THE WITTEN GENUS AND EQUIVARIANT ELLIPTIC COHOMOLOGY
MATTHEW ANDO AND MARIA BASTERRA
Abstract.We construct a Thom class in complex equivariant elliptic cohomo*
*logy extending the equivariant
Witten genus. This gives a new proof of the rigidity of the Witten genus,*
* which exhibits a close relationship
to recent work on non-equivariant orientations of elliptic spectra.
1.Introduction
Let T denote the circle group. If X is a T-space, let XT denote the Borel con*
*struction ET xT X. Let
HT denote Borel T-equivariant ordinary cohomology with complex coefficients. Ch*
*oose a generator of the
character group of T and so an isomorphism
HT(pt) ~=C[z]:
Let C be a lattice; and let C = C= be the associated elliptic curve. Grojnows*
*ki ([Gro94]; see also
[Ros99, GKV95]) has defined a T-equivariant elliptic cohomology functor E from *
*compact T-spaces to sheaves
of algebras over OC, equipped with a canonical isomorphism
E(X)0~=HT(X) C[z]OC0
of stalks at the identity of C. (For similar constructions for K-theory see [BB*
*M85 , DV93, BG94, RK99].)
The lattice determines a Weierstrass oe-function
oe: C ! C;
this is an odd holomorphic function which vanishes to first order at the points*
* of ; see x4. Its Taylor
expansion at the origin determines an orientation
OE:MSO ! HC
and so a Thom class OE(VT)2 HT(V; V0) OC 0for any oriented T-vector bundle V *
*, which we call the
C[z]
"equivariant oe-orientation". The associated genus is the Witten genus ([AHS98]*
*; see also x4).
If V is a vector bundle over a compact space X, then we define E(V ) to be th*
*e reduced E-cohomology
of the Thom space XV ; it is a sheaf of E(X)-modules, called the Thom sheaf of *
*V . If V is an even spin
T-vector bundle, then (see Proposition 2.16) E(V ) is an invertible sheaf of E(*
*X)-modules. More generally,
if T and V are even oriented T-vector bundles over X, and if V - T is a virtual*
* spin bundle, then we may
define E(V - T) to be the invertible sheaf
E(V - T) = E(T + T)-1E(X)E(V + T):
We recall that if W is a virtual spin vector bundle over X, then there is a c*
*haracteristic class p1_2(W) 2
H4(X; Z), twice which is the first Pontrjagin class. Our main result is the con*
*struction of a Thom class in
E-theory.
Theorem 1.1.Suppose that T acts non-trivially on a compact smooth manifold M. S*
*uppose that V and T
are oriented T-vector bundles of even rank over M such that
w2(V - T)T= 0 (1.2a)
p1_
2(V - T)T= 0: (1.2b)
Then the invertible sheaf of E(X)-modules E(V - T) has a global section , whose*
* stalk at the origin is the
equivariant oe-orientation.
___________
Date: version 4.10, August 2000.
The first author was supported by the NSF.
1
2 MATTHEW ANDO AND MARIA BASTERRA
Here T may be the tangent space of M, but need not be. If it is, and if we su*
*ppose that T is spin so that
Proposition 2.16 defines E(-T), then the Pontrjagin-Thom construction provides *
*a map of sheaves
* PT
E(V - T) i-!E(-T) --!E(*) ~=OC;
where i is the relative zero section. The class PTi*()0 is the equivariant Witt*
*en genus of M twisted by V
(see x4). Since PTi*() is a global section of OC, the twisted Witten genus is "*
*rigid":
Corollary 1.3.If T is the tangent bundle of M, the characteristic classes of T *
*and V satisfy (1.2), and T
(and so also V ) is spin, then the equivariant Witten genus of M twisted by V i*
*s constant.
This result was discovered by Witten, who gave a physical proof in [Wit87]. T*
*he first mathematical proofs
were given by Taubes and Bott-Taubes [Tau89, BT89]. Subsequently Kefeng Liu gav*
*e a shorter proof and
found many new cases [Liu94, Liu95b, Liu95a, GL96, Liu96a, Liu96b].
