ContemporaryMathematics A renormalized Riemann-Roch formula and the Thom isomorphism for the free 1 loop space 0 0 2 Matthew Ando and Jack Morava n a J Abstract. Let E be a circle-equivariant complex-orientable cohomology theory. We show that the fixed- pointformulaappliedtothefreeloopspaceofamanifoldX canbeunderstoodasaRiemann-Rochformula 4 forthequotient oftheformalgroupofE byafreecyclicsubgroup. Thequotient isnotrepresentable, but 1 (locallyatp)itsp-torsionsubgroupis,byap-divisiblegroupofheightonegreaterthantheformalgroupof E. ] T A . h I believe in the fundamental interconnectedness of all things. t —Dirk Gently [Ada88] a m [ 1. Introduction 1 v Let T denote the circle group, and, if X is a compact smooth manifold, let LX d=ef C∞(T,X) denote 1 its free loop space. The group T acts on LX, and the fixed point manifold is again X, considered as the 2 subspaceofconstantloops. Inthe1980’s,Wittenshowedthatthefixed-pointformulainordinaryequivariant 1 cohomology,applied to the free loop space LX of a spin manifold X, yields the index of the Dirac operator 1 (i.e. the Aˆ-genus) of X—a fundamentally K-theoretic quantity [Ati85]. He also applied the fixed-point 0 1 theorem in equivariant K-theory to a Dirac-like operator on LX to obtain the elliptic genus and “Witten 0 genus” of X [Wit88]—quantities associated with elliptic cohomology. / h Among homotopytheorists,these developments generatedconsiderableexcitement. The chromaticpro- at gram organizes the structure of finite stable homotopy types, locally at a prime p, into layers indexed by m nonnegativeintegers. The nth layeris detected by a family of cohomologytheoriesE ; rationalcohomology, n K-theory,andelliptic cohomologyaredetecting theoriesforthe firstthree layers[Mor85, DHS88, HS98]. : v i ThegeometryandanalysisrelatedtorationalcohomologyandK-theoryarereasonablywell-understood, X but forn≥2andfor elliptic cohomologyinparticular,verylittle is known. Witten’s workprovidesamajor r suggestion: for n=1 and n=2 his analysis gives a correspondence a analysis underlying E analysis underlying E n ↔ n−1 (1.0.1) applied to X applied to LX. This paper represents our attempt to understand why Witten’s procedure appears to connect the chro- maticlayersinthemannerof (1.0.1). Todothisweconsiderverygenerallythe fixed-pointformulaattached 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. 2000 Mathematics Subject Classification. Primary57R91,55N20; Secondary14L05, 19L10,55P92. Keywordsandphrases. Freeloopspace,fixed-pointformula,quotientsofformalgroups,Riemann-Roch,equivariantThom isomorphism,prospectra. BothauthorsweresupportedbytheNSF. (cid:13)c0000 (copyright holder) 1 2 MATTHEW ANDO AND JACK MORAVA Ourfirstresultisthatthefixed-pointformulaofasuitableequivariantextensionofE (Borelcohomology isfine,asistheusualequivariantK-theory)appliedtothe freeloopspaceyieldsaformulawhichisidentical to the Riemann-Roch formula for the quotient F/(qˆ) of the formal group law F by a free cyclic subgroup (qˆ) (compare formulae (3.2.2) and (4.2.3)). The quotient F/(qˆ) is not a formal group, so to understand its structure, we work p-locally and study its p-torsion subgroup F/(qˆ)[p∞]. We construct a group Tate(F) with a canonical map Tate(F)→F/(qˆ), whichinducesanisomorphismoftorsionsubgroupsinasuitablesetting. Oursecondresultisthatthegroup Tate(F)[p∞] is a p-divisible group, fitting into an extension F[p∞]→Tate(F)→Q /Z p p of p-divisible groups. If the height of F is n, then the height of Tate(F)[p∞] is n+1, but its ´etale quotient has height 1. In a sense we make precise in §5.3, it is the universal such extension. Thusthefixed-pointformulaonthefreeloopspaceinterpolatesbetweenthechromaticlayersinthesame waythatp-divisiblegroupsofheightn+1with´etalequotientofheight1interpolatebetweenformalgroupsof heightnandformalgroupsofheightn+1. Thisisdiscussedinmoredetail,fromthehomotopy-theoreticpoint 