Contemporary Mathematics
A renormalized Riemann-Roch formula and the Thom isomorphism for the free
loop space
Matthew Ando and Jack Morava
I believe in the fundamental interconnectedness of all things.
_Dirk Gently [Ada88 ]
1.Introduction
Let T denote the circle group, and, if X is a compact smooth manifold, let L*
*X def=C1 (T; X) denote
its free loop space. The group T acts on LX, and the fixed point manifold is ag*
*ain X, considered as the
subspace of constant loops. In the 1980's, Witten showed that the fixed-point f*
*ormula in ordinary equivariant
cohomology, applied to the free loop space LX of a spin manifold X, yields the *
*index of the Dirac operator
(i.e. the ^A-genus) of X_a fundamentally K-theoretic quantity [Ati85]. He also *
*applied the fixed-point
theorem in equivariant K-theory to a Dirac-like operator on LX to obtain the el*
*liptic genus and "Witten
genus" of X [Wit88 ]_quantities associated with elliptic cohomology.
Among homotopy theorists, these developments generated considerable exciteme*
*nt. The chromatic pro-
gram gives organizes the structure of finite stable homotopy types, locally at *
*a prime p, into layers indexed by
nonnegative integers. The nth layer is detected by a family of cohomology theor*
*ies En; rational cohomology,
K-theory, and elliptic cohomology are detecting theories for the first three la*
*yers [Mor85 , DHS88 , HS98].
The geometry and analysis related to rational cohomology and K-theory are re*
*asonably well-understood,
but for n 2 and for elliptic cohomology in particular, very little is known. W*
*itten's work provides a major
suggestion: for n = 1 and n = 2 his analysis gives a correspondence
analysis underlying Enanalysis underlying En-1
applied to X $ applied to LX. (1.0*
*.1)
This paper represents our attempt to understand why Witten's procedure appea*
*rs to connect the chro-
matic layers in the manner of (1.0.1). To do this we consider very generally th*
*e fixed-point formula attached
to a complex-oriented theory E with formal group law F. We recall that for n > *
*0, such a theory detects
chromatic layer n if the formal group law F has height n.
Our first result is that the fixed-point formula of a suitable equivariant e*
*xtension of E (Borel cohomology
is fine, as is the usual equivariant K-theory) applied to the free loop space y*
*ields a formula which is identical
to the Riemann-Roch formula for the quotient F=(^q) of the formal group law F b*
*y a free cyclic subgroup
(^q) (compare formulae (3.2.2)and (4.2.3)).
___________
Much of the work for this paper was carried out at Johns Hopkins University,*
* where the first author was supported by an
NSF Postdoctoral Fellowship.
Supported by the NSF.
cO0000 (copyr*
*ight holder)
1
2 MATTHEW ANDO AND JACK MORAVA
The quotient F=(^q) is not a formal group, so to understand its structure, w*
*e work p-locally and study
its p-torsion subgroup F=(^q)[p1 ]. We construct a group Tate(F) with a canonic*
*al map
Tate(F) ! F=(^q);
which induces an isomorphism of torsion subgroups in a suitable setting. Our se*
*cond result is that the group
Tate(F)[p1 ] is a p-divisible group, fitting into an extension
F[p1 ] ! Tate(F) ! Qp=Zp
of p-divisible groups. If the height of F it n, the height of Tate(F)[p1 ] is n*
* + 1, but itsetale quotient has
height 1. In a sense we make precise in x5.3, it is the universal such extensio*
*n.
Thus the fixed-point formula on the free loop space interpolates between the*
* chromatic layers in the same
way that p-divisible groups of height n+1 withetale quotient of height 1 interp*
*olate between formal groups of
height n and formal groups of height n+1. This is discussed in more detail, fro*
*m the homotopy-theoretic point
of view, in our earlier paper [AMS98 ] with Hal Sadofsky; this paper is a kind*
* of continuation, concerned
with analytic aspects of these phenomena. We show that Witten's construction i*
*n rational cohomology
produces K-theoretic genera because of the exponential exact sequence
0 ! Z ! C ! Cx ! 1 (1.0*
*.2)
expressing the multiplicative group (K-theory) as the quotient of the additive *
*group (ordinary cohomology)
by a free cyclic subgroup; while his work in K-theory produces elliptic genera *
*because of the Tate curve
0 ! qZ ! Cx ! Cx=qZ ! 1 (1.0*
*.3)
(where q is a complex number with |q| < 1), expressing the elliptic curve Cx=qZ*
* as the quotient of the
multiplicative group by a free cyclic subgroup.
These analytic quotients have already been put to good use in equivariant to*
*pology. Grojnowski con-
structs from equivariant ordinary cohomology a complex T-equivariant elliptic c*
*ohomology using the elliptic
curve C= which is the quotient of the complex plane by a lattice; and Rosu uses*
* Grojnowski's functor
to give a striking conceptual proof of the rigidity of the elliptic genus. Groj*
*nowski's ideas applied to the
multiplicative sequence (1.0.3)give a construction of complex T-equivariant ell*
*iptic cohomology based on
equivariant K-theory; details will appear elsewhere. Completing this circle, R*
*osu has used the quotient
(1.0.2)to give a construction of complex equivariant K-theory [Gro94 , Ros99, R*
*K99 ].
Several of the formulae in this paper involve formal infinite products; see *
*for example (3.2.2)and (4.2.3).
On the fixed-point formula side, the source of these is the Euler class of the *
*normal bundle of X in LX
(3.1.2). From this point of view, the problem is that the bundle is infinite-d*
*imensional, so it does not have
a Thom spectrum in the usual sense. However, has a highly nontrivial circle ac*
*tion, which defines a locally
finite-dimensional filtration by eigenspaces. Following the program sketched in*
* [CJS95 ], we construct from
this filtration a Thom pro-spectrum, whose Thom class is the infinite product.
1.1. Formal group schemes. In this paper (especially in section 5) we shall *
*consider formal schemes
in the sense of [Str99, Dem72 ]. A formal scheme is a filtered colimit of affin*
*e schemes. For example the
"formal line"
^A1def=colimspecZ[x]=xn
n
is a formal scheme. Note that an affine scheme is a formal scheme in a trivial *
*way. An important feature of
this category which we shall use is that it has finite products. For example,
^A1x ^A1= colimspecZ[x]=(xn) Z[y]=(ym ) :
In particular a formal group scheme means an abelian group in the category of f*
*ormal schemes. A formal
group scheme whose underlying formal scheme is isomorphic to the formal scheme *
*^A1is called a commutative
one-dimensional formal Lie group. We shall simply call it a formal group.
The first reason for considering formal schemes is that formal groups are no*
*t quite groups in the category
of affine schemes, because a group law
F(s; t) = s + t + . .2.R[[s; t]]
RIEMANN-ROCH AND THE FREE LOOP SPACE *
* 3
over a ring R gives a diagonal
R[[s]] -!R[[s; t]] ~=R[[s]]^ R[[t]]
only to the completed tensor product.
The second reason for considering formal schemes is that, if G is an affine *
*group scheme, then its torsion
subgroup Gtorsis a formal scheme (the colimit of the affine schemes G[N] of tor*
*sion of order N), but not in
general a scheme.
If X is a formal scheme over R, and S is an R-algebra, then XS will denote t*
*he resulting formal scheme
over S.
