On chiral differential operators over homogeneous spaces
Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman
Introduction
The notion of an algebra of chiral differential operators (cdo for short) ov*
*er
a smooth algebraic variety X has been studied in [GII]. (This notion has been
invented and first studied, in a different language, by Beilinson and Drinfeld,*
* cf.
[BD1], Chapter 3, x8). In the present note we consider some examples in more
detail. We shall work over the ground field C.
First, we give a classification of cdo over X in the following cases: X = G *
*is an
affine algebraic group; X = G=N or G=P where N is a unipotent subgroup and P
is a parabolic subgroup and G is simple (the extension to the case of a semisim*
*ple
G being straightforward).
Before we describe the result, let us explain some terminology and notation.*
* For
a smooth algebraic variety X, an algebra of cdo over X is by definition a Zaris*
*ki
sheaf V of Z0 -graded vertex algebras on X such that
(a) if Alg(V) = (A; T ; ; @; fl; <; >; c) is the sheaf of vertex algebroids *
*associated
with V (see [GII], x2), then the corresponding extended Lie algebroid (A; T ; ;*
* @)
(op. cit., 1.1, 1.4) is identified with (OX ; X ; 1X; dDR ) where X denotes t*
*he
tangent bundle and dDR the de Rham differential;
(b) the adjunction morphism UAlg(V) -! V is an isomorphism. Here U is the
functor of vertex envelope defined in op. cit., x9.
For each Zariski open U X we can consider the category (a groupoid in fact)
of cdo over U, or, what is the same, the groupoid of vertex algebroids (defined*
* in
[GII], x3) over U satisfying (a) above. When U varies, we get a sheaf of groupo*
*ids
DiffchXover X _ the gerbe of chiral differential operators. As usual, (U; Diffc*
*hX)
will denote the sections over U; a generic object of this category will be some*
*times
denoted DchU; the set of isomorphism classes of cdo over U will be denoted by
ss0((U; DiffchX)).
Let G be an affine connected algebraic group, g the corresponding Lie algebr*
*a.
For each symmetric ad-invariant bilinear form (; ) 2 (S2g*)g we construct a cdo
DchG;(;)over G such that if G is semisimple, then the correspondence (; ) 7! Dc*
*hG;(;)
gives rise to a bijection
(S2g*)g -~! ss0((G; DiffchG)) (I1)
We have a canonical embedding of vertex algebras
i(;): Vg;(;),! DchG;(;) (I2)
where Vg;(;)denotes the vacuum module of the Kac-Moody algebra bgat level (; ).
This embedding is induced by the embedding of g into TG := (G; G ) as left
invariant vector fields.
Typeset by AM S-T*
*EX
1
2
Let (; )g;(K) denote the Killing form on g. Let us define the dual level by
(; )o = -(; )g;(K)- (; ) (I3)
Using the embedding g ,! TG by means of right invariant vector fields one can
construct a canonical dual embedding of vertex algebras
io(;): Vg;(;)o,! DchG;(;) (I4)
It is characterised by the requirement that the images of i(;)and io(;)commute *
*in
an appropriate sense (see Theorem 2.5 and Corollary 2.6). This beautiful fact w*
*as
communicated to us by B.Feigin, E.Frenkel and D.Gaitsgory. We give a proof using
the language of [GII].
Let us pass to homogeneous spaces. Assume that G is simple. Let N G
be a unipotent group. The classification of cdo over G=N is the same as over G;
namely, for each level (; ) one can define a cdo DchG=N;(;)such that the corres*
*pondence
(; ) 7! DchG=N;(;)induces a bijection
(S2g*)g -~! ss0((G=N; DiffchG=N)) (I5)
The sheaves DchG=N;(;)are constructed using the BRST (or quantum Hamiltonian)
reduction of the corresponding cdo's on G. More precisely,
DchG=N;(;)= H0BRST (Ln; ss*DchG;(;)) (I6)
where the rhs denotes the BRST cohomology of the loop algebra Ln := n[T; T -1];*
* n :=
Lie(N). For the precise definition see x3.
Let B G be a Borel subgroup. We show that there exists a unique, up to
a unique isomorphism, cdo DchG=Bon the flag space G=B. Again this cdo may be
constructed using the BRST reduction. Namely,
DchG=B= H0BRST (Lb; h; ss*DchG;crit) (I7)
Here DchG;critis by definition the cdo DchG;(;)criton the critical level (; )cr*
*it= -(; )g;(K)=2.
For the definition of the relative BRST cohomology in the rhs we again refer the
reader to the main body of the note, see x4. (A more explicit construction of t*
*he
sheaf DchG=Bfor G = SL(n), using vertex operators, has been suggested in [MSV],
5.9-5.10.)
The embeddings (I4) induce canonical morphisms of vertex algebras
Vg;(;)o-! DchG=N;(;); Vg;(;)crit-! DchG=B (I8)
Taking the spaces of sections over a big cell, we get another construction of F*
*eigin-
Frenkel Wakimoto modules, cf. [FF1] - [FF3].
3
Finally, if P G is parabolic but not Borel, we show that (G=P ; DiffchG=P) *
*is
empty. The classification of cdo over homogeneous spaces is exactly reflected i*
*n the
BRST world: namely the square of the corresponding BRST charge is zero at all
levels for G=N, only at the critical level for G=B and is never zero for G=P .
