THE SIGMA ORIENTATION FOR ANALYTIC CIRCLEEQUIVARIANT ELLIPTIC
COHOMOLOGY
MATTHEW ANDO
Abstract.We construct a canonical Thom isomorphism for Toriented BO<8>b*
*undles in Tequivariant
analytic elliptic cohomology, which is natural under pullback of vector *
*bundles and exponential under
Whitney sum. It extends in the rational case the nonequivariant sigma or*
*ientation of Hopkins, Strickland,
and the author. The construction relates the sigma orientation to the rep*
*resentation theory of loop groups
and Looijenga's weighted projective space, and sheds light even on the no*
*nequivariant case. Rigidity
theorems of WittenBottTaubes including generalizations by Kefeng Liu fo*
*llow.
Contents
1. Introduction *
* 1
2. Notation *
* 4
3. Principal bundles with an action of the circle *
* 6
4. Cohomology *
* 9
5. Rational cohomology of principal bundles with compact connected structure*
* group 11
6. Degreefour characteristic classes and theta functions *
* 14
7. Equivariant elliptic cohomology *
* 22
8. Equivariant elliptic cohomology of Thom spaces *
* 24
9. A Thom class *
* 26
10. The sigma orientation *
* 30
11. A conceptual construction of the equivariant sigma orientation *
* 31
References *
* 36
1. Introduction
1.1. Summary. Let E be an even periodic, homotopy commutative ring spectrum, le*
*t C be an elliptic curve
over SE = specß0E, and let t be an isomorphism of formal groups
t : bC~=spfE0(CP1 ),
so that (E, C, t) is an elliptic spectrum. In [AHS01], to which we refer the re*
*ader for the terminology in this
paragraph, Hopkins, Strickland, and the author construct a canonical map of hom*
*otopy commutative ring
spectra
MU<6> oe(E,C,t)!E
(or even, conjecturally, MO<8> ! E) called the sigma orientation.
Let T be the circle group. We expect that the sigma orientation has an equiva*
*riant extension
(Tequivariant MO<8>) oeT(ET,C,t)!ET, (1*
*.1)
to a multiplicative map of Tequivariant spectra. Note however that the constru*
*ction of oeT would require us
among other things to say what Tspectra we have in mind for the domain and cod*
*omain.
___________
Date: Version 5.17, January 2002.
Supported by NSF grant DMS_0071482.
1
2 MATTHEW ANDO
Geometrically, a map oeT as in (1.1)would be equivalent to specifying, for ea*
*ch virtual TBO<8> vector
bundle V (whatever that means) over a Tspace X, a trivialization fl(V ) of ET(*
*XV ) as an ET(X)module.
The trivialization would be natural, in the sense that
fl(f*V ) = f*fl(V )
if f : X0! X is a map of Tspaces, and exponential, in the sense that
fl(V V 0) = fl(V ) fl(V 0)
under the isomorphism 0 0
ET(XV V) ~=ET(XV ) ET(XV ).
ET(X)
One expects that equivariant elliptic cohomology comes with a completion isomor*
*phism
ET(XV )^ ~=E((XT)VT), (1*
*.2)
and that this isomorphism carries fl(V ) to the Borelequivariant sigma orienta*
*tion of V . (The notation XT
will denote the Borel construction ET xT X.)
In this paper we carry out this program for the complexanalytic equivariant *
*elliptic cohomology of
Grojnowski; along the way we gain some insight into what is meant by the domain*
* and codomain in (1.1).
Let A1anbe a lattice in the complex plane, and let C be the analytic variet*
*y C = A1an= . Grojnowski
defines a functor ET from finite TCW complexes to sheaves of Z=2graded OCalg*
*ebras, equipped with a
natural isomorphism (1.2)([Gro94]; for a published account see [Ros01]). We rec*
*all that the Weierstrass
sigma function oe = oe(z, ) gives rise to a map of ring spectra
MSO !HC
whose restriction to MU<6> is the sigma orientation for the elliptic spectrum a*
*ssociated to the elliptic curve
C (see Example 5.7 and [AHS01, x2.7]).
If V is a Tequivariant vector bundle over X, then we write ET(V ) for the re*
*duced equivariant elliptic
cohomology of its Thom space; it is an ET(X)module. If V is a spin bundle, the*
*n ET(V ) is an invertible
ET(X)module. More generally, if V = V0 V1 is a virtual spin bundle, then we m*
*ay define
ET(V ) = ET(V0) ET(V1)1.
ET(X)
([Ros01, AB00]; see x8).
Definition 1.3. Let V be a Tvector bundle over X. A Torientation on V is a ch*
*oice of orientation on
the fixed subbundle WA for each closed subgroup A of T. A Toriented vector bu*
*ndle is a Tvector bundle
equipped with a Torientation. The Toriented vector bundles over X form a mon*
*oid under direct sum,
whose Grothendieck group is the group of Toriented virtual bundles over X.
(It is a result of Bott and Samelson ([BS58] see [BT89], Lemma 3.7, and x6.4)*
* that a Tequivariant spin
bundle is Torientable.)
Theorem 1.4 (10.1). Let V be a Toriented virtual bundle over X, with the prope*
*rty that
w2(VT) = 0
(1*
*.5)
c2(VT) = 0.
(Here c22 H4(BSpin, Z) is the generator whose restriction to BSU is the second *
*Chern class.) Then there
is a canonical trivialization fl(V ) of ET(V ), whose value in ET(V )0is (VT).*
* Moreover we have
fl(V V 0) = fl(V ) fl(V 0)
under the isomorphism
ET(V V 0)~=ET(V ) ET(X)ET(V 0).
It is natural in the sense that
fl(f*V ) = f*fl(V ),
if f : X0! X is a map of Tspaces.
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 3
Already in [AB00], Maria Basterra and the author showed that, under the hypot*
*heses of the Theorem,
there is a global section fl(V ) of ET(V ), whose value in the stalk at the ori*
*gin of C is the Borelequivariant
sigma orientation of V ; earlier Rosu [Ros01] did the same for the orientation *
*associated to the Euler formal
group law. However, those papers do not address the trivialization, exponential*
*, and naturality properties
of the classes they construct.
The naturality is particularly hard to discern. Both of the papers [Ros01, AB*
*00] closely follow [BT89] in
their implementation of the "transferä rgument, and so all three papers depend*
* on meticulous choices for
the characters of the action of T on V XT and of T[n] on V T[n]and for the or*
*ientations of V Tand V T[n],
along with surprising and not particularly intuitive results concerning the com*
*patibility of these choices. In
this paper we eliminate all of those choices, and show that any choice will do,*
* with no effect on the resulting
Thom class, which is completely determined by the choices of orientations of th*
*e bundles V T[n]and V T. The
argument is indifferent to the parity of n.
From a practical point of view there are two important new ingredients. The f*
*irst is a careful account of
the choice of characters (rotation numbers) for the action of A T on the vect*
*or bundle V XA . This makes
it easy to study the effect on our constructions of varying the characters and *
*for that matter the subgroup
A. The second is a systematic use of the geometry of the affine Weyl group of G*
*, and its associated theory
of theta functions [Loo76].
More important than any single practical improvements was the conceptual prog*
*ress in understanding the
relationship between the sigma orientation and Looijenga's work on root systems*
* and elliptic curves. Let
G = Spin(2d) with maximal torus T and Weyl group W. Let ~T= hom(T, T) be the la*
*ttice of cocharacters.
Let VC be the complex abelian variety
VC = ~T C.
Looijenga constructs a line bundle A over (VC=W) (see also x6.2). The sigma fun*
*ction oe defines a global
holomorphic section oed of A, whose zeroes define an ideal sheaf I on VC, such *
*that oed is an trivialization of
A I.
