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

32 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
2.2. Sketch of Proof of Theorem 2.2. Recall that Theorem 2.2 is stated for q ≥ 0,
hence in this subsection all Dirac operators will be q–graded with q ≥ 0.
Proposition 2.7. Let A 0 , . . . , A k ∈ b Ψ • ( )
cl,Cl q M; W . Assume that for all but one index
j 0 the indicial family is independent of λ and commutes with the actions of E 1 , . . . , E q
and E q+1 = −Γ. For the possible exception j 0 we assume that A j0 is proportional to Ḋt.
Then
where ε = |A k |(|A 0 | + . . . + |A k−1 |).
b 〈A 0 , . . . , A k 〉 = (−1) ε b 〈A k , A 0 , . . . , A k−1 〉, (2.31)
b 〈A 0 , . . . , A k 〉 =
=
k∑
b 〈A 0 , . . . , A j , 1, A j+1 , . . . , A k 〉
j=0
k∑
(−1) ε j b 〈1, A j , . . . , A k , A 0 , . . . , A j−1 〉,
j=0
(2.32)
where ε j = (|A 0 | + . . . + |A j−1 |)(|A j | + . . . + |A k |).
For j < k
b 〈A 0 , . . . , A j−1 , [D 2 , A j ], A j+1 , . . . , A k 〉
= b 〈A 0 , . . . , A j−2 , A j−1 A j , A j+1 , . . . , A k 〉
− b 〈A 0 , . . . , A j−1 , A j A j+1 , A j+2 , . . . , A k 〉.
(2.33)
Similarly, for j = k
b 〈A 0 , . . . , A k−1 , [D 2 , A k ]〉
= b 〈A 0 , . . . , A k−2 , A k−1 A k 〉
− (−1) |A k|(|A 0 |+...+|A k−1 |) b 〈A k A 0 , . . . , A k−1 〉.
(2.34)
Note that these formulæ are the same as in Getzler-Szenes [GeSz89, Lemma 2.2].
In particular there is no boundary term. The proof proceeds exactly as the proofs of
[Get93a, Lemma 6.3] (1),(2), (4), and we omit the details. We only note that one has
to make heavy use of the following lemma in order to show the vanishing of certain
terms:
Lemma 2.8 (Berezin Lemma). Let K ∈ L 1 Cl q
(H ). Then for j < q
Tr(αE 1 · . . . · E j K) = 0.
Proof. If j + q is odd then moving α past E 1 · . . . · E j K and using the trace property
gives
Tr(αE 1 · . . . · E j K) = − Tr(E 1 · . . . · E j Kα)
= − Tr(αE 1 · . . . · E j K) = 0.
(2.35)