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

CONNES-CHERN CHARACTER AND ETA COCHAINS 29
By carefully tracing all the signs and i–factors involved in the graded form valued
Clifford calculus, as well as due to the various conventions, it turns out that separating
(2.10) into its scalar and 1-form parts, using [Get93a, Lem. 2.5]
b 〈〈A 0 , · · · , A k 〉〉 =
= b 〈A 0 , · · · , A k 〉 − i
k∑
j=0
b 〈A 0 , · · · , A j , dt ∧ Ḋt, A j+1 , · · · , A k 〉,
(2.11)
one obtains Eqs. (2.6) and (2.7).
However, for completeness, we will give a more direct argument in Subsection 2.2,
without using operator valued forms. The proof below follows the lines of the standard
proof for the JLO cocycle representing the Chern character of a θ-summable Fredholm
module (cf. [JLO88], [GeSz89]).
2.1. Cocycle and transgression formulæ for the even/odd b-Chern character
(without Clifford covariance). Recall from Remark 2.1 that for q = 0 and k even
resp. q = −1 and k odd b Ch • (D) is the b-analogue of the even, resp. odd, JLO Chern
character. We shall relate the ungraded (q = −1) case to the graded case with q = 1.
Starting with an ungraded Dirac operator D t acting on the Hilbert space H, put
( )
1 0
˜H := H ⊕ H, α := , ˜Dt :=
0 −1
( )
0 Dt
. (2.12)
D t 0
Then ˜D t is odd with respect to the grading operator α and it anti-commutes with
E 1 :=
( )
0 1
. (2.13)
−1 0
Note that
˜D t = αE 1
(
Dt ⊗ I 2
)
(2.14)
with I 2 being the 2 × 2 identity matrix.
Proposition 2.4. Let D t be ungraded (q = −1) and let ˜D t = αE 1 (D t ⊗ I 2 ) be the
associated 1–graded (q = 1) operator. Then for k odd
b Ch k (˜D t ) = 1 √ π b Ch k (D t ),
b /ch k−1 (˜D, ˙ ˜D t ) = 1 √ π b /ch k−1 (D t , Ḋt).
(2.15)
Needless to say that these formulæ are valid as well for Ch • and /ch • .