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 hπ ⊆ TE, i.e. a complement of the vertical bundle T vπ = 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)