aa r X i v : . [ m a t h . A T ] J u l AN INTEGRAL LIFT OF THE Γ -GENUS JACK MORAVA
Abstract.
The Hirzebruch genus of complex-oriented manifolds asso-ciated to Euler’s Γ-function lifts to a homomorphism of ring-spectra as-sociated to a family of deformations of the Dirac operator, parametrizedby the homogeneous space Sp / U. Introduction
Kontsevich, in his early work on deformation quantization [12 § χ ∞ : M U ∗ → C [ v ](of graded rings, with a book-keeping indeterminate v ) having no very im-mediate integrality properties, but classical function theory [ § Q [˜ ζ (odd)] generated over the rationals by nor-malized zeta-values, usually expected to be transcendental. The principalresult here [ § M U Γ / / M U ∧ M Sp KO ∼ = [ 12 ] / / Sp / U ∧ KO[ ]of ring-spectra provides a lift of χ ∞ , via the compositionKO ∗ (Sp / U) ch / / H ∗ (Sp / U , Q [ v ± ]) / / H ∗ ( B U , Q [ v ± ]) / / C [ v ± ]which sends primitive generators of H ∗ (Sp / U , Q ) to odd ζ -values.It was the appearance of these periods (and their relation to the theory ofmixed Tate motives in algebraic geometry) that precipitated much of theinterest in the Γ-genus. They appear in the lift as generic parameters fora family of deformations of a Dirac operator over the homogeneous spaceSp / U. This seems to have interesting connections with [10] and [17].
Date : 15 July 2012.1991
Mathematics Subject Classification.
I’d like to thank Professor Hirzebruch for interest and conversation aboutthis material, and Peter Landweber and Ulrike Tillmann for helpful corre-spondence; but I owe special thanks to Bob Stong, for watching over myshoulder as I wrote. 1.
Coigns of vantage
It’s useful to distinguish a coordinate z at a point x of a space X fromthe corresponding parametrization of a neighborhood U ∋ x : the formeris a nice function X ⊃ U z / / A sending x to 0 in some commutative ring A , while the latter is the map z : Spec A → U ⊂ X it defines (assuming we’re in a context where this makes sense). For example, at the point x = [1 : 1] of the projective line, we have acoordinate [ u : 1] u − z which defines the parametrization z [1 + z : 1]of a neighborhood of [1 : 1]. Similarly,[ q : 1] q − := z is a coordinate at [1 : 0] = ∞ ∈ P , while[ x : 1] x := z is a coordinate at [0 : 1] = 0. An abelian group germ G at x ∈ X is the germ of a function G : U × U, x × x → U, x satisfying identities such as G ( x, G ( y, z )) = G ( G ( x, y ) , z ) , G ( x, x ) = G ( x , x ) = x, & c ;if G is suitably analytic, then a coordinate z at x associates to G , the formalgroup law ( z ◦ G )( z × z ) := z + G z ∈ A [[ z , z ]] . For example, the additive group germ G a ( x, y ) = x + y at [0 : 1] ∈ P defines z , z z + z , while the multiplicative group germ G m ( u, v ) = uv at [1 : 1]defines z + G m z = z + z + z z
1N INTEGRAL LIFT OF THE Γ-GENUS 3 (with coordinates as above). Different choices of coordinate (for fixed G and x ) define, in general, distinct (but isomorphic) formal group laws: forexample, if t ∈ A × then z = t − ( u −
1) associates the formal group law z , z z + z + tz z . to the multiplicative group at [1 : 1]. The introduction of such a variable t suggests the consideration offamilies, or deformations, of group laws: u, v uv − t ( u − v − t , to satisfy the axioms) is aninteresting example. With coordinate as above, the associated group law z , z z + z + (1 + t ) z z − tz z ;is (strictly) isomorphic to + G m , under the coordinate change z → (1 + t ) − log (cid:20) t − (cid:21) ( z ) ∈ Q [ t ][[ z ]] ;note that the fractional linear transformation fixes [1 : 1]. Similarly, exp A ( z ) := 2 sinh z/ z + A z = z (1 + z ) / + z (1 + z ) / ∈ Z [ ][[ z , z ]] , which is a specialization (at δ = − , ǫ = 0) of the formal group law z + E z = z R ( z ) + z R ( z )1 − ǫz z defined by Jacobi’s quartic Y = R ( X ) := 1 − δX + ǫX . The focus of this note is the group germ G ∞ : [ q : 1] , [ q : 1] [Γ(log ∞ ( q − ) + log ∞ ( q − )) : 1]at ∞ ∈ P ( R ) defined by the expansionexp ∞ ( z ) := z exp( γz − X k ≥ ζ ( k ) k ( − z ) k ) ∈ R [[ z ]]of the entire function Γ( z ) − near 0 (with log ∞ ( z ) denoting its formal com-position inverse): thus z + G ∞ z = Γ(log ∞ ( z ) + log ∞ ( z )) − = z + z + 2 γz z + · · · ∈ R [[ z , z ]]with z k = q − k . Ohm’s law for parallel resistors , in comparison, defines agroup germ [ q : 1] , [ q : 1] [1 : q − + q − ] a.k.a. the harmonic mean of Archytas of Tarentum JACK MORAVA at ∞ , which (because xyx + y is not differentiable at (0 , Characteristic classes and Hirzebruch genera