Besides giving an appealing relationship with equivariant elliptic cohomology*
*, our construction has the
virtue of axiomatizing the properties of the oe-function on which it depends. W*
*e construct global sections of
E(V - T) for a class of functions described in x3; Theorem 1.1 is a special ca*
*se of Theorem 5.1.
We were led to the functions of x3 by the work of the first author, Hopkins,*
* and Strickland. A theta
function such as we consider descends to a " structure" on the elliptic curve C*
*, in the sense of [Bre83].
Ando-Hopkins-Strickland have shown that, if E is a (non-equivariant) elliptic c*
*ohomology theory with elliptic
curve C, then maps of ring spectra MO<8> ! E are given by structures on the id*
*eal sheaf IbC(0) of the
origin in the formal group bCof C.
The Weierstrass oe-function given in x4 descends to the unique structure on *
*the ideal sheaf IC(0),
and so gives a structure on IbC(0) by restriction. It follows that the oe-orie*
*ntation is the unique natural
orientation from MO<8> to elliptic spectra. Theorem 1.1 establishes a fundament*
*al relationship between
these results and the rigidity theorems for equivariant elliptic genera. It se*
*ems reasonable to hope that
a (rational) equivariant analogue of the methods of [AHS98] will produce a func*
*torial equivariant MO<8>
orientation to the rational equivariant elliptic spectra of Greenlees, Hopkins,*
* and Rosu [GHR99 ], without
the elaborate calculations given below. We will return to that problem in anoth*
*er paper.
This paper was also inspired by the work of Ioanid Rosu [Ros99]. Rosu and the*
* first author observed that
rigidity theorems follow from the existence of a global section of the Thom she*
*af in Grojnowski's equivariant
elliptic cohomology. Rosu constructed a Thom class whose value at the origin gi*
*ves the Ochanine genus. In
fact, he showed that the "transfer" argument of Bott and Taubes is the essentia*
*l ingredient in the construction
of the Thom section. The rigidity of the Ochanine genus and Rosu's analysis req*
*uire only that the bundles
in question be spin bundles of even rank. This paper represents a first attempt*
* to understand the Thom
classes whose whose rigidity require the restrictions (1.2).
Our formulation leads to proofs of many of the rigidity theorems in the liter*
*ature, including the other
genera considered in [Wit87, BT89] and the twisted loop group genera of [Liu94]*
*. In the interest of brevity,
we shall return to these issues at another time.
2.Equivariant elliptic cohomology
In this section, we briefly review Grojnowski's equivariant elliptic cohomolo*
*gy, following [Gro94] and
[Ros99].
The elliptic curve. We fix a complex elliptic curve C over C, with an analytic *
*isomorphism
C ~=C=
We write ss for the covering map
C ss-!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 , and MV for the sheaf of meromorphic functions.
Both C and C are abelian topological groups. If G is an abelian topological g*
*roup and g 2 G, when we
write og for the translation map; and if V G is an open set, then we write
V - g def=o-g(V ):
THE WITTEN GENUS AND EQUIVARIANT ELLIPTIC COHOMOLOGY *
* 3
Definition 2.1.An open set U of C is small if ss-1U is a union of connected com*
*ponents each isomorphic
to U.
If U is small and V is a component of ss-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 ss-1U containing 0. This
determines a C[z]-algebra structure on OU, and a C[z; z-1] structure on OU|U\0:
Adapted open cover of an elliptic curve. Now suppose that X is a compact T-mani*
*fold. If a is a point
of C, then we define
( T[k]
Xa = X a is of order exactlyk inC
XT otherwise:
Let N 1 be an integer.
Definition 2.2.A point a 2 C is special of level N for X if XNa 6= XT.
If V is a T-bundle over a compact T-space X, then it is convenient to make a *
*few extra special points.
Over each component F of XT, there are integer mj and a decomposition
M R
V |F ~=V (0) + V (mj);
where V (0) is the summand of V |F on which T acts trivially, and V (mj) is a c*
*omplex vector bundle on
which z 2 T acts by zmj. The mj are called exponents or rotation numbers of V a*
*t F. Let V +denote the
one-point compactification of V .
