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

24 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
Definition 1.8 (cf. [Get93a, Sec. 5]). Let q be a natural number. By a degree q b-
Clifford module over M one then understands a Z 2 -graded complex vector bundle W →
M together with a Hermitian metric 〈−, −〉, a Clifford action c = c l : b T ∗ M ⊗W → W ,
and an action c r : W ⊗ Cl q → W such that both actions are graded and unitary and
supercommute with each other. A b-Clifford superconnection on a degree q b-Clifford
module W over M is a b-superconnection
b A : b Ω • (M, W ) := Γ ∞( M, Λ • ( b T ∗ M) ⊗ W ) → b Ω • (M, W )
which supercommutes with the action of Cl q , satisfies
[ b
A, c(ω) ] Z 2
= c (b ∇ω ) for all ω ∈ b Ω 1 (M),
and is metric in the sense that
〈 b
Aξ, ζ 〉 + 〈 ξ, b Aζ 〉 = d 〈ξ, ζ〉 for all ξ, ζ ∈ b Ω • (M, W ).
Here, and in what follows, b ∇ denotes the Levi-Civita b-connection belonging to g b . In
the remainder of this article, we assume that a b-Clifford superconnection is always of
product form near the boundary. This means that over Y s for some s with 0 < s < 2
the superconnection has the form
b A |Y s = η ∗ ∇ ∂ + η ∗ ω ∂ ∧ −,
where η : Y → ∂M is the boundary projection from Appendix A, ∇ ∂ is a metric
connection on the restricted bundle W |∂M and ω ∂ ∈ Ω •( ∂M; End ( ))
W |∂M . Recall that
the pull-back covariant derivative ( η ∗ ∇ ∂) on W |Y is uniquely defined by requiring for
ξ ∈ Γ ∞( Y, W ) that
(
{r η ∗ ∇ ∂) ∂ξ
ξ = , if V = r ∂ ,
∂r ∂r
V
∇ ∂ ξ, if V = Ṽ ◦ η for some Ṽ ∈ Ṽ Γ∞ (∂M, T ∂M).
The Dirac operator associated to a degree q b-Clifford module W and a b-Clifford
superconnection b A is defined as the b-differential operator
D := c l ◦ b A : Γ ∞ (M, W ) → Γ ∞( M, Λ • ( b T ∗ M) ⊗ W ) → Γ ∞ (M, W ).
In this paper the term “Dirac operator” will always refer to the Dirac operator
associated to a Clifford (super)connection in the above sense. Such Dirac operators
are automatically formally self-adjoint. By a Dirac-type operator we understand a
first order differential operator such that the principal symbol of its square is scalar
(cf. [Tay96]).
Note that the b-metric on M and the metric structure on W give rise to the Hilbert
space H = L 2 (M; W ) of square integrable sections of the b-Clifford module. By
assumption,
(
Cl q acts on L 2 (M; W ), hence by Eq. (1.68) one obtains a supertrace Str q :
LCl 1 q L 2 (M; W ) ) → C. Similarly the b-trace gives rise to a b-supertrace
b Str q : b