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 in rational cohomology produces K-theoretic genera because of the exponential exact sequence 0→Z→C→C× →1 (1.0.2) expressingthe multiplicative group(K-theory)as the quotient of the additive group(ordinarycohomology) by a free cyclic subgroup;while his work in K-theoryproduces elliptic generabecause ofthe exact sequence 0→qZ →C× →C×/qZ →1 (1.0.3) (where q is a complex number with |q|<1), expressing the Tate elliptic curve C×/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 topology. Grojnowski con- structs fromequivariantordinarycohomologyacomplexT-equivariantelliptic cohomologyusing the elliptic curve C/Λ 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. Grojnowski’s ideas applied to the multiplicative sequence (1.0.3) give a construction of complex T-equivariant elliptic cohomology based on equivariant K-theory; details will appear elsewhere. Completing this circle, Rosu has used the quotient (1.0.2) to give a construction of complex equivariant K-theory [Gro94, Ros99, RK99]. Severaloftheformulaeinthispaperinvolveformalinfiniteproducts;seeforexample(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 ν of X in LX (3.1.2). Fromthis pointofview,the problemis thatthe bundle ν is infinite-dimensional,soit doesnothave aThomspectrumintheusualsense. However,ν hasahighlynontrivialcircleaction,whichdefinesalocally finite-dimensional filtration by eigenspaces. Following the programsketched in [CJS95], we construct from this filtration a Thom pro-spectrum, whose Thom class is the infinite product. In the particular cases of the additive and multiplicative formal groups (n = 1,2 above), one can also control the infinite products by replacing them with products which converge to holomorphic functions on C; this construction of elliptic functions goes back to Eisenstein. We are grateful to Kapranov for pointing out to us that Eistenstein considered the the analogous problem for n > 2. In [Eis44] he described the difficulty ofinterpreting suchinfinite products. He wentonto hint that he perceiveda useful approach,and concluded the following. Die Functionen, zu welchen man auf diesen Wege gefu¨hrt wird, scheinen sehr merkwu¨rdige Eigenschaften zu besitzen; sie er¨offnen ein Feld, auf dem sich Stoff zu den reichhaltigsten Untersuchungen darbietet, und welches der eigentliche Grund und Boden zu sein scheint, auf welchem die schwierigsten Theile der Analysis und Zahlentheorie ineinander greifen. RIEMANN-ROCH AND THE FREE LOOP SPACE 3 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 affine schemes. For example the “formal line” Aˆ1 d=efcolimspecZ[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, Aˆ1×Aˆ1 =colimspec Z[x]/(xn)⊗Z[y]/(ym) . In particular a formal group scheme means an abel(cid:0)ian group in the catego(cid:1)ry of formal schemes. A formal groupschemewhoseunderlyingformalschemeisisomorphictotheformalschemeAˆ1 iscalledacommutative one-dimensional formal Lie group. We shall simply call it a formal group. Thefirstreasonforconsideringformalschemesisthatformalgroupsarenotquitegroupsinthecategory of affine schemes, because a group law F(s,t)=s+t+···∈R[[s,t]] over a ring R gives a diagonal R[[s]]−→R[[s,t]]∼=R[[s]]⊗ˆR[[t]] only to the completed tensor product. The secondreasonforconsideringformalschemesisthat,ifGisanaffinegroupscheme,thenitstorsion subgroupG is a formal scheme (the colimit of the affine schemes G[N] of torsionof order N), but not in tors general a scheme. If X is a formalscheme overR, andS is anR-algebra,then X will denote the resulting formal scheme S over S. 2. The umkehr homomorphism and an ungraded analogue 2.1. Let E be a complex-oriented multiplicative cohomology theory with formal group law F, and let h: X → Y be a proper complex-oriented map of smooth finite-dimensional connected manifolds, of fiber dimension d = dimX −dimY. The Pontrjagin-Thom collapse associates to these 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 our attention to even periodic cohomology theories E. The examples show (3.3) that this amounts to measuring quantities relative to the vacuum. 