This introduction would not be complete without mentioning that this note re*
*lies
heavily on the ideas of B.Feigin and E.Frenkel. This note started from our atte*
*mpts
to find a proof of 2.5 and 2.6. Our sincere gratitude goes to D.Gaitsgory who h*
*ad
communicated these facts to us and told us that he had known their proofs. We
are also grateful to H.Esnault for a crucial remark 4.1.1.
This work was done while the authors were visiting Institut des Hautes Etudes
Scientifiques in Bures-sur-Yvette and Max-Planck-Institut f"ur Mathematik in Bo*
*nn.
We are grateful to these institutes for the hospitality.
x1. Chiral differential operators over an algebraic group
Perfect vertex algebroids over constants
The discussion below is nothing but the specification of [GII], xx1 - 4 to t*
*he case
A = C.
1.1. Let g be a Lie algebra. We shall need two complexes connected with g,
both concentrated in nonnegative degrees. The first one, C.(g) = C.(g; C), is *
*the
cochain complex of g with trivial coefficients. Thus, by definition
Ci(g) = (ig)* = the space of skew symmetric polylinear maps f : gi -! C,
i 0.
The differential d : Ci-1(g) -! Ci(g) acts as
X
df(o1; : :;:oi) = (-1)p+q+1 f([op; oq]; o1; : :;:bop; : :;:boq;(:1:;*
*:oi):1:1)
1p = f(o1; : :;:oi) (1:1:*
*3)
We shall identify Ci(g) with its image in "Ci(g).
One checks that the embeddings (1.1.3) are compatible with the differentials*
*, so
that one has an embedding of complexes C.(g) ,! "C.(g).
1.2. Let us consider the groupod Algg of vertex algebroids of the form
A = (C; g; g*; @; fl; <; >; c) where T = (C; T; ; @) = (C; g; g*; 0) is a perfe*
*ct extended
Lie algebroid over C, in the sense of [GII], 1.2, with T = g. Note that the l*
*ast
object is uniquely defined by the Lie algebra g = T ; we must have = g*, the "*
*Lie
derivative" action of T on must be the coadjoint one, and a C-linear derivation
@ : C -! must be zero.
Turning to the axioms of a vertex algebroid, op. cit., 1.4, we see that for *
*A as
above, <; > : g x g -! C is a symmetric bilinear map (which may be regarded as
an element of "C2(g)), c 2 "C3(g), (A1) implies that fl = 0, (A2) and (A3) hold*
* true
automatically, (A4) takes the form
<[o1; o2]; o3> + = + 1:2:1)
and (A5) takes the form
dc = 0 (1:2:*
*2)
where d is the differential in "C.(g) given by (1.1.2).
So, an object of Algg has a form
Ag;<>;c= (C; g; g*; 0; 0; <; >; c) (1:2:*
*3)
where <; > 2 "C2(g)Z=2Z; c 2 "C3(g) satisfy (1.2.1) and (1.2.2).
The vertex envelope
Vg;<;>;c= UAg;<;>;c (1:2:*
*4)
(see [GII], x9) is generated by the fields o (z) (o 2 g) and !(z) (! 2 g*) of c*
*onformal
weight 1, subject to OPE
0> [o; o 0](w) - c(o; o 0)(w)
o (z)o 0(w) ~ _;c-! Ag;<;>0;c0
5
is by definition an element h 2 "C2(g) such that
+ = - 0 (1:2:*
*7)
and
dh = c - c0 (1:2:*
*8)
see [GII], Theorem 3.5. The composition of morphisms is induced by the addition
in "C2(g).
1.3. As a corollary, we have a canonical bijection
ss0(Algg) = H3(g) (1:3:*
*1)
More precisely, for a 3-cocycle c 2 C3;cl(g) we have a vertex algebroid
Ag;c:= Ag;0;c (1:3:*
*2)
and the correspondence c 7! Ag;cinduces the bijection (1.3.1).
The enveloping algebra Vg;c:= UAg;cis generated by the same fields as in 1.2,
subject to OPE 0 0
o (z)o 0(w) ~ [o;_o_](w)_-_c(o;_o_)(w)_z - w (1:3:*
*3)
and (1.2.6).
Let us define another interesting class of objects of Algg. Namely, each sym*
*met-
ric ad-invariant bilinear form (; ) 2 (S2g*)g gives rise to an object
eAg;(;):= Ag;(;);0 (1:3:*
*4)
The enveloping algebra Vg;c := UAg;c is generated by the same fields as in 1.2,
subject to OPE 0 0
o (z)o 0(w) ~ _(o;_o_)_(z+-[w)2o;_o_](w)z - w (1:3:*
*5)
and (1.2.6).
It is easy to see that we have an isomorphism
f(;): eAg;(;)~-!Ag;c(;) (1:3:*
*6)
given by a map h(;): g -! g* where
= 1_2(o1; o2) (1:3:*
*7)
and the cocycle c(;)is defined by
c(;)(o1; o2; o3) = ([o1; o2]; o3) (1:3:*
*8)
6
Cf. [GII], Theorem 4.5.
