DELOCALISED EQUIVARIANT ELLIPTIC COHOMOLOGY
I. Grojnowski
Feb 1994
Abstract. In this paper we construct an equivariant elliptic cohomology t*
*heory
over C. As defined, the level k equivariant cohomology of a point with re*
*spect to a
compact group G is just the span of the characters of the loop group dLGa*
*t level k.
In this paper we construct a `delocalised' equivariant elliptic cohomology o*
*ver
C. The construction here suffers from several obvious disadvantages_it is unwie*
*ldy
to work with, and clearly misses completely the point of elliptic cohomology [L*
*a].
However, it suffices for the construction of the elliptic affine algebras (see *
*below),
and does produce the `correct' results.
The result was inspired by [BG] (which in turn is a child of [BBM]). We esse*
*n-
tially define equivariant elliptic cohomology by using the Chern character E(X *
*xT
ET ) ! H(X xT ET ), using the topology of the abelian variety ET to avoid compl*
*e-
tions. This construction produces non-trivial bundles on ET (for example ES1(P1*
*)),
and, compatibly, the Gysin homomorphism we define involves twisting the bundles
still further.
In section 2.4 we define ES1(X), a sheaf over a fixed totally marked ellipti*
*c curve
E. The definition works by identifying a neighbourhood of the elliptic curve ar*
*ound
e 2 E with its tangent space, and taking the equivariant cohomology of Xe, the
fixpoints of e on X. In 2.6 this is generalised to an arbitrary compact connect*
*ed
group.
In section 2.5 we define the pushforward, or Gysin, homomorphism. This invol*
*ves
a choice of local coordinate on the elliptic curve, i.e. an analytic map s : U *
* E !
C, where U is a neighbourhood of 1. This data is precisely that of a complex
orientation in topology, and defines in a standard manner a homology theory with
pushforward maps [A]. When this standard pushforward is applied locally on the
elliptic curve, the effect of this is to twist the sheaf ES1(X), so that ß : X *
*! Y
induces a map from a twist of ES1(X) to ES1(Y ).
This behaviour is forced upon us, as elliptic cohomology satisfies a `locali*
*ty'
property: the Mayer-Vietoris exact sequence. As ES1(X) is usually a non-trivial
sheaf, if we can partition X into contractible pieces, a long exact sequence mu*
*st
necessarily twist some of the cohomology of the pieces in order to produce ES1(*
*X).
This is indeed what happens (consider, for example ES1(P1)).
It is immediate from definitions that all standard properties of a cohomology
theory hold; in section 3 we mention a few that are specific to elliptic cohomo*
*logy.
______________
Typeset by AM S-T*
*EX
1
2 I. GROJNOWSKI
Finally, we consider what happens when we vary the elliptic curve. It turns *
*out
that the fixpoints Xe(z,ø)make sense for a point on the universal elliptic curv*
*e (no
extra marking is necessary). It follows that if the given fixed complex orienta*
*tion
s is defined on a curve with some marking, than so is our cohomology theory. For
example, if s is the orientation class of [La], the S1-equivariant cohomology i*
*s a
sheaf over the universal elliptic curve with a marked point of order 2. Taking *
*the
formal neighbourhood of 1 on this marked universal curve, we recover the usual
elliptic cohomology E(X xT ET ) of [La]. (See also [M] for an explanation of why
this is the `wrong' object.)
However, instead of this one may take global sections on the universal curve*
* of
tensor products of EG (X) with certain line bundles (see section 3). These glo*
*bal
objects are the `correct' definition of a level k elliptic cohomology theory. I*
*n sharp
contrast to HG (X) and KG (X) they are finite dimensional.
This note is part of a project of the author to construct certain generalisatio*
*ns
of quantum groups and Hecke algebras which I call elliptic affine algebras. The*
*se
algebras depend on a marked elliptic curve as well as a point on the curve (the
analog of q in the quantum group), and theta constants occur in the structure
coefficents of the algebra.
