Table Of ContentConformai Field Theory and Cohomologies
of the Lie Algebra of Holomorphic Vector Fields
on a Complex Curve
Boris L. Feigin
Institute of Solid State Physics, USSR Academy of Sciences, Chernogolovka
142432 Moscow Region, USSR
Introduction
In this text we shall deal with only one aspect of the conformai field theory - the
concept of the modular functor. Recall the main structure of the conformai field
theory. Let M be a complex curve. First of all the theory associates to M a finite
dimensional vector space H(M). The construction of H(M) depends on the choice
of a complex structure on M, but really the correspondence M -> H (M) is topo
logical by nature. Our approach to the definition of H(M) is the following. Let
Dif(JVf) be the group of all C°°-diffeomorphisms of M. The group Dif(M) acts on the
space of all complex structures on M.
If a group G acts on a manifold S and n is a point of S we can define G - the
n
stability group of 77. If G acts transitively on S, then S = G/G . It is possible to define
n
the object "G„" for a non-transitive action so that we have always S ^ G/"G".
Assume that a group H acts transitively on S and H - the stability group of 77;
n
define G„ as the following diagram in the category of groups: G -> H — H . In other
n
words, G is the "intersection" G and H in H.
n n
So we can assign to a complex structure "com" on M the "subgroup" Dif (M)
com
c Dif(M). It can be shown, that for different complex structures com! and com
2
the "subgroups" Dif (M) and Dif (M) are "conjugate" in some "derived" sense.
conil com2
We want to think about the space H(M) as the space of cohomologies of
the group Dif (M) with coefficients in some one-dimensional representation.
com
This representation is related to the value of the central charge of the theory. In
this text we shall give some arguments in favour of this fact. Really, we shall
construct the Lie algebra of the group Dif (M), and realize H(M) as the space of
com
"integrable" homologies of this Lie algebra with coefficients in some distinguished
one-dimensional representation.
So, from this point of view, a minimal conformai field theory is the investigation
of the group of symmetries of a complex structure on a surface. By the same method
we can explore the symmetries of other geometrical structures. For example, the
Wess-Zumino theory is attached to the group of holomorphic transformations of
a vector bundle on M,
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
(O The Mathematical Society of Japan, 1991
72 Boris L. Feigin
=
In more general cases we have a sheaf of associative algebras on M (e.g.
conformai field theory associated with a W-algebra). The modular functor is the
homologies of this object.
The contents of the text are clear from the titles of the sections. The main result
is a theorem of the Section VI.
Now a few words about the references. Information about continuous coho
mologies of the Lie algebras of vector fields on smooth manifolds is contained in
the book: D.B. Fuchs, Cohomologies of infinite dimensional Lie algebras, Plenum
Publishing Corporation, New York 1986.
Conformai field theories, associated with Kac-Moody algebras are discussed by
Akihiro Tsuchiya, Kenji Ueno, and Yasuhiko Yamada in the article Conformai field
theory on universal family of stable curves with gauge symmetries (Advanced Studies
in Pure Math. 19 (1989) 459-565).
The Segal's article in these Proceedings is devoted to the notion of modular
functor. Determinant bundles on the moduli space of curves are explored in: A.A.
Beilinson and V. A. Schechtman, Determinant bundles and Virasoro algebras, Comm.
Math. Phys. 118(1988)651.
Minimal conformai field theories were constructed in: A.A. Bela vin, A.M.
Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333.
The theory of singular support of representations of a Lie algebra is developped
in O. Gabber, The integrability of the characteristic variety, Amer. J. Math. 103
(1981)445.
In the article: B.L. Feigin and B.L. Tsygan, Riemann-Roch theorem and Lie
algebra cohomology I, Proceedings of the winter school on geometry and physics,
Srni, 9-16 January, 1988, pp. 15-52, you can find the calculations that may be useful
for generalization of the results of this text on the manifolds of dimension greater
than one.
Ideas of deformation theory are in the works of Gerstenhaber, Quillen, Halperin,
Stasheff, and others.
