Transverse geometry The space of leaves of a foliation (V, F) can be described in terms of (M, Γ), with M = complete transversal and Γ = holonomy pseudogroup. The natural transverse coordinates form the crossed product algebra A Γ M := C c (M) Γ, consisting of finite sums of monomials of the form f U φ, f C c (F M), φ Γ, with the product f U φ g U ψ = (f g φ) U ψ φ. How to find a geometric structure = spectral triple that is invariant under the holonomy? D cannot be taken elliptic, unless the foliation admits a transverse Riemannian structure.
Foliation Transversals 10
A. Connes & H.M., The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), Part I Diff + (M)-invariant structure First, one replaces M by P M = F + M/SO(n), where F + M = J 1 (M) = GL + (n, R) principal bundle of oriented frames on M. The sections of π : P M M are precisely the Riemannian metrics on M. Canonical structure on P M: the vertical subbundle V T (P M), V = Ker π, has GL + (n, R)- invariant Riemannian metric, since its fibers = GL + (n, R)/SO(n). The bundle N = T (P M)/V has tautological Riemannian structure: every point q P M is an Euclidean structure on T π(q) (M) = N q via π.
Hypoelliptic signature operator The hypoelliptic signature operator D on P M is uniquely determined by Q = D D, Q = (d V d V d V d V ) γ V (d H + d H ), acting on H P M = L ( V N, vol P M ) ; d V = vertical exterior derivative, γ V = grading for the vertical signature, d H = horizontal exterior differentiation with respect to a torsion-free connection, vol P M = Diff + (M)-invariant volume form. If n 1 or (mod 4), one takes P M S 1 dimension of the vertical fiber be even. so that the
Theorem 1. The operator Q is selfadjoint and so is D defined by Q = D D. Moreover, (A Γ P M, H P M, D) is a (nonunital) spectral triple with simple dimension spectrum Σ P = {k Z +, k p := n(n+1) + n}. Proof By means of adapted pseudodifferential calculus = a version of ΨDO for Heisenberg manifolds: λ ξ = (λ ξ v, λ ξ n ), ξ = (ξ v, ξ n ), λ R +, ξ = ( ξ v 4 + ξ n ) 1/4, σ (x, λ ξ) = λ q σ (x, ξ), σ = q homogeneus. In particular, the residue density of R Ψ DO = 1 (π) p n ξ =1 σ p(r)(q, ξ) d ξ dq.
Example (codimension 1): S 1 / Diff(S 1 ) H = L (F S 1 S 1, ds dθ dα) C Q = s α γ 1 + 1 i e s θ γ + ( s α 1 ) γ 3, 4 where γ 1, γ, γ 3 are the Pauli matrices [ ] [ ] [ 0 1 0 i 1 0 γ 1 =, γ 1 0 = γ i 0 = 0 1 the dimension spectrum is Σ = {0, 1,, 3, 4}. The components of the Chern character are {ϕ 1, ϕ 3 } and are given by: ϕ 1 (a 0, a 1 ) = Γ ( ) 1 1! Γ ( 3 (a 0 [Q, a 1 ](Q ) 1/ ) ) ) (a 0 [Q, a 1 ](Q ) 3/ ) + 1 ( 5 3! Γ (a 0 [Q, a 1 ](Q ) 5/ ) 1 ( ) 7 4! Γ (a 0 3 [Q, a 1 ](Q ) 7/ ) ] ;
ϕ 3 (a 0, a 1, a, a 3 ) = ( ) 1 3 3i Γ (a 0 [Q, a 1 ][Q, a] [Q, a 3 ](Q ) 3/ ) ) 1 ( 5 4! Γ (a 0 [Q, a 1 ][Q, a ] [Q, a 3 ](Q ) 5/ ) 1 ( ) 5 3 4 Γ (a 0 [Q, a 1 ] ([Q, a ][Q, a 3 ](Q ) 5/ 1 ( ) 5 4 Γ (a 0 [Q, a 1 ][Q, a ] [Q, a 3 ](Q ) 5/ ). The computation is purely symbolical, but requires the symbol σ 4, hence about 103 terms! It eventually yields the following result: (ϕ 1 ) (1) (a 1, a 1 ) = 0, a 0, a 1 A; in fact, each of the 4 terms turns out to be 0; on the other hand 1 (ϕ 3 ) (1) = 1 π 3/ ( µ + bψ), where µ(f 0 U ϕ0, f 1 U ϕ1,..., f 3 U ϕ3 ) = 0, ϕ 0 ϕ 1 ϕ ϕ 3 1 = f 0 ϕ 0 (df 1 ) (ϕ 0 ϕ 1 ) (df ) (ϕ 0 ϕ 1 ϕ ) (df 3 ).
Underlying algebraic structure W.l.o.g. can assume M = R n, with the flat connection; {X k ; 1 k n}, {Y j i ; 1 i, j n} horizontal, resp. vertical vector fields. The operator Q is built of these vector fields, and the cocycle involves iterated commutators of them acting on A Γ F M. E.g. in case n = 1, acting as Y = y y and X = y x, Y (f U ϕ ) = Y (f) U ϕ, X(f U ϕ ) = X(f) U ϕ. However, while Y acts as derivation Y (ab) = Y (a) b + a Y (b), a, b A Γ. X satisfies instead X(ab) = X(a) b + a X(b) + δ 1 (a) Y (b).