2. The umkehr homomorphism and an ungraded analogue
2.1. Let E be a complex-oriented multiplicative cohomology theory with form*
*al group law F, and
let h: X ! Y be a proper complex-oriented map of smooth finite-dimensional conn*
*ected manifolds, of
fiber dimension d = dimX - dimY . The Pontrjagin-Thom collapse associates to th*
*ese data an "umkehr"
homomorphism [Qui71]
h!: E*(X) ! E*-d(Y ):
We will be concerned with similar homomorphisms in certain infinite-dimensional*
* contexts. In order to do
so, we systematically eliminate the shift of -d in the degree by restricting ou*
*r attention to even periodic
cohomology theories E. The examples show (3.3)that this amounts to measuring qu*
*antities relative to the
vacuum.
2.2. Even periodic ring theories. Let E be a cohomology theory. If X is a sp*
*ace, then E*(X) will
denote its unreduced cohomology; if A is a spectrum, then E*(A) will denote its*
* cohomology in the usual
sense. These notations are related by the isomorphism E*(X) ~=E*(1 X+), where X*
*+ denotes the union
of X and a disjoint basepoint. The reduced cohomology of X will be denoted eE(*
*X). Let * denote the
one-point space.
A cohomology theory E with commutative multiplication is even if Eodd(*) = 0*
*. It is periodic if E2(*)
contains a unit of E*(*). If E is an even periodic theory, then we write E(X) f*
*or E0(X) and E for E0(*). We
sometimes write XE = specE(X) for the spectrum, in the sense of commutative alg*
*ebra, of the commutative
ring E(X).
A space X is even if H*(X) is a free abelian group, concentrated in even deg*
*rees. In that case the
natural map
colimFE ! XE; (2.2*
*.1)
where F is the filtered system of maps of finite CW complexes to X, is an isomo*
*rphism. This gives XE the
structure of a formal scheme. The functor X 7! XE from even spaces to formal sc*
*hemes over E preserves
finite products and coproducts: if X and Y are two even spaces, then
(X x Y )E ~=XE x YE ~=specE(X)^ E(Y ):
Here ^ refers to the completion of the tensor product with respect to the topol*
*ogy defined by the filtrations
of E(X) and E(Y ).
2.3. Orientations and coordinates. Let P def=CP1 be the classifying space fo*
*r complex line bundles.
Let m:P x P ! P be the map classifying the tensor product of line bundles. It i*
*nduces a map
PE x PE mE--!PE;
which makes PE a formal group scheme over E. Of course it is a formal group: le*
*t i:S2 ! P be the map
classifying the Hopf bundle. A choice of element x 2 eE(P) such that v = i*x 2 *
*eE(S2) ~=E-2(*) is a unit is
called a coordinate on PE. There is then an isomorphism
E(P) ~=E[[x]];
4 MATTHEW ANDO AND JACK MORAVA
which determines a formal group law F over E by the formula
F(x; y) = m*x 2 E(P x P) ~=E[[x; y]]:
Any even-periodic cohomology theory E is complex-orientable. An orientation on*
* E is a multiplicative
natural transformation
MU ! E:
These correspond bijectively with elements u 2 eE2(P) such that
i*u = 2(1); (2.3*
*.1)
where is the suspension isomorphism [Ada74 ]. A coordinate x thus determines a*
*n orientation u = v-1x.
Definition 2.3.2.We shall use the notation (E; x; F) to denote an even perio*
*dic cohomology theory E
with coordinate x and group law F. We shall call such a triple a parametrized t*
*heory.
2.4. Thom isomorphism. An orientation u 2 eE2(P) gives the usual Thom classe*
*s and characteristic
classes for complex vector bundles. If k is an integer, let k_denote the trivia*
*l complex vector bundle of rank
k. If X is a connected space and V is a complex vector bundle of rank d over X,*
* then we write
XV def=1 (P(V 1_)=P(V ))
for the suspension spectrum of its Thom space, with bottom cell in degree 2d. W*
*e write ffVusualfor the Thom
isomorphism
ffVusual:E*(X) ~=E*+2d(XV ):
In the same way, a coordinate x 2 eE(P) gives rise to a Thom isomorphism
ffV :E(X) ~=E(XV ):
If v = i*x 2 eE(S2) is the associated orientation, the isomorphisms ffusualand *
*ff are related by the formula
ffV = vrankVffVusual:
Remark 2.4.1.One effect of condition (2.3.1)is that ffd_usualcoincides with *
*the suspension isomorphism
ffd_usual= 2d:E*(X) ~=E*+2d(Xd_):
The Thom isomorphism ff defined by a coordinate chooses v 2 eE(S2) ~=E(*1_) as *
*ff1_. Thus ffusualmay be
viewed as a composition of Thom isomorphisms
d_)-1 ffV
ffVusual:E(Xd_) (ff----!E(X) --!E(XV ):
If i :1 X+ ! XV denotes the zero section, then we write
eusual(Vd)ef=i*ffVusual(1) 2 E2d(X)
e(V )def=i*ffV(1) 2 E(X)
for the usual and degree-zero Euler classes of V ; these are related by the for*
*mula
e(V ) = vrankVeusual(V ):
If U(n) denotes the unitary group and T is its maximal torus of diagonal mat*
*rices, then the map
E(BU(n)) ! E(BT) ~=E((BT)n)
is the inclusion of the ring of invariants under the action of the Weyl group W*
*. The coordinate gives an
isomorphism
E(BT) ~=E((BT)n) ~=E[[x1; :::; xn]];
RIEMANN-ROCH AND THE FREE LOOP SPACE *
* 5
with W acting as the permutation group n on the xi's. Thus we can define degree*
*-zero Chern classes ciin
E(BU(n)) by the formula
Xn nY
cizn-i= (z + xi): (2.4*
*.2)
i=0 i=0
If F is the group law resulting from the coordinate x, then we call the cithe "*
*F-Chern classes".
Returning to the map
X h-!Y;
as in (2.1), we can now define an umkehr map
E(X) hF--!E(Y );
using the degree-zero Thom isomorphism ff. We write F to indicate the dependenc*
*e on the coordinate.
Definition 2.4.3.If X is any manifold, we denote by pX the map
X
X p--!*:
If E is an even periodic theory with group law F, then its F-genus is the eleme*
*nt pXF(1) of E.
2.5. The Riemann-Roch formula. The Riemann-Roch formula compares the umkehr *
*homomor-
phisms hF and hG of two coordinates with formal group laws F and G, related by *
*an isomorphism
: F ! G:
The book of Dyer [Dye69 ] is a standard reference.
Proposition 2.5.1.If h:X ! Y is a proper complex-oriented map of fiber dimen*
*sion 2d, then
2 3
dY x
hG(u) = hF 4u . ___j_5; (2.5*
*.2)
j=1(xj)
where the xiare the terms in the factorization
Yd
zd+ c1zd-1+ : :+:cd = (z + xi)
j=1
of the total F-Chern class of the formal inverse of the normal bundle of h. *
* |___|
Remark 2.5.3.Changing the coordinate by a unit u 2 E multiplies the umkehr h*
*omomorphism by ud;
by such a renormalization, we can always assume that is a strict isomorphism.
2.6. The fixed-point formula.
Notations for circle actions. Let T denote the circle group R=Z. If X is a T*
*-space then we write
XT def=ET xX
T
for the Borel construction and XT for the fixed-point set. Let T* = Hom[T; Cx] *
*be the character group of
T; we will also write ^T= T*- {1} for the set of nontrivial irreducible represe*
*ntations. For k 2 T*, let C(k)
be the associated one-dimensional complex representation. There is then an asso*
*ciated complex line bundle
C(k)Tover BT.
It is convenient to choose an an isomorphism T* ~=Z; this determines, in par*
*ticular, an isomorphism
BT ~=CP1 . For k 2 Z we have C(k) = C(1)k and C(k)T= C(1)kT. If ^q2 E(BT) is t*
*he Euler class of
C(1)T, then the Euler class of C(k)Tis [k](^q).
