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

CONNES-CHERN CHARACTER AND ETA COCHAINS 31
2.1.2. q = −1. Now let D be ungraded and put ˜D, α, E 1 as in Eqs. (2.12), (2.13), (2.14).
Then by Proposition 2.4 we have
b Ch k (D t )(a 0 , . . . , a k ) = √ π b Ch k (˜D t )(a 0 , . . . , a k ), (2.21)
b /ch k (D t , Ḋt)(a 0 , . . . , a k ) = √ π b /ch k (˜D t , ˜D ˙
t )(a 0 , . . . , a k ). (2.22)
In the collar of the boundary, we write as usual D = Γ d + D dx ∂, and thus
( ) ( )
0 Γ d 0
˜D =
Γ 0 dx + D∂
. (2.23)
D ∂ 0
} {{ } } {{ }
=: Γ e =: D f ∂
˜D ∂ is 2–graded with respect to
( )
0 1
E 1 = , E
−1 0 2 = −˜Γ =
Note that
For even k we have
( )
0 −Γ
. (2.24)
−Γ 0
αE 1 E 2 = −Γ ⊗ I 2 , ˜D∂ = αE 1 (D ∂ ⊗ I 2 ). (2.25)
(
Str 2 a0 e −σ 0D g 2
∂,t
[˜D∂,t , a 1 ] · . . . · [˜D ∂,t , a k ]e −σ kD g 2)
∂,t
= 1
4π Tr( αE 1 E 2 (αE 1 ) k( ) )
a 0 e −σ 0D 2 ∂,t [D∂,t , a 1 ] · . . . · [D ∂,t , a k ]e −σ kD 2 ∂,t ⊗ I2 (2.26)
= − 1
2π Tr( )
Γa 0 e −σ 0D 2 ∂,t [D∂,t , a 1 ] · . . . · [D ∂,t , a k ]e −σ kD 2 ∂,t .
With respect to the grading given by −iΓ we can now write
−1
2π Tr( Γ·) = 1
2πi Str 0 . (2.27)
Together with (2.21) and (2.22) we have thus proved:
Proposition 2.6. Let M be an odd dimensional compact manifold with boundary with
an exact b-metric and let D be an ungraded Dirac operator. Writing D in a collar of
the boundary (in cylindrical coordinates) in the form
D =: Γ d
dx + D ∂, (2.28)
D ∂ is a graded Dirac type operator with respect to the grading operator −iΓ. Furthermore,
we have
b b Ch k−1 (D t ) + B b Ch k+1 (D t ) = 1
2 √ πi Chk (D ∂ ) ◦ i ∗ , (2.29)
d b Ch k (D t ) + b b /ch k−1 (D t ,
dt
Ḋt) + B b /ch k+1 (D t , Ḋt)
= − 1
2 √ πi /chk (D ∂ t , Ḋ∂ t ) ◦ i ∗ .
(2.30)