})) in terms of affine and projective connections, Σ_{r} being a compact Riemann surface punctured in r points This is done using the cocycles which have been evidenced by V. Ovsienko and C. Roger in [13] and globalizing them by their transformation property. AMS classification: 17B56, 17B65, 17B66, 17B68

Introduction

The continuous cohomology of Lie algebras of C^{∞}-vector fields has been studied by I. M. Gelfand, D. B. Fuks, R. Bott, A. Haefliger and G. Segal in some outstanding papers [4], [9], [1].

B. L. Feigin [2] and N. Kawazumi [11], whose work is continued in [18], studied Gelfand-Fuks cohomology of Lie algebras of holomorphic vector fields Hol(Σ) on an open Riemann surface Σ. Kawazumi calculated the cohomology spaces H^{∗}(Hol(Σ), F_{λ}(Σ)) of Hol(Σ) with values in the space of (holomorphic) λ-densities on Σ, using a well known theorem of Goncharova, cf [3]. He expressed the generators of the cohomology spaces in terms of the nowhere-vanishing holomorphic vector field ∂ which exists on open Riemann surfaces, trivializing the holomorphic tangent bundle.

In this article, we give explicit formulae for the generators of H^{2}(Hol(Σ_{r}), F_{λ}(Σ_{r})) where Σ_{r} = Σ \ {p_{1}, …, p_{r}}, Σ being a compact Riemann surface, in terms of affine and projective connections. This is done using the cocycles which have been evidenced by V. Ovsienko and C. Roger in [13] and globalizing them by their transformation property.

The main reason to look for explicit formulae is the search for a generalization of the Krichever-Novikov algebras [12] to extensions of Hol(Σ_{r}) by F_{λ}(Σ_{r}), cf [13] for the case of V ect(S^{1}).

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag Acknoledgements: The author thanks V. Ovsienko and C. Roger for the statement of the

problem and usefull discussions on the subject of this paper. He thanks also D. Millionshshikov for illuminating conversations and his formula for the Krichever-Novikov cocycle.

1. Preliminairies, statement of the result

In this section, we state the theorems of Kawazumi and of Ovsienko-Roger which are the starting point of our work.

Let V ect(S^{1}) denote the Lie algebra of differentiable vector fields on the circle S^{1}, and F_{λ} the V ect(S^{1})-module of λ-densities, using the action

L_{f}a = (fa′ + λf′a)(dx)^{λ},  (1)

where f ∈ V ect(S^{1}) and a ∈ F_{λ} are both represented by their coefficient function. Here F_{λ} is identified with the space of C^{∞} functions on the circle S^{1} and only the V ect(S^{1})-action changes with λ. Thus, λ can be any complex number.

Theorem 1.1. (Theorem 3, [13]) The cohomology groups H^{2}(V ect(S^{1}), F_{λ}) are non-zero only for λ = 0, 1, 2, 5, 7. They are two-dimensional for λ = 0, 1, 2 and one-dimensional for λ = 5, 7.

The cohomology classes are represented by generators, denoted by c_{λ} or c̄_{λ} corresponding to the different values of λ, which read explicitly

c̄_{0}(f, g) = | \begin{matrix}f & g \\ f′ & g′ \end{matrix} |
c_{0}(f, g) = c_{GF} (f, g)
c_{1}(f, g) = | \begin{matrix}f′ & g′ \\ f′′ & g′′ \end{matrix} | dx
c̄_{1}(f, g) = | \begin{matrix}f & g \\ f′′ & g′′ \end{matrix} | dx
c_{2}(f, g) = | \begin{matrix}f′ & g′ \\ f′′′ & g′′′ \end{matrix} | (dx)^{2}
c̄_{2}(f, g) = | \begin{matrix}f & g \\ f′′′ & g′′′ \end{matrix} | (dx)^{2}
c_{5}(f, g) = | \begin{matrix}f′′′ & g′′′ \\ f^{(IV)} & g^{(IV)} \end{matrix} | (dx)^{5}
c_{7}(f, g) = (2 | \begin{matrix}f′′′ & g′′′ \\ f^{(V I)} & g^{(V I)} \end{matrix} | − 9 | \begin{matrix}f^{(IV)} & g^{(IV)} \\ f^{(V)} & g^{(V)} \end{matrix} |) (dx)^{7}

In the theorem, c_{GF} denotes the Gelfand-Fuks cocycle, being a cocycle with values in the trivial module ℂ ⊂ F_{0}. Its explicit formula is

c_{GF} (f, g) = ∫_{S^{1}} | \begin{matrix}f′ & g′ \\ f′′ & g′′ \end{matrix} | (x) dx.