6 MATTHEW ANDO AND JACK MORAVA
Equivariant cohomology.
Definition 2.6.1.Let (E; x; F) be a parametrized theory. A T-equivariant coh*
*omology theory ET is
an extension of (E; x; F) if
(1)There is a natural transformation
E(X=T) ! ET(X);
which is an isomorphism if T acts freely on X. In particular the coefficie*
*nt ring ET(*) is an algebra
over E(*), and so it is 2-periodic.
(2)There is a natural forgetful transformation
ET(X) ! E(X):
If X is a trivial T-space then the composition
E(X) ! ET(X) ! E(X)
is the identity.
(3)ET has Thom classes and so Euler classes for complex T-vector bundles, whi*
*ch are multiplicative and
natural under pull-back. If V=X is such a bundle, then we write eT(V ) 2 E*
*T(X) for its (degree-zero)
Euler class. These are compatible with the Thom isomorphism in E in the se*
*nse that, if the T-action
on V=X is trivial, then
eT(V ) = e(V ):
(4)If L1 and L2 are complex T-line bundles, then
eT(L1 L2) = eT(L1) +F eT(L2):
Definition 2.6.2.If ET is equivariantly complex oriented as above, a homomor*
*phism ET ! ^ETof
multiplicative T-equivariant cohomology theories is a suitable localization if
(1)E^T(*) is flat over ET(*),
1.When k 6= 0, eT(C(k)) maps to a unit of ^ET(*), and
(2)The fixed-point formula (2.6.4)holds for ^ET.
In order to state the fixed-point formula, we need the following observation*
* of [AS68 ].
Lemma 2.6.3.Let ET be a suitable theory. Let S be a compact manifold with tr*
*ivial T-action, and let
V be a complex T-vector bundle over S. If the fixed-point bundle V Tis zero, th*
*en eT(V ) is a unit of ET(S).
Proof. Recall [Seg68] that the natural map
M
V (k) C(k) -!V
k2^T
is an isomorphism, where V (k) def=Hom[C(k); V ] is the evident vector bundle o*
*ver S with trivial T-action.
By applying the ordinary splitting principle to V (k), we are reduced to the ca*
*se that V = L C(k), where
L is a complex line bundle over S with trivial T-action. If ET is suitable then*
* the Euler class of V is
eT(L C(k)) = e(L) +F eT(C(k)):
Since S is a compact manifold, e(L) is nilpotent in E(S), so eT(V ) is a unit o*
*f ET(S)_because eT(C(k)) is a
unit of ET(*). *
* |__|
Now suppose that M is a compact almost-complex manifold with a compatible T-*
*action. Let
j :S ! M
denote the inclusion of the fixed-point set; it is a complex-oriented equivaria*
*nt map, with T-equivariant
normal bundle . The fixed-point formula which we require in Definition 2.6.2 is*
* the equation
*
pMF(u) = pSF _j_u_e: (2.6*
*.4)
T()
RIEMANN-ROCH AND THE FREE LOOP SPACE *
* 7
By Lemma 2.6.3 eT() is a unit of ET(S), so this localization theorem is a corol*
*lary to the projection formula
jFj*(x) = x . eT()
for the umkehr of the inclusion of the fixed-point set.
Example 2.6.5.The Borel extension
EBorel(X) def=E(XT)
of an even periodic ring theory has Thom classes eT(V ) = e(VT) for complex T-v*
*ector bundles, and the
localization defined by inverting the multiplicative subset generated by eT(C(k*
*)); k 6= 0 will be suitable.
Example 2.6.6.Let KT denote the usual equivariant K-theory. Then KT = Z[q; *
*q-1], where q is
the representation C(1), considered as a vector bundle over a point. The Euler*
* class of a line bundle is
eT(L) = 1 - L, so ^q= 1 - q. The group law is multiplicative:
Gm (x; y) = x + y - xy: (2.6*
*.7)
We have [k](^q) = 1 - qk, and consequently
^K(X) def=KT(X) KT Z((q))
is suitable.
Example 2.6.8.If
H*T(X) def=H*(XT; Q[v; v-1])
is Borel cohomology with two-periodic rational coefficients, and ^q= e(C(1)T), *
*then HT(*) ~=Q[[^q]], and
e(C(k)T) = k^q: The rational Tate cohomology
^H*(X) def=H*T(X)[^q-1]
is suitable.
Example 2.6.9.More generally, two-periodic ^K(n)T (with n finite positive) i*
*s suitable: if [p]K(n)(X) =
Xpn and k = k0ps with (k0; p) = 1 then
ns pns
[k](^q) = [k0](^qp) = k0^q + . .2.Fp((^q))
has invertible leading term. Integral lifts of K(n) behave similarly; the Cohen*
* ring [AMS98 ] of Fp((^q))
defines a completion of the Borel-Tate localization.
These coefficient rings have natural topologies, which are relevant to the c*
*onvergence of infinite products
in Corollary 6.2.3.
3. Application to the free loop space
Let ^ETbe a suitable localization (2.6.2)of an equivariantly complex oriente*
*d cohomology theory, let X be
a compact complex-oriented manifold, and let LX be its free loop space. Since L*
*X is not finite-dimensional,
the existence of an umkehr homomorphism pLXFis not clear. However, T acts on LX*
* by rotations with fixed
set X of constant loops, and Witten discovered that the fixed-point formula (2.*
*6.4)for the F-genus pLXF(1)
of LX continues to yield interesting formulae. In this section we review his ca*
*lculation.
3.1. The normal bundle to the constant loops and its Euler class. One approx*
*imates the space
C1 (S1; C) by the sub-vector space of Laurent polynomials
M
C[T*] ~= C(k) ,! C1 (S1; C):
k2T*
The tangent space of LX is C1 (S1; TX). If p 2 X is considered as a constant lo*
*op, then the tangent space
to LX at p is the T-space TLXp ~=C1 (S1; TXp). It is a T-bundle with a Laurent *
*polynomial approximation
TXp C[T*]
8 MATTHEW ANDO AND JACK MORAVA
Thus the normal bundle of the inclusion of X in LX has approximation
M
' TX C(k): (3.1*
*.1)
k2^T
If
Yd
zd+ c1zd-1+ : :+:cd = (z + xi)
j=1
is the formal factorization of the total F-Chern class of TX, then
Yd Y
eT() = (xj+F [k]F(^q)); (3.1*
*.2)
j=1k6=0
where ^q= eT(C(1)).
3.2. The fixed point formula. Applying (2.6.4) to the inclusion
X -!LX
yields the formula
2 3
Yd Y 1
pLXF(1) = pXF4 ___________5: (3.2*
*.1)
j=1k6=0xj+F [k]F(^q)
Equation (3.2.1)requires some interpretive legerdemain. For example, the lea*
*ding coefficient of
Y
(x +F [k]F(^q))
k6=0
Q
is the objectionable expression k6=0(k^q); but, as physicists say, this quant*
*ity is not directly `observable'.
For this reason, we consider the the renormalized formal product
Y (x +F [k](^q))
F(x; ^q) def=x __________:
k6=0 [k](^q)
In section 6 below we provide a natural setting for such formal products.