One proceeds by applying the equivariant elliptic cohomology constructed here
to certain well known varieties to produce elliptic Hecke algebras, elliptic qu*
*antum
groups and their finite dimensional representations. The usual case is obtained*
* by
applying equivariant K-theory to these same varieties (see [G2]).
The construction detailed here produces certain `twisted' algebras: coherent
sheaves A on an abelian variety, equipped with a multiplication A A L ! A,
where L is a certain line bundle. The representations of A are easily described*
* as
in [G], even at points of finite order. The unfinished task is then to produce*
* an
honest algebra out of A, with a similar representation theory.
I believe I now understand how to do this, though at the glacial pace at whi*
*ch I
work it will take some time to get the details correct. In any case, the polite*
* interest
expressed by the people who have seen this note (which has been circulating sin*
*ce
February, 1994) suggest that it may be worth publishing as is.
Finally, it is worth mentioning what we have really done. We have produced a
theory such that the level k elliptic cohomology of a point, with respect to a *
*group
G, is precisely the span of the characters of the level k representations of gL*
*G.
We have done this purely finite dimensionally, by a cheap trick. However, it*
* is
morally clear how to do this in general. One must define a certain category of *
*gLG
equivariant vector bundles on the loopspace LX with the semi-infinite topology.
Pushforward maps then become Euler characteristics in semi-infinite cohomology
[FF]. I believe this is not too difficult to do rigorously.
DELOCALISED EQUIVARIANT ELLIPTIC COHOMOLOGY 3
2. Elliptic Cohomology
2.1
Fix ø 2 C, Im(ø ) > 0. Let E = Eø denote the marked elliptic curve C=(Z + ø *
*Z).
There is a continuous isomorphism of groups E ! S1 x S1, induced from the map
C ! S1 x S1, x + ø y 7! (e2ßix, e2ßiy).
Let T be a compact torus (product of S1's); Y (T ) = Hom(S1, T ) its lattice*
* of
cocharacters. Then T ~= Y (T ) Z S1; define ET = Y (T ) Z E and t = Y (T ) Z*
* C.
We identify t with both the complexified Lie algebra of T , and the tangent spa*
*ce
to ET at 1. Write Ot for the sheaf of complex valued analytic functions on t.
The map E ! S1 x S1 gives rise to a continuous isomorphism of groups ET !
T x T ; for e 2 ET denote its image under this map as (e1, e2).
If V is a small neighbourhood of 0 2 t, there is a neighbourhood U of 1 2 ET
and an isomorphism exp : V ! U with inverse log: U ! V . We will write log* for
the corresponding map from sheaves on V to sheaves on U.
Now suppose T acts on a topological space X. Define the fixpoint set Xe, for
e 2 ET to be Xe1,e2= {a 2 X | e1a = e2a = a}. For H a connected subgroup
of T , put X(H) = {a 2 X | stab(a)0 = H}. Here, stab(a)0 denotes the identity
component of the subgroup of T that fixes a. Then for x 2 t define the fixpoint*
* set
Xx as Xx = [H:x2(LieH)C X(H). If X is compact and T acts smoothly, then for
each e 2 ET there exists an open neighbourhood U of e such that Xf Xe for all
f 2 U. This is essentially a result of Mostow (see [BG,1.3]). Note that for e i*
*n a
small neighbourhood of 1 2 ET we have Xe = Xloge, and more generally, for f in
a small neighbourhood of e we have Xf = (Xe)log(f-e).
We will systematically use this fixpoint notation (though neither the Abelian
variety or the Lie algebra act, we have made perfect sense of their fixpoints).
For e 2 ET let te : ET ! ET , e07! e0+ e be the map `translation by e'.