The idea, which we discuss in more detail as Principle X in x11, is that if V*
* is a Tequivariant Gvector
bundle over a Tspace X, and if w2(VT) = 0, then A I defines an invertible sh*
*eaf of ET(X)modules
A(V ) I(V ). This sheaf has the properties
(1)there is a canonical isomorphism
I(V ) ~=ET(V )
of sheaves of ET(X)modules
(2)the ET(X)module A(V ) is trivial precisely when c2(VT) = 0.
Thus in the case that c2(VT) = 0, the section oedgives a trivialization of ET(V*
* ) which is the sigma orientation.
The letter A was chosen to stand for ä nomaly".
Early versions of this paper were attempts to prove Principle X and then to d*
*educe Theorem 1.4 from it.
However, for reasons we explain in x11, we were only occasionally able to convi*
*nce even ourselves of those
proofs. Eventually, detailed consideration of the consequences of Principle X l*
*ed us to the formulae in x6.4
and so to concrete proofs.
In x11 we do establish Principle X for a functor which captures the behavior *
*of the stalks of Grojnowski's
functor. It was also inspired by Greenlees's rational Tequivariant elliptic sp*
*ectra [Gre01] and by Hopkins's
work on characters in elliptic cohomology [Hop89]. Indeed we suspect that Green*
*lees's rational Tequivariant
elliptic spectra will admit a very natural proof of Principle X, and so give an*
* account of the rational circle
equivariant sigma orientation. Likely this will involve recasting the arguments*
* in this paper about the sigma
function and the affine Weyl group in terms of the algebraic theory of theta fu*
*nctions. We hope to turn to
those problems in the near future.
1.2. Plan. The plan of the paper is the following. We begin in x3 with a study *
*of the structure of principal G
bundles with an action of the circle. In x4 we discuss complexorientable cohom*
*ology theories in general and
ordinary and elliptic cohomology theories in particular. In x5 we interpret the*
* analysis of x3 in the presence
of a (rational) complexorientable cohomology theory. We begin x6 with an inter*
*lude (x6.1) on degreefour
characteristic classes. In x6.2 we recall a result essentially due to [Loo76], *
*that a degreefour characteristic
class , 2 H4(BG; Z) gives rise to a Wequivariant line bundle L(,) over (T~ C)*
*. We define a theta function
4 MATTHEW ANDO
of level , for G to be a Winvariant holomorphic section of the line bundle L(,*
*); the sigma function provides
the most important examples for us, and so we discuss it in x6.3.
Section 6.4 is the heart of the paper. In it we use the results of x3_6.3 to *
*construct some holomorphic
characteristic classes for Tequivariant principal Gbundles which are the buil*
*ding blocks of the Thom classes
in x9 and x10.
In x7 we recall the construction of Grojnowski's analytic Tequivariant ellip*
*tic cohomology associated to
a lattice A1an. In x8 we review the equivariant elliptic cohomology of Thom*
* complexes [Ros01, AB00],
recalling what is involved in constructing a global section of ET(V ), where V *
*is (virtual) Tequivariant Spin
vector bundle.
In x10 we construct the equivariant sigma orientation, proving Theorem 1.4. I*
*n x9 we prove the following
related result. Let G be a spinor group, let G0be a connected compact Lie group*
*. Let V be a Tequivariant
Gbundle over X, and let V 0be a Tequivariant G0bundle. Suppose that VT is a *
*Gbundle, and that VT0is
a G0bundle, over XT. Suppose that ,0is a degreefour characteristic classes fo*
*r G0, with the property that
c2(VT) = ,0(VT0).
Suppose that `0is a theta function for G0of level ,0.
Theorem 1.6 (9.5). A Torientation on V determines a canonical global section f*
*l of ET(V )1, whose value
in ET(V )10g is `0(VT0) (VT)1.
In particular, suppose that V 0is a Spin(2d0) Tvector bundle over X, and V i*
*s a Spin(2d) Tvector bundle.
If
w2(VT0) = 0 = w2(VT),
then the Borel constructions are again spinor bundles. Suppose that `0is the ch*
*aracter of a representation
of LSpin(2d0) of level k: then it is a theta function of level kc2 for Spin(2d0*
*). If
c2(VT) = kc2(VT0),
then Theorem 1.6 gives a global section fl of ET(V )1. The PontrjaginThom co*
*nstruction for the map
ß : X ! * gives map
(ETß)*ET(V )1 ! ET(*) = OC
of OCmodules which takes fl to the equivariant Witten genus of V twisted by th*
*e characteristic class `(VT0).
Since the global sections of OC are the constants, we have the following result*
* of [Liu96].
Corollary 1.7. Under these conditions, the equivariant Witten genus of V twiste*
*d by `(VT0) is constant.
If x6.4 is the heart of the paper, then x11 is the soul. There we discuss the*
* conceptual framework which led
to the results in this paper. We give a conceptual construction of the equivari*
*ant sigma orientation, which
even illuminates the nonequivariant case. Because the characters of representa*
*tions of the loop group LG
are sections of the line bundle A, it illuminates the relationship between the *
*sigma orientation, equivariant
elliptic cohomology, and representations of loop groups which was studied in [A*
*nd00].
1.3. Acknowledgments. My first debt is to Maria Basterra. Our work on [AB00] le*
*d directly to the results
in this paper, and I have very much enjoyed and benefitted from our collaborati*
*on. Maria was to have been
an author of this paper as well, and I have reluctantly accepted her request to*
* withdraw her name from it.
I thank Alejandro Adem for inviting me to visit Madison; it was during the eigh*
*thour roundtrip drive that
the the function F (6.15)was discovered, along with its properties as described*
* in x6.4. John Greenlees,
Haynes Miller, and Jack Morava offered useful comments and advice. I thank Hayn*
*es Miller, Jack Morava,
and Amnon Neeman for encouraging me to record the ideas in x11; I hope they do *
*not regret the result. My
work on elliptic cohomology has been profoundly influenced by Mike Hopkins, and*
* I am grateful to him for
his work on the subject and his generosity to me in particular.
2.Notation
2.1. Abelian groups. Let C be a category with finite products. The category AC *
*of abelian groups in C is
an additive category. In fact AC is tensored over the subcategory of finitely g*
*enerated free abelian groups.
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 5
That is, a finitely generated free abelian group F and an abelian group X of C *
*determine (naturally in F
and X) an object F X of AC, with a natural isomorphism
AC[F X, Y ] ~=(abelian groups)[F, AC[X, Y ]].
If A is an abelian group written additively, and M is an abelian group writte*
*n multiplicatively, then we
write ma for the element a m of A M. Similarly, if M0is an abelian group, u 2 M*
*0, and m 2 hom(M0, M)
then we may write um for m(u).
2.2. Lie groups and the group VX. In general, the letter G will stand for a com*
*pact Lie group with
maximal torus T with Weyl group W. We define
~Tdef=hom[T, T]
^Tdef=hom[T, T]
to be the lattices of cocharacters and characters. We write
c : G ! Aut(G)
for the action of G on itself by conjugation:
cgh = ghg1.
If X is an abelian group in any category, then we write VX for the tensor pro*
*duct
VX def=~T X,
which carries an action of the Weyl group W. If r is the rank of G, then VX is *
*isomorphic to Xr.
2.3. Elliptic curves. Fix ø in the complex upper half plane, and let be the l*
*attice
= 2ßiZ + 2ßiøZ.
We write Ganafor the analytic affine plane A1anwith its additive structure of a*
*belian topological group, and
we write z for the standard coordinate on Ga and also on A1, A1an, Gana, etc. W*
*e write Ganmfor the punctured
plane (A1an)x with its multiplicative group structure. For r 2 Q we write ur= e*
*rzand qr = e2ßirø.