Now, let Σ_{r} - as in the rest of this article - denote an open Riemann surface, obtained from the compact Riemann surface Σ by extraction of r points: Σ_{r} := Σ \ {p_{1}, …, p_{r}}.

Let Hol(Σ_{r}) denote the (infinite dimensional) Lie algebra of holomorphic vector fields on Σ_{r}. Let F_{λ}(Σ_{r}) be the space of sections of the bundle of holomorphic λ-densities. As all bundles on Σ_{r} are trivial, elements of F_{λ}(Σ_{r}) can be represented by holomorphic functions. Hol(Σ_{r}) still acts on F_{λ}(Σ_{r}) according to

L_{f}a = (fa′ + λf′a)(dz)^{λ},

where f ∈ Hol(Σ_{r}) and a ∈ F_{λ}(Σ_{r}) are both represented by their coefficient function, the (dz)^{λ} being the global section trivialising the bundle of λ-densities.

Recall that

H^{p}(Σ_{r}) = \{ \begin{matrix}ℂ & for & p = 0 \\ ℂ^{2g+r−1} & for & p = 1 \\ 0 & for & p ≥ 2 \end{matrix} ,

if g denotes the genus of Σ.

In [11], Kawazumi calculates the spaces H^{2}(Hol(Σ_{r}), F_{λ}(Σ_{r})) using the Res̆etnikov [14] spectral sequence. This sequence has as E_{2}-term the sheaf cohomology of a sheaf whose stalk at x ∈ Σ_{r} is the cohomology of Hol(Σ_{r}) with values in F_{λ}(Σ_{r})_{x}, the fibre of F_{λ}(Σ_{r}) at x.

Furthermore, Kawazumi uses the main result of his article to express the stated E_{2}-term as a tensor product of some “covariant derivative” cocycles with the formal version of his cohomology, namely H^{∗}(W_{1}, T_{λ}). Here, W_{1} is the Lie algebra of formal vector fields on the complex line and T_{λ} is the corresponding module of formal λ-densities. H^{∗}(W_{1}, T_{λ}) is explicitly given, thanks to the theorem of Goncharova, cf [3]. Thus, he obtains a (collapsing) spectral sequence for the wanted cohomology.

Let us state his result just for the dimensions of H^{2}(Hol(Σ_{r}), F_{λ}(Σ_{r})) for the different λ.

Theorem 1.2. (consequence of (9.7), [11])

dim H^{2}(Hol(Σ_{r}), F_{0}(Σ_{r})) = 2(2g + r − 1)
dim H^{2}(Hol(Σ_{r}), F_{1}(Σ_{r})) = 2g + r
dim H^{2}(Hol(Σ_{r}), F_{2}(Σ_{r})) = 2g + r
dim H^{2}(Hol(Σ_{r}), F_{5}(Σ_{r})) = 1
dim H^{2}(Hol(Σ_{r}), F_{7}(Σ_{r})) = 1

For all other values of λ, H^{2}(Hol(Σ_{r}), F_{λ}(Σ_{r})) is zero.

To understand these dimensions, recall from [11] (9.7) p.701 that for λ = 0, H^{2}(Hol(Σ_{r}), F_{λ}(Σ_{r})) is generated by some classes, which we denote c_{0}^{ω} and c̄_{0}^{ω}, each depending on an element ω ∈ H^{1}(Σ_{r}) = ℂ^{2g+r−1}. In the same manner, H^{2}(Hol(Σ_{r}), F_{1}(Σ_{r})) is generated by one family, denoted c̄_{1}^{ω}, and a cocycle c_{1}, H^{2}(Hol(Σ_{r}), F_{2}(Σ_{r})) is generated by a family, denoted c̄_{2}^{ω}, and a cocycle c_{2} and H^{2}(Hol(Σ_{r}), F_{5}(Σ_{r})) and H^{2}(Hol(Σ_{r}), F_{7}(Σ_{r})) are each generated by a cocycle c_{5} and c_{7}. We have chosen the same notation as in the above theorem of Ovsienko and Roger, but their cocycles would not give globally defined objects on a Riemann surface Σ_{r}. The explicit construction in terms of connections of these cocycles, resp. families of cocycles, is not known and the subject of this article.