2.2
Recall there is a functor, equivariant cohomology, from the category of pairs
(G, X), where G is a topological group and X a topological space on which G acts
continuously to the category of Z-graded super-commutative complex algebras,
(G, X) 7! HG (X), with the following properties:
i) HG (X) is a graded super-commutative algebra over HG = HG (point). If
T is a compact torus, then HT = S(t*) canonically, where S(t*) is the algebra
of polynomial functions on t, graded so the generators are in degree 2. If G is
compact connected, T G a maximal torus, W = NG (T )=T the Weyl group, then
HG = HWT, the W -invariant polynomial functions on t.
We often regard HG (X) as a sheaf over Spec(HG ). In the case G is compact
connected, we can also regard HG (X) as a W -equivariant sheaf on t.
ii) If G is compact and connected, HG (X) is determined by X and the Lie
algebra of G. We denote it Hg(X). Further, if G is a general compact group,
HG (X) = Hg(X)G=G0 , and if G ! G0 is a homotopy equivalence, where G' is an
arbitrary topological group, the induced map HG (X) ! HG0(X) is an isomorphism
of graded algebras.
iii) Let T be a compact torus. Then if x 2 t and U is a sufficiently small
neighbourhood of x, the inclusion i : Xx ,! X induces an isomorphism i* :
Ht(X) Ht (U, Ot) ~=Ht(Xx ) Ht (U, Ot). More generally, if x 2 g is semisim-
ple, where G is a possibly disconnected compact Lie group, then the inclusion
i : (Gx, Xx ) ,! (G, X) induces a map HG (X) HG HGx ! HGx (Xx ) which be-
4 I. GROJNOWSKI
comes an isomorphism when both sides are localised at x 2 SpecHGx . This is the
ön n-Abelian localisationö f Atiyah-Segal.
iv) If tx : t ! t, y 7! y + x is the translation by x map, then it induces a*
* functor
from sheaves on t to sheaves on t, denoted (as always) by t*x. Then t*xHt(Xx )*
* ~=
Ht(Xx ). We indicate the proof. Write t = h h0, where h is the line of multip*
*les
of x. Then Ht(Xx ) = Hh C Hh0(Xx ), and tx acts only on Hh.
More generally, if x 2 g is semisimple, then t*x: HGx (Xx ) ! HGx (Xx ) is an
isomorphism, as x is in the center of gx.
2.3
Let O = OET denote the sheaf of complex valued analytic functions on ET ;
(U, O) = (U) its sections over U. Recall that to specify a sheaf A of O-modul*
*es
on ET it is enough to give a (U, O) module AU for each U in some open cover of*
* ET
by sufficiently small sets, and for each U, V with U \V 6= ; a (U \V ) isomorp*
*hism
OEUV : (U \ V ) U AU ! (U \ V ) V AV such that if U, V, W are such that
U \ V \ W 6= ;, then OEV W OEUV = OEUW . Clearly it also suffices to only gi*
*ve this
glueing data OEUV when V U if the open cover is closed under finite intersec*
*tion.
Similarly, if A is a sheaf of Z=2-graded super-commutative O-algebras on ET ,
then elements ~UV 2 (U \ V ) U AxU(where AxUdenotes the commutative group
of invertible elements in the ring AU ) such that ~V W ~UV = ~UW defines an e*
*lement
[~] of H1(ET , Ax ) and a sheaf A[~], the "twistö f A by ~. Here, (U, A[~]) =*
* AU
and the glueing isomorphisms are OE0UV = ~UV OEUV . The isomorphism class of A[*
*~]
depends only on the class of [~] in H1(ET , Ax )
(A notational warning: t*e, log* refer to the pullback of sheaves. On the o*
*ther
hand, if ß : X ! Y is a map, we also write ß* to denote pullback in cohomology.
These two uses are married in the definition below, most particularly in 2.6).
2.4
Now define ET (X), a Z=2 graded sheaf of supercommutative algebras over O.
If e 2 ET , and U is a sufficiently small neighbourhood of e, define
(U, ET (X)) = t*elog*{Ht(Xe) Ht (Ot, log(t-e U)}.