A complex line bundle λ ∈ H ( X, C × ) has an associated class λ − dλ πi [ λ ] : H ( X, Z (1)) → H ( X, πi Z )corresponding to the coordinate [1 § § z = vx ∈ H even ( X, Z [ v ± ])on the Picard group of topological complex line bundles. Interpreting v asthe product of the Bott class with Deligne’s motive 2 πi reconciles some con-ventions of algebraic geometry with those of algebraic topology: for example π [ λ ]sin π [ λ ] vx/ vx/ . When the grading is of background interest, I’ll set v equal to 1. A (one-dimensional) formal group law over a Q -algebra A can bewritten uniquely as z + G z = exp G (log G ( z ) + log G ( z )) ;in that case let H G ( z ) := z exp G ( z ) ∈ A [[ z ]] × denote its Hirzebruch multiplicative series [8 § M (cid:16) i = n Y i =1 H G ( vx i ) (cid:17) [ M ] ∈ A [ v ]from (cobordism classes of) compact closed complex-oriented manifolds ofreal dimension 2 n , with Chern roots x i , defines a homomorphism χ G : M U ∗ → A [ v ]of graded rings: the Hirzebruch genus associated to the group law G . By atheorem of Mishchenko,log G ( v ) = X n ≥ χ G ( P n − ( C )) n ∈ A [[ v ]] ;the deformation of the multiplicative group in § χ − t genus (defined on smooth projective complexvarieties by V X ( − p ( − t ) q dim C H p,q dg ( V ) v dim C V ) . N INTEGRAL LIFT OF THE Γ-GENUS 5
The coordinate rescaling v t − / v sends its logarithm to X n ≥ [ n ]( t ) v n n (with Gaussian t n/ − t − n/ t / − t − / = [ n ]( t )), and its formal group law to X, Y X + Y + ( t / + t − / ) vXY − vXY (which is symmetric under the involution t /t ). I’ll refer below to M SO , M U, and M Sp as the cobordism theories of R , C , and H -oriented manifolds, respectively.The Pontryagin classes p SO t ( V ) = X k ≥ p SO k ( V ) t k := X k ≥ ( − k c k ( V ⊗ C ) t k of a real vector bundle V are defined in terms of the Chern classes of itscomplexification; if V was complex to begin with, then c t ( V ⊗ C ) = X k ≥ c k ( V ⊗ C ) t k = c t ( V ) · c t ( V )equals Y (1 − x i t ) = X ( − k e k ( x i ) t k , which expresses the Pontryagin classes p SO k ( V ) = e k ( x i )in terms of elementary symmetric functions of the Chern roots x i of V ⊗ C .If H G ( z ) := ˆ H G ( z ) is an even power series, then the associated genus χ G ofa C -oriented manifold M can be evaluated in terms of Pontryagin classes,since Y ˆ H G ( x i ) := H G ( p SO k )for some polynomial H G ; this factors χ G through a homomorphism M U / / M SO ˆ χ G / / A [ v ] . The complex vector bundle underlying a quaternionic vector bundle V , onthe other hand, can be decomposed as the sum of a complex bundle withits conjugate. In that case we have p SO t ( V ) = p SO t ( W ⊕ W ) = p SO t ( W ) (at least, with coefficients in a Z [ ]-algebra). The symplectic Pontryaginclasses of V are defined by p Sp t ( V ) = X ( − k c k ( V ) t k JACK MORAVA [20], so p Sp t ( V ) = p SO t ( W ), hence p SO t ( V ) = ( p Sp t ( W )) . Since p SO t ( V ) canbe expressed in terms of the power sums P x ki = s SO k of the Chern roots of V ⊗ C as exp( X s SO2 k t k k ) , we have s SO2 k := s k ( V ⊗ C ) = 2 s k ( V ) := 2 s Sp2 k (in terms of the Chern roots of the complex structure underlying a quater-nionic structure on V ). Rewriting the logarithm of Weierstrass’s product formula for Γ, wehave Γ(1 + z ) = exp( − γz + X k> ζ ( k ) k ( − z ) k ) ;from this, and the duplication formulaΓ( z )Γ(1 − z ) = π sin πz it follows that x/ x/ X k ≥ ζ (2 k )(2 πi ) k x k k ) , with rational coefficients ζ (2 k )(2 πi ) k = − B k k )! . The ˆ A -genus of an oriented manifold (corresponding to the group law in § Y (cid:16) vx i / vx i / (cid:17) = exp( − X B k (2 k )! s SO2 k k v k )on its fundamental class. If the manifold is H -oriented, this characteristicclass equals the product Y (cid:16) x i / x i / (cid:17) / (now taken over the Chern roots of the complex bundle underlying the H -oriented structure). Proposition.