The construction of the modular functor in the Virasoro case is contained in:
B.L. Feigin and D.B. Fuchs, Cohomology of some nilpotent subalgebras of the
Virasoro and Kac-Moody Lie algebras, IGP, vol. 5, no. 2 (1988).
I. Definition of the Lie Algebra of Vector Fields
on a Complex Manifold
Let M be a complex manifold of dimension n. What is the Lie algebra of holo
morphic vector fields on Ml
Let T be the tangent bundle of M, T = T0'-1 0 T~uo - the Hodge decomposi
tion of the complexification of T The space of C00 sections F(T_1'°) of the bundle
T-1'0 is closed with respect to the bracket. Locally, an element of r(T~lt0) has the
form
ZiJ - MZ1> •••' Zn> Zl> •••J Zn ) ^ ~• OZ:
Conformai Field Theory and Cohomologies of the Lie Algebra 73
Denote by TiJ the bundle (/\"'T_1'0) ® (/\~-,T,0*~1), where /,; are integers. In
particular F1,1 is the bundle of volume forms. Consider the Dolbeault resolution
of the sheaf of holomorphic vector fields on M:
0 _• T-1'0 -^ T~ul ^ T"1'2 -> • • • T-1'" -• 0.
The bracket on the space F(T~1,0) is extended in a natural way on the space
F(T-1'0) © T(T~U1) © • • • © r(T~l>n). (Since elements of r(T~ui) are (0, l)-forms
with values in the bundle T~1,0). It is easy to see that this extended bracket defines
a structure of differential. Lie superalgebra on the following complex
o -• r(T~U0) -+ r(T~ul) ^ r(T~U2) -• • • • r(T~Un) -+ o.
We shall call this differential superalgebra the Lie algebra of holomorphic vector
fields on M since it is the space of global sections of the resolution of the sheaf of
holomorphic vector fields on M. We shall denote this complex by F(Lie(M)). It is
clear that for affine M the algebra F(Lie(M)) is equivalent to Lie(M), where Lie(M)
is the Lie algebra of holomorphic vector fields on M. Another approach to the
definition of the Lie algebra of vector fields on a manifold is the following. Choose
a covering {U i e 1} of M, where each C/ is affine. Let us associate to any subset
h f
h < ii < " ' < h °f Jtne Lie algebra Lie(£/ n l// n • • • n U ). Using this set of data
fi 2 ti
we can define by standard method a cosimplicial object in the category of Lie
algebras. We omit details of the construction, but it is possible to show that such
cosimplicial Lie algebras for different coverings are equivalent in some sense. In
particular, we can use the covering consisting of all open sets. So, we shall call this
cosimplicial object the Lie algebra of vector fields on M and denote it by Lie^(M).
The differential Lie superalgebra F(Lie(M)) defines in the standard way a co
simplicial Lie algebra which is equivalent to Lie (M).
zJ
Remark. Let A be a group, £ an >4-bundle on M, End(£) the bundle of endo-
morphisms of £. Denote by End(^) ® T0,i the bundle of (0, /)-forms with values in
End(£). The Dolbeault complex of F End(£):
0 -> r(End({) ® T°'°) ^ F(End(0 ® T0'1) ^ • • • -• F(End(£) ® T°>") -• 0
has an obvious structure of differential Lie superalgebra. This object is the natural
candidate for the Lie algebra of the endomorphisms of the bundle. We can also give
the cosimplicial version of this definition, using coverings of M. Note that all
statements in this text have their counterpart for the algebra FEnd(^), or for the
cosimplicial version End^(^).
II. Continuous Cohomologies of Lie(Af)
Using the methods of the Gel'fand-Fuchs theory of the continuous cohomologies
of the Lie algebra of vector fields on a smooth manifold it is possible to define
and to calculate the continuous cohomologies of the differential Lie superalgebra
FLie(M). First we recall main definitions and results from the Gel'fand-Fuchs
theory.
74 Boris L. Feigin
=
Let AT be a C00-manifold and L(N) be the Lie algebra of C°°-vector fields on N;
L(N) is a topological Lie algebra with respect to the C°°-topology. Continuous
cohomologies of L(N) are defined as the cohomologies of the subcomplex of the
standard cohomological complex of L(N) which consists of continuous cochains.