1.4. Given (; ) 2 (S2g*)g, consider a vertex algebroid
Ag;(;):= (C; g; g*; 0; 0; (; ); 0) (1:4:*
*1)
Its vertex envelope Vg;(;):= UAg;(;)is generated by fields o (z) (o 2 g) of con*
*formal
weight 1, subject to OPE (1.3.5).
The correspondence o . T n 7! o(n) defines on Vg;(;)a structure of the vacuum
module over the Kac-Moody algebra bg= g[T; T -1] C . 1 at level (; ).
We have an obvious embedding of vertex algebroids Ag;(;),! Ag;0which induces
an embedding of vertex algebras
Vg;(;),! eVg;(;) (1:4:*
*2)
1.5. If the Lie algebra g is semisimple then the correspondence (; ) 7! c(;)*
*induces
a bijection
(S2g*)g -~! H3(g) (1:5:*
*1)
Therefore, in this case the algebroids Aeg;(;)form a complete set of representa*
*tives
of isomorphism classes in Algg. In other words,
1.5.1. if g is semisimple then the correspondence (; ) 7! Aeg;(;)induces a b*
*ijection
(S2g*)g -~! ss0(Algg) (1:5:1:*
*1)
Passing to a group
1.6. Let G = Spec(A) be an affine algebraic group, g the corresponding Lie
algebra . The tangent bundle G is trivial, so the obstruction c(DiffchG) to t*
*he
existence of an algebra of chiral do over G, DchG2 (G; DiffchG) (cf. [GII], Cor*
*ollary
7.11) vanishes.
By op. cit., x4, the set of isomorphism classes of chiral do over G, ss0((G;*
* DiffchG))
is a nonempty torseur under the "Chern-Simons group" H3DR(G) = H3(G; C).
In fact the groupoid (G; DiffchG) has a distinguished object DchG;0, so that*
* we
have a canonical bijection
ss0((G; DiffchG)) -~! H3DR(G) (1:6:*
*1)
This is a consequence of the following general construction.
1.7. Let Ag;<;>;cbe an arbitrary object of Algg. Let us apply to it the push*
*out
construction of [GII], 1.10 with respect to the structre morphism C -! A. Here
the morphism g -! T := DerC(A) is defined as the embedding of left invariant
7
vector fields, and the map fl : A x g -! := 1(A) is set to be zero. This way
we get a vertex A-algebroid AG;<;>;c. Its enveloping algebra
DchG;<;>;c= UAG;<;>;c (1:7:*
*1)
obviously belongs to (G; DiffchG).
We have a canonical embedding
Vg;<;>;c,! DchG;<;>;c (1:7:*
*2)
We shall use the notations AG;(;):= AG;(;);0; AG;c := AG;0;c; AG;0 := AG;0*
*;0
and DchG;(;), etc. for the corresponding enveloping algebras.
If AG;0 = (A; T; ; dDR ; fl0; <; >0; c0) and ! 2 3;cl(A) is a closed 3-form *
*then we
can form a vertex algebroid
.
AG;! := AG;0 + ! = (A; T; ; dDR ; fl0; <; >0; c0 + !) (1:7:*
*3)
The correspondence ! 7! AG;! induces the bijection (1.6.1).
If c 2 C3;cl(g) is a 3-cocycle with trivial coefficients then by definition
AG;c = AG;!c (1:7:*
*4)
where !c 2 3;cl(A) is the left invariant 3-form on G corresponding to c.
1.8. Corollary. Assume that G is reductive. Then the correspondence c 7! AG;c
induces a bijection
H3(g) -~! ss0((G; DiffchG)) (1:8:*
*1)
Indeed, one knows that for a reductive group the correspondence c 7! !c gives
rise to an isomorphism H3(g) -~! H3DR(G).
1.9. Corollary. Assume that G is semisimple. Then the correspondence (; ) 7!
AG;(;)induces a bijection
(S2g*)g -~! ss0((G; DiffchG)) (1:9:*
*1)
This follows from 1.8 and the remarks 1.5.
1.10. Note that for an arbitrary G and (; ) 2 (S2g*)g one has a canonical
embedding
i(;): Vg;(;),! DchG;(;) (1:10:*
*1)
It is the composition of (1.4.2) and (1.7.2).
8
x2. Dual embedding
2.1. Let G = Spec(A) be a smooth affine connected algebraic group with the
Lie algebra g. Pick a symmetric ad-invariant bilinear form ("level") (; ) 2 (S2*
*g*)g.
Let (; )(K) denote the Killing form on g,
(x; y)(K) = trg(adx . ady) (2:1:*
*1)
Let us pick a base {oi} of g. In terms of structure constants
[oi; oj] = cijpop (2:1:*
*2)
the form (2.1.1) is given by
(oi; oj)(K) = cipqcjqp (2:1:*
*3)
Let us define the dual level (; )o 2 (S2g*)g by
(; )o = -(; )(K) - (; ) (2:1:*
*4)
Define the critical level (; )critby (; )crit= (; )ocrit, i.e.
(; )crit= - 1_2(; )(K) (2:1:*
*5)
If we want to stress the dependence on g, we shall write (; )g;(K); (; )g;crit.
2.2. We have two commuting left actions of G on itself: the left multiplicat*
*ion,
(g; x) 7! gx and the right one, (g; x) 7! xg-1 .