Partial results are known: namely, the holomorphic version of the Gelfand-Fuks cocycle leading to a meromorphic version of the Virasoro algebra appeared in work of Krichever and Novikov, further developped by Schlichenmaier and Sheinman, cf [16]. It reads (cf [17] where we applied Poincaré duality to write it as an integral over Σ_{r}):

c_{0}^{ω}(f, g) = \frac{c}{24πi} ∫_{Σ_{r}} (\frac{1}{2} | \begin{matrix}f & g \\ f′′′ & g′′′ \end{matrix} | − 2R | \begin{matrix}f & g \\ f′ & g′ \end{matrix} |) dz ∧ ω̄,

where ω ∈ H^{1}(Σ_{r}) and R is a projective connection. Recall that for a Stein manifold (in particular for an open Riemann surface) the subcomplex of holomorphic forms calculates all the de Rham cohomology, cf [6] p.449.

On this form, we already see how such a cocycle is constructed: the Gelfand-Fuks cocycle serves as symbol, and then one adds terms to have a globally defined 1-form. In other words, the 1-form without the term involving R is globally defined only with respect to an atlas of charts from P Sl(2; ℂ), to define it for a general holomorphic atlas, one has to use a projective connection.

Affine connections are more general than projective connections, namely, a manifold supporting an affine connection admits also a projective connection. These connections come from the corresponding structures, an affine (projective) structure being an (holomorphic) atlas such that the chart transitions are in the subgroup of affine (resp. projective) transformations. One sees that an affine structure is in particular a projective structure. We will state some well known facts about these objects in the next section.

Let us state the main result of this article:

Theorem 1.3. Let Σ_{r} be an open Riemann surface, Hol(Σ_{r}) the Lie algebra of holomorphic vector fields on Σ_{r} and F_{λ}(Σ_{r}) the space of holomorphic λ-densities.

The spaces H^{2}(Hol(Σ_{r}), F_{λ}(Σ_{r})) for λ = 0, 1, 2, 5, 7 are generated by the classes c̄_{0}, c_{0}^{ω}, c_{1}, c̄_{1}^{ω}, c_{2}, c̄_{2}^{ω}, c_{5}, c_{7}, where the subscript indicates the value of λ and the superscript the dependence on (the class of) a holomorphic 1-form ω on Σ_{r}.

The explicit formulae are given in section 2 (in terms of affine and projective connections) and in section 4 (in terms of the covariant derivative).

Note that the theorem asserts in particular that the formulae in section 2 and in section 4 for c̄_{0}, c_{0}^{ω}, c_{1}, c̄_{1}^{ω}, c_{2}, c̄_{2}^{ω}, c_{5}, c_{7} coïncide.

2. Transformation behavior

In this section, we shall calculate the correction terms in order to make the cocycles of theorem 1 globally defined geometrical objects.

Let X, Y denote holomorphic vector fields on Σ_{r}. Let U_{α}, U_{β} ⊂ Σ_{r} be open subsets such that U_{α} ∩ U_{β} ≠ Ø. Let X and Y be given by local coefficient functions f_{α}, g_{α} in U_{α} and f_{β}, g_{β} in U_{β}. Denote by z_{α} and z_{β} local coordinates in U_{α} and U_{β}, and by h(z_{α}) = z_{β} the holomorphic change of coordinates. We have

f_{β} = \frac{∂h}{∂z_{α}} f_{α},

and similarly for g_{β}. Denote \frac{∂h}{∂z_{α}} just by h′.