The fixed point formula suggests that we define the equivariant F-genus of L*
*X to be
2 3
Yd x
p"LXF(1) def=pXF4 ___j____5: (3.2*
*.2)
j=1F(xj; ^q)
3.3. Examples.
The additive group law. When F is the additive group law F becomes
Y x
Ga(x; ^q)= x (__ + 1)
k6=0k^q
Y x2
= x (1 - ____22):
k>0 k ^q
This is the Weierstrass product for ss-1^qsin^q-1ssx, so for the theory ^Hof (2*
*.6.8), formula (3.2.2) gives
2 3
dY x =2
"pLXF(1) = (2ssi=^q)d4 ___j____5 [X] :
j=1sinh(xj=2)
RIEMANN-ROCH AND THE FREE LOOP SPACE *
* 9
This is just the ^A-genus of X, up to a normalization depending on the dimensio*
*n of X. In [Ati85], Atiyah
rewrites the formal product
Y
x (x + k^q)
k6=0
as
!2
Y Y x2
x k __2- ^q2
k>0 k>0 k
Q
and invokes zeta-function renormalization [Den92 ] to replace ( k>0k)2 with 2s*
*s, yielding
Y x2
2ssx (__2- ^q2) ;
k>0k
specializing ^qto i then gives the classical expression. From our point of view*
* it's natural to think of the
Chern class ^qof C(1) as the holomorphic one-form z-1dz on the complex projecti*
*ve line, and thus to identify
^qwith its period 2ssi with respect to the equator of CP1 as in x2.1 of [Del89]*
*: the `Betti realization' of the
Tate motive Z(n) is (2ssi)nZ C.
The multiplicative group law. In the case of the equivariant K-theory ^Kof e*
*xample (2.6.6), the Euler
class of a line bundle L is eT(L) = 1 - L. Writing q for the generator C(1) of *
*T*, the Euler class of C(1) is
^q= eT(C(1)) = 1 - q:
The multiplicative group law (2.6.7)gives
eT(L) +Gm [k](^q) = 1 - qkL;
and so the formal product Gm(x; ^q) becomes
Y (1 - Lqk)(1 - Lq-k)
Gm(x; ^q)= (1 - L) ________________k-k
! k>0 (1 - q )(1 - q )
Y Y (1 - Lqk)(1 - L-1qk) (3.3*
*.1)
= L (1 - L) ________________k2:
k>0 k>0 (1 - q )
Aside for the powers of L, this is essentially the product expansion for the We*
*ierstrass oe function
Y (1 - qkL)(1 - qkL-1)
oe(L; q) = (1 - L) ________________k22 Z[L; L-1][[q]]
k>0 (1 - q )
(see for example [MT91 ] or p. 412 of [Sil94]). The infinite factor is objectio*
*nable: in the product (3.2.2)
defining the hypothetical ^K-genus of LX this factor contributes an infinite po*
*wer of topTX, but if c1X = 0
(e.g. if X is Calabi-Yau) and we are careful with the product, we can replace G*
*m with . The resulting
invariant is the Witten genus [Wit88 , AHS98 ]. Segal [Seg88] replaces the form*
*al product
Y
(1 - qkL)(1 - q-kL)
k6=0
which arises in the multiplicative case with
Y Y
( q-kL) (1 - qkL)(1 - qkL-1):
k>0 k>0
He eliminates the infinitePproduct of L's by assuming topTX trivial and he uses*
* zeta-function renormaliza-
tion to replace q- k with q-1=12.
10 MATTHEW ANDO AND JACK MORAVA
4.A Riemann-Roch formula for the quotient of a formal group by a free subg*
*roup
The starting point for this paper was the discovery that the formal products*
* (3.2.1)and (3.2.2)which
arise in applying the fixed point formula to study the F-genus of the free loop*
* space are precisely the same as
those obtained from the Riemann-Roch theorem for the quotient of F by a free cy*
*clic subgroup. We explain
this in section 4.2, after briefly reviewing finite quotients of formal groups,*
* following [Lub67 , And95 ].
4.1. The quotient of a formal group by finite subgroup. In this section, we *
*assume that F is a
formal group law over a complete local domain R of characteristic 0 and residue*
* characteristic p > 0. If A
is a complete local R-algebra, the group law F defines a new abelian group stru*
*cture on the maximal ideal
mA of A. We will refer to (mA; +F) as the group F(A) of A-valued points of F.
If H is a finite subgroup of F(R), then Lubin shows that there is a formal g*
*roup law F=H over R,
determined by the requirement that the power series
Y
fH (x) def= (x +F h) 2 R[[x]] (4.1*
*.1)
h2H
is a homomorphism of group laws F -fH-!F=H; in other words there is an equation
F=H(fH (x); fH (y)) = fH (F(x; y)):
The main point is that the power series fH is constructed so the kernel of fH a*
*pplied to F(R) is the subgroup
H.
Q
The coefficient f0H(0) = h6=0h of the linear term of fH (x) is not a unit o*
*f R, and so fH is an isomorphism
of formal group laws only over R[f0H(0)-1]: Over this ring, we might as well re*
*place fH with the strict
isomorphism
Y x +F h fH (x)
gH (x) def=x ______= _____0;
h6=0 h fH (0)
and define G to be the formal group law
-1 -1
G(x; y) = gH F(gH (x); gH (y))
over R[f0H(0)-1]; then F=H and G are related by the isomorphism
t(x) = f0H(0)x:
If F is the group law and R the ring of coefficients of a parametrized theor*
*y, then the Riemann-Roch
formula (2.5.2)for a compact complex-oriented manifold X is the equation
2 3
dY x
pXG(u) = pXF4u ___j__5 (4.1*
*.2)
j=1gH (xj)
over R[f0H(0)-1].
4.2. The case of a free cyclic subgroup. Now suppose that (E; x; F) is a par*
*ametrized theory. Let
R be the even periodic theory defined by the formula
R(X) def=E(BT x X):
The projection BT ! * gives a natural transformation E(X) ! R(X). In particular*
* the coordinate x 2 E(P)
gives a coordinate x 2 R(P). The group law F is just the group law F, considere*
*d over the E-algebra R.
What extra structure is available over R? A character 2 T* gives a map BT !*
* P, and so an R-valued
point u() of PE. As varies through the group of characters, these points assem*
*ble into a homomorphism
of groups
u:T* ! F(R): (4.2*
*.1)
RIEMANN-ROCH AND THE FREE LOOP SPACE *
* 11
It is easy to see that this is the inclusion of a subgroup. As usual we write ^*
*q= u(1) for the generator of T*
using the isomorphism T* ~=Z. We write (^q) for the subgroup u(T*) generated by*
* ^q.
The analogue of Lubin's formula (4.1.1)is
Y
f(^q)(x) = (x +F [k](^q)); (4.2*
*.2)
k2Z
and the resulting Riemann-Roch formula
2 3
Yd Y 1
pXF=(^q)(u) = pXF4u ___________5
j=106=kxj+F [k]F(^q)
has right-hand side identical to the formal product (3.2.1)arising from the fix*
*ed-point formula.
Of course this does not avoid the problems of applying the fixed-point formu*
*la to the free loop space.
From this point of view, the trouble is thatQthe quotient object F=(^q) isn't a*
* formal group; for example the
coefficient of x is the product (4.2.2)is k6=0(k^q) This particular problem is*
* fixed by using
Y x +F [k]F(^q)
g(^q)(x)= x __________
06=k2T*[k]F(^q)
= F(x; ^q):
The Riemann-Roch formula in this case is
2 3
dY x
pXG(u) = pXF4u ____j___5 (4.2*
*.3)
j=1F(xj; ^q)
with right-hand-side identical to the renormalized genus (3.2.2).
However, expanding such Weierstrass products as formal power series is still*
* highly nontrivial, as the
examples above have shown.
5.The structure of F=(^q)
Suppose now that F is a formal group law over a complete local ring E with r*
*esidue field k of characteristic
p > 0, and that F is its pullback over the power series algebra R = E[[^q]]. T*
*he coordinate defines a
homomorphism
u:Z! F(R)
(5.0*
*.4)
u(n)= [n]F(^q);
but there is no reason to expect the cokernel of this homomorphism to be a form*
*al group. However, F=(^q)
certainly makes sense as a group-valued functor of complete local R-algebras.