Let C be the elliptic curve
C = Gana= ~=Ganm=qZ.
We write " for the covering map
Gana"!C.
If V is an open set in a complex analytic variety, then we write OV for the she*
*af of holomorphic functions
on V .
If A is an abelian topological group and a 2 A, when we write øa for the tran*
*slation map; and if V G
is an open set, then we write
V  g def=øg(V ).
Definition 2.1. An open set U of C is small if it is connected and "1U is a un*
*ion of connected components
V with the property that
"V : V ! U
is an isomorphism.
If U is small and V is a component of "1U, then the covering map induces an *
*isomorphism
OU ~=OV.
In particular, if U contains the origin of C, then there is a unique component *
*V of "1U containing 0. This
determines a C[z]algebra structure on OU, and a C[z, z1] structure on OUU\0.
6 MATTHEW ANDO
2.4. Ringed spaces. GinzburgKapranovVasserot [GKV95 ] have proposed that the *
*Tequivariant elliptic
cohomology associated to an elliptic curve C should be a covariant functor
Egkv: (Tspaces) ! (schemes)=C.
Grojnowski's Tequivariant elliptic cohomology is a contravariant functor
ET : (finite TCW complexes) ! (Z=2graded OCalgebras) (2*
*.2)
(see x7 and [Gro94, Ros01, AB00]). These are meant to be related by the formula
ET(X) = f*OEgkv(X), (2*
*.3)
where
f : Egkv(X) ! C
is the structural map. However, Grojnowksi's functor can not quite be of the fo*
*rm (2.3), since in (2.2)OC
is the sheaf of holomorphic functions on the analytic space C = Gana= . However*
*, it does give a covariant
functor
(Tspaces) ! (ringed spaces)=C.
Precisely, we have the following.
Definition 2.4. By a (super, or Z=2graded) ringed space we shall mean a pair (*
*X, OX ) consisting of a
space X and a sheaf OX of Z=2graded rings on X. A map of ringed spaces
f = (f1, f2) : (X, OX ) ! (Y, OY)
consists of a map of spaces f1: X ! Y and a map of sheaves of Z=2graded algebr*
*as over Y
f2: OY ! (f1)*OX .
The resulting category of ringed spaces will be denoted R. If X = (X, OX ) is a*
* ringed space and U is an
open set of X, then we may write X(U) in place of OX (U).
If X is a finite TCW complex, then
XET = (C, ET(X))
is a ringed space, and this defines a covariant functor
(  )ET : (TCW complexes) ! R=C.
We have found this point of view to be extremely helpful, and so we have adopte*
*d it in writing this paper.
3. Principal bundles with an action of the circle
3.1. Abundles over trivial Aspaces. Let A be a closed subgroup of the circle *
*T. Let G be connected
compact Lie group, and let ß : Q ! Y be a principal Gbundle over a connected s*
*pace Y . Suppose that A
acts on Q=Y , fixing Y . The group of automorphisms of Q=Y is the ((Q xG,cG)=Y*
* ), and an action of A
on Q=Y is equivalent to a section a 2 ((Q xG,chom[A, G])=Y ).
Every Gorbit in hom[A, G] intersects hom[A, T] nontrivially, and so for x 2 *
*Y we may choose a homo
morphism mx 2 hom[A, T] such that
a(x) = [p, mx]
for some p 2 Q; the square brackets indicate the class in the Borel constructio*
*n of the element (p, mx) 2
Q x hom[A, G]. The choice of mx determines p only up to the centralizer Z(mx) i*
*n G of the homomorphism
mx. Since hom[A, T] is discrete, we may choose mx to be constant on Y .
Definition 3.1. A reduction of the action of A on Q is a homomorphism
m : A ! T
such that, for all x 2 Y , there is a p 2 Q such that
a(x) = [p, m].
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 7
The terminology is justified by the following observation. Let
Q(m) = {p 2 Q[p, m] = a(ß(p))}.
Then ßQ(m): Q(m) ! Y is a principal Z(m) bundle over Y , the reduction of the *
*structure group of Q to
Z(m). In other words, we have given a factorization
BZ(m);; (3*
*.2)
Q(m)xxxx
xxx 
xx fflffl
Y ___Q_//_BG.
Let
W(m) = {w 2 Wwm = m}
be the stabilizer of m. One sees that T is a maximal torus of Z(m), with Weyl g*
*roup W(m).
Example 3.3. Suppose that G = U(n) is the unitary group with its maximal torus *
*T = (z1, . .,.zn) of
diagonal matrices, and suppose that A = T. Every homomorphism
m : T ! T
is conjugate in U(n) to one of the form
m(z) = (zm1, . .,.zm1, zm2, . .,.zm2, . .,.zmr, . .,.zmr),
where mi2 Z. Let dj be the multiplicity of mj; then the centralizer is the bloc*
*kdiagonal matrix
2 3
U(d1)
Z(m) = 4 . . . 5,
U(dr)
with Weyl group
W(m) = d1x . .x. dr n = W.
If V is a Tequivariant complex vector bundle over a connected trivial Tspace *
*Y , and if this m is a reduction
of the action of T on the principal bundle of V , then the reduction of the str*
*ucture group to Z(m) corresponds
to the decomposition of V as the direct sum
V ~=V (m1) . . .V (mr),
where T acts on the fiber of V (mj) by the character zmj.
The composition
g(m) : A x Z(m) mxZ(m)!T x Z(m) ! Z(m)
is a group homomorphism, and so we have a map
BA x BZ(m) Bg(m)!BZ(m).
The following Lemma will be used directly to prove Lemma 6.5, a calculation o*
*f degreefour characteristic
classes. Moreover, the algebrogeometric form of this diagram after applying a *
*complexorientable cohomol
ogy theory (see Lemma 5.1) captures the essential point of the "transfer argume*
*ntö f [BT89], as we explain
in Remark 11.7.
Lemma 3.4. (1)The diagram
BT6x6BTMMn
BmxBTnnnnn MMM
nnnn MMMM
nn M&&M
BA x BT BT
 
 
fflffl Bg(m) fflffl
BA x BZ(m)_________________//BZ(m)
commutes.
8 MATTHEW ANDO
(2)The map Bg(m) classifies the principal Z(m)bundle EZ(m)A = EAxA,mEZ(m) o*
*ver BAxBZ(m).
Proof.The first part is easy. For the second part, it suffices to construct a m*
*ap of principal Z(m)bundles
over BA x BZ(m). The map
Eg(m) : EA x EZ(m) ! EZ(m)
factors through EA xA,mEZ(m), and gives the desired map. (Note that the map Eg(*
*m) is obtained by
constructing EA and EZ(m) functorially as spaces with actions on the same side,*
* say the left. In forming
EA xA,mEZ(m), one makes A act on the right of EA by the inverse.)
If m0: A ! T is another reduction of the action of A on Q, then m and m0diffe*
*r by conjugation in G,
and we have for some g in G a commutative diagram
Q(m0) 0
Y___________//_GGBZ(m;); (3*
*.5)
 GGGG wwww 
 GGG wwww 
Q(m) wGGGGw 
 cg www GGG 
 www GGG 
fflfflww Q ##Gfflffl
BZ(m) ___________//BG.
3.2. The case of a connected centralizer.
Lemma 3.6. If the centralizer Z(m) is connected, then the element g in the diag*
*ram (3.5)may be taken to
be in the normalizer NGT of T in G.
