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

CONNES-CHERN CHARACTER AND ETA COCHAINS 5
When M is even-dimensional and D ∂ is invertible, the equality between the above
limit and the limit as t → ∞ yields, for any n = 2l ≥ m, the identity
∑ 〈
κ 2k (D), ch 2k (p F ) − ch 2k (p E ) 〉 + 〈 B b T/ch n+1
∞ (D), ch n(p F ) − ch n (p E ) 〉 +
0≤k≤l
where
+ 〈 T/ch n ∞ (D ∂), ch n (p F∂ ) − ch n (p E∂ ) 〉 =
∫
= Â( b ∇ 2 g) ∧ ( ch • (p F ) − ch • (p E ) ) ∫
−
b M
+ 〈 B T/ch n ∞ (D ∂), T/ch n−1
(h) 〉 ,
ch 2k (p) =
∂M
Â(∇ 2 g ∂
) ∧ T/ch •
(h)
{
tr 0 (p), for k = 0,
(−1) k (2k)!
(
tr
k! 2k (p −
1
) ⊗ p⊗2k) , for k > 0.
2
(0.14)
Like the Atiyah-Patodi-Singer index formula [APS75], the equation (0.14) involves
index and eta cochains, only of higher order. Moreover, the same type of identity
continues to hold in the odd-dimensional case. Explicitly, it takes the form
(−1) n−1
2
( n−1
2
)
!
( 〈B b T/ch n+1
∞ (D), (V −1 ⊗ V ) ⊗ n+1
〉
2 − (U −1 ⊗ U) ⊗ n+1
2 +
〈
T/ch
n
∞ (D 〉 )
∂), (V −1
∂
⊗ V ∂ ) ⊗ n+1
2 − (U −1
∂
⊗ U ∂ ) ⊗ n+1
2 =
∫
= Â( b ∇ 2 g) ∧ ( ch • (V ) − ch • (U) ) ∫
− Â(∇ 2 g ∂
) ∧ T/ch •
(h)
b M
+ 〈 B T/ch n ∞ (D ∂), T/ch n−1
(h) 〉 .
∂M
(0.15)
The relationship between the relative pairing and the Atiyah-Patodi-Singer index
theorem can actually be made explicit, and leads to interesting geometric consequences.
Indeed, under the necessary assumption that M is even-dimensional, we show (cf.
Theorem 7.6) that the above pairing can be expressed as follows:
〈[D], [E, F, h]〉 = Ind APS D F − Ind APS D E + SF(h, D ∂ ); (0.16)
here Ind APS stands for the APS-index, and SF(h, D ∂ ) denotes the spectral flow along
the path of operators ( ) h(s);
D + ∂ D
+
∂
is the restriction of c(dx)−1 D ∂ to the positive half
spinor bundle and c(dx) denotes Clifford multiplication by the inward normal vector.
On applying the A-P-S index formula [APS75, Eq. (4.3)], the pairing takes the explicit
form
∫
〈[D], [E, F, h]〉 = Â( b ∇ 2 g) ∧ ( ch • (F ) − ch • (E) )
b M
(
)
(0.17)
− ξ(D +,F ∂
∂
) − ξ(D +,E ∂
∂
) + SF(h, D ∂ ),
where
ξ(D +,E ∂
∂
) = 1 (
2
η(D +,E ∂
∂
)
) + dim Ker D +,E ∂
∂
. (0.18)