In section 5.1, we show by construction that the torsion subgroup of F=(^q) *
*has a natural approximation
by a representable functor. We construct a formal group scheme Tate(F) over R t*
*ogether with a natural
transformation
Tate(F)tors! F=(^q)tors
of group-valued functors, which is an isomorphism if ^qhas infinite order in F.*
* The formal group scheme
Tate(F)torsis our model for F=(^q).
Because E has finite residue characteristic, we work p-locally: the formal g*
*roup scheme Tate(F)[p1 ] is
a p-divisible group in the sense of [Tat67, Dem72 ]. The p-torsion subgroup G[p*
*1 ] of a formal group G of
finite height is a connected p-divisible group, but Tate(F)[p1 ] is not; its ma*
*ximaletale quotient is a constant
height-one p-divisible group Qp=Zp, and its connected component is just the p-d*
*ivisible group F[p1 ] of F.
In other words, there is an extension
F[p1 ] -!Tate(F)[p1 ] -!Qp=Zp
12 MATTHEW ANDO AND JACK MORAVA
of p-divisible groups over R. We will see in (5.3.1)that it is in fact the univ*
*ersal example of such an extension,
and it follows that if En is the ring which classifies lifts of a formal group *
*G of height n over an algebraically
closed field k [LT66 ], then R = En[[^q]] represents the functor which classifi*
*es lifts of a p-divisible group of
height n + 1 with connected component G[p1 ]. Thus ^qmay be viewed as a "Serre-*
*Tate parameter" in the
sense of [Kat81].
5.1. A model for the torsion subgroup of F=(^q). One difficulty in represent*
*ing the quotient F=(^q)
over E[[^q]] is that the subgroup (^q) does not act universally freely on F: co*
*nsider, for example, the spe-
cialization ^q= 0. It turns out that, as long as one restricts to the torsion s*
*ubgroup of F=(^q), this is the
only obstruction. By freeing up the action, we are able to construct a represen*
*table functor whose torsion
subgroup coincides with that of F=(^q) whenever ^qis of infinite order.
Following [KM85 , x8.7], let Tate(F) be the scheme over R = E[[^q]] defined*
* by the disjoint union
[
Tate(F) = FR x {a}:
a2Q\[0;1)
If A is a complete local R-algebra, then
Tate(F)(A) = {pairs(g; a) withg 2 F(A) anda 2 Q \ [0; 1):}
This has a group structure given by
(
(g; a) . (h; b) = (g +F h; a + b) if a + b < 1
(g +F h -F ^q; a + bi-f1)a + b 1.
By construction, Tate(F)(A) is the quotient in the exact sequence
0 -! Z -! F(A) x Q -!Tate(F)(A) -!0
n 7! ([n](^q); n) (5.1*
*.1)
(x; a)7! (x -F [[(a)](^q); ](a));
where [(a) and ](a) are the integral and fractional parts of the rational numbe*
*r a. Equivalently, Tate(F) is
the pushout of Z ! Q ! Q=Z along the homomorphism u:
Z ----! Q ----! Q=Z
? ? ?
u?y ?y ?y=
FR ----! Tate(F)----! Q=Z:
Thus Tate(F) is a kind of homotopy quotient of F by Z.
Proposition 5.1.2.Projection onto the first factor in the construction above*
* defines a natural trans-
formation
Tate(F)tors! (F=(^q))tors;
which is an isomorphism if ^qis not in F(A)tors.
Proof. Let A be a complete local R algebra, and suppose that ^qis not torsio*
*n in F(A). We see from
(5.1.1) that there is an isomorphism
Tate(F)(A) ~=F=(^q)(A) x Q
which is compatible with the projection and clearly induces an isomorphism
Tate(F)tors(A) ~=(F=(^q))tors(A):
*
*|___|
Remark 5.1.3.The Proposition holds in any context in which ^qis not torsion.*
* Another source of
examples is p-adic fields. Let S = E[1_p]((^q)). Let L be a complete nonarchime*
*dean field with norm k - k.
Given a continuous map S ! L, the group law F defines a group structure on the *
*set {v 2 L|kvk < 1}; and
RIEMANN-ROCH AND THE FREE LOOP SPACE *
* 13
we write F(L) for this group. The subgroup generated by ^qis necessarily free, *
*and the argument shows that
there is an isomorphism of groups
Tate(F)tors(L) ~=(F=(^q))tors(L):
Remark 5.1.4.If the ring E is the ring of coefficients in an even periodic r*
*ing theory, then the preceding
construction could be carried out with R = E(BT) and the homomorphism u (4.2.1)*
*. The result is an
extension
PE ! Tate(PE) ! (T* Q)=T*
of group schemes over R. Proposition 5.1.2 becomes an isomorphism
Tate(PE)tors~=((PE)R=T*)tors:
5.2. Notation. If A is an abelian group, then AT = specTA is the resulting c*
*onstant formal group
scheme over T. The category of test rings is the category of Artin local E-alge*
*bras with residue field k.
Definition 5.2.1.If G is a formal group scheme over a ring T, and A is an ab*
*elian group, let Hom[A; G]
be the group of homomorphisms
AT -!G
of formal group schemes over T. Similarly, let Ext[A; G] be the set of isomorph*
*isms classes of extensions of
formal group schemes
G -!X -!AT:
If G is a formal group scheme over E, let Hom_[A; G] and Ext_[A; G] be the func*
*tors from test rings to groups
such that
Hom_[A; G](T) def=Hom[AT; GT]
and
Ext_[A; G](T) def=Ext[AT; GT]:
Now if G is a formal group over E, pulling back over G defines a natural poi*
*nt
:G -!G x G = GG
and hence a homomorphism
u:ZG -!GG:
It is clear that this gives an isomorphism G ~=Hom_[Z; G].
Equivalently, if F is a formal group law over E, then R = E[[^q]] pro-repres*
*ents the functor Hom_[Z; F]
on the category of test rings, with universal example u(n) = [n]F(^q) (5.0.4). *
*Similarly, if E is the ring of
coefficients of an even periodic ring theory, then R = E(BT) pro-represents the*
* functor Hom_[T*; PE], with
the homomorphism u of (4.2.1)as the universal example.
5.3. Universal properties.
A universal extension. A continuous homomorphism of E-algebras from R to a t*
*est algebra T defines
an extension
FT -!(Tate(F))T -!(Q=Z)T
and hence an extension
F[p1 ]T -!(Tate(F)[p1 ])T -!(Z[1_p]=Z)T :
of torsion subgroups.
Lemma 5.3.1.The ring R pro-represents the functor Ext_[Qp=Zp; F], with Tate(*
*F) as the universal ex-
ample.
14 MATTHEW ANDO AND JACK MORAVA
Proof. In the exact sequence
Hom_[Q; F] -!Hom_[Z; F] -!Ext_[Q=Z; F] -!Ext_[Q; F];
the first and last terms are zero because p acts nilpotently on F and as an iso*
*morphism on Q. |___|
A universal p-divisible group. Any p-divisible group over a field k is natu*
*rally an extension
0-! -!et (5.3*
*.2)
of a connected group by anetale group. If the residue field k is algebraically *
*closed, then the sequence (5.3.2)
has a canonical splitting [Dem72 , p. 34]. We will be interested in the case wh*
*en ethas height one, which
is to say that it is isomorphic to the constant group scheme (Qp=Zp)k.