Write Ht(Xe)U-e for Ht(Xe) Ht (Ot, log(t-e U).
If f 2 U, and V U is a small enough neighbourhood of F , define OEUV :
(U, ET (X)) U V ! (V, ET (X) as the composition of the following iso-
morhisms
(U, ET (X)) U V ~= t*elog*{Ht(Xe)U-e (log(t-eU,Ot)) (log(t-e V ), Ot)}
~i* * e log(f-e)
= te log{Ht((X ) )V -e)}
~= t*flog*{t*log(e-f)Ht(Xf )V -e}
~= t*flog*{Ht(Xf )V -f} = (V, ET (X))
where i denotes the inclusion (Xe)log(f-e)= Xf ,! Xe. We have i* is an isomor-
phism by localisation in equivariant cohomology (2.2,iii), and the last line is*
* an
isomorphism by (2.2,iv). It is clear that OEUV satisfy the cocycle condition, *
*and so
by the discussion above we have defined a sheaf over ET .
Similarly, if ß : X ! Y is a T -map, then ß* : Ht(Y e) ! Ht(Xe), e 2 ET , in*
*duces
a map of O-algebras, also denoted ß*, ß* : ET (Y ) ! ET (X). (This is a map of
sheaves by naturality of ß* and by (2.2,iv) above; the diagram chase is omitted*
*).
DELOCALISED EQUIVARIANT ELLIPTIC COHOMOLOGY 5
We remark that if L is a T -equivariant local system on X, or even a complex
in DT (X), the derived category of T -equivariant sheaves on X, then the same
procedure serves to define elliptic cohomology with coefficents in L, ET (X, L)*
* and
ß* : ET (Y, ß*L) ! ET (X, L) (see [Lu]), as the localisation theorem (2.2,ii) i*
*s still
true in this case.
2.5 exp
Consider the local ring at 1 of ET (X); ET (X)1 - -! Ht(X)0 = Ht(X) Ht
(Ot)0 ,! H(X xT ET ), where BT = ET=T is the classifying space of T . In the
case X is a point, we may define s(x) = s exp = exp*(s) 2 H(BS1) as an orientat*
*ion
class, and regard it as an element of (U, ES1) for sufficiently small U. Here*
* s :
U E ! C is a local coordinate. The usual machinary of algebraic topology
means that from this data we get Gysin morphisms ßE! : Ht(X)0 ! Ht(Y )0 for
ß : X ! Y a proper weakly complex oriented map [A], as well as Todd classes
s(x)=x and a Riemann-Roch isomorphism relating ßE! and ßH!, where ßH! is the
usual Gysin morphism in cohomology.
Now let ß : X ! Y be a proper weakly complex oriented map. If e 2 ET , denote
~ß: Xe ! Y ethe induced map. This is still a proper weakly complex oriented map.
The map ß defines a cohomlogy class ~(ß) = [ß] 2 H1(ET , ET (X)x ) as follow*
*s.
Let e 2 U, f 2 V U, e 6= f be points on ET and small neighbourhoods containing
them. Let i : Xf ,! Xe be the inclusion. Then i*iE!: Ht(Xf )U-e ! Ht(Xf )U-e *
*is
well defined, and i*iE!1, which we write as e(Xe=Xf ), the Euler class of Xf ,!*
* Xe
gives an invertible element of Ht(Xf )V -f. (It is invertible as for each conn*
*ected
component X0 of Xf , the normal bundle in Xe to X0 does not contain the trivial*
* T -
bundle, and the orientation s(x) is a local coordinate on E; i.e. an analytic f*
*unction
with an isolated simple zero at 0 2 t).