Let T = Derk(A) denote the Lie A-algebroid of vector fields over G. The above
two actions induce two embeddings of Lie algebras
iL : g ,! T; iR : g ,! T (2:2:*
*1)
such that
[iL (x); iR (y)] = 0; for allx; y 2 g (2:2:*
*2)
Below we shall identify g with its image under iL , i.e. write simply x instea*
*d of
iL (x). We shall also use the notation xR := iR (x) (x 2 g).
Embedding iL induces an isomorphism of left A-modules
A k g -~! T (2:2:*
*3)
Thus, {oi} form an A-base of T . In particular
oiR = aijoj (2:2:*
*4)
for some invertible matrix (aij) over A.
9
The commutation relations
[oi; ojR] = 0 (2:2:*
*5)
are equivalent to the identities
oi(ajs) + cipsajp = 0 (2:2:*
*6)
true for all i; j; s.
Let us write down the relations
[oiR; ojR] = [oi; oj]R (2:2:*
*7)
in coordinates. We have
[oiR; ojR] = [oiR; ajsos] = oiR(ajs)os = aipop(ajs)os (2:2:*
*8)
due to (2.2.5). Plugging this into (2.2.7) we get
aipop(ajs) = cijqaqs (2:2:*
*9)
for all i; j; s.
2.3. We set := 1A=k= HomA (T; A), and denote by <; > : T x -! A the
canonical A-bilinear pairing. Let {!i} be the A-base of dual to {oi}. The Lie
algebra T acts on by the Lie derivative.
We have oi(!j) = ffijs!s where
ffijs= = oi() - <[oi; os]; !j> = -cisj= csij
Thus,
oi(!j) = csij!s (2:3:*
*1)
Similarly,
oiR(!j) = 0 (2:3:*
*2)
2.4. Recall that we have an embedding of vertex algebras
i(;): Vg;(;),! DchG;(;) (2:4:*
*1)
see 1.10. More precisely, it is induced by an embedding of conformal weight 1
components
jL : g = Vg;(;)1-! DchG;(;)1= T (2:4:*
*2)
defined by a composition
g ,! T ,! T
where the first arrow is iL and the second one sends x to (x; 0).
10
The fact that jL induces a map of vertex algebras (2.4.1) simply means that *
*we
have the identities in DchG;(;)
jL (o )(1)jL (o 0) = (o; o 0); jL (o )(0)jL (o 0) = jL ([o;(o20*
*]):4:3)
for all o; o 02 g.
2.5. Theorem. (B.Feigin - E.Frenkel, D.Gaitsgory) (i) There exists a unique
embedding
jR : g ,! DchG;(;)1 (2:5:*
*1)
such that
(a) the composition of (2.5.1) with the canonical projection DchG;(;)1-! T i*
*s equal
to iR ;
(b) for all o; o 02 g and n 0
jL (o )(n)jR (o 0) = 0 (2:5:*
*2)
(ii) We have
jR (o )(1)jR (o 0) = (o; o 0)o (2:5:*
*3)
and
jR (o )(0)jR (o 0) = jR ([o; o 0]) (2:5:*
*4)
for each o; o 02 g.
2.6. Corollary. (B.Feigin - E.Frenkel, D.Gaitsgory) The map (2.5.1) induces
an embedding of chiral algebras
jR : Vg;(;)o,! DchG;(;) (2:6:*
*1)
The images of jL and jR commute in the following sense:
jL (x)(n)jR (y) = 0 (2:6:*
*2)
for each x 2 Vg;(;); y 2 Vg;(;)oand n 0.
2.7. Proof of (2.5). (i) As usual, we denote jL (o ) simply by o . We are lo*
*oking
for jR (o ) in the form
jR (oi) = oiR+ biq!q (2:7:*
*1)
for some biq2 A. We have, by [GII], 1.4 (A2)
oiR(1)oj = = = aip(op; oj) - opoj(aip)
Using (2.2.6) and (2.1.4),
-opoj(aip) = op(cjspais) = -cpupcjspaiu = cupscjspaiu = (ou; oj)(K)aiu;
11
so
oiR(1)oj = -(op; oj)oaip (2:7:*
*2)
On the other hand,
(biq!q)(1)oj = = bij
Therefore the condition
jR (oi)(1)oj = 0 (2:7:*
*3)
defines the matrix (biq) uniquely: we must have
jR (oi) = oiR+ (op; oq)oaip!q (2:7:*
*4)
2.8. Let us prove that
oi(0)jR (oj) = 0 (2:8:*
*1)
We have
oi(0)jR (oj) = oi(0) ojR+ (os; ou)oajs!u
On the one hand,
oi(0)ojR = oi(0)(ajqoq) = oi(0)(ajq(-1)oq) = oi(ajq)(-1)oq + ajq(-1)[oi; *
*oq] = 0
using (2.2.6). On the other hand,
o js o js o js js
oi(0) (os; ou) a !u = (os; ou) oi(a !u) = (os; ou) oi(a )!u + a oi(!u)*
* =
using (2.2.6) and (2.3.1)
= -(os; op)ociqsajq!p-(os; ou)ocipuajs!p = -([oi; oq]; op)oajq!p-(os; [oi; op])*
*oajs!p = 0
due to the invariance of the form (; )o. This proves (2.8.1).