2.2. Even periodic ring theories. Let E be a cohomology theory. If X is a space, 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∗(Σ∞X ), where X denotes the union + + of X and a disjoint basepoint. The reduced cohomology of X will be denoted E(X). Let ∗ denote the one-point space. A cohomology theory E with commutative multiplication is even if Eodd(∗)=e0. It is periodic if E2(∗) containsaunitofE∗(∗). IfE isanevenperiodictheory,thenwewriteE(X)forE0(X)andE forE0(∗). We sometimeswriteX =specE(X)forthespectrum,inthesenseofcommutativealgebra,ofthecommutative E ring E(X). A space X is even if H (X) is a free abelian group, concentrated in even degrees. In that case the ∗ natural map colimF →X , (2.2.1) E E 4 MATTHEW ANDO AND JACK MORAVA where F is the filtered system of maps of finite CW complexes to X, is an isomorphism. This gives X the E structure of a formal scheme. The functor X 7→ X from even spaces to formal schemes over E preserves E finite products and coproducts: if X and Y are two even spaces, then (X ×Y) ∼=X ×Y ∼=specE(X)⊗ˆE(Y). E E E Here ⊗ˆ refersto the completionof the tensorproduct with respect to the topologydefined by the filtrations of E(X) and E(Y). 2.3. Orientationsand coordinates. LetP d=efCP∞ betheclassifyingspaceforcomplexlinebundles. Let m: P ×P →P be the map classifying the tensor product of line bundles. It induces a map P ×P −m−→E P , E E E which makes P a formal group scheme over E. Of course it is a formal group: let i: S2 → P be the map E classifying the Hopf bundle. A choice of element x∈E(P) such that v =i∗x∈E(S2)∼=E−2(∗) is a unit is called a coordinate on P . There is then an isomorphism E E(P)∼=eE[[x]], e which determines a formal group law F over E by the formula F(x,y)=m∗x∈E(P ×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∈E2(P) such that i∗u=Σ2(1), (2.3.1) e where Σ is the suspension isomorphism [Ada74]. A coordinate x thus determines an orientation u=v−1x. Definition 2.3.2. We shalluse the notation(E,x,F) to denote anevenperiodic cohomologytheoryE with coordinate x and group law F. We shall call such a triple a parametrized theory. 2.4. Thom isomorphism. An orientationu∈E2(P) gives the usual Thom classes and characteristic classes for complex vector bundles. If k is an integer, let k denote the trivial complex vector bundle of rank k. If X is a connected space and V is a complex vectoer bundle of rank d over X, then we write XV d=efΣ∞(P(V ⊕1)/P(V)) forthe suspensionspectrumofits Thomspace,withbottomcellindegree2d. We write αV forthe Thom usual isomorphism αV : E∗(X)∼=E∗+2d(XV). usual In the same way, a coordinate x∈E(P) gives rise to a Thom isomorphism αV : E(X)∼=E(XV). e If v =i∗x∈E(S2) is the associated orientation, the isomorphisms α and α are related by the formula usual αV =vrankVαV . e usual Remark 2.4.1. One effect of condition (2.3.1)is that αd coincides with the suspensionisomorphism usual αd =Σ2d: E∗(X)∼=E∗+2d(Xd). usual The Thom isomorphism α defined by a coordinate chooses v ∈ E(S2) ∼= E(∗1) as α1. Thus α may be usual viewed as a composition of Thom isomorphisms αV : E(Xd)−(−α−d)−−→1 E(X)−α−→VeE(XV). usual RIEMANN-ROCH AND THE FREE LOOP SPACE 5 If ζ: Σ∞X →XV denotes the zero section, then we write + e (V)d=efζ∗αV (1)∈E2d(X) usual usual e(V)d=efζ∗αV(1)∈E(X) for the usual and degree-zero Euler classes of V; these are related by the formula e(V)=vrankVe (V). usual If U(n) denotes the unitary group and T is its maximal torus of diagonal matrices, 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[[x ,...,x ]], 1 n with