The genus of complex-oriented manifolds defined by the mul-tiplicative series H G ∞ ( x ) = Γ(1+[ λ ]) = (cid:16) x/ x/ (cid:17) / exp( i γ π x + X ζ (odd)(2 πi ) odd x odd odd ) ∈ C [[ x ]] agrees on the image of M Sp in M U with the ˆ A -genus. [Because the odd terms in the exponential cancel, for a bundle of the form W ⊕ W .] N INTEGRAL LIFT OF THE Γ-GENUS 7
Note that the Witten genus [14] H W ( x ) = x/ x/ Y n ≥ [(1 − q n e x )(1 − q n e − x ] − can be written similarly, in terms of Eisenstein series, asexp( X k G k ( q ) x k k ) ;but this deformation of the ˆ A -genus is an even function of x . The elementary symmetric functions e n and the corresponding powersums s n are related by e ( z ) = X n ≥ e n z n := Y k ≥ (1 + x k z ) = exp( − X n ≥ s n n ( − z ) n ) . The assignment x k /k [5, 9, 14 I § s k to ζ ( k ) if k >
1, and s to γ . The formalpower series Exp ∞ ( z ) = z · e ( z )thus specializes to exp ∞ ( z ) under this mapping, defining a lift G ∞ of G ∞ to a formal group law over the polynomial algebra Z [ e n | n ≥ Z , it is of additive type, and is in fact theuniversal such formal group law.Similarly H G ∞ ( z ) = X k ≥ h k ( − z ) k , in terms of the complete symmetric functions h k .3. The Real structure of M U In the homotopy-commutative diagram M U Γ [ 12 ] ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ / / S [ BU + ] ∧ H Z / / S [Sp / U + ∧ B Sp + ] ∧ H Z [ ] ζ (even) (cid:15) (cid:15) S [Sp / U + ] ∧ M Sp O O / / S [Sp / U + ] ∧ KO[ ] * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ / / S [Sp / U + ] ∧ H Q [ v ± ] ζ (odd) (cid:15) (cid:15) M Sp O O ˆ A / / KO O O / / H C [ v ± ] of spectra, the diagonal composition represents the Γ -genus. JACK MORAVA
Here S [ G + ] is the suspension ring-spectrum defined by an H -space G , such as the fiber Sp / U ( ∼ ΩSp ∼ B (U / O)) of the quaternionifi-cation map B U → B Sp. Note that the inclusion of the fiber into B U makes S [ B U + ] (and hence M U) into S [Sp / U + ]-modules).The two vertical maps at the lower left side of the diagram are the obvioussmash products with the unit S → S [Sp / U + ], while the horizontal mapsacross the middle of the diagram are smash products with the ˆ A -genus,regarded as defined by the index of a Dirac operator on an H -oriented man-ifold, followed by the Chern character on KO. The top left-hand map is justthe total characteristic number homomorphisms of Boardman and Quillen,and can alternately be described as the composition M U ∗ → M U ∗ ⊗ S ∗ → Z ⊗ S ∗ = S ∗ of the total Landweber-Novikov operation with Steenrod’s cycle map1 ∈ H ( B U , Z ) → H ( M U , Z ) = [ M U , H Z ] . The (related) upper left-hand vertical and upper right-hand horizontal mapsare more interesting. An element of M Sp ∗ (Sp / U + ) can be interpreted asthe bordism class of an H -oriented manifold M , equipped with a map toSp / U, and if we regard M as merely complex-oriented, then the productcomposition M → Sp / U + ∧ B U + → B U + defines a new complex orientation on M , and thus a ring homomorphism M Sp ∗ (Sp / U + ) → M U ∗ . By [3], this is in fact an isomorphism away from the prime (2); similarly,the composition Sp /U + ∧ B Sp + → Sp / U + ∧ B U + → B U + defines an isomorphism H ∗ (Sp / U , Z [ ]) ⊗ Z [ 12 ] H ∗ ( B Sp , Z [ ]) ∼ = H ∗ ( B U , Z [ ])of Hopf algebras, which is the upper right-hand map.Since the diagonal maps are defined by the diagram, only the right-handvertical maps remain to be constructed, but that is the content of § H ∗ ( B U , Q ) map to normalized zeta-values s k ˜ ζ ( k ) := (2 πi ) − k ζ ( k ) if k > ,
7→ − γ π · i if k = 1 . This is factored into two steps: ζ (even) : s k B k k (2 k )! ∈ Q can be interpreted as defining the ˆ A -genus, while ζ (odd) : s k +1 ( − k +1 (2 π ) − k − ζ (2 k + 1) · i . N INTEGRAL LIFT OF THE Γ-GENUS 9