Evidently oi(n)jR (oj) = 0 for n 2. This proves part (i) of the theorem.
2.9. Let us compute jR (oi)(1)jR (oj). We have
R o ip R o js
jR (oi)(1)jR (oj) = oi + (op; oq) a !q (1)oj + (os; ou) a !u
This is a sum of four terms.
I := oiR(1)ojR = =
using [GII] (1.8.3)<;>
= aipajs(op; os) - aiposop(ajs) - ajsopos(aip) - op(ajs)os(aip)
Using (2.2.6) and (2.1.4) we see that
-aiposop(ajs) = -ajsopos(aip) = op(ajs)os(aip) = (op; os)(K)aipajs;
12
whence
I = -(op; os)oaipajs
Next,
o ip R o ip js o ip jq
II := (op; oq) a !q (1)oj = (op; oq) = (op; oq) a a
Similarly,
III := oiR(1)(os; ou)oajs!u = II
and evidently
IV := (op; oq)oaip!q (1)(os; ou)oajs!u = 0
Adding up we get
jR (oi)(1)jR (oj) = (op; oq)oaipajq (2:9:*
*1)
Let us differentiate this expression. We have
ip jq ip jq
os{(op; oq)oaipajq} = (op; oq)o os(a )a + a os(a ) =
su iu jq sv ip jv o iu jq o ip*
* jv
= (op; oq)o -cp a a -c qa a = -([os; ou]; oq) a a -(op; [os; ov]) a *
*a = 0
Therefore, (2.9.1) is a constant. It may be computed by noticing that the matrix
(aij), considered as a function on the group G, is equal to the identity at the*
* identity
of the group. Hence (2.9.1) is equal to (oi; oj)o, which proves (2.5.3).
2.10. Let us compute
jR (oi)(0)jR (oj) 2 DchG;(;)1= T
We have
jR (oi)(0)jR (oj) = jR (oi)(0) ajqoq + (os; ou)oajs!u (2:10:*
*1)
Let us compute the first summand. Using (2.8.1), we have
jR (oi)(0)(ajqoq) = (jR (oi)(0)ajq)oq = aipop(ajq)oq = [oiR; ojR] = [oi;(oj*
*]R9:2)
by (2.8) and (2.7).
On the other hand
o js R o js o R js
jR (oi)(0) (os; ou) a !u = oi(0) (os; ou) a !u = (os; ou) oi (a !u) =
by (2.3.2) and (2.2.9)
= (os; ou)oaipop(ajs)!u = (os; ou)ocijqaqs!u (2:10:*
*3)
Adding up (2.10.2) and (2.10.3) we see that
jR (oi)(0)jR (oj) = jR ([oi; oj])
which proves (2.5.4) and part (ii) of the theorem. 4
13
11. Proof of (2.6). The first claim is a reformulation of (2.5.3) and (2.5.4*
*).
The second claim is a trivial consequence of (2.5.2) and two Borcherds' form*
*ulas
Xn n
x(n)y(-1)z = y(-1)x(n)z + (x(j)y)(n-1-j)z
j=0 j
(n 0), cf. [GII] (0.5.12), and
X
(x(-1)y)(n)z = x(-1-j)y(n+j)z + y(-n-1-j) x(j)z
j0
cf. [GII] (0.5.4). 4
14
x3. BRST
3.1. Recall the definition of the BRST reduction due to Feigin. See [F]; t*
*he
definition in the language of vertex algebras was given in [FF4] (cf. Section 4*
*), cf.
also [FF3], Appendix A; for a more modern treatment see [BD1], 3.7, [BD2], 7.13.
Let a be a finite dimensional Lie algebra. Choose a base {ai} in a; denote *
*the
structure constants
[ai; aj] = cijpap (3:1:*
*1)
Recall that the Killing form (; )(K) : a x a -! C is given by
(ai; aj)(K) = cipqcjqp (3:1:*
*2)
Let a be the space a with the reversed parity; denote by {OEi = ai} the corre-
sponding base and by {OE*j} the dual base of a* given by
= ffiij (3:1:*
*3)
Let CBRST (La) denote a graded vertex superalgebra generated by odd fields OEi*
*(z)
of conformal dimension 1 and odd fields OE*i(z) of conformal dimension 0 with O*
*PE
OEi(z)OE*j(w) ~ __ffiij_z - w (3:1:*
*4)
We shall identify the spaces a and a* with their obvious images in CBRST (La)1
and CBRST (La)0 respectively.
Let us introduce an odd element Da 2 CBRST (La) of conformal dimension 1 by
Da = - 1_2cijpOEpOE*iOE*j (3:1:*
*5)
P
Thus, we have the corresponding field Da(z) = Da;nz-n-1 and we set
da := Da;0 (3:1:*
*6)
The pair (CBRST (La); da) may be regarded as a chiral analogue of the Chevalley
cochain complex C(a). However, in the chiral case the square d2amay be nonzero.
It is easy to compute it. Namely, let us write down the OPE Da(z)Da(w) using
Wick theorem. We have
(ai; aj)(K)OE*i(z)OE*j(w)
Da(z)Da(w) ~ _____________________(z - w)2 (3:1:*
*7)
Therefore Z
d2a= (ai; aj)(K) OE*i(w)0OEj(w) (3:1:*
*8)
15
3.2. Corollary. If the Lie algebra a is nilpotent then d2a= 0.
Indeed, the Killing form of a nilpotent Lie algebra is zero.