The procedure of calculation of the continuous cohomologies H*(L(N)) is divided
into two parts - "local" and "global". The aim of the "local" part is the calculation
of the continuous cohomologies of the Lie algebra W of formal vector fields on R",
n
where n = dim N. An element of W is a vector field on the formal neighbourhood
n
of the origin in R". Gel'fand and Fuchs proved, that H*(W ) is isomorphic to the
n
algebra of cohomologies of some topological space X . The space X may be
n n
described as a preimage of 2n-skeleton of the base space of the universal GL(n, (C)-
bundle in the total space of this bundle. It is important for us that X is a GL(n, <C)-
n
space and therefore a GL(n, R)-space, since GL(n, R) c GL(n, C).
The aim of the "global" part of the calculation is to "sew" the local calculations
at all points of N. To do this let us associate with the tangent bundle of AT the bundle
Q:U -> N, of which each fiber is isomorphic to X . The main theorem on continuous
n
cohomologies asserts that the algebra Hf(L(N)) is isomorphic to the cohomologies
of the topological space of all sections of the bundle Q. Note that in the most
important case (for us) where n = 1 the space X is homotopic to the three-
n
dimensional sphere S3.
The differential Lie superalgebra F(Lie(M)) is also equipped with C00-topology.
So we can define the continuous cohomologies and calculate them using the same
technique. Let us formulate the result in the case dim M = 1.
Proposition. In the case dim M = 1, the algebra FT*F(Lie(M)) is isomorphic to the
algebra of cohomologies of the space of all continuous maps Hom(M, S3) from M (as
a topological space) into S3.
It is not hard to find out the structure of the algebra if*(Hom(M, S3), (C).
Consider the natural map 6 : M x Hom(M, S3) -> S3; 9(m,f) = f(m), where msM,
f e Horn (M, S3). The corresponding map between cohomologies gives us a map
C ^ H3(S3, <C) -> H*(M, C) (g) tf *(Hom(M, S3), <C) which we can rewrite as a set of
three maps 0 : H(M, <C) -> H3~i{Rom(M, S3), C), i = 0, 1, 2. It can be proved that
t t
the algebra if*(Hom(M, S3), C) is a free skew-commutative algebra and the images
of 0 6 , 9 are the space of generators. Now we shall write down the explicit
l9 2 2
formulas for the cochains in the standard complex of F(Lie(M)) which represent
the classes corresponding to the images of {0}.
f
We need some preparations. First we recall the definition of holomorphic
projective connections on a one-dimensional manifold, A projective connection
defines the following holomorphic differential operator of order three:
D:r(T-1'°)->/XT2'0)
such that in a neighbourhood of an arbitrary point three independent solutions of
the equation D(q>) = 0 constitute the Lie algebra which is isomorphic to sl (C). This
2
Conformai Field Theory and Cohomologies of the Lie Algebra 75
operator is defined by the projective connection up to a constant. We can fix this
constant if we suppose that in the local coordinate Z the highest symbol of D is
/ d\3
equal to [ —— j .An operator D with these properties defines a projective connection.
Remark. Fix a closed curve y e M without self-intersections. After restriction on y
the operator D defines the operator of the second hamiltonian structure in the KdV
theory. Note that a projective connection provides the family of operators:
Di'.r^^^^T2^).
Locally, the kernel of the operator D, consists of antiholomorphic tensors of type
(0,1) with values in the kernel of D = D. We suppose that all operators D have the
0 f
same symbol. It is easy to see that D, is adjoint to D^. (Recall that for any
x {
differential operator U : F -> F, where F, F are vector bundles on a manifold, the
adjoint operator acts from the bundle F* ® Vol into F* ® Vol, where Vol is the
bundle of volume forms, and * is a sign for the dual bundle.)
Let us consider the following maps:
¥ :A3r(T-1'0)-^r(T0'0),
0
¥ :A2r(T^°)-^r(T1'0).