W acting as the permutationgroup Σ on the x ’s. Thus we can define degree-zeroChern classes c in n i i E(BU(n)) by the formula n n c zn−i = (z+x ). (2.4.2) i i i=0 i=0 X Y If F is the group law resulting from the coordinate x, then we call the c the “F–Chern classes”. i Returning to the map h X −→Y, as in (2.1), we can now define an umkehr map E(X)−h−→F E(Y), using the degree-zero Thom isomorphism α. We write F to indicate the dependence on the coordinate. Definition 2.4.3. If X is any manifold, we denote by pX the map pX X −−→∗. If E is an even periodic theory with group law F, then its F–genus is the element pX(1) of E. F 2.5. The Riemann-Roch formula. The Riemann-Roch formula compares the umkehr homomor- phisms h and h of two coordinates with formal group laws F and G, related by an isomorphism F G θ: 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 dimension 2d, then d x j h (u)=h u· , (2.5.2) G F  θ(x ) j j=1 Y where the x are the terms in the factorization   i d zd+c zd−1+...+c = (z+x ) 1 d i j=1 Y 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∈E multiplies the umkehr homomorphismby ud; by such a renormalization,we can always assume that θ is a strict isomorphism. 2.6. The fixed-point formula. 6 MATTHEW ANDO AND JACK MORAVA Notations for circle actions. Let T denote the circle group R/Z. If X is a T-space then we write X d=efET×X T T for the Borel construction and XT for the fixed-point set. Let T∗ = Hom[T,C×] be the character group of T; we will also write Tˆ =T∗−{1} for the set of nontrivial irreducible representations. For k ∈T∗, let C(k) be the associated one-dimensionalcomplex representation. There is then an associatedcomplex line bundle C(k) over BT. T It is convenient to choose an an isomorphism T∗ ∼= Z; this determines, in particular, an isomorphism BT ∼= CP∞. For k ∈ Z we have C(k) = C(1)⊗k and C(k) = C(1)⊗k. If qˆ∈ E(BT) is the Euler class of T T C(1) , then the Euler class of C(k) is [k](qˆ). T T Equivariant cohomology. Definition 2.6.1. Let (E,x,F) be a parametrized theory. A T-equivariant cohomology theory E is T an extension of (E,x,F) if (1) There is a natural transformation E(X/T)→E (X), T which is an isomorphism if T acts freely on X. In particular the coefficient ring E (∗) is an algebra T over E(∗), and so it is 2-periodic. (2) There is a natural forgetful transformation E (X)→E(X). T If X is a trivial T-space then the composition E(X)→E (X)→E(X) T is the identity. (3) E hasThomclassesandsoEulerclassesforcomplexT-vectorbundles, whicharemultiplicative and T naturalunder pull-back. If V/X is such a bundle, then we write e (V)∈E (X) for its (degree-zero) T T Eulerclass. These arecompatible withthe ThomisomorphisminE in the sense that, ifthe T-action on V/X is trivial, then e (V)=e(V). T (4) If L and L are complex T-line bundles, then 1 2 e (L ⊗L )=e (L )+ e (L ). T 1 2 T 1 F T 2 Definition 2.6.2. If E is equivariantly complex oriented as above, a homomorphism E → Eˆ of T T T multiplicative T-equivariant cohomology theories is a suitable localization if (1) Eˆ (∗) is flat over E (∗), T T 1. When k 6=0, e (C(k)) maps to a unit of Eˆ (∗), and T T (2) The fixed-point formula (2.6.4) holds for Eˆ . T In order to state the fixed-point formula, we need the following observation of [AS68]. Lemma 2.6.3. Let E be a suitable theory. Let S be a compact manifold with trivial T–action, and let T V be a complex T–vector bundle over S. If the fixed-point bundle VT is zero, then e (V) is a unit of E (S). T T Proof. Recall [Seg68] that the natural map V(k)⊗C(k)−→V Mk∈Tˆ RIEMANN-ROCH AND THE FREE LOOP SPACE 7 is an isomorphism, where V(k) d=ef Hom[C(k),V] is the evident vector bundle over S with trivial T–action. By applying the ordinary splitting principle to V(k), we are reduced to the case that V =L⊗C(k), where L is a complex line bundle over S with trivial