Now, it is easy to transform derivatives on the coefficient functions:

f′_{β} = \frac{1}{h′} (h′′f_{α}) + f′_{α},

and

f′′_{β} = \frac{1}{h′} \{(\frac{h′′′}{h′} − \frac{(h′′)^{2}}{(h′)^{2}} ) f_{α} + \frac{h′′}{h′} f′_{α} + f′′_{α}\} .

Observe that this kind of manipulations is particularly well suited for being treated by MAPLE.

Denote by S := S(h) the Schwarzian derivative of h, i.e. the expression

S = \frac{h′′′}{h′} − \frac{3}{2} (\frac{h′′}{h′} )^{2} .

It is easy to show and well known that we have:

f′′′_{β} = \frac{1}{(h′)^{2}} (f′′′_{α} + S′f_{α} + 2Sf′_{α}) .

Now recall some generalities on affine and projective structures, resp., connections, cf. [7] §9 p. 164 and [15] p. 137–138:

Definition 2.1. Let {U_{α}, z_{α}} be a covering of Σ_{r} by coordinate charts and z_{β} = h(z_{α}) the coordinate transitions for non-empty U_{α} ∩ U_{β}.

A (holomorphic) projective connection is a family of holomorphic functions R_{α} on U_{α} such that for non-empty U_{α} ∩ U_{β}, we have

R_{β}(h′)^{2} = R_{α} + S.

In the same way, we have

Definition 2.2. A (holomorphic) affine connection is a family of holomorphic functions T_{α} on U_{α} such that for non-empty U_{α} ∩ U_{β}, we have

T_{β}h′ = T_{α} + \frac{h′′}{h′} .

(Observe that h′ ≠ 0.) There is a 1-1 correspondance between connections and the corresponding structures, see [7] thm. 19, p. 170. See also [5], section 2, for a brief summary on these structures. Affine connections (thus affine structures, projective structures and projective connections) exist on any open Riemann surface, cf [8]. This is in contrast to compact Riemann surfaces where affine connections exist only for genus 1, see [7] p. 173.

We use these objects to compensate extra terms arising from the transition behaviour of the cocycles of theorem 1. One arrives at the following results (the first two are trivial; in the following, R denotes a projective connection and T an affine connection):

• c̄_{0}(f, g) is a well-defined global vector field, so c̄_{0}^{ω}(f, g) := c̄_{0}(f, g)ω is a welldefined global function

• c_{0}^{ω}(f, g) is a well-defined global (constant) function

• c_{1}(f, g) := | \begin{matrix}f′ & g′ \\ f′′ & g′′ \end{matrix} | − T | \begin{matrix}f & g \\ f′′ & g′′ \end{matrix} | + (R − \frac{1}{2}T^{2}) | \begin{matrix}f & g \\ f′ & g′ \end{matrix} | is a well-defined

global 1-form

• c̄_{1}(f, g) := | \begin{matrix}f & g \\ f′′ & g′′ \end{matrix} | − T | \begin{matrix}f & g \\ f′ & g′ \end{matrix} | is a well-defined global function, so

c̄_{1}^{ω}(f, g) := c̄_{1}(f, g)ω is a well-defined global 1-form

• c_{2}(f, g) := | \begin{matrix}f′ & g′ \\ f′′′ & g′′′ \end{matrix} |−T | \begin{matrix}f & g \\ f′′′ & g′′′ \end{matrix} |−(2T R−R′) | \begin{matrix}f & g \\ f′ & g′ \end{matrix} | is a well-defined

global 2-form

• c̄_{2}(f, g) := | \begin{matrix}f & g \\ f′′′ & g′′′ \end{matrix} | − 2R | \begin{matrix}f & g \\ f′ & g′ \end{matrix} | is a well-defined global 1-form, so

c̄_{2}^{ω}(f, g) := c̄_{2}(f, g) ⊗ ω is a well-defined global quadratic differential

• c_{5}(f, g) := | \begin{matrix}f′′′ & g′′′ \\ f^{(IV)} & g^{(IV)} \end{matrix} |+R′′ | \begin{matrix}f & g \\ f′′′ & g′′′ \end{matrix} |+3R′ | \begin{matrix}f′ & g′ \\ f′′′ & g′′′ \end{matrix} |+2R | \begin{matrix}f′′ & g′′ \\ f′′′ & g′′′ \end{matrix} |+

