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

CONNES-CHERN CHARACTER AND ETA COCHAINS 33
If j + q is even then, since j < q, E q anti commutes with αE 1 · . . . · E j K and hence
similarly
Tr(αE 1 · . . . · E j K) = − Tr(E 2 q αE 1 · . . . · E j K) = Tr(E q αE 1 · . . . · E j KE q )
= Tr(E 2 q αE 1 · . . . · E j K) = − Tr(αE 1 · . . . · E j K) = 0. □
We will make repeated use of the equations (2.31)–(2.34). Now we can proceed as
for a θ–summable Fredholm module. Following [GBVF01, p. 451] we start with the
supercommutator
∫
([ ]) b Str q Dt , a 0 e −σ 0D 2 t [Dt , a 1 ] . . . [D t , a k ]e −σ kD 2 t dσ. (2.36)
∆ k
As in [Get93a, bottom of p. 37] one shows, using Proposition 1.9 and the fact that
∫
e
−λ 2 dλ = √ π, that this supercommutator equals
〈a 0,∂ , [D ∂ t , a 1,∂ ], . . . , [D ∂ t , a k,∂ ]〉 D ∂
t
. (2.37)
It is important to note that here we are in the case q + 1, where the grading is the
induced grading on the boundary and E q+1 = −Γ.
For convenience we will write D instead of D t . Expanding the supercommutator
(2.36) on the other hand gives
b 〈[D, a 0 ], . . . , [D, a k ]〉
k∑
+ (−1) j−1 b 〈a 0 , [D, a 1 ] . . . , [D, a j−1 ], [D 2 , a j ], . . . , [D, a k ]〉,
j=1
(2.38)
where we have used [D 2 , a j ] = [D, [D, a j ]] Z 2.
We can now calculate the effect of b and B on b Ch.
B b Ch k+1 (D)(a 0 , . . . , a k )
k∑
= (−1) kj b 〈1, [D, a j ], . . . , [D, a k ], [D, a 0 ], . . . , [D, a j−1 ]〉
j=0
k∑
=
b 〈[D, a 0 ], . . . , [D, a j−1 ], 1, [D, a j ], . . . , [D, a k ]〉
j=0
= b 〈[D, a 0 ], . . . , [D, a k ]〉,
(2.39)
where we used (2.32) twice. Thus, the first summand in (2.38) equals
B b Ch k+1 (D)(a 0 , . . . , a k ).