1
Here the operator iF is given by the formula: ^(a, b) = (Da)b — a(Db), a,
1
b G r(T~1,0). The map WQ is characterized up to a constant by the property: *F is
0
a r(T_1'0)-homomorphism. (Note that the spaces T(TiJ) are F(T~1'°)-modules.)
The explicit formula for W \
Q
J\ J2 J3
V (U U , U ) = Det fi fi fi
0 U 2 3
/ '1ft Jft2f J3fit
l/je/Xr-1-0). U = f{z,z)^,
t t
where z is a local coordinate on M, and ""' means the derivation —. The value of
oz
the determinant does not depend on the choice of a coordinate. We need also the
maps W , W .
2 3
¥ : A2r(T~^°) ® T(T~U1) -+ T^T0'1),
2
^. ( -i,o) /^r-i.i) _> r(Tul).
r T 8
The formula for !F is similar to the formula for *P namely ^(fl, b) = D (a)-b —
3 l9 0
a • D (b), a e r(T~u°), b e T(T~U 1). The map ¥ is a r(r_1'°)-homomorphism and
t 2
is given by the same determinant as W , where U = f —, U = f — and l/ =
Q 1 t 2 2 3
~ ÖZ ÖZ
1,3 3z
=26= Rnric T Fpioin
Proposition. There is a map ç from the standard homological complex of the Lie
algebra F(Lie(M)) into the de Rham complex of M. The construction of <p is clear
from the following diagram. In the top of the diagram is a fragment of the homo-
logical complex and the de Rham complex is on the bottom.
A°r(T-^°) r(T~U0) A2r(T-^0)- A3r(T~U0)
©
rrCT""1'0)®/^-1'1) r^2F(T-1'0)®F(T-1'1) Vo
<P
r(T1>°)<
© Ti I
±r{Tul)
F(T°'°)
Note that the de Rham complex here is reflected and shifted.
Now let ß be an element of H^M). We can assign a continuous cochain of the
Lie algebra F(Lie(M)) by the formula u -> </?, <p(u)}, where u is a chain, and < , >
is the integration of cp(u) over the cycle ß. It is easy to see that this construction
gives us the desired map H(M) -• H3'l(r(Uc(M))).
(
Remark 1. The following part of our diagram
*- A2r(T~U0) «- A3r(T~U0) <-
0 r(Tl,0) r(T°>°) o
is a morphism from the standard complex of the Lie algebra F(T"1,0) into the
Dolbeault complex of M. For non-compact M the continuous cohomologies of
F(T_1'°) is infinite-dimensional at least in dimension three. This map of complexes
induces the map V* -> H3(r(T~1,0)) where Fis the space of holomorphic functions
on M. But for compact M we have dim /^(/"(T"1,0)) < oo for arbitrary i. Let us
formulate the important corollary from the results of this section.
Corollary. The dimension of the space H1(F(Lie(M))) is equal to one if M is compact.
c
It means that the algebra F(Lie(M)) has a non-trivial one-dimensional representation.
Remark 2. If U is a finite dimensional Lie algebra and v : U -> End(F) is a finite
dimensional representation of U, then the map a i-> tr v(a), a e U is a 1-cocycle of U.