Proof.Let g 2 G be such that
cgm = m0
Then m0(A) cg(T) \ T, so both T and cg(T) are maximal tori in Z(m0). Since Z(*
*m0) is connected, there
is an element h 2 Z(m0) such that
chcg(T) = T,
so hg 2 NGT. Since h 2 Z(m0), we have
m0= chm0= chcgm.
Example 3.3 shows that Z(m) is connected if G is a unitary group. Bott and S*
*amelson have shown
([BS58]; see Proposition 10.2 of [BT89]) that Z(m) is connected if G is a spino*
*r group.
Lemma 3.7. If G is simple and simply connected then the centralizer Z(m) is con*
*nected.
3.3. Nested fixedpoint sets. Now suppose that B A is a larger closed subgrou*
*p of T (primarily, we
shall be interested in the case that B = T), and that the bundle Q=Y is actuall*
*y an equivariant Bbundle
over Y .
Lemma 3.8. If
m : A ! T
is a reduction of the action of A on Y , then the action of B on Q induces on Q*
*(m) the structure of a
Bequivariant Z(m)bundle over Y . If moreover the Borel construction QB is a p*
*rincipal Gbundle over YB,
then the Borel construction Q(m)B is a principal Z(m)bundle over YB.
Proof.This follows from the fact that B is abelian.
If F Y Bis a connected component of the subspace of Y fixed by the action o*
*f B, then we may choose
a reduction
mF : B ! T
of the action of B on QF.
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 9
Lemma 3.9. The restriction
mFA : A ! T
is a reduction of the action of A on Y .
Proof.The action of A on Q=Y is a section a of (Q xchom[A, G])=Y . The restrict*
*ion aF records the action
of A on QF. The action of B on QF is a section b of (QF xchom[B, G])=F, and *
*with the obvious notation
we have
bA = aF.
4.Cohomology
4.1. Ordinary cohomology. If R is a commutative ring, let HR denote ordinary co*
*homology with coeffi
cients in R. If X is a space, then we write
XHR def=spec(HR2*(X))
for scheme over specR associated to the cohomology of X. We may also view XHR a*
*s the ringed space with
underlying space spec(HR2*(X)) and structure sheaf associated to
HReven(X) HRodd(X).
We shall write H for HC, cohomology with complex coefficients.
4.2. Equivariant cohomology. If X is a space with a circle action, then (XT)HR *
*is a scheme over BTHR ,
which we denote XTHR.. We choose a generator of the character group of T, and w*
*rite z for the resulting
generator of HZ2(BT); this gives an isomorphism
BTHR ~=(Ga)R (4*
*.1)
of group schemes over specR. We shall use (4.1)to view XTHR as a scheme over A1*
*R.
We recall [Qui71] that equivariant cohomology satisfies a localization theore*
*m.
Theorem 4.2. If X has the homotopy type of a finite TCW complex (e.g. if X is *
*a compact Tmanifold),
then the natural map
XTTH!XTH
induces an isomorphism over specC[z, z1] BTH .
Holomorphic cohomology. Let A1anbe the analytic complex plane, so OA1anis the s*
*heaf of holomorphic
functions on C. Because of the natural maps
A1an! A1C! A1~=BTHZ
we may view z as a function on A1an. Given a Tspace X we define the holomorphi*
*c cohomology of X to be
the sheaf of super OA1analgebras given by
H(X; U) def=HC(XT) OA1an(U).
C[z]
We view H(X) as the structure sheaf of a ringed space (2.4)XH over A1an, namely*
* the the pullback in the
diagram
XH ! XTH ! XTHZ
?? ? ?
y ?y ?y
A1an! A1C ! A1Z.
The stalk of holomorphic cohomology. Let
A1an,0= spec(OA1an,0)
be the local scheme associated to the stalk of OA1anat the origin. The stalk of*
* H(X) at the origin is
H(X)0~=H(XT) (OA1an,0).
C[z]
We write XTH,0for the resulting scheme over A1an,0.
10 MATTHEW ANDO
Periodic Borel cohomology. Let ^Hdenote periodic ordinary cohomology with compl*
*ex coefficients: that is,
`
^H= 2kH
k2Z
so
ß*^H= C[v, v1]
with v 2 ß2^H. Then
H^0(BT) ~=C[[z]]
so spf^H0(BT) = (A1an)^0= (A1C)^0, and ^H0(XT) is the ring of formal functions *
*on the pullback XTH^in the
diagram of formal schemes
XTH^ ! XTH,0
?? ?
y ?y
(A1an)^0!A1an,0.
Combining all these, we have a collection of forms of ordinary cohomology
XTH^ ! XTH,0 ! XH ! XTH ! XTHZ
?? ? ? ? ?
y ?y ?y ?y ?y (4*
*.3)
(A1C)^0!A1an,0!A1an! A1C! A1Z.
4.3. Generalized cohomology.
Even periodic ring spectra. A ring spectrum E will be called "even periodic" if*
* ßoddE = 0 and ß2E contains
a unit of ß*E.
If E is an even periodic ring spectrum, and if X is a space, then we shall wr*
*ite E*X for the unreduced
cohomology of X. As in [AHS01], we write XE for the formal scheme
XE = colimF XspecE0F
over SE = specE0(*); the colimit is over the compact subsets of X.
An even periodic ring spectrum is always complexorientable. In particular
PE def=BTE
is a (commutative, onedimensional) formal group over SE.
Borel cohomology. If E is an even periodic ring spectrum, and X is a space with*
* a circle action, then we
write
XTE def=(XT)E^
for the formal scheme associated to the even Borel ^Etheory of X. The natural *
*map
XT ! BT
induces a map
XTE ! PE.
For example, let HPZ denote periodic ordinary cohomology with integer coefficie*
*nts. Then PHPZ = bGa.
Elliptic spectra. We recall [AHS01] that an elliptic spectrum is a triple (E, C*
*, t) consisting of
(1)an even periodic ring spectrum E,
(2)a (generalized) elliptic curve C over SE, and
(3)an isomorphism of formal groups
t : bC~=PE.
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 11
Rational elliptic spectra. Let C be an elliptic curve over a Qalgebra R, and l*
*et !_= 0* 1C=R. If ! is a
trivialization of !_, then there is a canonical isomorphism of formal groups ov*
*er R
Cblog!!bG
~= a
with the property that 0*log*!dz = 0*dz.
Let x(!_) be the functor from rings to sets which given by
x(!_)(T) = {(j, !)  j : specT ! specR, ! a trivializatonjof*!*
*_}.
If ! is a trivialization of !_, then the trivialization of j*!_are of the form *
*vj*! for u 2 Tx , so
x(!_) ~=specR[u, u1].
Let S = O( x(!_)).
The logarithm then gives a canonical isomorphism of formal groups
bCSlogbC!(bGa)S
by the formula
(c, !) 7! (log!c, !).
If HS[v, v1] is the even periodic ring spectrum such that
(HS[v, v1])*X def=H*(X; S[v, v1])
then
PHS[v,v1]= (bGa)S,
and we have an elliptic spectrum
(HS[v, v1], C, logbC).
Alternatively, given over R a trivialization ! of !_we have the elliptic spectr*
*um (HR[v, v1], C, log!).
Complex elliptic spectra. The projection " : Gana! C induces an isomorphism of *
*formal groups
b": bGa! bC.
There is a unique cotangent vector ! such that
"*! = 0*dz.
We have
log!= ("b)1,
and so an elliptic spectrum
(H^, C, b"1) = (H^, C, log!). (4*
*.4)
5.Rational cohomology of principal bundles with compact connected structure g*
*roup
Let E be an rational even periodic ring spectrum. Let G be a connected compac*
*t Lie group. Let T be a
maximal torus of G, with Weyl group W. The natural isomorphism
~T T ! T
induces a Wequivariant isomorphism
VPE ~=BTE
of formal groups over SE. Moreover [Bor55] the natural map
BTE=W ! BGE
is an isomorphism. We shall repeatedly use the resulting isomorphism
BGE ~=VPE=W.