Then if ~ß : Xf ! Y f is the induced map, ~ß*e(Y e=Y f) . e(Xe=Xf )-1 is an
invertible element of Ht(Xf )V -e, so applying t*elog*(i*)-1 to it gives an ele*
*ment
~UV 2 (U \ V ) U (U, ET (X)x ). It is clear that if W V , W a neigbourho*
*od
of f0 that ~V W ~UV = ~UW and so (~UV ) defines a cohomology class, ~(ß).
Now we define ß! : ET (X)~(ß) ! ET (Y ) by defining, for e 2 U, U sufficient*
*ly
small, (U, ß!) := t*elog*^ßE!, where ^ß: Xe ,! Y e. It follows from the defin*
*itions
and the "excess intersection formula" in a generalised cohomology theory that t*
*his
is well defined (again we leave the diagram chase to the reader), and a map of
ET (Y )-modules.
2.6
More generally, suppose G is a compact connected Lie group, T ,! G the maxi-
mal torus, W = NG (T )=T the Weyl group. We define EG (X), a Z=2-graded sheaf
on ET =W as follows. Write p : ET ! ET =W for the canonical projection. Then if
U ET is a small open neighbourhood, e 2 U is such that W eU = U, wU \ U = ;
if w 62 W ewe define
we W
(pU, EG (X)) = ( w2W=We t*welog*{HGwe (Xwe HWwet (log(t-we U), Ot)W })
~=t*elog*{HGe (Xe) HWe (log(t U), O )We }.
t -e t
Observe that HGe = HWet, and that neighbourhoods of this form cover ET , as W is
finite. If V U is a neighbourhood of f (so W f W e) such that W fV = V and
wV \V = ; if w 62 W f, define OEUV : (pU, EG (X)) UWe ( V )Wf ! (pV, EG (X*
*))
as the composition of the obvious maps and the maps induced by i : (Gf, Xf =
6 I. GROJNOWSKI
(Xe)log(f-e)) ,! (Ge, Xe) and translation t*log(e-f)as in (2.4). Then OEUV is*
* an
isomorphism of rings over (pV, ET =W ) = ( V )Wf by non-Abelian localisation
(2.2,iii), and the cocycle condition is satisfied.
Similarly, if f : (G, X) ! (H, Y ) is a map of (compact connected groups, sp*
*aces)
inducing a map TG ! TH of the maximal tori of G to that of H, and hence a map
f~ : ETG =WG ! ETH =WH , we get an induced map of sheaves f* : f~*EH (Y ) !
EG (X). Further, if h : (H, Y ) ! (K, Z) is another such map of (groups,spaces)*
* we
have h*f* = (fh)*. The obvious diagram chases required to verify this are omitt*
*ed.
Finally, by repeating word for word the discussion in (2.5) we see that for a
proper weakly complex oriented map of G-spaces ß : X ! Y we have a cohomology
class ~(ß) 2 H1(ET =W, EG (X)x ) and maps of Z=2-graded OET=W -modules ß! :
EG (X)~(ß) ! EG (Y ). (This is not a ring homomorphism!).
One may check that this is functorial in the appropriate sense; i.e. that *
*if
ß0 : Y ! Z is also a proper weakly complex oriented0map of G-spaces, then
ß*~(ß0) . ~(ß) = ~(ß0ß) and (ß0ß)! : EG (X)~(ß ß)! EG (Z) is the composite of ß*
*0!
and the map induced from ß!.
3. Remarks
3.1 We leave to the reader the task of making a systematic list of the prope*
*rties of
EG . Most follow immediately from the definitions and the corresponding propert*
*ies
of HG , and are obvious analogues of the usual properties of a cohomology theor*
*y.
The following remarks are some indications of properties more particular to EG .
3.2 Let X, X0 be smooth projective G-spaces, with Hodd(X) = Hodd(X0) = 0.
Then the natural morphism EG (X) EG EG (X0) ! EG (X x X0) is an isomorphism
if and only if the centralizer of every pair of commuting semisimple elements o*
*f G is
connected (see [HKR]). (This is immediate from the definition of EG , as with t*
*hese
hypotheses on X, X0 such a Kunneth theorem holds in equivariant cohomology
for arbitrary connected G). Essentially the only groups G with this property a*
*re
products of GLn's and tori.