We can interpret this 1-cocycle as an element of the space H°(U, U*). This construc
tion can be easily generalized. The function a"\->tr(v(an))/nl is an element tr^n) of
the space H°(U,Sn(U*)). In other word, the representation v defines a functional
tr e H°(U, 5*(U*)), tr = ^tx^ Let us consider the Lie superalgebra U © Ql =
v v
U ® C[£], where C[<^] is a Grassmann algebra in one variable; U © £U is a graded
Conformai Field Theory and Cohomologies of the Lie Algebra 77
Lie superalgebra with deg(tf) = 0, deg(^U) = — 1. It is evident that H*(U © ÇU) =
H*(U, S*(U*)). So we have tr G H°(U © £U). This idea can be applied to the Lie
v
algebra F Lie(M). Namely, using Gel'fand-Fuchs technique it is possible to find out
the continuous cohomologies of the algebra F Lie(M) © ÇT Lie(M). This problem
is also divided into two parts: "local" and "global". The local part consists of the
calculation of the algebra H*(W © ^W) where W is the algebra of formal vector
1 x x
fields on a line. The answer is the following:
fl*(Wi © tW) S H*(Wi) © Hf(W Wf) © H*(W S2W?) © • • • ;
x c u U
HkWJ^V ifi = 0,3 and H'(Wi)sO if/ ^0,3;
c
H2(W SW?) s H3(Wi, S'Wl*) s C
U
and in other dimensions the cohomologies are equal to zero (i ^ 0). The multi
plication in the space H*(W ®^W ) is trivial. The cohomologies of the algebra
1 X
T Lie(M) © CT Lie(M) are the same as those of the space of all continuous maps of
M (as a topological space) into Y, where Y is the space satisfying H*(Y, <C) ^
H*(W © £W ). In particular, we can conclude that H °(F Lie(M) © £F Lie(M)) is
1 X C
an algebra of polynomials in an infinite set of variables a a ,..., G i/°(F Lie(M) ©
l9 2 c
^FLie(M)). Therefore we can associate with a representation v of FLie(M) an
analog of trace - an element of the space (£>[_&, cr ,...].
x 2
III. Description of a 1-Cocycle of the Cosimplicial
Lie Algebra hie (M)
A
In this section we shall suppose that M is compact and one-dimensional.
Choose the following simple Zarisky covering of M. Let p be a point of M,
V = M{p), U is a small neighborhood ofp; {U U } is a covering and let Lie (M)
1 2 l9 2 0
be the following diagram of Lie algebras:
Lie([/ ) Lie(l/ )
1 2
M
Liefl/j n U )
2
We shall work with this diagram in the category of Lie algebras instead of the full
object Lie^M). A 1-cocycle of Lie (M) is the following set of data: {v v , Q}, here
0 l5 2
Q is a 2-cocycle of the Lie algebra Lie([/ n U ), v is a 1-cochain of Lie(£/j) and
1 2 t
7r*£ = dv where d is the differential in the standard cohomological complexe. Let
i9
z be a local coordinate at p, and Q is a standard Virasoro-type cocycle:
8 fd\-n, (d2fidf2 82f*dfi
e
ldz'hdz) - KeS'=°l W Tz~ ~Wz Tz
It can be proved that the restrictions of Q on Lie(l/ ) and Lie(U ) are homologous
1 2
to zero. It means that we can find y and v such that dv = Q; these equations
x 2 {
78 Boris L. Feigin
(i = 1, 2) have only one solution since LT1(Lie(l/)) = 0. Let us write down the
i
formula for Q without using a local coordinate.
Let us fix a projective connection on M and the corresponding operator D. The
expression for the cochain: Q(U U) = Res (Du •u —u - Du ), u e Lie([/ n U ).
U 2 p 1 2 1 2 t 1 2
It is evident that § is homologous to Q. The restriction of Q onto Lie(LQ is zero since
in this case (Du )u — u (Du ) is holomorphic on U ifu u e Lie(C/).
1 2 1 2 h iy 2 i
So, we have constructed the class in the space f/1(Lie (M)). Really this is the
0
same class as we constructed earlier for F(Lie(M)). Note that our construction of
the cochain of the algebra F(Lie(M)) depends on the choice of projective connec
tions, but the corresponding cohomology classes are equal. The same is true
for the algebras Lie (M) and Lie^(M). In this section we have constructed a map
0
H2(IàQ(U1 n U2)) -> i/1(Lie0(M)) ^ if1(Lie^(M)). It enables us to associate a one-
dimensional representation of the algebra Lie,(M) to any value of the central charge
z
of the Virasoro algebra.
It is not hard to calculate the whole space of cohomologies of Lie^(M). The
result is: H*(Lie (M)) s tf *(Lie„(M)) s tf*(F(Lie(M))). This is a consequence of
0 c
the following fact. Let N be an affine one-dimensional manifold and Lie(Af) is a Lie
algebra of algebraic vector fields on N.
