Connes-Chern Character for Manifolds with Boundary and ETA ...

6 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
Comparing this expression with the local form of the pairing (0.13) leads to a generalization
of the A-P-S odd-index formula [APS76, Prop. 6.2, Eq. (6.3)], from trivialized
flat bundles to pairs of equivalent vector bundles in K-theory. Precisely (cf. Corollary
7.9), if E ′ , F ′ are two such bundles on a closed odd-dimensional spin manifold N, and
h is the homotopy implementing the equivalence of E ′ with F ′ , then
∫
ξ(D F ′
g ) − ′ ξ(DE′ g ) = Â(∇ 2 ′ g ′) ∧ T/ch • (h) + SF(h, D g ′) , (0.19)
N
where D g ′ denotes the Dirac operator associated to a Riemannian metric g ′ on N;
equivalently,
∫ 1
0
1 d (
η(ph(t) D g ′ p h(t) ) ) ∫
dt =
2 dt
N
Â(∇ 2 g ′) ∧ T/ch •(h) , (0.20)
where p h(t) is the path of projections joining E ′ and F ′ , and the left hand side is the
natural extension of the real-valued index in [APS76, Eq. (6.1)].
Let us briefly comment on the main analytical challenges encountered in the course
of proving the results outlined above. In order to compute the limit as t ↘ 0 of the
Chern character, one needs to understand the asymptotic behavior of expressions of
the form
b 〈A 0 , A 1 , . . . , A k 〉 √ tD
∫
:=
b Tr ( )
A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 . . . A k e −σ (0.21)
ktD 2 dσ,
∆ k
where ∆ k denotes the standard simplex {σ 0 + . . . + σ k = 1, σ j ≥ 0} and A 0 , . . . , A k
are b–differential operators. The difficulty here is twofold. Firstly, the b–trace is a
regularized extension of the trace to b–pseudodifferential operators on the non-compact
manifold M \∂M (recall that the b-metric degenerates at ∂M). Secondly, the expression
inside the b-trace involves a product of operators. The Schwartz kernel of the product
A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 · . . . · A k e −σ ktD 2
does admit a pointwise asymptotic expansion (see
[Wid79], [CoMo90], [BlFo90]), namely
(
A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 · . . . · A k e −σ ktD 2) (p, p)
n∑
j−dim M−a
(0.22)
=: a j (A 0 , . . . , A k , D)(p) t 2 + O p (t (n+1−a−dim M)/2 ),
where a =
k ∑
j=0
j=0
ord A j .
However, this asymptotic expansion is only locally uniform
in p; it is not globally uniform on the non-compact manifold M \ ∂M. A further
complication arises from the fact that the function a j (A 0 , . . . , A k , D) is not necessarily
integrable. Nevertheless, a partie finie-type regularized integral, which we denote by