Tate showed [Tat67] that the functor G 7! G[p1 ] is an equivalence between t*
*he categories of formal
groups of finite height and connected p-divisible groups. Let's fix a one-dime*
*nsional formal group 0 of
height n over the algebraically closed field k and define to be the product ex*
*tension
0! ! (T* Qp=Zp)k :
Definition 5.3.3.If G is a p-divisible group over k, and if T is a test ring*
*, then a lift of G to R is a
pair (F; ffi) consisting of a p-divisible group F over R and an isomorphism
Fk ffi-!~G
=
of p-divisible groups over k. An equivalence of lifts (F; ffi) and (F0; ffi0) i*
*s an isomorphism f :F ! F0 such
that
ffi = ffi0fk:
The set of isomorphism classes of lifts of G to R will be denoted LiftsG(R). As*
* R varies, LiftsG(R) defines
a functor from test rings to sets.
Lubin and Tate construct a formal power series algebra En over the Witt ring*
* of k which pro-represents
the functor Lifts0. There is an even periodic cohomology theory with En as ring*
* of coefficients, and the
universal lift F of 0 as formal group.
Theorem 5.3.4.The ring R = En(BT) pro-represents Lifts, with universal examp*
*le Tate(F)[p1 ].
Proof. Let T be a test ring. Suppose that (H; ffl) is a lift of . Then (H0; *
*ffl0) is a lift of 0. According
to [LT66 ], there is a unique pair (f; a) consisting of a map f :En ! T and an *
*isomorphism
a:(FT; fflunivT) ~=(H0; ffl0)
of lifts of 0. On the other hand, ffletinduces an isomorphism
Het= (Het(k))T ~=(et(k))T = (T* Qp=Zp)T:
Assembling these gives an extension
FT ! H ! (T* Qp=Zp)T
which defines an isomorphism
Lifts~=Ext_[F; T* Qp=Zp] :
R pro-represents the right-hand side by Lemma 5.3.1. *
* |___|
Remark 5.3.5.The analogous result for the ordinary multiplicative group Gm i*
*s described in [KM85 ,
x8.8]. Closely related examples occur in [AMS98 ], which are motivated by pur*
*ely homotopy-theoretic
questions about Mahowald's root invariant.
RIEMANN-ROCH AND THE FREE LOOP SPACE *
* 15
5.4. Examples. If L 2 A((q)) is a unit, the formal product defining the Weie*
*rstrass function of3.3
defines an element oe(L; q) 2 A((q)). The functional equation oe(qL; q) = (-L)-*
*1oe(L; q) then implies that
the modified product
oe[L; r] def=q-[(r)([(r)+1)=2(-L)[(r)oe(L; q)
(where [(r) is the integral part of r 2 Q) satisfies the identity oe[qL; r +1] *
*= oe[L; r] and can thus be regarded
as a function from a localization of Tate(Gm ) to the Z((q))-line. Since oe(L; *
*0) = 1 - L we can think of the
modified function as a deformation of the usual coordinate at the identity of t*
*he multiplicative group.
Similarly, if we regard ^qas an element of the locally compact field C, the *
*modified sine function
s[x; r] def=ss-1^qsin^q-1(ssx - r)
satisfies the identity s[x + ^q; r + 1] = s[x; r] and so can be interpreted as *
*a function from Tate(Ga)(A) to A.
It is also a deformation of the usual coordinate on the additive group, in that*
* s[x; 0] ! 0 as ^q! 1.
In general, however, there seems to be no reason to expect that a coordinate*
* on F will extend to a
coordinate on Tate(F): our construction yields a group object, but not a group *
*law. In the two examples
above, we do have (something like) coordinates, which define interesting genera*
*: ordinary cohomology leads
to the ^A-genus, suitably normalized, if r = 0; but if r 6= 0 2 Q=Z, Cauchy's t*
*heorem (applied to a small
circle C around the origin) yields
2 3
_1_Z pX 4j=dY___^q-1ssxjz_5 z-d-1dz = (___ss_)dO(X) :
2ssiC Ga j=1sin^q-1ss(xjz + r^q) ^qsinssr
The K-theory genus extends similarly, but the resulting function is just (-1)dq*
*-d[(r)([(r)+1)=2times the
standard elliptic genus.
Any multiplicative cohomology theory E can be described as taking values in *
*a category of sheaves
over specE(*), and the Borel extension of such a theory takes values in sheaves*
* over spec(E(*)[[^q]]). The
construction in 5.1 of the Tate group as a disjoint union of copies of such aff*
*ines implies that a theory E
with formal group F has a natural extension to an equivariant theory taking val*
*ues in a category of sheaves
over the group object Tate(F). Similarly, a suitable localization of the Borel *
*extension defines an equivariant
theory taking values in sheaves over a suitable localization of Tate(F). This r*
*esembles (but is easier than)
the constructions of [GKV , Gro94, RK99 ], for here we're only patching togeth*
*er Borel extensions.]
6.Prospectra and equivariant Thom complexes
Cohen, Jones, and Segal show that pro-objects in the category of spectra are*
* the appropriate context in
which to study Thom complexes, and so umkehr maps, for semi-infinite vector bun*
*dles. In this section we
observe that the ideas of this paper fit very naturally into their framework.
6.1. Thom prospectra. If V and W are complex vector bundles over a space X, *
*we extend the
notation for Thom isomorphisms in 2.4.1 by writing
ffVWdef=ffV O (ffW )-1:E(XW ) ! E(XV ) :
`Desuspending' by V W then gives a homomorphism
ff-W-V:E(X-V ) ! E(X-W ) :
The inclusions of a filtered vector bundle
V : 0 = V0 V1 . . .
define maps
iVn: XVn-1! XVn
of Thom spectra, which desuspend to define a pro-object
Vn
X-V def={. .!.X-Vn i--!X-Vn-1! : :}:
16 MATTHEW ANDO AND JACK MORAVA
in the category of spectra. The E-cohomology of X-V is the colimit
E(X-V) def=colimnE(X-Vn);
as in the appendix to [CJS95 ].
Lemma 6.1.1.On cohomology, the homomorphism induced by iVn is e(Vn=Vn-1)ff-V*
*n-Vn-1. |___|
Example 6.1.2.For any integer n, let
M
In = C(k);
n|k|>0
thus I = colimnIn is a filtered T-vector bundle over a point. More generally, i*
*f V is a complex vector bundle
over a space X, let
IV def=V I
be the corresponding filtered T-vector bundle. In this notation the Laurent app*
*roximation (3.1.1)to the
formal normal bundle of the constant loops in LX is ITX.
We write jn for the map
jn = iInV:X-InV ! X-In-1V:
Example 6.1.3.If Vn is the sum of n copies of C(1), considered as a T-vector*
* bundle over a point, then
the Borel construction on V is the Thom prospectrum for CP11constructed in [CJS*
*95 ].
6.2. Equivariant cohomology. Suppose now that ^ETis a suitable extension of *
*an equivariantly ori-
ented theory ET, and let X be a finite complex with trivial T-action, as above.*
* As n varies, the Thom
isomorphisms
ffInV:ET(X) ET-!~(X-InV)
=
are not compatible with the maps jn; but this can be cured over ^ETby a suitabl*
*e renormalization. By (2.6.3)
the class
Y
un(V ) def=e(V In) = eT(V C(k)):
0<|k|n
is a unit of ^ET(X), and the homomorphism
!n(V ):E^T(X) -!^ET(X-InV)
defined by the formula !n(V ) = un(V )ff-InV is an isomorphism.
Theorem 6.2.1.The diagram
^ET(X) _______ E^T(X)
? ?