For example, a principal Gbundle Q over X is classified by a map
X Q!BG
12 MATTHEW ANDO
whose effect in Etheory
XE QE!VPE=W
is an XEvalued point of VPE=W.
5.1. Periodic cohomology of circleequivariant principal bundles. We interpret *
*the analysis of x3
in Etheory. Suppose that G is a connected compact Lie group, and that Q is a T*
*equivariant principal
Gbundle over a connected Tspace Y . Suppose that a closed subgroup A of the c*
*ircle acts trivially on Y .
Suppose that
m : A ! T
is a reduction of the action of A on Q=Y with connected centralizer Z(m).
Applying Ecohomology to the diagram (3.2)yields
VPE=W(m)99
Q(m)Essss 
sss 
sss fflffl
YE __QE_//_VPE=W
The multiplication
BT x BT ! BT
induces the addition map
VPE x VPE +!VPE
whose restriction to
(VPE)W(m)x VPE +!VPE
factors to give a translation map
(VPE)W(m)x (VPE)=W(m) +!(VPE)=W(m).
In E cohomology, Lemma 3.4 implies the following result. It is used in the case*
* A = T to construct the
commutative diagram (9.2)and so prove Lemma 9.4. It is also implies the commuta*
*tivity of the diagram
(11.6), which captures the essence of the "transfer formulaö f BottTaubes [BT*
*89]; see Remark 11.7.
Lemma 5.1. The diagram
~=
BAE x BZ(m)EO____//_(VPE)W(m)xO(VPE)=W(m)
BAExQ(m)E +
 Q(m)AE fflffl
BAE x YE____________//(VPE)=W(m)
commutes. If moreover the Borel construction QT is a principal Gbundle over YT*
*, then the diagram
BAE^x YE^_Q(m)AE__//_(VPE)=W(m)55
jjj
 jjjj
 jjjjj
fflfflQ(m)TEjjjjj
YTE
commutes.
5.2. Holomorphic characteristic classes. Let f be a Winvariant holomorphic fun*
*ction on VGana. By
the splitting principle, the Taylor expansion of f at the origin defines a clas*
*s in ^H(BG). Suppose that Q is a
principal Gbundle over X, with the property that QT is a principal Gbundle ov*
*er XT. Then we get a class
f(QT) 2 ^H(XT).
The following result is due to Rosu.
Lemma 5.2. The class f(QT) is in fact an element of H(X; A1an). Similarly, if f*
* 2 (OVGana)0, then
f(QT) 2 H(X)0.
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 13
Proof.Proposition A.6 of [Ros01] proves this result in the case that G = U(n) a*
*nd
Y
f(z) = g(zj),
j
where g 2 OA1an,0and z = (z1, . .,.zn) 2 Anan~=VGana. The same argument works i*
*n the indicated generality.
An important example of a holomorphic characteristic class is the euler class*
* associated to a üm ltiplicative
analytic orientation". A power series
f(z) = z + higher terms2 ^H(BT) ~=C[[z]]
satisfying
f(z) = f(z)
determines a multiplicative orientation (map of homotopy commutative ring spect*
*ra)
OE : MSO ! ^H,
characterized by the property that if
V = L1+ . .+.Ld
is a sum of complex line bundles, then its euler class is
Y
eOE(V ) = f(c1Lj).
j
If V is an oriented vector bundle, we write OE(V ) 2 ^H(V ) = ^H(XV ) for the*
* resulting Thom class. It is
multiplicative in the sense that
OE(V W) = OE(V ) ^ OE(W) (5*
*.3)
under the isomorphism
(X x Y )V W~=XV ^ Y W.
Definition 5.4. The orientation OE is analytic if f is contained in the subring*
* OA1an,0 C[[z]] of germs of
holomorphic functions at 0; equivalently, if there is a neighborhood U of 0 in *
*C on which the power series f
converges to a holomorphic function.
Lemma 5.2 implies the following.
Corollary 5.5 ([Ros01]). If OE is analytic and V is an oriented Tvector bundle*
* over a compact Tspace X,
then the euler class eOEassociated to OE satisfies
eOE(VT) 2 H(X)0.
If denotes the standard Thom isomorphism, then
OE(VT) = eOE(VT)_e (VT),
(VT)
and the ratio of euler classes is a unit in H(X)0. Of course multiplication by *
* (VT) induces an isomorphism
H(XT) ~=H(VT),
and so we have the following.
Corollary 5.6. There is a neighborhood U of the origin in A1ansuch that
OE(VT) 2 H(V ; U),
and such that multiplication by this class induces an isomorphism of sheaves
H(X)U OE!~=H(V )U.
In other words, for every open set U0 U, multiplication by OE(VT) induces an i*
*somorphism
H(X; U0) OE!~=H(V ; U0).
14 MATTHEW ANDO
Example 5.7. For example, let oe = oe(u, q) denote the expression
Y (1  qnu)(1  qnu1)
oe = (u1_2 u1_2) ________________n2. (5*
*.8)
n 1 (1  q )
This may be considered as an element of Z[[q]][u 1_2] which is a holomorphic fu*
*nction of (u1_2, q) 2 Cx x D,
where D = {q 2 C0 < q < 1}. Let H = {ø 2 C=ø > 0} be the open upper half pl*
*ane. We may consider oe
as a holomorphic function of (z, ø) 2 A1anx H by setting
ur= erz
qr= e2ßirø
for r 2 Q. It is easy to check using (5.8)that
oe(z)= oe(z) (5.*
*9a)
oe(z)= z + o(z2) (5.*
*9b)
_2_
oe(uqn)= (1)nunqn2oe(u). (5.*
*9c)
The equations (5.9)imply that the Taylor expansion of oe at the origin defines *
*a multiplicative analytic
orientation
MSO !H^. (5.*
*10)
Definition 5.11. If V is an oriented vector bundle, we write (V ) for the Thom*
* class associated to the
orientation (5.10), and oe(V ) for the associated euler class.
In [AHS01, x2.7] is it shown that is the sigma orientation associated to th*
*e elliptic curve C: that is, the
diagram
MSOOO____//_^H<
commutes.
6.Degreefour characteristic classes and theta functions
6.1. Degreefour characteristic classes. If G is a connected compact Lie group,*
* then by the splitting
principle the natural maps
H2(BG, Z)! H2(BT, Z)W ~=^TW
(6*
*.1)
H4(BG, Z)! H4(BT, Z)W ~=(S2^T)W ~=hom( 2~T, Z)W
are rational isomorphisms. Here, if M is an abelian group, then S2M and 2M den*
*ote degreetwo parts of
the symmetric and divided power algebras on M.
Without rationalizing, a degreefour characteristic class , 2 H4(BG, Z) gives*
* rise to a homomorphism
2~TI!Z.
We shall abuse notation and also write I for the bilinear map
T~x ~Tfl1xfl1! 2~TI!Z. (6*
*.2)
We shall say that the characteristic class , is positive definite if the pairin*
*g I is so. We also write OE for the
quadratic function given by the composition
2~T_I_//_ZOO>>"
""
fl2"""
"""OE
~T
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 15
(If A and B are abelian groups, then a function
f : A ! B
is quadratic if
0 = f(0)
0 = f(x + y + z)  f(x + y)  f(x + z)  f(y + z) + f(x) + f(y) *
*+ f(z)
f(x)= f(x).
The function
A fl2! 2A
is the universal quadratic function out of A).
