Second-order scalar operators

For Heisenberg-elliptic differential operators P, the principal Heisenberg symbol and its action on the Bargmann–Fock spaces can be explicitly computed by an algorithmic procedure. Theorem 6.3.1 then gives an explicit (M, E, ϕ)-cycle that corresponds to [P]. We will illustrate this by computing the geometric K-cycle for the Heisenberg-elliptic operatorsPγ=∆H+iγT. The procedure is essentially the same for alldifferential Heisenberg-elliptic operators, but it is most easily illustrated by an explicit example.

Because of Darboux’s theorem the contact manifoldXcan locally be identified with an open subset of the Heisenberg group R²ⁿ⁺¹. Let Xj, Yj and T be the standard right-invariant vector fields on the Heisenberg group with [Xj, Yj]=T. In such local coordinates, the operatorPγ can be written as

Pγ=

n

X

j=1

−(X_j²+Y_j²)+iγT+...,

where we ignore first order terms in Xj and Yj. These lower-order terms may appear because the sublaplacian ∆_H=P

−(X_j²+Y_j²)+... is unique only up to lower-order terms.

We have J X_j=Y_j, and so Z_j=(X_j−iYj)/√

2. A simple computation shows that 2Z_jZ_j=X_j²+Y_j²+i[X_j, Y_j], and so

The symmetric powers ofH^1,0 are eigenspaces ofπ+(Px). To see this, observe that π+(Px)z^α= (2|α|+n−γ(x))z^α,

wherez^α=z₁^α1... z_n^αnand|α|=α1+...+α_n. Therefore, on Sym^jH^1,0, the operatorπ+(P_x) acts as the scalar

a_j(x) = 2j+n−γ(x).

In the dual Bargmann–Fock representation the roles ofZ andZ are reversed. So, from

n

The representation ofσ_H(P) on the conjugate Bargmann–Fock spaces (H^BF₋ )_x is then

we obtain an order-zero operator whose principal symbols act on the symmetric powers ofH^1,0 by the scalaraj/p

1+|aj|², and similarly forH^0,1. For large values of j these renormalized scalars are close to 1, and so they contribute only trivial summands to the K-cycle and can be ignored. We can use the non-normalized scalars, because as vector bundle automorphisms they are homotopic to their normalized versions. The result is the following formula inK-homology.

Proposition 6.5.1. Let Pγ=∆γ+iγT be a Heisenberg-elliptic operator. Then the element in K-homology represented by Pγ is

∞

Here the vector bundles Sym^jH^1,0 and Sym^jH^0,1 represent elements in the K-theory group K⁰(X), while the functions 2j+n−γ and 2j+n+γ represent elements in odd K-theory K¹(X).

Corollary 6.5.2. The Chern character of the K-cycle [Pγ] is the Poincar´e dual of the cohomology class

Here the odd Chern character ch(f)∈H¹(X,Z)=[X, S¹]refers to the1-cocycle associated with the continuous map f:X!C^×∼S¹.

Example 6.5.3. In [17] we derived an explicit topological formula for the Fredholm index ofPγ in the simplest possible case whereX is a 3-manifold. We now see that this index formula, while correct, was incomplete. Let us calculate theK-cycle, and compare it with the index formula in [17].

On a 3-manifoldX we have ˆA(X)=1, while the bundleH^1,0 is a line bundle. Then Sym^jH^1,0=(H^1,0)^⊗j. Writingc1=c1(H^1,0), we have

the 1-cocycle that encodes the winding of the coefficientγaround the odd integer k, γ:X−!C\{odd integers}.

Here all cocycles are integer cocycles. The Poincar´e dual of the 3-cocycle X

kodd

kWk∪c1

2

is a 0-cycle whose image under the map H₀(X,Z)!H₀(pt,Z)=Zis just the Fredholm index of P_γ. This term in our K-cycle contains exactly the same information as the index formula of [17]. The Poincar´e dual of the termP

koddW_k is a 2-cycle. If we twist P_γ by a complex vector bundle F!X, then this 2-cycle will pair with c₁(F), and the curvature ofF will contribute to the index ofF⊗P_γ. The information contained in the termP

koddW_k cannot be derived or guessed from the formulas of [16] and [17]. The following formula highlights the gap between the result of the present paper and the earlier formula,

Example 6.5.4. The formula for theK-cycle ofP_γ easily extends to the case where P_γ acts on sections in a trivial vector bundle. WithX as above, letrbe a positive integer and letγ be aC^∞map fromX to theC-vector spaceM(r,C) of allr×rmatrices,

γ:X−!M(r,C).

Then—entirely analogous to the caser=1—a differential operator P_γ is given by Pγ= ∆H⊗Ir+iT⊗γ:C^∞(X,C^r)−!C^∞(X,C^r),

where (as usual)Ir is ther×r identity matrix. Pγ is elliptic in the Heisenberg calculus if and only if, for allx∈X,

γ(x)−λIris invertible for allλ∈ {...,−n−4,−n−2,−n, n, n+2, n+4, ...}.

As above,Pγ determines an element [Pγ] in KK⁰(C(X),C). TheK-cycle which solves the index problem for [Pγ] is (X⁺×S¹, E⁺, ϕ)t(X⁻×S¹, E⁻, ϕ), where (X⁺×S¹)t(X⁻×S¹) andϕare as above, and

E⁺=

N

M

j=0

(ν_2j+n^γ ⊗ϕ^∗Sym^jH^1,0),

E⁻=

N

M

j=0

(ν_−(2j+n)^γ ⊗ϕ^∗Sym^jH^0,1).

Hereν_2j+n^γ (resp.ν_−(2j+n)^γ ) is theC-vector bundle of fiber dimensionronX×S¹obtained by doing a clutching construction usingγ−(2j+n)Ir (resp.γ+(2j+n)Ir).

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Paul F. Baum

Pennsylvania State University University Park, PA 16802 U.S.A.

baum@math.psu.edu

Erik van Erp Dartmouth College 6188 Kemeny Hall Hanover, NH 03755 U.S.A.

jhamvanerp@gmail.com Received February 15, 2013