!n-1(V?)y ?y!n(V )
E^T(X-In-1V)-jn---!^ET(X-InV)
commutes; in particular, the maps !n(V ) assemble into a "Thom isomorphism"
~= -IV
!V :^ET(X) -!E^T(X ) :
*
*|___|
If V and W are two complex vector bundles over X, then (as with the usual Th*
*om isomorphism) we
define
~= -IV
!VW:^ET(X-IW ) -!E^T(X ):
RIEMANN-ROCH AND THE FREE LOOP SPACE *
* 17
Corollary 6.2.2.If the vector bundle V has rank d, then the relative isomorp*
*hism
!Vd:^ET(X-Id_) ! ^ET(X-IV)
is given by the formula
0 1
Yd Y x + [k](^q)
!Vd= @ _j_F_____A ff-IV-Id_;
j=1k6=0 [k](^q)
where the xj are the terms in the formal factorization (2.4.2)of the total F-Ch*
*ern class of V . |___|
The terms in this product are well-defined at any finite stage, if not in th*
*e limit. If ET is ^Hor ^K(and
c1(V ) = 0), then the infinite products converge and we have the
Corollary 6.2.3.If the vector bundle V has rank d, then the diagram
-Id_
E^T(Xd_)ffd_----!^ET(X-Id_)
? ?
^ffVd_?y ?y!Vd_
-IV
E^T(XV )ffV----!^ET(X-IV) :
commutes; where
^ffVd_def=F(V_)cffVd_:
d(V )
More precisely, the assertion is that these examples are naturally topologiz*
*ed, and that at any finite_
stage the diagram commutes modulo error terms which converge in this topology t*
*o zero as n grows. |__|
In other words, in a suitable theory with group law F the isomorphism !Vd_is*
* very much like a Thom
isomorphism ^ffVd_for a theory with F replaced by its extension Tate(F). From t*
*his point of view, the fact
that Witten's formula for the ff-genus of LX equals the ^ff-genus of X can be i*
*nterpreted as saying that the
inclusion j :X ! LX behaves as if there is a cohomological analogue
j*:E^T(X-ITX ) ! ^ET(LX)
of the Thom collapse map, with an associated umkehr jF = j*!TX satisfying the p*
*rojection formula
jFj*(x) = x . eT() :
Perhaps the intuition underlying the physicists' interest in elliptic cohomolog*
*y is that a reasonable equivariant
theory applied to the geometrical object X-ITX captures more information than t*
*hat theory does, when
applied directly to X. Thus the equivariant K-theory of this formal neighborhoo*
*d of X is the Tate elliptic
cohomology of X. This seems to be related to the recent construction (by Kontse*
*vich and others) of new
invariants for singular complex algebraic varieties, by considering them as var*
*ieties over the Laurent series
field C((q)).
6.3. Polarizations. In the preceding account, the role of the rational param*
*eter in the construction of
Tate(F) is geometrically unmotivated, because we have ignored some issues conne*
*cted with the polarization
[CJS95 ] of the loopspace.
Such a structure is an equivalence class of splittings of the tangent bundle*
* of LX into a sum of positive-
and negative-frequency components: if X is complex-oriented (e.g. symplectic, w*
*ith a choice of compatible
almost-complex structure), then the composition
LX ! L(BU) = U x BU ! U=SO
of the map induced by the classifying map for the tangent bundle of X with the *
*projection to a classifying
space for such splittings defines a canonical polarization.
18 MATTHEW ANDO AND JACK MORAVA
The universal cover of the free loopspace. If X is simply-connected, the fun*
*damental group of its free
loopspace will be isomorphic to H2(X; Z), which will be nontrivial in general. *
*There is thus good reason
to consider the simply-connected cover gLXof the loopspace: this can be defined*
* as the space of smooth
maps of a two-disk to X, modulo the relation which identifies two maps if their*
* restrictions to the boundary
circle agree, and if furthermore their difference, regarded as an element of th*
*e deck-transformation group
ss2(X) = H2(X; Z), is null-homotopic. The circle acts on gLXby rotating loops, *
*as does the fundamental
group of LX, and in general the fixed-point set
gLXT~=H2(X; Z) x X
will have many components. The choice of a basepoint defines a lift of the cano*
*nical polarization to a map
gLX! R x SU=SO
which restricts to a locally constant map
H2(X; Z) x X ! R x SU=SO ! R
defined by evaluating ff 2 H2(X; Z) on the first Chern class of X. We think of *
*the polarization as defining
the zero-frequency modes in the Fourier decomposition of small loops near a fix*
*ed-point component, so that
shifting by fi 2 H2(X; Z) gives an isomorphism
T T
TLgXx;ff~=TLgXx;ff+fi C() :
T-equivariant Picard groups and orientations. We can regard the cohomology o*
*f a space Y as a sheaf of
rings over the zero-dimensional scheme
(ss0Y )Z def=specH0(Y; Z) :
Similarly, the set of equivalence classes of T-line bundles over a T-space X is*
* naturally isomorphic to
H2(YT; Z), which can be interpreted as a sheaf of groups over (ss0Y )Z, the fib*
*er above component Yibeing
the constant group scheme
coker[H2T(*; Z) ! H2T(Yi)] ;
when the circle action on X is trivial, this is just a complicated way of index*
*ing the summands of H2(Y; Z).
An orientation on a complex-oriented cohomology theory E with formal group F*
* defines a natural
homomorphism from the Picard group of line bundles over Y to the group of E(Y )*
*-valued points of F. This
suggests that a complex orientation for the Tate extension of an equivariantly *
*complex-oriented theory should
be defined as a natural transformation from the Picard group of T-line bundles *
*over X to Tate(F)(E(Y )),
both regarded as schemes over (ss0Y )Z; by restriction such a transformation de*
*fines a map
ss0Y ! Q=Z :
In other words, such a generalized orientation assigns to a component of Y and *
*a T-line bundle over it, a
characteristic class in the cohomology of the component, together with an r 2 Q*
* depending on the component,
which shifts by an integer when the bundle is twisted by a character.
T
When Y is gLX, the polarization defines a very natural map of this sort, whi*
*ch sends ff to .
In more general situations, e.g. in Givental's work [Vo99 ] on the quantum coh*
*omology of a symplectic
manifold (X; !) the evaluation map ff 7! plays a similar role; but in t*
*his generalization, ! no longer
needs to be an integral class.
7. Concluding remarks
The parts of this paper which deal principally with the fixed point formula *
*on the free loop space are
formulated in terms of a general equivariant cohomology theory ET, but those wh*
*ich relate to the quotient
F=(^q) use only the group law F of the theory R(X) = E(BT x X), and so essentia*
*lly use the Borel theory
EBorel. We do this because in that case we can be more specific about our const*
*ructions. In good cases one
can hope to do better.
RIEMANN-ROCH AND THE FREE LOOP SPACE *
* 19
Specifically, suppose that the multiplication homomorphism T x T ! T induces*
* a map
ET ! ETxT
and that one has an isomorphism ETxT(*) ~=ET(*) ET(*). The upshot is a group s*
*cheme G = specET(*)
E
over E, such that the formal group PE associated to E(P) is the completion of G*
*. Indeed [GKV , Gro94],
one expects in general that there is an abelian group scheme G which is the mor*
*e fundamental object in
T-equivariant E-theory, with ET(*) as structure sheaf.
In any case, as remarked in x5.2, over GG = G x G there is a natural map
ZG ! GG; (7.0*
*.1)
and the natural map GG ! Hom[Z; G] is an isomorphism. One could consider the gr*
*oup Tate(G) over G,
fitting into a short exact sequence
GG -!Tate(G) -!Q=Z:
The group Tate(PE) considered in the main text would then arise from Tate(G) by*
* completing in both copies
of G.