Definition 2.3.A point a in C is special of level N for V if it is special for *
*V +or if for some fixed component
F of X there is a rotation number mj of V such that mjNa = 0.
In any case, for fixed N the set of special points of level N is a finite sub*
*set of the torsion subgroup of C.
Mostly we shall fix an N, and say simply that a is special.
Definition 2.4.An open cover {Ua}a2C of C is adapted to X or V (to level N) if *
*it satisfies the following.
1)a is contained in Ua for all a 2 C.
2)If b 2 Ua is special then b = a.
3)If a and b are both special and a 6= b, then the intersection Ua\ Ubis empt*
*y.
4)If b is ordinary, then Ua\ Ubis non-empty for at most one special a.
5)Each Ua is small.
Lemma 2.5. C has an adapted open cover, and any two adapted open covers have a *
*common refinement. |___|
Equivariant cohomology. We write HT for Borel T-equivariant ordinary cohomology*
* with complex coef-
ficients: so if X is a T-space then
H*T(X) = H*(XT; C):
We choose a generator of the character group of T, and write z for the resultin*
*g generator of H2(BT; Z);
this gives a generator of H2T(*) which we also call z. We shall often consider *
*H*T(*) ~=C[z] as a subring of
the ring OC(C) of global analytic functions on C.
We recall [AB84] that HT satisfies a localization theorem.
Theorem 2.6.The natural map
HT(X) -!HT(XT)
induces an isomorphism
HT(X) C[z; z-1] ~=HT(XT) C[z; z-1]:
C[z] C[z]
|_*
*__|
4 MATTHEW ANDO AND MARIA BASTERRA
Elliptic cohomology of a space. The equivariant elliptic cohomology of X is a s*
*heaf of OC-algebras over
C. Let {Ua}a2C be an adapted open cover of C.
For each a 2 C, we define an sheaf of OC|Ua-algebras by the formula
E(X)a(U) = HT(Xa) OC(U - a) (2.7)
C[z]
for U Ua. Here OC(U - a) is a C[z]-algebra via the C[z]-structure on OC|Ua-a. *
*The ring OC(U) acts by
the formula
g . (x y) = x yo*ag:
If a 6= b and Ua\ Ub is not empty, then by the definition (2.4)of an adapted *
*cover, at least one of Ua
and Ub, suppose Ub, contains no special point. In particular we have Xb = XT an*
*d so an isomorphism of
C[z]-algebras
HT(Xb) ~=H*(Xb) C[z]: (2.8)
C
Lemma 2.9. If a 6= b, U Ua\ Ub, and b is not special, then the inclusion
i:Xb -!Xa
induces an isomorphism
*1
HT(Xa) OC(U - a) i---!HT(Xb) OC(U - a):
C[z] C[z]
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. *
*In particular,_z_is a unit in
OC(U - a). The localization theorem (2.6)gives the result. *
* |__|
We then define
~=
OEab:E(X)a|Ua\Ub-! E(X)b|Ua\Ub
as the composition (for U Ua\ Ub)
*1
HT(Xa) OC(U - a)i---!HT(Xb) OC(U - a)
C[z] C[z]
~= * b
-! H (X ) OC(U - a)
C
-1ob-a----!H*(Xb) O (U - b) (2.10)
C C
~= b
-! HT(X ) OC(U - b):
C[z]
The first map is an isomorphism by Lemma 2.9. The rest of the isomorphisms are *
*tautologies, although
some use the isomorphism (2.8).
The cocycle condition
OEabOEbc= OEac
needs to be checked only when two of a; b; c are not special; and in that case *
*it follows easily from the
equation
oc-bob-a= oc-a:
We shall write E(X) for the resulting sheaf over C. One then has the following.*
* It was certainly known to
Grojnowski [Gro94], but for a detailed account the reader may wish to consult [*
*Ros99].