T-action. If E is suitable then the Euler class of V is T e (L⊗C(k))=e(L)+ e (C(k)). T F T Since S is a compact manifold, e(L) is nilpotent in E(S), so e (V) is a unit of E (S) because e (C(k)) is a T T T unit of E (∗). T 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 equivariant map, with T–equivariant normal bundle ν. The fixed-point formula which we require in Definition 2.6.2 is the equation j∗u pM(u)=pS . (2.6.4) F F e (ν) (cid:18) T (cid:19) ByLemma2.6.3e (ν)isaunitofE (S),sothislocalizationtheoremisacorollarytotheprojectionformula T T j j∗(x)=x·e (ν) F T for the umkehr of the inclusion of the fixed-point set. Example 2.6.5. The Borel extension def E (X) = E(X ) Borel T of an even periodic ring theory has Thom classes e (V) = e(V ) for complex T-vector bundles, and the T T localization defined by inverting the multiplicative subset generated by e (C(k)),k 6=0 will be suitable. T Example 2.6.6. Let K denote the usual equivariant K-theory. Then K = Z[q,q−1], where q is T T the representation C(1), considered as a vector bundle over a point. The Euler class of a line bundle is e (L)=1−L, so qˆ=1−q. The group law is multiplicative: T G (x,y)=x+y−xy. (2.6.7) m We have [k](qˆ)=1−qk, and consequently Kˆ(X)d=efK (X)⊗ Z((q)) T KT is suitable. Example 2.6.8. If H∗(X)d=efH∗(X ;Q[v,v−1]) T T is Borel cohomology with two-periodic rational coefficients, and qˆ = e(C(1)T), then HT(∗) ∼= Q[[qˆ]], and e(C(k) )=kqˆ. The rational Tate cohomology T Hˆ∗(X)d=efH∗(X)[qˆ−1] T is suitable. Example 2.6.9. Moregenerally,two–periodicKˆ(n) (withnfinitepositive)issuitable: if[p] (X)= T K(n) Xpn and k =k ps with (k ,p)=1 then 0 0 [k](qˆ)=[k ](qˆpns)=k qˆpns +···∈F ((qˆ)) 0 0 p has invertible leading term. Integral lifts of K(n) behave similarly; the Cohen ring [AMS98] of F ((qˆ)) p defines a completion of the Borel-Tate localization. Thesecoefficientringshavenaturaltopologies,whicharerelevanttotheconvergenceofinfiniteproducts in Corollary 6.2.3. 8 MATTHEW ANDO AND JACK MORAVA 3. Application to the free loop space LetEˆ beasuitablelocalization(2.6.2)ofanequivariantlycomplexorientedcohomologytheory,letX be T acompactcomplex-orientedmanifold,andletLX beits freeloopspace. SinceLX isnotfinite-dimensional, the existence ofanumkehrhomomorphismpLX is notclear. However,TactsonLX byrotationswithfixed F set X of constant loops, and Witten discoveredthat the fixed-point formula (2.6.4) for the F-genus pLX(1) F of LX continues to yield interesting formulae. In this section we review his calculation. 3.1. The normal bundle to the constant loops and its Euler class. Oneapproximatesthe space C∞(S1,C) by the sub-vector space of Laurent polynomials C[T∗]∼= C(k)֒→C∞(S1,C). k∈T∗ M The tangent space of LX is C∞(S1,TX). If p∈X is consideredas a constant loop, then the tangent space toLX atpistheT–spaceTLX ∼=C∞(S1,TX ). ItisaT-bundlewithaLaurentpolynomialapproximation p p TX ⊗C[T∗] p Thus the normal bundle ν of the inclusion of X in LX has approximation ν ≃ TX⊗C(k). (3.1.1) Mk∈Tˆ If d zd+c zd−1+...+c = (z+x ) 1 d i j=1 Y is the formal factorization of the total F-Chern class of TX, then d e (ν)= (x + [k] (qˆ)), (3.1.2) T j F F j=1k6=0 Y Y where qˆ=e (C(1)). T 3.2. The fixed point formula. Applying (2.6.4) to the inclusion X −→LX yields the formula d 1 pLX(1)=pX . (3.2.1) F F  x + [k] (qˆ) j F F j=1k6=0 Y Y   Equation (3.2.1) requires some interpretive legerdemain. For example, the leading coefficient of (x+ [k] (qˆ)) F F k6=0 Y is the objectionable expression (kqˆ); but, as physicists say, this quantity is not directly ‘observable’. k6=0 For this reason, we consider the the renormalized formal product Q (x+ [k](qˆ)) def F Θ (x;qˆ) = x . F [k](qˆ) k6=0 Y 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 LX to be d x p˜LX(1)d=efpX j . (3.2.2) F F  Θ (x ;qˆ) F j j=1 Y   RIEMANN-ROCH AND THE FREE LOOP SPACE 9 3.3. Examples. The