3.3. Let (; ) : a x a -! C be an arbitrary symmetric invariant bilinear fo*
*rm
("level"). Recall that the vertex algebra Va;(;)is generated by even fields ai(*
*z) of
conformal weight 1, subject to OPE
ai(z)aj(w) ~ _(ai;_aj)_(z+-[w)2ai;_aj](w)_z - w (3:3:*
*1)
3.4. Lemma. The rule
ai 7! cipqOEqOE*p (3:4:*
*1)
defines an embedding of vertex algebras
Va;(;)a;(K),! CBRST (La) (3:4:*
*2)
3.5. Let M be a vertex module over Va;-(;)a;(K). Let us introduce a space
CBRST (La; M) := CBRST (La) M (3:5:*
*1)
According to the previous lemma this space is canonically a graded (by conformal
weight) Va;0-supermodule. This space is also graded by "fermionic charge"
CBRST (La; M) = p2Z CpBRST (La; M) (3:5:*
*2)
where we assign to OEi (resp. OE*i; m 2 M) the charge -1 (resp., 1; 0).
Introduce an odd element Da;M 2 CBRST (La; M) of conformal weight 1 and
fermionic charge 1 by
Da;M = OE*i ai+ Da 1 (3:5:*
*3)
It follows from (3.1.7) that
Da;M (z)Da;M (w) ~ 0 (3:5:*
*4)
Therefore, setting Z
da;M := Da;M (z) (3:5:*
*5)
we get a differential
d2a;M = 0 (3:5:*
*6)
By definition d increases the fermionic charge by 1.
The pair (CBRST (La; M); da;M ) is called the BRST complex of La with coeff*
*i-
cients in M, and its cohomology H*BRST (La; M) is called the BRST cohomology.
3.6. Example. Let N be a unipotent algebraic group with the Lie algebra n.
Consider a Vn;0-module DchN;0(note that according to 1.6 this algebra represent*
*s a
16
unique isomorphism class of cdo's over N). Inside the loop algebra Ln = n[T; T *
*-1]
consider two Lie subalgebras: n- and n+ , generated by all elements o T n (o 2*
* n)
with n < 0 and n 0 respectively. Then DchN;0is a free n- -module and a cofree
(i.e. the dual module is free) n+ -module.
It follows that
HiBRST (Ln; DchN;0) = 0 (i 6= 0); H0BRST (Ln; DchN;0) = C (3:6:*
*1)
cf. [FF3], p. 178.
17
x4. Homogeneous spaces
4.1. Let X be a smooth variety. We have an exact triangle
[2;3>X-! [2X-! C (4:1:*
*1)
where
[2X: 0 -! 2X -! 3X -! : : : (4:1:*
*2)
(2X sitting in degree 0) and
C : 0 -! 4X=d3X -! 5X -! : : : (4:1:*
*3)
(4X=d3X sitting in degree 2). It follows that
4.1.1.1 the canonical map
Hi(X; [2;3>X) -! Hi(X; [2X) (4:1:*
*4)
is injective for i = 2 and bijective for i = 0; 1.
4.2. Let G be a simple algebraic group. In this section we shall discuss the*
* chiral
differential operators on homogeneous spaces G=G0 where G0 = N _ a unipotent
subgroup, G0= P _ a parabolic but not minimal parabolic, or G0= B _ a Borel
subgroup.
The case G=N
4.3. Consider the projection ss : G -! X := G=N. The variety X is quasiaffin*
*e,
therefore we have
Hi(X; [2X) = Hi(X; [2X) = Hi+2DR(X) (i 1) (4:3:*
*1)
On the other hand,
H*DR(X) = H*(X; C) (4:3:*
*2)
by Grothendieck's theorem, (cf. [Gr], Theorem 1').
The projection ss is an affine morphism which is a Zariski locally trivial b*
*undle
with fiber N isomorphic to an affine space, so ss* : H*(X; C) -~! H*(G; C). *
*It
follows that
ss* : Hi(X; [2X) -~! Hi(G; [2G) (i 1) (4:3:*
*3)
We have a short exact sequence
0 -! G=X -! G - ! ss*X -! 0 (4:3:*
*4)
______________1
this remark is due to H.Esnault
18
and the vector bundles G ; G=X are trivial (a base of global sections of G=X *
*is
given by left invariant vector fields coming from the Lie algebra n := Lie(N)).
Therefore we have
ss*c(DiffchX) = ss*ch2(X ) = ch2(G ) = 0 (4:3:*
*5)
hence
c(DiffchX) = 0 (4:3:*
*6)
by 4.1.1.
Therefore,
4.4. the groupoid (X; DiffchX) is nonempty. The set of isomorphism classes
ss0((X; DiffchX)) is a torseur under H3DR(X) = H3DR(G).