Proposition. The natural pairing between the spaces H(Lie(N)) and FTJ(F(Lie(N)))
t
is non-degenerate for arbitrary i. The algebra H?(r(Lk(N))) is free and skew-
commutative with one generator in degree three and r generators in degree 2, where
r = dimff1(iV,C).
IV. Homologies of the Lie Algebra of Holomorphic
Vector Fields on a Compact One-dimensional Manifold M
As in the previous section we choose the diagram (*) as a model for the Lie algebra
of holomorphic vector fields Lie^(M). Let us fix a one-dimensional representation
of the diagram Lie (M) which we denote by D, c G C. It means that the correspond
0 c
ing element H1(Lie (M)) is the image of the class a G H2(Lie(U n U )) which is
0 1 2
represented by the cocycle:
We shall denote by Vir(l/ n U ) = Lie(C/ n U ) © (CC the central extension of
1 2 1 2
Lie(t/ n U ), where C is the central element, and the cocycle a corresponds to the
1 2
value c = 1.
The diagram (*) is the same as the set of mappings:
Lie(l/ )
1
Lie (M) Lie(C/ n U ) (**)
0 1 2
^Lie(t/ )
2
Conformai Field Theory and Cohomologies of the Lie Algebra 79
Informally speaking, Lie (M) is an intersection of the subalgebras Lie([/ ) and,
0 1
Lie([7 ) in Lie^ n U ). By the standard arguments in the spirit of the Frobenius
2 2
reciprocity law the homologies if,(Lie (M), D) are isomorphic to
0 c
H(Lie([/ ), Ind(Lie (M), Q; Lie([/ ))).
f 1 0 c 1
Here we denote by the symbol Ind(y4, M; B) the induced representation of the Lie
algebra B from the representation M of subalgebra A c B. It is evident from the
diagram (**) that the representation Ind(Lie (M), \3 ; Lie(f/ )) is isomorphic to the
0 C 1
restriction onto Lie(C/ ) of the Vir(U n t/ ^representation Ind(Lie((/ ) © (DC, l ;
1 î 2 2 c
Vir([/ n U )). Here l is the one-dimensional module of the algebra Lie(C/ ) © (CC,
1 2 c 2
where Lie([/ ) acts in trivially on l and C acts by multiplication with ceC.lt is
2 c
easy to see that if the genus of M is greater than one, then the representation
Ind(Lie (M), •«,; Lie(t/ )) is free over the universal enveloping algebra of Lie([/ ).
0 1 1
Now we are ready to formulate the result about the homologies but first we recall
the standard notations about the representations of the Virasoro algebra. The
algebra Vir is the central extension of the Lie algebra of vector fields on a punctured
disk around the origin in C. Let C be the central element of Vir. There are two
important subalgebras Vir+ and Vir+ in Vir : Vir+ = \f() — Q CC|/(0) = 0 [ and
z
— ( d) I dz J
Vir+ = y(z)—-©(CC>, where / is regular at z — 0. Denote by l the one-
c
dimensional Vir+-module, where C acts by multiplication with c G C; l denotes the
hc
one-dimensional Vir+-module, whose corresponding character is f(z)—\-^hf'(0),
dz
Ch->c, for h, ceC.We also put M = Ind(Vir+, l ; Vir) and M = Ind(Vir+, 1,, ;
c c htC >C
Vir). Usually M and M are called a Verma module and a parabolic Verma module
htC c
respectively.
We can attach to an arbitrary point p G M the Virasoro algebra Vir(p) which is
the central extension of the Lie algebra of holomorphic vector fields in a punctured
neighborhood U(p) of p. The algebras Vir+(p) and Vir+(p) consist of regular vector
fields and regular vector fields which are zero at p respectively. So, for any p e M
we have the Vir(p)-modules M (v) and M(p). It is easy to see that the representa
Kc c
tion M(p) is isomorphic to Ind(Lie(t/ ) © CC, l ; Vir^ n U )).
c 2 c 2
Proposition. Suppose that the genus of M > 1. Then for any i ^ 0 H(Lie (M), D) £
{ 0 c
H,(Lie^(M), DJ = 0;H (Lie (M), U) S H (Lie^(M), D) = M(p)/(Lie(t/ )M(p));
o o 0 c c 1 c
the last term is the space of Lie(t/ ) - coinvariants in the module M(p).