Proposition 2.11.The sheaf E(X) is a sheaf of analytic OC-algebras, which is in*
*dependent up_to_canonical
isomorphism of the choice of adapted open cover. *
* |__|
THE WITTEN GENUS AND EQUIVARIANT ELLIPTIC COHOMOLOGY *
* 5
Cohomology of the Thom space. Now suppose that V is a T vector bundle over X. L*
*et E(V ) denote
the reduced cohomology of the one-point compactification of V . In the case th*
*at V is spin, we give an
explicit cocycle which exhibits E(V ) as an invertible sheaf of E(X)-modules. T*
*he main tool which makes
this possible is the following.
Lemma 2.12. Let V be a SpinT-vector bundle of even rank over X. For all n, the *
*fixed bundle V T[n]over
XT[n]is orientable.
Proof.This is Lemma 10.1 in [BT89]. *
* |___|
Let OE be a multiplicative analytic orientation on HC with (equivariant) eule*
*r class e. We define a class
[OE; V ] 2 H1(C; E(X)x) as follows.
Choose a cover of C adapted to V . For each special point a of C, choose an o*
*rientation on V aand on V T.
Suppose that a; b are two points of C, such that Ua \ Ub is non-empty: we may*
* suppose that b is not
special. Recall that there is a tautological isomorphism (2.7)
E(X)(U) ~=HT(Xb) OC(U - a):
C[z]
Lemma 2.13. There is a unit e(a; b) 2 E(X)(U) such that e(V b)e(a; b) = e(V a|X*
*b). Moreover we have
e(a; b)e(b; c) = e(a; c) (2.14)
whenever that makes sense.
Proof.If a is not special then e(a; b) = 1. Otherwise, we have a 62 U so 0 62 U*
* - a, and so z is a unit in
OC(U -a). On each component F of Xb, there are integers mj6= 0 and complex vect*
*or bundles V (mj) over
F such that
M R
V a|F = V b|F V (mj):
mj
Here T acts on V (mj) fiberwise by the character u 7! umj.
Let f(z) = z + higher termsbe the characteristic series of the orientation OE*
*, so that
e(L) = f(c1L)
for L a complex line bundle. Let xj;1; : :;:xj;djbe the roots of the total Cher*
*n class of V (mj). Since F is
compact, the xi;jare nilpotent. We have
Y Y
e(a; b)|F= f(mjz + xj;i)
j i
Y d P
= ( mjj)z dj((1 + N) + higher terms)in;z
j
where N is a nilpotent class in H*(Xb). The result follows, since the mj are no*
*n-zero and z is a unit in
OC(U - a).
The cocycle condition (2.14)is easy, because as usual the equation needs only*
* to be verified_when at most
one of a, b, and c is special. *
* |__|
Let [OE; V ] 2 H1(C; E(X)x) be the cohomology class defined by the e(a; b). L*
*et E(X)[OE;Vd]enote the
resulting invertible sheaf of E(X)-modules over C: Explicitly, the sheaf E(X)[O*
*E;Vi]s assembled from the
sheaves
Ea(X)(U) = HT(Xa) OC(U - a)
C[z]
over Ua, using the formula
6 MATTHEW ANDO AND MARIA BASTERRA
Ea(X)(U)~=HT(Xa) OC(U - a)
C[z]
-i*1--!H b
T(X ) C[z]OC(U - a)
-e(a;b).---!H b
T(X ) C[z]OC(U - a)
(2.15)
~=H*(Xb) OC(U - a)
C
1o*b-a* b
-----!H (X ) OC(U - b)
C
~=HT(Xb) OC(U - b):
C[z]
It will be important in x5 that we allowed the orientation of V Tto vary with*
* the special point a. It fact,
the resulting sheaf E(X)[OE;Vi]s independent of the choices, up to canonical is*
*omorphism.
Proposition 2.16.If V is a spin T-bundle of even rank, then the Thom isomorphis*
*m OE induces an iso-
morphism
E(X)[OE;V~]=E(V )
of sheaves of E(X)-modules.
