4 1
In the general case the assumptions above are satisfied locally on M so that a0
can be determined locally be the same formula.
1.1.2.3. Let us now consider a smooth family of fiber-wise Dirac operators (Db)b∈B
on a smooth fiber bundle π : E B with even-dimensional fibers. Then the
heat kernel method above can be generalized in order to compute a de Rham
representative of the Chern character of the index of the family. Thus let ch :
K∗(B) H∗(B, Q) be the Chern character, and dR : H∗(B, Q) HdR(B) be the
de Rham map.
The main idea is due to Quillen and known under the name super-connection
formalism (our general reference for all that is [7]). If we fix a horizontal distribu-
tion T TE, i.e. a complement of the vertical bundle T = ker(dπ), and a
connection on V , then we obtain an unitary connection ∇u on the bundle of Hilbert
spaces (L2(Eb, V|Eb ))b∈B . We define the family of super-connections
St := tD +
∇u
.
For t 0 the curvature
St
2
=
t2D2
+ higher degree forms
is a differential form on B with values in the fiber-wise differential operators. Its
exponential is a form on the base with coefficients in the fibre-wise smoothing
operators. Thus Trse−St
2
is a differential form on B.
The generalization of the McKean-Singer formula asserts now that for all t 0
dTrse−St
2
= 0 , (1.2)
(2πi)− deg /2[Trse−St
2
] = dR(ch(index((Db)b∈B))) ,
where deg is the Z-grading operator on differential forms on B.
1.1.2.4. The integral kernel of e−St
2
again has an asymptotic expansion of the form
(1.1) with locally determined coefficients ak which are now differential forms on B
with values in fiber-wise densities.
Compared with the case of a single operator the situation is now more compli-
cated because of the following. The differential form Trse−St
2
depends on t, but we
have a transgression formula which is formally a consequence of (1.2)
(1.3)
Trse−St
2

Trse−Ss
2
= −d
t
s
Trs
∂Su
∂u
e−Su
2
du .
Thus in order to use the local asymptotic expansion of the heat kernel in the
computation of a de Rham representative of the Chern character of the index we
would like to require that the limit s 0 of the integrals in (1.3) converges.
1.1.2.5. As observed by Bismut this is the case for families of compatible Dirac
operators if one modifies the superconnection to the Bismut superconnection
At := tD +
∇u
+
1
4t
c(T ) ,
where c(T ) is the Clifford multiplication with the curvature of the horizontal dis-
tribution which can be considered as a two form on B with values in the vertical
vector fields.
In this case the limit
lim(2πi)−
t→0
deg /2Trse−At
2
=: Ω(Egeom)
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