(2RR′−3(R′)^{2}) | \begin{matrix}f & g \\ f′ & g′ \end{matrix} |−2RR′ | \begin{matrix}f & g \\ f′′ & g′′ \end{matrix} |−R′ | \begin{matrix}f & g \\ f^{(IV)} & g^{(IV)} \end{matrix} |−4R^{2} | \begin{matrix}f′ & g′ \\ f′′ & g′′ \end{matrix} |−

2R | \begin{matrix}f′ & g′ \\ f^{(IV)} & g^{(IV)} \end{matrix} | is a well-defined global 5-form

Note that the assignment of holomorphic 1-forms ω to certain cocycles gives exactly the number of generators which is needed to generate the cohomology spaces. The formula for c_{7} is given in the appendix.

These cocycles do not depend on the choice of the connections.

3. Cocycle property

Now, we have globalized the cocycles to individual cochains or families of cochains. But it is not clear whether the terms that we added will disturb the cocycle property. This is what we check in this section.

By writing explicitly the cocycle identity for the different expressions which we considered in the preceeding section to globalize the cocycles (the action depends on λ (cf equation (1) in the preliminairies)), we get the following result:

(1) | \begin{matrix}f & g \\ f′ & g′ \end{matrix} | is a cocycle for any value of λ

(2) | \begin{matrix}f & g \\ f′′ & g′′ \end{matrix} | is a cocycle only for λ = 1

(3) | \begin{matrix}f & g \\ f′′′ & g′′′ \end{matrix} | is a cocycle only for λ = 2

(4) | \begin{matrix}f′ & g′ \\ f′′ & g′′ \end{matrix} | is a cocycle only when taking trivial action

(5) | \begin{matrix}f′ & g′ \\ f′′′ & g′′′ \end{matrix} | is a cocycle for any value of λ

(6) | \begin{matrix}f & g \\ f^{(IV)} & g^{(IV)} \end{matrix} | is never a cocycle

(7) | \begin{matrix}f′ & g′ \\ f^{(IV)} & g^{(IV)} \end{matrix} | is never a cocycle

(8) | \begin{matrix}f′′ & g′′ \\ f′′′ & g′′′ \end{matrix} | is a cocycle only for λ = 3

(9) | \begin{matrix}f′′′ & g′′′ \\ f^{(IV)} & g^{(IV)} \end{matrix} | is a cocycle only for λ = 5

The 6th and the 7th expressions arise as terms in c_{5}(f, g). Note that in this formal calculation, f, g can be interpreted as vector fields on the circle or on the open Riemann surface. In the latter case, the expressions are not globally defined geometric objects.

It is thus obvious that c̄_{0}^{ω}, c_{0}^{ω}, c_{1}, c̄_{1}^{ω} c_{2}, c̄_{2}^{ω} are well-defined, global 2cocycles for cohomology with values in F_{λ} with λ = 0, 0, 1, 1, 2 and 2 respectively.

For c_{5} and c_{7}, we will take a different point of view.

4. Formulation in terms of the covariant derivative

The fundamental fact which assures the validity of our work is the existence of affine structures on open Riemann surfaces.

These connections are flat integrable connections in the sense of differential geometry, thus we can talk about associated covariant derivatives. The covariant derivative associated to the affine connection reads locally (on a λ-density ϕ)

∇ϕ = ϕ′ − λΓϕ.

In general, Γ plays the role of the trace of the Christoffel symbols; in higher dimensions, we have

∇_{i}ϕ = ∂_{i}ϕ − λΓ_{ij}^{j}ϕ.

Actually, Γ is nothing else than what we called before the affine connection T. ∇ϕ is a globally defined object. On \frac{1}{2} -densities, one can exhibit a particular convenient choice of a projective connection associated to an affine connection:

∇^{2}(ϕ(dz)^{− \frac{1}{2}} ) = ∇((ϕ′ + \frac{1}{2}Γϕ)(dz)^{\frac{1}{2}} )
= (ϕ′′ + \frac{1}{2}(Γϕ)′ − \frac{1}{2}Γ(ϕ′ + \frac{1}{2}Γϕ))(dz)^{\frac{3}{2}}
= (ϕ′′ − \frac{1}{4}Γ^{2}ϕ + \frac{1}{2}Γ′ϕ)(dz)^{\frac{3}{2}}
= (∂^{2} + \frac{1}{2}R)ϕ(dz)^{− \frac{1}{2}}

where we have put R = −\frac{1}{2}Γ^{2} + Γ′.