Proof.Choose a cover adapted to both V and X. Suppose that U Ua\Ub, a is speci*
*al, and b is not. Then
the diagram
HT(Xa) OC(U - a)--OE--!HT(V a; V0a) OC(U - a)
C[z]? ? C[z]
e(a;b)i*?y ?yi*
HT(Xb) OC(U - a)--OE--!HT(V b; V0b) OC(U - a)
C[z]? ? C[z]
?y ?y
HT(Xb) OC(U - b)--OE--!HT(V b; V0b) OC(U - b)
C[z] C[z]
commutes (all the arrows are isomorphisms). The left column describes the sheaf*
* E(X)[OE;V,]while_the right
column describes E(V ). *
* |__|
Remark 2.17.If V and T are even T-bundles and w2(V - T) = 0, then we can define*
* E(X)[OE;V -T]and
obtain an isomorphism E(X)[OE;V -T]~=E(V - T), as explained in the introduction.
3. Theta functions
We consider meromorphic functions : C ! C with the following properties. Firs*
*t of all, we require that
there is an integer N such that the zeroes and poles of are contained in N-1. *
*Moreover, we ask that
have the following properties.
(-z) = -(z) (3.1*
*a)
(0) = 0 (3.1*
*b)
0(0)6= 0 (3.1*
*c)
_)
(z + ) = c()efl(z+2(z); (3.1*
*d)
where is a point of the lattice ; fl is a complex number depending on ; and
c: -!{1}
THE WITTEN GENUS AND EQUIVARIANT ELLIPTIC COHOMOLOGY *
* 7
is a function (not a necessarily a homomorphism) satisfying
c(`) = c()`:
Note that the transformation formula implies that
2_)
(z + `) = c(`)efl(`z+`(2z): (3.2)
The equations (3.1a_3.1c) imply that the Taylor expansion of at the origin d*
*efines an orientation
MSO ! HC:
Explicitly, the genus associated to this orientation is
Z Yd
M 7! _xj_;
Mj=1(xj)
where M is an oriented manifold of dimension 2d, and the xj are the roots of th*
*e total Pontrjagin class of
M; see xA.
4. Examples
The Witten genus. One example of a theta function satisfying (3.1)is the Weiers*
*trass oe-function. We
describe a variant associated to the Witten genus.
Let oe denote the expression
Y (1 - qnu)(1 - qnu-1)
oe = (u1_2- u-1_2) ________________n2: (4.1)
n1 (1 - q )
This may be considered as an element of Z[[q]][u1_2] which is a holomorphic fun*
*ction of (u1_2; q) 2 Cx x D,
where D = {q 2 C|0 < |q| < 1}.
Let h = {o 2 C| Imo > 0} be the open upper half plane. We may consider oe as *
*a holomorphic function
of (z; o) 2 C x h by setting
u1_2= ez_2
q= e2ssio:
For any complex elliptic curve C, there is a o 2 h such that C is analytically *
*isomorphic to C= with
= 2ssiZ + 2ssioZ. For our purposes, it is sufficient to fix o, and consider oe*
* as a function of z alone.
The formula (4.1)permits one to check directly that oe(z) is of the form (3.1*
*). For example, the function
c is given by the formula
(
c() = 1 2 2
-1 otherwise.
The equations (3.1)in this case are descent data for the "-structure" on the id*
*eal sheaf IC(0) of the origin
in C, in the sense of Breen [Bre83]; see particularly x3.12.
If V is a complex vector bundle, let rV = rankV - V denote the associated vir*
*tual bundle of rank 0. Let
SkV denote the k symmetric power of V , and let
X
Sym t(V ) = tkSkV:
k0
This extends to an operation
K(X) ! (1 + tK(X)[[t]])x
because of the formula
Symt(V W) = SymtV . SymtW:
Formula (4.1)shows that the genus associated to oe is
O
M 7! bA(M; Sym qn(rTM C));
n1
8 MATTHEW ANDO AND MARIA BASTERRA
which is equivalent to formula (27) in [Wit87]; see also [AHS98].