1 c
Now we fix a set of distinct points p p,..., p of M and let W(p p,..., p) =
u 2 n u 2 n
A^\{Pi» P2> - - • -> Pn}' We want to describe the induced representation Ind(Lie^(Af),
•: Lie W(p p ,...,p )). Let Z„ be the sum Lie U(p)® Lie U(p )©• • • Lie U(p )
c u 2 n x 2 n
and let Z„ be the central extension of the Lie algebra Z which corresponds to the
n
cocycle:
80 Boris L. Feigin
/l(Zl)^7 + fz{z2)dz~ -+ " '+ /"(z")â7'
2
9 (^ + 9 (z )^ + - + g (z )±
1 2 2 2 n a
^KfC-g'i-fi-g'n, ce€,
where z is a locai coordinate at p. It is clear that the algebra Z acts on the space
t t n
M(px) ® M(p ) ® • • • ® M(Pn\ The Lie algebra Lie W(p p,..., p) is a sub-
c c 2 c u 2 n
algebra of Z , and the imbedding 9 : Lie W(p p ,..., p) -> Z sends a vector field
n l9 2 n M
a G L\Q(W(p , p ,..., p)) to the set of the Laurent expansions of a at the points p
1 2 n l9
p, ..., p. It may be shown that the imbedding 9 has a lift to an imbedding
2 n
9 : LIQ(W(P1, p2,..., pn)) -> Zn. Using the same arguments as in the beginning of
this section we can see that the module Ind(Lie^(M), Dc; LìQ(W(P1, p2,..., pn))) is
isomorphic to the restriction of the Z„-module M(Pi) ® M(p ) ® • • • ® M(p ) on
c c 2 c n
LiG(W(p ,p ,...,p )).
1 2 n
It is easy to define the (cosimplicial) Lie subalgebra Lie^M, p p,..., p) a
l9 2 n
Lie (M) which consists of holomorphic vector fields on M with zeros at the points
zf
Each point p defines the homomorphism cp(hj) : IàQ (M,p p,..., p )-> C,
t A u 2 n
he<L which assigns to a vector field in a neighbourhood of p the number by the
{
rule: (pihMf^)—) = hf(0). We also have a map \x ; Lie^(M, p p,..., p ) ->
t u 2 n
Lie^(M) -> C, where the last arrow corresponds to the representation Q- The tensor
c
product of all these 1-dimensional modules gives us the representation Oh,...h c
l t ni
of the algebra Lie^(M, p p,..., p). Consider the imbedding
l9 2 n
Lie„(M, Pi, p ,..., p ) -> UQ(W(p p,..., p ))
2 n u 2 n
and the corresponding induced representation.
Proposition. 1) The module Ind(Lie^(M,p ,p ,..., p), \J ', UQ(W(P J,
1 2 n hl hniC U 2
• • • » Pn))) is isomorphic to the restriction of the representation of the Lie algebra Z in
n
the space M (Pi) ® M (p ) ® • • • ® M (p ) on UQ(W(P, p,..., p )).
hitC h2tC 2 hntC n 1 2 n
2) For i / 0, Hi(Ue (M, p p,..., p), D ..., ) = 0 if the genus g(M) > 1 or
A u 2 n hu hniC
g(M) = 1 and n>\ or g(M) = 0 and n>3.
3)H (UQ (M, ,p ,...,p ),n c) = M (p )®M (p )(g)-'-®
0 A Pl 2 n hl hni hliC 1 h2tC 2
M (p )/LiQ(W(p, p,..., p ))(M (Pi) ® M (p ) ® • • • ® M (p )).
hntC n i 2 n huC h2tC 2 hntC n
Remark. Let y be a small circle around the point p G M and D(y) is the group of
t t (
diffeomorphisms of y with itself; D(y) is a real infinite-dimensional Lie group, and
t t
let LieR(y) be the Lie algebra of the group D(y). Denote by Lieft;;) = C ® Lie^fyj)
f t R