4.5. Let DchG;(;)be the sheaf of chiral differential operators on G of level*
* (; ) where
(; ) : g x g -! C (4:5:*
*1)
is a fixed symmetric invariant bilinear form on g = Lie(G), cf. 1.7. The form
(4.5.1) is a scalar multiple of the Killing form
(; ) = c(; )g;(K); c 2 C (4:5:*
*2)
The Killing form on g restricts to zero on n (since the trace of a nilpotent en*
*do-
morphism is zero). Therefore we have the canonical embedding of vertex algebras
Vn;0,! VG;(;),! DchG;(;) (4:5:*
*3)
so that the sheaf DchG;(;)becomes a sheaf of Vn;0-modules.
Applying the BRST construction 3.5 to M = ss*DchG;(;)and a = n we get a
sheaf of BRST complexes CBRST (Ln; ss*DchG;(;)) and BRST cohomology sheaves
H*BRST (Ln; ss*DchG;(;)) over X. They are sheaves of Z0 -graded vertex supera*
*lge-
bras.
4.6. Theorem. We have
HiBRST (Ln; ss*DchG;(;)) = 0 (i 6= 0) (4:6:*
*1)
The sheaf H0BRST (Ln; ss*DchG;(;)) is an algebra of chiral differential operato*
*rs over
X.
The correspondence
DchG;(;)7! H0BRST (Ln; ss*DchG;(;)) (4:6:*
*2)
induces a bijection of the sets of isomorphism classes
ss0((G; DiffchG)) -~! ss0((X; DiffchX)) (4:6:*
*3)
19
We shall use the notation DchX;(;)for the cdo H0BRST (Ln; ss*DchG;(;)).
Note that higher direct images Riss*DchG;(;)are trivial for i > 0 since the *
*morphism
ss is affine and the sheaves DchG;(;)admits a filtration whose qutients are coh*
*erent
(in fact, locally free) OG -modules.
4.7. Proof (sketch). Locally on X the projection ss : G -! X is isomorphic to
the direct product U x N -! U. If D is an algebra of chiral differential operat*
*ors
on U x N then D -~! DU DN where DU (resp. DN is an algebra of differential
operators on U (resp. N). Now the first claim of the theorem follows from (3.6.*
*1).
The second claim is a corollary of 4.4. 4
4.8. Corollary. The dual embedding
jR : VG;(;)o-! DchG;(;) (4:8:*
*1)
defined in Corollary 2.6 induces a canonical morphism of vertex algebras
io(;): VG;(;)o-! DchX;(;) (4:8:*
*2)
Indeed, we know that (4.8.1) commutes with the left action of bghence with t*
*he
BRST differential.
In particular, the Kac-Moody algebra bgat level (; )o acts canonically on Wa*
*ki-
moto modules which may be defined as the spaces of sections (U; DchX;(;)) where*
* U
is a big cell. This is a result due to Feigin-Frenkel obtained in [FF1] - [FF3*
*] in a
different way.
The case G=B
4.9. Let B G be a Borel subgroup, ss : G -! X := G=B. X is a smooth
projective variety and we have
Hp(X; qX) = 0 (p 6= q) (4:9:*
*1)
and
Hi(X; iX) = H2i(X; C) (4:9:*
*2)
It follows that H2(X; [2) = H4(X; C), and [GII], Theorem 7.5 says that the image
of c(DiffchX) in H4(X; C) is equal to
2ch2(X ) := c21(X ) - c2(X ) (4:9:*
*3)
where ci(X ) 2 H2i(X; C) are the Chern classes of the tangent bundle X .
20
4.9.1. Lemma. We have
ch2(X ) = 0 (4:9:1:*
*1)
Proof. The space H2(X; C) may be identified with the complexification of the
root lattice of G. The classical theorem by J. Leray says that that the cohomol*
*ogy
algebra H*(X; C) is equal to the quotient of the symmetric algebra of the space
H2(X; C) modulo the ideal generated by the subspace of invariants of the Weyl
group W having positive degree, cf. [L], Theoreme 2.1 b).
P The class [X ] in the Grothendieck group of vector bundles is equal to the s*
*um
[Lff] where ff runs throughPall negative roots, and Lffis the line bundle wi*
*th
c1(Lff) = ff. Hence ch2(X ) = ff2; this element is invariant under the actio*
*n of
W , therefore its image in H4(X; C) is zero. 4
This lemma implies that
c(DiffchX) = 0 (4:9:*
*4)
by 4.1.1. On the other hand, it follows from 4.1.1 and (4.9.1) that
Hi(X; [2;3>X) = 0 (i = 0; 1) (4:9:*
*5)
Thus, by [GII], loc. cit. we get
4.10. Theorem. The groupoid (X; DiffchX) is nonempty and trivial. In
other words, there exists a unique, up to a unique isomorphism, algebra of chir*
*al
differential operators DchXover X. 4
4.11. Let us construct the algebra DchXusing BRST reduction. Consider the
sheaf DchG;(;)as in 4.5. Let (; )b denote the restriction of (; ) to b := Lie(B*
*).
We have a canonical embedding of vertex algebras
Vb;(;)b,! DchG;(;) (4:11:*
*1)
We have
(; )g;(K)|bxb = 2(; )b;(K) (4:11:*
*2)
Therefore, to get the minus Killing on b we have to start from -(; )g;(K)=2, i.*
*e.
from the critical level on g. Thus, by construction 3.5,
4.11.1. the BRST complex CBRST (Lb; ss*DchG;(;)) is defined iff (; ) = (; )*
*g;crit, i.e.
on the critical level, cf. (2.1.5).