From the definitions it follows that
OE(a + b)= OE(a) + I(a, b) + OE(b)
OE(na)= n2OE(a)
(6*
*.3)
OE(wa)= OE(a)
I(wa, wb)= I(a, b)
for a, b 2 ~T, n 2 Z, and w 2 W.
There are a variety of ways to express the relationship between the character*
*istic class , and the map I.
For example, suppose that Q0 and Q1 are two principal Gbundles over X, given a*
*s maps
X Qi!BT.
Then we get a new principal Gbundle as the composition
X ! X x X Q0xQ1!BT x BT ~!BT.
The effect of Qiin cohomology
XHZ (Qi)HZ!BTHZ ~=VGa.
is an XHZvalued point of VGa. Equation (6.3)implies that
,(~(Q0x Q1)) = ,(Q0) + I((Q0)HZ, (Q1)HZ) + ,(Q1),
where we have extended I to a bilinear map
VGax VGa ! Ga.
As another example, suppose that A T is a closed subgroup with character gr*
*oup Z=nZ (n 0), and
suppose that m : A ! T is a homomorphism. If
__m, __m02 hom(T, T) = ~T
satisfy
__m __0
A = m A = m,
then
__m0= __m+ nffi
for some ffi 2 ~T, and equation (6.3)implies that
OE(__m0) OE(__m) mod n
We write OE(m) for the class of OE(__m) in Z=n. If z is the chosen generator of*
* HZ2BT, write also z for the
induced generator of HZ*BA ~=(Z=nZ)[z]. With these conventions
(Bm)*, = OE(m)z2. (6*
*.4)
We write ^Ifor the map
^I: ~T! ^T
which is the adjoint of (6.2). Note that we have
I^(wa)(wb) = I(wa, wb) = I(a, b) = ^I(a)(b).
16 MATTHEW ANDO
It follows that if a 2 ~TWthen
I^(a) 2 ^TW! H2(BG, Q).
defines a rational characteristic class of principal Gbundles, which we also d*
*enote ^I(a).
Now suppose that Q is a Tequivariant principal Gbundle over a connected tri*
*vial Tspace Y . Suppose
as usual that QT is a principal Gbundle over YT = BT x Y . Let m 2 ~Tbe a redu*
*ction of the action of T on
Q=Y . Then Q(m) is a principal Z(m)bundle over Y , and we have the following.
Lemma 6.5. In H4(YT, Q) = H4(BT x Y, Q) we have
,(QT) = OE(m)z2+ ^I(m)(Q(m))z + ,(Q).
Proof.By the splitting principle, we may suppose that we have a factorization
BT==
Q zzzz
zz 
zz fflffl
Y __Q_//_BG.
Lemma 3.4 implies that the map
BT x Y BmxQ!BT x BT ~!BT
classifies QT. It follows that
,(QT)= ,(~(Bm x Q)HQ)
= ,(Bm) + I((Bm)HQ, QHQ) + ,(Q)
= OE(m)z2+ ^I(m)(Q)z + ,(Q).
Example 6.6. Let TSO(2d)~=Td be the standard maximal torus in SO(2d) (the image*
* under the map U(d) !
SO(2d) of the torus of diagonal matrices), and let T be its preimage in Spin(2d*
*). If m = (m1, . .,.md) 2
Zd ~=(TSO(2d))_, then there is a lift in the diagram
;T;x
x 
x 
xx m fflffl
T ____//_TSO(2d)
P
precisely when miis even, that is
X
~T~={(m1, . .,.md) 2 Zd mieven}.
The function X
OE(m1, . .,.md) = 1_2 m2i (6*
*.7)
therefore defines a quadratic map
OE : ~T! Z
with associated bilinear form X
I(m, m0) = mim0i.
It is not hard to check using (6.7)that it is the quadratic form associated to *
*c22 HZ4(BSpin(2d)).
Now suppose that
V = L1 . . .Ld
is a Tequivariant Spin(2d) bundle, written as a sum of Tequivariant complex l*
*ine bundles, over a trivial
Tspace Y . Let xi= c1Li, and suppose that T acts on Liby the character mi. In *
*order that VT be a spin
bundle, we must have X
0 = w2(VT) = (miz + xi) (mod 2).
i
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 17
In that case, Lemma 6.5 says that
` X ' X
c2(VT) = 1_2 m2i z2+ ( mixi)z + c2V
in H4(Y x BT).
6.2. Theta functions. Recall that q = e2ßiø, and that C is the elliptic curve
C = Gana= ~=Ganm=qZ
Following [Loo76], we define a line bundle over VC by the formula
anx C
L = L(,) def=_____VGm___________^. (6*
*.8)
(u, ~) ~ (uqm , uI(m)qOE(m)~)
for m 2 ~T.
Remark 6.9. The identity map of VGanmx C induces for w 2 W an isomorphism of li*
*ne bundles
L ~=w*L
over VC, which is certainly compatible with the multiplication in W, so L desce*
*nds to a line bundle A(,)
on VC=W.
A theta function for G of level , is a Winvariant holomorphic section of L(,*
*).
Definition 6.10. A theta function for G of level , is a function
X
` = an(u)qn 2 (Z[T^])((q)) (6.*
*11)
n>>1
which for u = ez and q = e2ßiøis a holomorphic function of (z, ø) 2 VA1anx H, a*
*nd which satisfies
`(uqm )= u^I(m)qOE(m)`(u) (6.1*
*2a)
`(uw)= `(u) (6.1*
*2b)
for m 2 ~Tand w 2 W, where OE and I are the quadratic form and bilinear map ass*
*ociated to the characteristic
class ,.
Remark 6.13. There is a good deal of redundancy in the definition. Looijenga st*
*udies formal series of the
form (6.11)which transform according to (6.12). One has to be careful to identi*
*fy a group formal series
which is closed under the operations implied by (6.12). If , is positive defini*
*te, then every such formal theta
function defines a holomorphic function of (z, ø) (see [Loo76]).
6.3. The sigma function and the basic representation of LSpin(2d). If G is a si*
*mple and simply
connected Lie group, then there is a unique generator , of H4(BG; Z) ~=Z such t*
*hat the associated pairing
I is positive definite. If V is a representation of LG of level k in the sense *
*of [PS86], then its character Ø is
a theta function for G of level k, [Kac85]. The most important example for us i*
*s the basic representation of
LSpin(2d), whose character is a theta function associated to the characteristic*
* class c2 of Spin(2d). Up to
a factor, it is the euler class of the sigma orientation (5.11).
It is useful to be more explicit about this euler class. As in Example 6.6, l*
*et TSO(2d)be the image under
the map U(d) ! SO(2d) of the torus of diagonal matrices. For u = (u1, . .,.ud) *
*2 T write
dY
oed(u) = oe(ui). (6.*
*14)
i=1
The product expression (6.14)and the fact (5.9a)that oe is odd imply that oed(u*
*w) = oed(u) for w 2 W,
and so oed is a Winvariant function on VGana(with zeroes precisely at the poin*
*ts V ) and so by Lemma 5.2
defines a holomorphic characteristic class for oriented vector bundles of rank *
*2d. If V is such a vector bundle,
then oe(V ) = oed(V ).
18 MATTHEW ANDO
The fractional powers of u in the expression (5.8)for oe prevent oed from bei*
*ng a theta function for SO(2d),
but if G = Spin(2d) then the formula
0 1
Y 1_Y Y (1  qnui)(1  qnu1)
oed(u) = (1)d( ui)2 @ (1  ui) _________________ni2A
i i n 1 (1  q )
shows that oed 2 (Z[T^])[[q]]. The formula (5.9c)implies that, if I and OE are *
*the pairing and quadratic form
associated to the generator c22 H4(BG; Z) as in Example 6.6, then
oed(uqm ) = u^I(m)qOE(m)oed(u),
Q
so oed is a theta function for Spin(2d) of level c2. Up to the factor n(1  qn*
*)2d, it is the character of the
socalled äb sic" representation of LSpin(2d) [Kac85, PS86, Liu96].
