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

52 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
Theorem 5.3. Under the assumptions of the previous Proposition 5.2 we have an
asymptotic expansion
b Tr
(A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 ...A k e −σ ktD 2)
= ∑ (−t) |α|
σ α 1
0 (σ 0 + σ 1 ) α 2
...(σ 0 + ... + σ k−1 ) αk·
α!
α∈Z k + ,|α|≤n
=
· bTr ( A 0 ∇ α 1
D A 1...∇ α k
D A )
ke −tD2 +
( (
k∏
+ O σ −a )
j/2
j t
(n+1−a−dim M)/2)
,
n∑
∫
j=0
b M
j=1
j−dim M−a
a j (A 0 , ..., A k , D)d vol t 2 +
( (
k∏
+ O σ −a )
j/2
j t
(n+1−a−dim M)/2)
.
j=1
(5.11)
Remark 5.4. The O-constant in (5.11) is independent of σ ∈ ∆ k . However, the
factor (∏ k
)
j=1 σ−a j/2
j inside the O() causes some trouble because it is integrable over
the standard simplex ∆ k only if a 1 , ..., a k ≤ 1. We do not claim that this factor is
necessarily there. It might be an artifact of the inefficiency of our method. Cf. also
Remarks 3.3, 3.6.
Proof. The strategy of proof we present here can also be used to prove Proposition 5.2.
Again by the comparison Theorem 3.2 we may assume that D is the model Dirac
operator and A 0 , ..., A k ∈ b Diff cpt ((−∞, 0) × ∂M; W ).
Using Proposition 1.6 we have
b Tr
(A 0 e −σ 0tD 2 A 1 ...A k e −σ ktD 2)
(
= − b Tr x [ d
dx , A ] )
0e −σ 0tD 2 A 1 ...A k e −σ ktD 2
= −
k∑ (
Tr
j=0
xA 0 e −σ 0tD 2 A 1 ...[ d
dx , A j]...A k e −σ ktD 2) .
(5.12)
[ d , A dx j] is again in b Diff cpt ((−∞, 0) × ∂M; W ) and its indicial family vanishes. Hence
by Proposition 3.8 all summands on the right are of trace class. Cf. also the comment
at the end of the proof of Theorem 3.9.
It therefore suffices to prove the ) claim for the summands on the right of (5.12), i.e.
for Tr
(xA 0 e −σ 0tD 2 A 1 ...A k e −σ ktD 2 where at least one of the A j has vanishing indicial
family.