This is the situation in K-theory. Over the ring KT = Z[q; q-1] one has a ho*
*momorphism
u:Z ! GmKT
n 7! qn
and this gives rise to a group Tate(Gm ) over KT. Over the completion KT ! K(BT*
*) = Z[[^q]] one recovers
the group Tate(Gm ) considered in the main text. On the other hand, the group *
*Tate(Gm )torsbecomes
isomorphic to the torsion subgroup of the classical Tate curve Tate(q) already *
*over the suitable localization
K^= Z((q)) of (2.6.6), where u is the inclusion of a sub-groupscheme. See [KM85*
* , x8.8] for details.
Interestingly enough, in the case of K-theory this is the solution to the pr*
*oblem of the infinite products.
We have purposefully written the equivariant Euler class of L C(k) as e(L) +F *
*[k]F(^q), as in any finite
situation the formula
1 - L1L2= (1 - L1) + (1 - L2) - (1 - L1)(1 - L2)
makes it possible to calculate the Euler class of a vector bundle using the for*
*mal multiplicative group, i.e.
the right-hand side. However, in order to calculate the infinite product Gm(x; *
*^q) (3.3.1), one is forced to
use the left-hand side. Thus from this point of view the renormalization is han*
*dled by knowing about the
global multiplicative group, instead of merely its formal completion.
It seems reasonable to hope that the rich theory of T-equivariant ring spect*
*ra will provide additional
examples of such ET. If ET is a complex-oriented T-equivariant ring spectrum, t*
*hen the cohomology ET(CP)
(where CP is the space of lines in the ambient T-universe) carries the structur*
*e of a T-equivariant formal
group law (in the sense of [CGK ]). This is, among other things, a formal grou*
*p law F and a homomorphism
v :T* ! F(ET(CP)):
In the situation we are considering, F is the completion of the group G = specE*
*T, and the homomorphism
v is obtained from the homomorphism u by completion.
References
[Ada88]Douglas Adams. Dirk Gently's holistic detective agency. Pocket Books, 19*
*88.
[Ada74]J. Frank Adams. Stable homotopy and generalised homology. Univ. of Chica*
*go Press, 1974.
[And95]Matthew Ando. Isogenies of formal group laws and power operations in the*
* cohomology theories En. Duke Math. J.
79:423-485, 1995
[AHS98]Matthew Ando, Michael J. Hopkins, and Neil P. Strickland. Elliptic spect*
*ra, the Witten genus, and the theorem of
the cube, 1998. Preprint available at hopf.math.purdue.edu.
[AMS98]Matthew Ando, Jack Morava, Hal Sadofsky. Completions of Tate Z=(p)-cohom*
*ology of periodic spectra. Geometry
and Topology 2:145-174, 1998.
[Ati85]Michael F. Atiyah. Circular symmetry and stationary phase approximation.*
* Asterisque, 132:43-60, 1985.
[AS68]M. F. Atiyah and G. B. Segal. The index of elliptic operators. II. Annals*
* of Mathematics, 87:531-545, 1968.
20 MATTHEW ANDO AND JACK MORAVA
[CJS95]R. L. Cohen, J. D. S. Jones, and G. B. Segal. Floer's infinite-dimension*
*al Morse theory and homotopy theory. In The
Floer memorial volume, pages 297-325. Birkh"auser, Basel, 1995.
[CGK]Michael Cole, Igor Kriz, John Greenlees. Equivariant formal groups. Prepri*
*nt available at hopf.math.purdue.edu
[Del89]Pierre Deligne. Le groupe fondamental de la droite projective moins troi*
*s points. In Galois groups over Q. ed. Y. Ihara
et al (Berkeley, CA, 1987), 79-297, Springer, New York, 1989.
[Dem72]Michel Demazure. Lectures on p-divisible groups, Lecture Notes in Mathem*
*atics 302. Springer, 1972.
[Den92]Christopher Deninger. Local L-factors of motives and regularized determi*
*nants. Inventiones Math> 107:135-150, 1992.
[DHS88]Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith. Nilpotence *
*and stable homotopy theory I. Annals of
Mathematics, 128:207-241, 1988.
[Dye69]Eldon Dyer. Cohomology theories. W. A. Benjamin, Inc., New York-Amsterda*
*m, 1969. Mathematics Lecture Note
Series.
[GKV]Viktor Ginzburg, Mikhail Kapranov, Eric Vasserot. Elliptic algebras and eq*
*uivariant elliptic cohomology Preprint avail-
able at xxx.lanl.gov
[Gro94]Ian Grojnowski. Delocalized equivariant elliptic cohomology. Unpublished*
* manuscript, 1994.
[HS98]Michael J. Hopkins and Jeffrey H. Smith. Nilpotence and stable homotopy t*
*heory II. Annals of Mathematics, 148:1-49,
1998
[Kat81]Nicholas M. Katz. Serre-Tate local moduli. In Surfaces algebriques: semi*
*naire de geometrie algebrique d'Orsay 1976-
1978, Lecture Notes in Mathematics 868 Springer, 1981.
[KM85]Nicholas M. Katz and Barry Mazur. Arithmetic moduli of elliptic curves, A*
*nnals of Mathematics Studies 108. Princeton
University Press, 1985.
[LT66]Jonathan Lubin and John Tate. Formal moduli for one-parameter formal lie *
*groups. Bull. Soc. math. France, 94:49-60,
1966.
[Lub67]Jonathan Lubin. Finite subgroups and isogenies of one-parameter formal L*
*ie groups. Annals of Math., 85:296-302,
1967.
[Mor85]Jack Morava. Noetherian localizations of categories of cobordism comodul*
*es. Annals of Mathematics, 121:1-39, 1985.
[MT91]B. Mazur and J. Tate. The p-adic sigma function. Duke Math. J., 62(3):663*
*-688, 1991.
[Qui71]Daniel M. Quillen. Elementary proofs of some results of cobordism theory*
* using Steenrod operations. Advances in
Mathematics, 7:29-56, 1971.
[Ros99]Ioanid Rosu. Equivariant elliptic cohomology and rigidity. Preprint avai*
*lable at hopf.math.purdue.edu
[RK99]Ioanid Rosu and Allen Knutson. Equivariant K-theory and equivariant cohom*
*ology. Preprint available at
hopf.math.purdue.edu
[Seg68]Graeme Segal. Equivariant K-theory. Inst. Hautes Etudes Sci. Publ. Math.*
* No., 34:129-151, 1968.
[Seg88]Graeme Segal. Elliptic cohomology. Seminaire Bourbaki, (695), 1988.
[Sil94]Joseph Silverman. Advanced topics in the arithmetic of elliptic curves, *
*Graduate Texts in Mathematics 151. Springer,
1994.
[Str99]Neil P. Strickland. Formal schemes and formal groups. In J. P. Meyer, J.*
* Morava, and W. S. Wilson, editors, Homotopy-
invariant algebraic structures: in honor of J. M. Boardman, Contemporary mathe*
*matics 239 American Mathematical Society,
1999.
[Tat67]John Tate. p-divisible groups. In T.A. Springer, editor, Proceedings of *
*a Conference on Local Fields (Driebergen).
Springer, 1967.
[Vo99]Claire Voisin Mirror symmetry. Translated from the 1996 French original b*
*y Roger Cooke, American Mathematical
Society, Providence, RI, 1999.
[Wit88]Edward Witten. The index of the Dirac operator in loop space. In Peter S*
*. Landweber, editor, Elliptic curves and
modular forms in algebraic topology, Lecture Notes in Mathematics 1326 Springe*
*r, 1988
The University of Illinois at Urbana-Champaign
E-mail address: mando@math.uiuc.edu
The Johns Hopkins University
E-mail address: jack@math.jhu.edu