6.4. A useful holomorphic characteristic class. Suppose that Q is a Tequivaria*
*nt principal Gbundle
over a connected Tspace Y . Suppose that T[n] T acts trivially on Y . Let
m : T[n] ! T
be a reduction of the action of T[n] on Q=Y .
Let , 2 H4(BG; Z) be a positive definite class, with associated quadratic for*
*m OE and bilinear map I. Let
` be a theta function of level , for G (Definition 6.10). Let C be the elliptic*
* curve Gana=(2ßiZ + 2ßiøZ). Let
a be a point of C of order n. Choose a point ~a2 Ganasuch that ä(~) = a.
We are going to define a holomorphic function
F = F(`, m, _a) 2 O(VGana)W(m),
and so a holomorphic characteristic class of principal Z(m)bundles (see x5.2).*
* To give a formula for F it is
convenient to define
A= e2ßi~a,
and recall that we have set
qr= e2ßirø
ur= erz
for r 2 Q.
Since na = 0 in C, there are unique integers ` and k such that
n~a= 2ßi` + 2ßiøk.
We choose an extension ~mmaking the diagram
T`O~m`//`TO==_
 ___
 _m_
OO__
T[n]
commute. With these choices, the formula for F is
__m)k_OE(__m)_m
F(z) = uk_n^I(An `(uA ). (6.*
*15)
Remark 6.16. The factors preceding the ` are closely related to the line bundle*
* V 1_nwhich appears in [BT89]
and [AB00].
Lemma 6.17. F is independent of the choice of lift __m.
Proof.Let __m0be another choice. Let F0 be the function defined using __m0. Sin*
*ce __m0and __mboth restrict to
m on A, there is a 2 ~Tsuch that
__m0= __m+ n .
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 19
We have
__m0)k_OE(__m0)_m0
F0(z)= uk_n^I(An `(uA )
__m+nk)_OE(__m+n_)mk
= uk_n^I( An `(uA q )
__m)k^I(k)_OE(__m)kI(__m,k)2OE()k^I()kI(,__m)k2OE(_)m
= uk_n^I(u An A q u A q `(uA )
= F(z).
Lemma 6.18. F is invariant under W(m).
Proof.Suppose w 2 W(m). We have
wm = m
so
__m= w__m+ n
for some 2 ~T. The proof is now similar to the proof of Lemma 6.17.
The dependence of F(`, m, ~a) on the lift _acan be calculated as follows. Let*
* _a0be another lift. Then
_a0= ~a+ ~
for some ~ 2 2ßiZ + 2ßiøZ, so letting _0
B = ea
we have
B = Aqffi
for some ffi 2 Z. Thus 0
Bn = qk
with k0= k + nffi, and so the quantity
_0_
w(a, q 1_n) def=A1q k_n= B1q kn
is1an nthroot of unity which is independent of the choice of lift of a; in fact*
* it is the Weil pairing of a and
q _nin the curve C. Because it is an nthroot of unity, the quantity
__m)
w(a, q 1_n)OE(m)def=w(a, q 1_n)OE(
is independent of the lift __mof m.
Lemma 6.19. 1
F(`, m, _a0) = w(a, q _n)ffiOE(m)F(`, m, _a).
Proof.If F0= F(`, m, _a0) then
_0_^__ k0_ __ __
F0(z)= uknI(m)B nOE(m)`(uBm )
__m)ffi^I(__m)k_OE(__m)ffiOE(__m)ffi_kOE(__m)ffi2OE(__m)*
*_mffi__m
= uk_n^I(u An A q q `(uA q n)
__m)ffi^I(__m)k_OE(__m)ffiOE(__m)ffi_kOE(__m)ffi2OE(__m)*
*ffi^I(__m)2OE(__m)ffi2OE(__m)_m
= uk_n^I(u An A q q u A n q `(uA )
__m)k_OE(__m)1k_ffiOE(__m)_m
= uk_n^I(An (A q n) `(uA )
__m) _
= w(a, q 1_n)ffiOE(F(`, m, a).
Lemma 6.18 implies that the Taylor series expansion of F(`, m, _a) defines a *
*class in H^(BZ(m)) =
H^(BT)W(m), which we also denote F(`, m, _a). Let `(Q, m, _a) 2 H(Y ; A1an) be *
*the holomorphic cohomology
class given by the formula
`(Q, m, _a) = Q(m)*F(`, m, _a) (6.*
*20)
(using Lemma 5.2 to conclude that the class `(Q, m, _a) is in fact holomorphic).
20 MATTHEW ANDO
Lemma 6.21. If the centralizer Z(m) is connected, for example if G is unitary o*
*r spin, then the class
`(Q, m, _a) is independent of the reduction m of the action of T[n] on Q=Y .
Proof.Let m0be another reduction of the action of T[n] on Q=Y . By Lemma 3.6, t*
*here is an element w 2 W
such that
m0= wm.
If __m: T ! T is a lift of m, then w__mis a lift of m0. Using this lift to defi*
*ne F(`, m0, _a), we have
w*F(`, m0, _a)(z)= F(`, m0, _a)(w(z))
__m)k_OE(w__m)ww__m
= (uw)k_n^I(wAn `(u A )
__m)k_OE(__m)_m
= uk_n^I(An `(uA )
= F(`, m, _a)(z).
Since the diagram
Q(m)
YT____//_GBZ(m)
GG
GGG w
Q(m0)GG##Gfflffl
BZ(m0)
commutes, we have
`(Q, m0,=_a)Q(m0)*F(`, m0, _a)
= Q(m)*w*F(`, m0, _a)
= Q(m)*F(`, m, _a)
= `(Q, m, _a).
The results of this section justify the following.
Definition 6.22. Suppose that Q is a Tequivariant principal Gbundle over a co*
*nnected Tspace Y on
which T[n] acts trivially. Suppose that QT is a principal Gbundle over YT, and*
* suppose also that for some
(equivalently every) reduction
m : T[n] ! T
of the action of T[n] on Q=Y , the centralizer Z(m) is connected. Let C be the *
*elliptic curve Gana=2ßiZ+2ßiøZ.
Let a be a point of C of order n. Let ` be a theta function for G of level ,, a*
*nd let _abe a point of A1anwhose
image in C is a. We define `(Q, _a) 2 H(Y ; A1an) to be the holomorphic cohomol*
*ogy class
`(Q, _a) = `(Q, m, _a),
where m is any reduction of the action of T[n] on Q=Y .
Lemma 6.23. If _a0= _a+ 2ßis + 2ßiffiø is another lift of a, then
`(Q, _a0) = w(a, q 1_n)ffiOE(m)`(Q, _a),
where again m is any reduction of the action of T[n] on Q=Y .