Let us denote the algebra DchG;(;)g;critby DchG;crit.
4.12. Let Vb;critdenote the vacuum module on the critical level Vb;(;)b;crit*
*. Let
M be a vertex algebra equipped with a morphism of vertex algebras Vb;crit-! M.
We have a decomposition
b = h n (4:12:*
*1)
21
where n b (resp. h b) the maximal nilpotent (resp. the Cartan) subalgebra; n *
*is
spanned by elements eff, ff being a positive root. Let us define a vertex subal*
*gebra
CBRST (Lb; h; M) CBRST (Lb; M) (4:12:*
*2)
The vector space CBRST (Lb; M) is spanned by all the monomials
h1;i1. : :.:h*1;j1. : :.:eff1;k1. : :.:e*ff01;l1. : : :m(4:12:*
*3)
where the indices ip; jp; kp; lp denote the conformal weight, ip; kp 1; jp; l*
*p 0,
m 2 M is a vector of weight 2 h*.
By definition, the subspace CBRST (Lb; h; M) is spanned by all monomials (4*
*.12.3)
such that
(a) all jp 1;
P P
(b) ffp - ff0q+ = 0.
It is a vertex subalgebra. One checks that the BRST differential d in CBRST *
* (Lb; M)
preserves CBRST (Lb; h; M).
We define the relative BRST cohomology H*BRST (Lb; h; M) as the cohomology
of CBRST (Lb; h; M) with respect to this differential. It is canonically a ve*
*rtex
algebra.
4.13. Applying the previous definition to M = ss*DchG;critwe get the BRST
cohomology sheaves H*BRST (Lb; h; ss*DchG;crit) on X.
4.14. Theorem. We have
HiBRST (Lb; h; ss*DchG;crit) = 0 (i 6= 0) (4:14:*
*1)
and the sheaf of vertex algebras H0BRST (Lb; h; ss*DchG;crit) is canonically is*
*omorphic
to DchX.
The proof is the same as that of Theorem 4.6.
4.15. Corollary. We have a canonical isomorphism of vertex algebras
H*(X; DchX) = H*BRST (Lb; h; (G; DchG;crit)) (4:15:*
*1)
Indeed, as we have already remarked, sheaves of cdo DchGon G have a filtrati*
*on
whose graded quotients are vector bundles; but the vatiety G and the morphism ss
are affine, hence Hi(G; DchG) = Riss*DchG= 0 for i > 0.
4.16. Corollary. The dual embedding
jR : VG;crit-! DchG;crit (4:16:*
*1)
defined in Corollary 2.6 induces a canonical morphism of vertex algebras
icrit: VG;crit-! DchX (4:16:*
*2)
22
Indeed, we know that (4.16.1) commutes with the left action of bghence with *
*the
BRST differential.
In particular, the Kac-Moody algebra bgon the critical level acts canonicall*
*y on
Wakimoto modules which may be defined as the spaces of sections (U; DchX) where
U is a big cell. This is a result of Feigin-Frenkel proven in [FF1] - [FF3] in *
*a different
way.
The case G=P
4.17. Let P G be a parabolic but not Borel, p = Lie(P ), ss : G -! X :=
G=P . The discussion of 4.9 applies as it is, except for Lemma 4.9.1, which is
replaced by
4.17.1. Lemma. We have
ch2(X ) 6= 0 (4:17:1:*
*1)
Hence c(DiffchX) 6= 0, and we get
4.18. Theorem. The groupoid (X; DiffchX) is empty. In other words, there
is no cdo over X. 4
4.19. Let us see how this fact is reflected in the BRST world. We would li*
*ke
to get a sheaf of cdo over X as the BRST cohomology of Lp with coefficients in
some sheaf ss*DchG;(;). However, no form (; ) on G restricts to the Killing for*
*m on p,
which implies that the square of the BRST differential in CBRST (Lp; ss*DchG;(*
*;)) is
always nonzero. Thus, the BRST cohomology is not defined, which is compatible
with 4.18.
References
[BD1] A. Beilinson, V. Drinfeld, Chiral algebras, Preprint.
[BD2] A. Beilinson, V.Drinfeld, Quantization of Hitchin integrable system and
Hecke eigensheaves, Preprint.
[F] B.L. Feigin, Semiinfinite homology of Lie, Kac-Moody and Virasoro algebr*
*as,
Uspekhi Matem. Nauk, 39, no. 2(236) (1984), 195-196 (Russian).
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Uspekhi Matem. Nauk, 43, no. 5(263) (1988), 227-228 (Russian).
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Teaneck, NJ, 1990.
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[FF3] B.L. Feigin, E. Frenkel, Affine Kac-Moody algebras and semi-infinite f*
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operators. II, math.AG/0003170.
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V.G.: Department of Mathematics, University of Kentucky, Lexington, KY
40506, USA; vgorb@ms.uky.edu
F.M.: Department of Mathematics, University of Southern California, Los An-
geles, CA 90089, USA; fmalikov@mathj.usc.edu
V.S.: IHES, 35 Route de Chartres, 91440 Bures-sur-Yvette, France; vadik@ihes*
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