δ 1 (f U ϕ 1) = y d dx δ 1 is a derivation, ( log dϕ dx ) fu ϕ 1. δ 1 (ab) = δ 1 (a) b + a δ 1 (b), but its higher commutators with X ( δ n (f U ϕ 1) = y n dn dx n log dϕ fu dx ϕ 1, n 1, satisfy more complicated Leibniz rules. ) All this information can be encoded in a Hopf algebra H 1. As algebra = universal enveloping algebra of the Lie algebra with presentation [Y, X] = X, [Y, δ n ] = n δ n, [X, δ n ] = δ n+1, [δ k, δ l ] = 0, n, k, l 1.
The coproduct is determined by Y = Y 1 + 1 Y, X = X 1 + 1 X + δ 1 Y δ 1 = δ 1 1 + 1 δ 1, (δ 3 ) = δ 3 1 + 1 δ 3 + + δ δ 1 + 3δ 1 δ + δ1 δ 1; the antipode is determined by S(Y ) = Y, S(X) = X + δ 1 Y, S(δ 1 ) = δ 1 and the counit is ε(h) = constant term of h H 1. The canonical trace τ Γ on A Γ satisfies τ Γ (h(a)) = δ(h) τ Γ (a), h H 1, a A. where δ H 1 is the character δ(y ) = 1, δ(x) = 0, δ(δ n ) = 0.
While S Id, the δ-twisted antipode, is involutive: S = Id. S(h) = δ(h (1) ) S(h () ), Finally, the cochains {ϕ 1, ϕ 3 } can be recognized as belonging to the range of a certain cohomological characteristic map. More precisely, requiring the assignment χ Γ (h 1... h n )(a 0,..., a n ) = τ Γ (a 0 h 1 (a 1 )... h n (a n )), to induce a characteristic homomorphism χ Γ : HC Hopf (H 1) HC (A Γ ), practically dictates the definition of the Hopf cyclic cohomology. [ A. Connes & H.M., Hopf algebras, cyclic Cohomology and the transverse index theorem, Commun. Math. Phys. 198 (1998)]
H = Hopf algebra over a field k containing Q, (δ, σ) =modular pair: δ H character, and σ H, (σ) = σ σ, ε(σ) = 1, with δ(σ) = 1. One also requires S = Id. Then the following is a (co)cyclic structure: H (δ,σ) = C n 1 H n : δ 0 (h 1... h n 1 ) = 1 h 1... h n 1 δ j (h 1... h n 1 ) = h 1... h j... h n 1 1 j n 1 δ n (h 1... h n 1 ) = h 1... h n 1 σ σ i (h 1... h n+1 ) = h 1... ε(h i+1 )... h n+1 0 i n τ n (h 1... h n ) = S(h 1 ) (h... h n σ).
Equivalence of characteristic maps [Gelfand-Fuchs-Bott-Haefliger] = Hopf J M := {j 0 (ψ); ψ : Rn M }, π 1 : J M J 1 M = F M projection with cross-section σ (u) = j 0 (exp x u), u F x M given by connection ; a GL n (R), ϕ Γ σ R a = R a σ and σ ϕ = ϕ 1 σ ϕ. Define by σ (ϕ 0,..., ϕ p ) : p F M J M σ (ϕ 0,..., ϕ p )(t, u) = σ (ϕ0,...,ϕ p ;t) (u), where (ϕ 0,..., ϕ p ; t) = p 0 t i ϕ i; σ (ϕ 0 ϕ,..., ϕ p ϕ)(t, u) = ϕ 1 σ (ϕ 0,..., ϕ p )(t, ϕ(u)).
C (a n ) = Gelfand-Fuchs Lie algebra cohomology complex of a n = Lie algebra of formal vector fields on R n. For ϖ C q (a n ), define η Ω m c (F M), C p,m (ϖ)(ϕ 0,..., ϕ p ), η = ( 1) m(m+1) p F M η σ (ϕ 0,..., ϕ p ) ( ϖ) C (ϖ) = C p,m (ϖ) : C (a n ) C ( Γ; Ω c (F M)) ; defines a map of (total) complexes, C (dϖ) = (δ + )C (ϖ). For the relative (to SO n ) cohomology, one constructs similarly a homomorphism H (a n, SO n ) H ( Γ; Ω c(p M) ), which can be followed by Connes map Φ Γ : HΓ (P M) HC (A Γ P M ), yielding χ Γ GF : H (a n, SO n ) HC (A Γ P M ).
Composing χ Γ GF with the natural restriction P HC (A Γ P M ) P HC (C c (P M) one recovers the Pontryagin classes of M as images of the universal Chern classes c i1 c ik H (a n, SO n ), i 1 +... + i k n. From Hopf cyclic to cyclic : M = R n χ τ (h 1... h n )(a 0,..., a n ) = τ(a 0 h 1 (a 1 )... h n (a n )), inducing characteristic homomorphism χ Γ Hopf : HC Hopf (H n, SO n ) HC (A Γ P M ) (1). Theorem. There is a canonical isomorphism κ n : H (a n, SO n ) P HC Hopf (H n, SO n ), such that χ Γ Hopf κ n = χ Γ GF.
Summary: Transverse Index Theorem Theorem 3. There are canonical constructions for the following entities: a Hopf algebra H n with modular character δ, and with (δ, 1) modular pair in involution; a co-cyclic structure for any Hopf algebra with a modular pair in involution (δ, σ); an isomorphism κ n between the Gelfand-Fuks cohomology H GF (a n), resp. H GF (a n, SO n ), and HP (H n ; C δ ), resp. HP (H n, SO n ; C δ ); an action of H n on A Γ (F R n ), inducing a characteristic map χ Γ : HP (H n, SO n ; C δ ) HP (1) (A Γ(P R n )) = H (P R n Γ EΓ); a class L n H GF (a n, SO n ), such that ch (A Γ (P R n ), H(P R n ), D) (1) = (χ Γ κ n)(l n ).