Remark 4.2.The sigma function (4.1)is related to the classical Weierstrass oe-f*
*unction oeWeierstrassin for
example [Sil86] by the formula
2
oeWeierstrass(z) = eazoe(z);
where a is a constant. It is not hard to check that these define the same invar*
*iants of MO<8>-manifolds.
The oe-function (4.1)also arises as the p-adic oe-function of the Tate curve [M*
*T91 ].
The Ochanine genus. Let C ~=C= be an elliptic curve, and let p; q; r be the poi*
*nts of order 2 of C. Let
P; Q; R be representatives of these points, such that Q + R = P, and let s be t*
*he function
s(z) = oe(z)oe(-Q)oe(-R)oe(z_-oP)e(z:- Q)oe(z - R)oe*
*(-P)(4.3)
Then s is the pull-back to C of the unique meromorphic function f on C with div*
*isor
divf = (0) + (p) - (q) - (r)
and df0= dz0. It is odd, and the resulting genus is the elliptic genus of Ochan*
*ine [Och87] genus for the pair
consisting of the lattice and the point p. It is customary to consider s as a *
*theta function for the lattice
0= { + nP| 2 ; n 2 Z}:
The function s is not quite periodic with respect to this lattice, but satisfies
s(z + + nP) = (-1)ns(z)
for 2 . Explicitly, if = 2ssiZ + 4ssioZ and P = 2ssio, then using (4.1)in (4.*
*3)gives the formula
Y (1 - qn)(1 - qn-1)
s(z) = 1_-_1 +________________nn-1:
n1 (1 + q )(1 + q )
It is customary to consider s as a theta function for the lattice
0= { + nP| 2 ; n 2 Z}:
The function s is not quite periodic with respect to this lattice, but satisfies
s(z + + nP) = (-1)ns(z)
for 2 . Theorem 5.1 for this function is due to Rosu [Ros99].
5.The equivariant Thom class
We fix a meromorphic function satisfying the conditions of x3. We write OE *
*for the resulting Thom
isomorphism.
Suppose that M is a compact smooth T-manifold, and that W is a virtual orient*
*ed T-bundle over M.
The Thom isomorphism OE gives a generator OE(W)Tof HT(W), and so a generator of
HT(W) OC 0
C[z]
which we shall also denote OE(W)T.
Theorem 5.1.If T and V are oriented T-vector bundles of even rank over M such t*
*hat w2(V - T) = 0,
and either the function fl of (3.1d)is identically zero or the equations (1.2)h*
*old, then the invertible sheaf
of E(X)-modules E(V - T) has a global section , such that
0= OE(V - T)T
under the isomorphism
HT(V - T) OC 0~=E(V - T)0:
C[z]
THE WITTEN GENUS AND EQUIVARIANT ELLIPTIC COHOMOLOGY *
* 9
The proof will occupy the rest of this section, and to give it we fix a cover*
* of C adapted to V and T of
level N, where N is large enough that the zeroes and poles of are contained in*
* N-1.
Let us first indicate precisely what it is we must construct. Since w2(V - T)*
* = 0, Proposition 2.16 shows
that it is equivalent to construct a global section of the sheaf E(X)[OE;V,-T]w*
*hose value in
E(X)[OE;V0-T]~=HT(X) OC(U0)
C[z]
is 1. The formula (2.15)for this sheaf shows that such a global section is asse*
*mbled from sections a 2
E(X)a(Ua) which satisfy the formula
b= o*b-ae(a; b): (5.2)
on U Ua\ Ub. As usual, it will suffice to suppose that b is not special. In or*
*der to give the formula for ,
we introduce the following notations and results.
Suppose that a is a special point of order n. The structure of the formula fo*
*r a depends on the parity
of n, but the two cases share many components. Let h = n=2. In the following, t*
*erms which involve h are
simply absent in the case that n is odd.
Let be the lattice point na. Let fl = fl() be the constant in the translatio*
*n formula (3.1d)for , and
let S(x) = eflx.
Let P be a component of the submanifold XT[n]of points fixed by T[n]. Let F *
*P be a component of
XT. We have decompositions
M
T|P = T0 Th TrR (5.3)
0