Complex conjugation on M U is represented by the coordinate change z [ − z ) on the formal group, which corresponds to complex conjugationon the value group of the Γ-genus. In other words, the Γ-genus is naturally Z -equivariant, with respect to the Galois action defined by the Real struc-ture on complex cobordism.Away from (2), the Landweber-Novikov algebra of cobordism operations isan enveloping algebra of a Z -graded Lie (NB not super-Lie) algebra. Theodd part corresponds, in classical Lie theory, to the tangent space of thesymmetric space associated to the complexification of a real Lie group; itacts transitively on Spec H ∗ (Sp / U , Q ), cf. [3, 17].4. Closing remarks
The index map M Sp → KO dates back to Conner and Floyd’s 1968work on the relation of cobordism to K -theory, but seems to have receivedremarkably little attention: it is surely represented geometrically by a Diracoperator on H -oriented manifolds, but the question of a nice constructionseems not to have caught the differential geometers’ attention. In view ofthis, I have not tried to define an explicit family of deformations of such anoperator over Sp / U. R. Lu [8] has proposed an analytic interpretation of a variant of theΓ-genus of a complex-oriented M as a T -equivariant Euler class of its freeloopspace, following Atiyah ([2]; see also [1]). Lu’s construction depends ona choice of polarization U / O / / B Gl res ∼ B ( Z × B O) (cid:15) (cid:15) LM O O ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ / / LB U ∼ B ( L U) ∼ B ( Z × B U)of the tangent bundle of LM : that is, a lift of the map classifying its tan-gent bundle, to the restricted Grassmannian defined by writing loops in thetangent space as a sum of something like positive and negative-frequencycomponents. Since M is complex-oriented, such a lift exists, but is not ingeneral unique: it can be twisted by a map LM → U / O ∼ Ω(Sp / U)[6 §
2, 7, 10]. The free loops on a map α : M → Sp / U ∈ KO ( X ) thus definea twist L ( α ) : LM → L (Sp / U) ∼ U / O × Sp / U → U / O ;its restriction to the subspace M of constant loops defines a map to Sp / U ∼ ΩSp which acts naturally on U / O ∼ Ω(Sp / U), and it seems reasonable toexpect that Lu’s class for the polarized manifold (
M, L ( α )) can be expressed in terms of Γ ( M ) evaluated at suitable values s k ( α ) of the deformationparameters. I have also not tried to pin down the two-local properties of Γ , whichseem quite interesting. Away from (2), Sp / U is closely related [4] to B Sp( Z ) + ,which is in turn related (via Siegel) to the K -theory spectrum of the sym-metric monoidal category of Abelian varieties. This suggests that one mighthope to see in the Γ-genus, some homotopy-theoretic residue of the interme-diate Jacobians of complex projective manifolds. Kontsevich’s original remarks were motivated by questions of quantiza-tion, and nothing in the discussion above says much about that: homotopytheory is often revealing about the bones of a subject, without resolving thesurrounding analytical structures.It is intriguing that the points 0 , , ∞ on the projective line seem to havenaturally associated genera and cohomology theories: the additive group atzero is related to de Rham theory, and the multiplicative group at one to K -theory. The association of the point at infinity with the Kontsevich genussuggests it might be related to a Galois theory of asymptotic expansions,along lines suggested by Cartier, Connes, Kreimer, Marcolli, and others. References
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Department of Mathematics, Johns Hopkins University, Baltimore, Mary-land 21218
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