An important case of the preceding constructions is that G = Spin(2d), and oe*
*d is the character of the
basic representation of LG as in x6.3. Let p : G ! SO(2d) be standard the doubl*
*e cover. Let P=Y be the
resulting Tequivariant SO(2d)bundle over Y , and let V be the associated vect*
*or bundle. We recall from
[BT89] that Lemma 3.7 implies that the subvector bundle V T[n]is orientable. E*
*xplicitly, if m is a reduction
of the action of T[n] on Q=Y , then pm is a reduction of the action of T[n] on *
*P=Y . If V T[n]has rank 2k,
then the map
Y !BO(2k)
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 21
classifying V T[n]factors through the map Y ! BZ(pm) classifying P(pm), and we *
*have the solid diagram
BZ(m)c`c`H`` ` ``//BSO(2k)
 HHHH 
 HH 
Q(m) HH 
Bp vY HH 
 vvv HHVHT[n]H
 vvv HH 
fflfflP(pm)vv$$fflfflH
BZ(pm) ___________//_BO(2k)
Since (by Lemma 3.7) the centralizer Z(m) is connected, there is a dotted arrow*
* making the diagram
commute. In other words, V T[n]is orientable, and a choice of orientation (V T*
*[n]) determines a map
f : BZ(m) ! BSO(k)
such that fQ(m) classifies (V T[n], (V T[n])).
Let oek 2 ^H(BSO(2k)) be the characteristic class associated to the sigma fun*
*ction as in x 6.3. We then
have two W(m)invariant holomorphic functions on VGana. One is f*oek, and the o*
*ther is F(`, m, _a).
Lemma 6.24. The ratio
*oek
R = __f______F(oe _
d, m, a)
is a W(m)invariant unit of OVGana0.
Proof.The poles of R occur at zeroes of F. Using the standard maximal torus of *
*SO(2d) then we may write
a typical element of VGanmas
(u1, . .,.ud) 2 (Ganm)d ~=VGanm.
In these terms, a lift __mof m is of the form
__m m m
u = (u 1, . .,.u d)
for some integers m1, . .,.md. We have
__m)k_OE(__m)dYm
F(oed, m, _a) = uk_n^I(An oe(ujA j).
j=1
The product of sigma functions contributes a zero near z = 0 if and only if mja*
* = 0 in C. Let j1, . .,.jk 2
{1, . .,.d} be the indices such that mjia = 0 in C; then
kY
f*oek = oe(ujiAmji).
i=1
So the zeroes of f*oek precisely cancel those of F(oed, m, _a).
Lemmas 5.2 and 6.24 together imply that R defines a holomorphic characteristi*
*c class for Z(m)bundles.
Corollary 6.25. The holomorphic characteristic class
R(V, V T[n], (V T[n]), _a) def=Q(m)*R 2 (H(Y )0)x
is independent of the reduction m, and satisfies
oe(V T[n], (V T[n])) = R(V, V T[n], (V T[n]), _a)oe(V, _*
*a).
22 MATTHEW ANDO
7.Equivariant elliptic cohomology
7.1. Adapted open cover of an elliptic curve. If X is a Tspace and if a is a p*
*oint of C, then we define
( T[k]
Xa = X a is of order exactlyk inC
XT otherwise.
Let N 1 be an integer.
Definition 7.1. A point a 2 C is special for X if Xa 6= XT.
If V is a Tbundle over a Tspace X, then it is convenient to consider a few *
*additional points to be special.
Suppose that F is a component of XT and
m : T ! T
is a reduction of the action of T on the principal bundle associated to V . If *
*we choose an isomorphism
T~~=Zr,
then we may view m as an array of integers (m1, m2, . .,.mr). These integers ar*
*e called the exponents or
rotation numbers of V at F. Let V +denote the onepoint compactification of V .
Definition 7.2. A point a in C is special for V if it is special for V +or if f*
*or some component F of XT
there is a rotation number mj of V such that mja = 0.
In either case, if X is a finite TCW complex, then the set of special points*
* is a finite subset of the torsion
subgroup of C.
Definition 7.3. An indexed open cover {Ua}a2C of C is adapted to X or V if it s*
*atisfies the following.
1)a is contained in Ua for all a 2 C.
2)If a is special and a 6= b, then a 62 Ua\ Ub.
3)If a and b are both special and a 6= b, then the intersection Ua\ Ub is e*
*mpty.
4)If b is ordinary, then Ua\ Ub is nonempty for at most one special a.
5)Each Ua is small (2.1).
Lemma 7.4. Let X be a finite TCW complex. Then C has an adapted open cover, an*
*d any two adapted
open covers have a common refinement.
7.2. Complex elliptic cohomology. Let
E^def=(H^, C, b"1) = (H^, C, log!)
be the equivariant elliptic spectrum associated to the elliptic curve C (see (4*
*.4)). Since this is just a form of
ordinary cohomology, we write
XE = XH .
Suppose that U C is a small open neighborhood of the identity in C. Suppose*
* that V A1anis the
component of "1U containing the origin. We let XTEU be the ringed space defin*
*ed as the pullback in the
diagram
XTEU ____//_XH V (7*
*.5)
 
 
fflffl("fflfflV)1
U _______//_V.
The diagram (4.3)shows that XTE^and XTEU are related by the formula
XTE^~=(XTEU)^0.
ANALYTIC EQUIVARIANT SIGMA ORIENTATION *
* 23
7.3. Equivariant elliptic cohomology. Grojnowski's circleequivariant extension*
* of ^Eis a contravariant
functor associating to a compact Tmanifold X a Z=2graded OCalgebra ET(X), wi*
*th the property that
ET(*) = OC.
It is equipped with a natural isomorphism
E(XT) A(X)!~=(ET(X))^0, (7*
*.6)
such that
A(*) = log!: (OC)^0~=^E(BT).
We shall write XET for the ringed space (C, ET(X)) (see (2.4)). We take this op*
*portunity to phrase the
account in [AB00] of the construction of ET(X) as the construction of a covaria*
*nt functor
X 7! XET
from finite TCW complexes to ringed spaces (2.4)over C, equipped with an ident*
*ification
(*)ET = C
and a natural isomorphism of formal schemes
XTE^A(X)!~=(XET)^0
such that
A(*) = log!: PE^= bGa~=bC.
If X = (X, OX ) is a ringed space and U is an open set of X, then we may write *
*X(U) in place of OX (U).
Let {Ua}a2C be an adapted open cover of C. For each a 2 C, we make a ringed s*
*pace XET,aover Ua as
the pull back in the diagram
XET,a____//_XTEUaa__//_XH V (7*
*.7)
  
  
fflffløa fflffl("Vfflffl)1
Ua ______//_Ua_a_____//V.
As in (7.5), V is the component of "1(Ua a) containing the origin. In other w*
*ords, let Va "1(Ua) be
the component containing the origin. For U Ua let V = Va\ "1(U  a), and let
XET,a(U) = H(Xa; V ),
considered as an OC(U)algebra via the isomorphism
1
U øa!U  a ("V)!V.
If a 6= b and Ua\ Ubis not empty, then by the definition (7.3)of an adapted c*
*over, at least one of Ua and
Ub, suppose Ub, contains no special point. In particular we have Xb = XT and so*
* an isomorphism
XbTEU~=XbEx U
i.e.
E(XbT) OU~=E(Xb) OU
C[z] C
for any small neighborhood U of the origin.
Lemma 7.8. If a 6= b, U Ua\ Ub, and b is not special, then the inclusion
i : Xb !Xa
induces an isomorphism
XbTEUa ~=XaTEUa.
Proof.If a is not special, then Xa = Xb and the result is obvious. If a is spec*
*ial, then it is not contained
in U (by the definition of an adapted cover), and so 0 is not contained in U  *
*a. The localization theorem
(4.2)gives the result.
24 MATTHEW ANDO
Let U = Ua\ Ub. We define
~=
_ab= _Xab: XET,aU !XET,bU
as the arrow making the diagram
XET,aUE_________//XaTEUa~=oXbTEUao_ (7*
*.9)
EEE  pppppp ~
 EEE  pppp =
 E""E øa fflfflwwpp fflffl
 U _____//_U oao__Xb^x (U  a)
  OO E OO
_ab  øba Xb^Exøba
   
  øb//  oo b
