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

30 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
Proof. Using Proposition 1.9 we find for k odd:
b 〈a 0 , [˜D t , a 1 ], . . . , [˜D t , a k ]〉 eDt
= b 〈(αE 1 ) k a 0 , [D t ⊗ I 2 , a 1 ], . . . , [D t ⊗ I 2 , a k ]〉 Dt⊗I2
∫
1
= √ b Tr ( )
(αE 1 ) k+1 a
∆ k 4π } {{ } 0 e −σ 0D 2 t [Dt , a 1 ] . . . [D t a k ]e −σ kD 2 t ⊗ I2
=1
= √ 1 b 〈a 0 , [D t , a 1 ], . . . , [D t , a k ]〉 Dt .
π
The calculation for b /ch k−1 (˜D t , ˙ ˜D t ) is completely analogous.
(2.16)
□
Now we are ready to translate (2.6) and (2.7) into formulæ for the standard even and
odd Chern character without Clifford action.
2.1.1. q = 0. A priori we are in the standard even situation without Clifford right
action. However, D ∂ is viewed as Cl 1 covariant with respect to the Clifford action given
by E 1 = −Γ. On the boundary, Γ gives a natural identification of the even and odd
half spinor bundle and with respect to the splitting into half spinor bundles D takes the
form:
( ) ( )
0 −1 d 0 A
D =
1 0 dx + A 0
(2.17)
} {{ } } {{ }
Γ
D ∂
;
A is an ungraded Dirac type operator acting on the positive half spinor bundle (it is the
operator whose positive spectral projection gives the APS boundary condition). In the
notation of Eq. (2.12), we have D ∂ = Ã, E 1 = −Γ. Thus, Proposition 2.4 and Theorem
2.2 give the following result.
Proposition 2.5. Let M be an even dimensional compact manifold with boundary with
an exact b-metric and let D t = f(t)D be as before. Writing D in a collar of the boundary
(in cylindrical coordinates) in the form
(
0 −
d
D =:
+ A
)
dx
d
+ A , (2.18)
dx
A is an ungraded Dirac type operator acting on the positive half spinor bundle restricted
to the boundary. Furthermore we have with A t = f(t)A
b b Ch k−1 (D t ) + B b Ch k+1 (D t ) = 1 √ π
Ch k (A t ) ◦ i ∗ , (2.19)
d b Ch k (D t ) + b b /ch k−1 (D t ,
dt
Ḋt) + B b /ch k+1 (D t , Ḋt)
= −√ 1 /ch k (A t , A ˙ t ) ◦ i ∗ .
π
(2.20)