Note that a Kunneth theorem holds in equivariant K-theory if and only if the
centraliser of every semisimple element is connected; i.e. if and only if G is *
*simply
connected, by a theorem of Steinberg. The usual proof of this fact (Kazhdan-
Lusztig, Hodgkins) relies on another theorem of Steinberg; clearly the `delocal*
*ised'
technique we use gives a different proof.
3.3 Let G be a compact group, with a fixed invariant non-degenerate symmetric
bilinear form on g. This data defines a line bundle L on ET with this form as *
*its
Chern class [Lo]. For example, if G is simple and simply connected, and L has
degree the order of the center of G, then the Weyl denominator for the affine K*
*ac-
Moody algebra bgis a section of Lg, where g is the dual Coxeter number for G
[Lo].. One may then consider a "level k" elliptic cohomlogy of X as (EG (X) Lk*
*).
3.4 Modularity.ijLet H = {ø 2 C | Im ø > 0}. Recall the group SL2Z acts on
C x H by abcd. (z, ø ) = (z=(cø + d), (aø + b)=(cø + d)). Hence SL2Z acts on *
*t x H
also. Denote by e(z, ø ) the image of (z, ø ) 2 t x H in ET ø = t C Eø. Then*
* if X
is a T -space, the fixpointsiXe(z,ø)dependjonlyionjthe orbit of (z, ø ) under S*
*L2Z.
(This is clear for -1 1 and 111 , which generate SL2Z). Hence the modular
properties of EG (X) depend only on the modular properties of s(z, ø ), the loc*
*al
coordinate around 1. We can thus regard equivariant elliptic cohomology as defi*
*ned
DELOCALISED EQUIVARIANT ELLIPTIC COHOMOLOGY 7
on the moduli of marked elliptic curves, with marking determined by the chosen *
*s.
For example, for s the orientation class of [La], EG (X) is defined over the cu*
*rve
S = H= 0(2). (In such a case, all sections should be interpreted as sections *
*over
the universal marked curve.)
References
[A] J. F. Adams, Stable homotopy and generalised homology, Chicago lectures *
*in mathematics,
University of Chicago press, 1974.
[BG] J. Block and E. Getzler, Equivariant cyclic homology and equivariant dif*
*ferential forms,
preprint 1993.
[BBM] P.Baum, J. L. Brylinski and R. MacPherson, Cohomologie 'equivariante d'e*
*localise, C.R.
Acad. Sci. Paris, Serie I 300 (1985), 605-608.
[EF] P. Etingof and I. Frenkel, Central extensions of current groups in two d*
*imensions, Com-
munications in Math Physics.
[G] I. Grojnowski, Representations of affine Hecke algebras (and affine quan*
*tum GLn) at
roots of unity, International Math. Research Notes 4 (1994), 215-217.
[G2] I. Grojnowski, Affinizing quantum algebras: from D-modules to K-theory, *
*Preprint, 1994.
[HKR] M. J. Hopkins, N. J. Kuhn and D. C. Ravenel, Generalised group character*
*s and complex
oriented cohomology theories, preprint.
[La] P. S. Landweber (Ed.), Elliptic curves and modular forms in algebraic to*
*pology, Springer
LNM 1326 (1988).
[Lo] E. Looijenga, Root systems and elliptic curves, Invent. Math 38 (1976), *
*17-32.
[Lu] G. Lusztig, Cuspidal local systems and graded Hecke algebras I, Inst. Ha*
*utes 'Etudes Sci.
Publ. Math 67 (1988), 145-202.
[M] H. Miller, The elliptic character and the Witten genus, Contemp. Math 96*
* (1989), 281-
289.
Yale University, New Haven, CT 06520