additive group law. When F is the additive group law Θ becomes F x Θ (x,qˆ)=x ( +1) Ga kqˆ k6=0 Y x2 =x (1− ). k2qˆ2 k>0 Y This is the Weierstrass product for π−1qˆsinqˆ−1πx, so for the theory Hˆ of (2.6.8), formula (3.2.2) gives d x /2 p˜LX(1)=(2πi/qˆ)d j [X]. F  sinh(x /2) j j=1 Y   This is just the Aˆ–genus of X, up to a normalization depending on the dimension of X. In [Ati85], Atiyah rewrites the formal product x (x+kqˆ) k6=0 Y as 2 x2 x k −qˆ2 k2 k>0 ! k>0(cid:18) (cid:19) Y Y and invokes zeta-function renormalization[Den92] to replace ( k)2 with 2π, yielding k>0 x2 Q 2πx ( −qˆ2); k2 k>0 Y specializing qˆ to i then gives the classical expression. From our point of view it’s natural to think of the ChernclassqˆofC(1)astheholomorphicone-formz−1dz onthecomplexprojectiveline,andthustoidentify qˆwith its period 2πi with respect to the equator of CP as in §2.1 of [Del89]: the ‘Betti realization’ of the 1 Tate motive Z(n) is (2πi)nZ⊂C. The multiplicative group law. In the case of the equivariant K-theory Kˆ of example (2.6.6), the Euler class of a line bundle L is e (L)=1−L. Writing q for the generator C(1) of T∗, the Euler class of C(1) is T qˆ=e (C(1))=1−q. T The multiplicative group law (2.6.7) gives e (L)+ [k](qˆ)=1−qkL, T Gm and so the formal product Θ (x,qˆ) becomes Gm (1−Lqk)(1−Lq−k) Θ (x,qˆ)=(1−L) Gm (1−qk)(1−q−k) k>0 Y (3.3.1) (1−Lqk)(1−L−1qk) = L (1−L) . (1−qk)2 ! k>0 k>0 Y Y Aside for the powers of L, this is essentially the product expansion for the Weierstrass σ function (1−qkL)(1−qkL−1) σ(L,q)=(1−L) ∈Z[L,L−1][[q]] (1−qk)2 k>0 Y (see for example [MT91] or p. 412 of [Sil94]). The infinite factor is objectionable: in the product (3.2.2) definingthehypotheticalKˆ-genusofLX thisfactorcontributesaninfinite powerofΛtopTX,butifc X =0 1 10 MATTHEW ANDO AND JACK MORAVA (e.g. if X is Calabi-Yau) and we are careful with the product, we can replace Θ with θ. The resulting Gm invariant is the Witten genus [Wit88, AHS98]. Segal [Seg88] replaces the formal product (1−qkL)(1−q−kL) k6=0 Y which arises in the multiplicative case with ( q−kL) (1−qkL)(1−qkL−1). k>0 k>0 Y Y He eliminates the infinite product ofL’s by assumingΛtopTX trivialandhe useszeta-function renormaliza- tion to replace q− k with q−1/12. P 4. A Riemann-Roch formula for the quotient of a formal group by a free subgroup The starting point for this paper was the discovery that the formal products (3.2.1) and (3.2.2) which ariseinapplyingthefixedpointformulatostudytheF-genusofthefreeloopspaceareprecisely thesameas those obtainedfromthe Riemann-Rochtheoremforthe quotientofF by a free cyclic 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 structure on the maximal ideal m of A. We will refer to (m ,+ ) as the group F(A) of A-valued points of F. A A F If H is a finite subgroup of F(R), then Lubin shows that there is a formal group law F/H over R, determined by the requirement that the power series def f (x) = (x+ h)∈R[[x]] (4.1.1) H F h∈H Y is a homomorphism of group laws F −f−→H F/H; in other words there is an equation F/H(f (x),f (y))=f (F(x,y)). H H H Themainpointisthatthepowerseriesf isconstructedsothekerneloff appliedtoF(R)isthesubgroup H H H. Thecoefficientf′ (0)= hofthelineartermoff (x)isnotaunitofR,andsof isanisomorphism H h6=0 H H of formal group laws only over R[f′ (0)−1]. Over this ring, we might as well replace f with the strict Q H H isomorphism x+ h f (x) def F H g (x) = x = , H h f′ (0) h6=0 H Y and define G to be the formal group law G(x,y)=g F(g−1(x),g−1(y)) H H H over R[f′ (0)−1]; then F/H and G are related by t(cid:0)he isomorphism (cid:1) H t(x)=f′ (0)x. H If F is the group law and R the ring of coefficients of a parametrized theory, then the Riemann-Roch formula (2.5.2) for a compact complex-oriented manifold X is the equation d x pX(u)=pX u j (4.1.2) G F  g (x ) H j j=1 Y   over R[f′ (0)−1]. H