Otherwise, this choice is justified by Γ = \frac{h′′}{h′} and S = \frac{h′′′}{h′} − \frac{3}{2} (\frac{h′′}{h′} )^{2}, giving also S = Γ′ − \frac{1}{2}Γ^{2}, cf [10] equation (10) p. 205.

Furthermore, we can set

L_{f}ϕ(dx)^{λ} = f∇ϕ + λ(∇f)ϕ,

because this action coïncides with the action defined in equation (1). We have [f, g] = f∇g − (∇f)g and a derivation property of ∇ on tensor products. This corresponds to the product formula for the derivative. With this in mind, we have the same rules of manipulation as before for computations which concerned only ordinary derivatives of functions on the circle.

In conclusion, it is clear that we can formulate all cocycles in terms of the covariant derivative:

• c_{1}(f, g) = | \begin{matrix}∇f & ∇g \\ ∇^{2}f & ∇^{2}g \end{matrix} | dz

• c̄_{1}(f, g) = | \begin{matrix}f & g \\ ∇^{2}f & ∇^{2}g \end{matrix} | dz^{0}

• c_{2}(f, g) = | \begin{matrix}∇f & ∇g \\ ∇^{3}f & ∇^{3}g \end{matrix} | dz^{2}

• c̄_{2}(f, g) = | \begin{matrix}f & g \\ ∇^{3}f & ∇^{3}g \end{matrix} | dz^{1}

• c_{5}(f, g) = | \begin{matrix}∇^{3}f & ∇^{3}g \\ ∇^{4}f & ∇^{4}g \end{matrix} | dz^{5}

• c_{7}(f, g) := (2 | \begin{matrix}∇^{3}f & ∇^{3}g \\ ∇^{6}f & ∇^{6}g \end{matrix} | − 9 | \begin{matrix}∇^{4}f & ∇^{4}g \\ ∇^{5}f & ∇^{5}g \end{matrix} |) dz^{7}

The cocycle c̄_{2} is the covariant derivative version of the Krichever-Novikov cocycle; we learnt this expression from D. Millionshshikov.

Obviously, this description is much simpler. To show at least in principle how the proof of this coïncidence looks like, take for example c̄_{2}. We have to calculate

∇^{3}f = ∇^{2}(f′ + Γf)
= ∇(f′′ + Γ′f + Γf′)
= f′′′ + Γ′′f + 2Γ′f′ − ΓΓ′f − Γ^{2}f′.

Note that in the first line, f is an element of F_{−1}(Σ_{r}), but (f′ + Γf) is an element of F_{0}(Σ_{r}), in the second line, (f′′ + Γ′f + Γf′) is an element of F_{1}(Σ_{r}) and the result is in F_{2}(Σ_{r}).

This gives

f(∇^{3}g) − g(∇^{3}f) = fg′′′ − gf′′′ + (2Γ − Γ^{2})(fg′ − gf′).

Identifying (2Γ−Γ^{2}) with (2Γ−Γ^{2}) = 2R, we get the coincidence of the covariant derivative expression with c̄_{2}.

As said before, it is clear that all covariant derivative expressions will be cocycles with values in the appropriate F_{λ} - the computations are straight forward.

5. Non-triviality of the cocycles

Let us scetch here an argument showing the non-triviality of the constructed cocycles (and their linear independence):

Choose an embedding S^{1} → Σ_{r}. The associated restriction of holomorphic vector fields (resp. holomorphic λ-densities) to S^{1} gives a map ϕ : Hol(Σ_{r}) → V ect(S^{1}) (resp. ψ : F_{λ}(Σ_{r}) → F_{λ}(S^{1})). These maps are injective Lie algebra homomorphisms with dense image (where the image is equipped with the induced topology from V ect(S^{1})), cf [12].

There is a commutative diagram:

[figure: 0; F_{λ}(S^{1}); V ect(S^{1}) × F_{λ}(S^{1}); V ect(S^{1}); 0; 0; F_{λ}(Σ_{r}); Hol(Σ_{r}) × F_{λ}(Σ_{r}); Hol(Σ_{r}); 0; ϕ; ψ]

The non-triviality now follows from the non-triviality (see [13]) of the corresponding cocycles for V ect(S^{1}).

A. The formula for c_{7}

Let T denote an affine connection on the open Riemann surface Σ_{r}. In this appendix, we give the formula for the global version of the cocycle c_{7} in terms of T.

c_{7}(f, g) = (2 | \begin{matrix}f′′′ & g′′′ \\ f^{(V I)} & g^{(V I)} \end{matrix} | − 9 | \begin{matrix}f^{(IV)} & g^{(IV)} \\ f^{(V)} & g^{(V)} \end{matrix} |)
+ (2T^{(V)} − 22T′T′′′ − 2T T^{(IV)} − 20(T′′)^{2} + 14(T′)^{3} + 42T T′T′′

+7T^{2}T′′′ − 21T^{2}(T′)^{2} − 7T^{3}T′′) | \begin{matrix}f & g \\ f′′′ & g′′′ \end{matrix} |

+ (10T^{(IV)} − 100T′T′′ − 10T T′′′

+70T (T′)^{2} + 35T^{2}T′′ − 35T^{3}T′) | \begin{matrix}f′ & g′ \\ f′′′ & g′′′ \end{matrix} |

+ (18T′′′ − 46(T′)^{2} − 18T T′′ + 28T^{2}T′ − 7T^{4}) | \begin{matrix}f′′ & g′′ \\ f′′′ & g′′′ \end{matrix} |
+ (−37T′′T^{(IV)} + 60T′(T′′)^{2} − 35T T′′T′′′ − 167T (T′)^{2}T′′ − 6T^{2}(T′′)^{2}

+32T^{3}T′T′′ + 16T^{5}T′′ + 33T T′T^{(IV)} + 35T^{2}T′T′′′ + 32T^{2}(T′)^{3} −32T^{4}(T′)^{2} + 4T^{(V)}T′ − 152(T′)^{2}T′′′ + 100(T′)^{4} − 2T^{2}T^{(V)}

+2T^{3}T^{(IV)} − 16T^{4}T′′′ + 135T′T′′′ + 36(T′′′)^{2}) | \begin{matrix}f & g \\ f′ & g′ \end{matrix} |

+ (27T′′T′′′ + 91(T′)^{2}T′′ − 27T (T′′)^{2} − 64T^{2}T′T′′ + 16T^{4}T′′

−9T T′T′′′ − 37T (T′)^{3} + 64T^{3}(T′)^{2} − 16T^{5}T′ − 18T′T^{(IV)}

+9T^{2}T^{(IV)} − 9T^{3}T′′′) | \begin{matrix}f & g \\ f′′ & g′′ \end{matrix} |

+ (−9T^{(IV)} − 14T^{2}T′′ + 14T^{3}T′ + 55T′T′′ − 28T (T′)^{2}

+9T T′′′) | \begin{matrix}f & g \\ f^{(IV)} & g^{(IV)} \end{matrix} |

+ (−36T′T′′′ + 236(T′)^{3} − 162T T′T′′ − 111(T′)^{2}T^{2} + 96T′T^{4}

+54T^{2}T′′′ − 54T^{3}T′′ − 16T^{6} − 72T′T′′ + 135(T′′)^{2}) | \begin{matrix}f′ & g′ \\ f′′ & g′′ \end{matrix} |

+ (−56T^{2}T′ + 92(T′)^{2} + 14T^{4} − 36T′′′ + 36T T′′) | \begin{matrix}f′ & g′ \\ f^{(IV)} & g^{(IV)} \end{matrix} |
+ (−5T^{2} + 10T′) | \begin{matrix}f′′′ & g′′′ \\ f^{(IV)} & g^{(IV)} \end{matrix} |
+ (−2T′′ + 2T T′) | \begin{matrix}f & g \\ f^{(V I)} & g^{(V I)} \end{matrix} |
+ (−9T T′′ + 9T′′′ − 9(T′)^{2}) | \begin{matrix}f & g \\ f^{(V)} & g^{(V)} \end{matrix} |
+ (−4T′ + 2T^{2}) | \begin{matrix}f′ & g′ \\ f^{(V I)} & g^{(V I)} \end{matrix} |
+ (−27T T′ + 27T′′) | \begin{matrix}f′ & g′ \\ f^{(V)} & g^{(V)} \end{matrix} |
+ (−45T′′ + 45T T′) | \begin{matrix}f′′ & g′′ \\ f^{(IV)} & g^{(IV)} \end{matrix} |

+ (18T′ − 18T^{2}) | \begin{matrix}f′′ & g′′ \\ f^{(V)} & g^{(V)} \end{matrix} |

is a well-defined global 7-form.

References

[1] Bott, R., G. Segal, The cohomology of the vector fields on a manifold, Topology 16 (1977), 285–298.

[2] Feigin, B. L., Conformal field theory and cohomologies of the Lie algebra of holomorphic vector fields on a complex curve, Proc. ICM, Kyoto, Japan (1990), 71–85.

[3] Fuks, D. B., “Cohomology of Infinite Dimensional Lie algebras,” Consultant Bureau, New York and London 1986.

[4] Gelfand, I. M., and D. B. Fuks, Cohomologies of Lie algebra of tangential vector fields of a smooth manifold, Funktsional. Anal. i Prilozhen. 3 (1969), 194–210.

[5] Ghys, E., Rigidité différentiable des groupes Fuchsiens, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 163–185.

[6] Griffith, P., and J. Harris, “ Principles of Algebraic Geometry, ” Wiley Interscience, John Wiley and Sons 1978.

[7] Gunning, R. C., “ Lectures on Riemann Surfaces,” Princeton Mathematical Notes 1966.

[8] —, Special Coordinate Coverings of Riemann Surfaces, Math. Ann. 170 (1967), 67–86.

[9] Haefliger, A., Sur la cohomologie de l’algèbre de Lie des champs de vecteurs, Ann. Sci. ENS, 4^{ème} série 9 (1976), 503–532.

[10] Hawley, N. S., and M. Schiffer, Half-Order Differentials on Riemann Surfaces, Acta Math. 115 (1966), 199–236.

[11] Kawazumi, N., On the complex analytic Gel’fand-Fuks cohomology of open Riemann surfaces, Ann. Inst. Fourier 43 (1993), 655–712.

[12] Krichever, I. M., and S. P. Novikov, Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons, Funktsional. Anal. i Prilozhen 21 (1987), 46–63.

[13] Ovsienko, V., and C. Roger, Generalizations of Virasoro group and Virasoro algebra through extensions by modules of tensor-densities on S^{1}, Indag. Mathem., N.S. 9 (1998), 277–288.

[14] Res̆etnikov, V. N., On the cohomology of two Lie algebras of vector fields on a circle (in Russian), Uspekhi Mat. Nauk 26 (1971), 231–232.

[15] Schlichenmaier, M., ,,Zwei Anwendungen algebraisch-geometrischer Methoden in der theoretischen Physik: Berezin–Toeplitz Quantisierung und globale Algebren der zweidimensionalen konformen Feldtheorie, “ Habilitationsschrift, Universität Mannheim.

[16] Schlichenmaier, M., and O. K. Sheinman, The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, Uspekhi Mat. Nauk 54 (1999), 213–250.

[17] Wagemann, F., Some remarks on the cohomology of Krichever-Novikov algebras, Lett. Math. Phys. 47 (1999), 173–177.

[18] —, Differential graded cohomology and Lie Algebras of holomorphic vector fields, Commun. Math. Phys. 208 (1999), 521–540.

Friedrich Wagemann Institut G. Desargues Université C. Bernard, Lyon 1 43, bd du 11 novembre 1918 F-69622 Villeurbanne cedex, France wagemann@desargues.univ-lyon1.fr

Received March 27, 2000 and in final form May 29, 2000