 
Summary: ADIABATIC LIMIT, HEAT KERNEL AND ANALYTIC TORSION
XIANZHE DAI AND RICHARD B. MELROSE
Introduction
The adiabatic limit refers to the geometric degeneration in which the metric
is been blown up along certain directions. The study of the adiabatic limit of
geometric invariants is initiated by E. Witten [39], who relates the adiabatic limit
of the invariant to the holonomy of determinant line bundle, the so called "global
anomaly". In this case the manifold is fibered over a circle and the metric is been
blown up along the circle direction. Witten's result was given full mathematical
treatment in [8], [9] and [13], see also [16]. In [4], J.M. Bismut and J. Cheeger
studied the adiabatic limit of the eta invariant for a general fibration of closed
manifolds. Assuming the invertibility of the Dirac family along the fibers, they
showed that the adiabatic limit of the invariant of a Dirac operator on the total
space is expressible in terms of a canonically constructed differential form, ~, on
the base. The BismutCheeger ~ form is a higher dimensional analogue of the
invariant and it is exactly the boundary correction term in the families index
theorem for manifolds with boundary, [5], [6]. The families index theorem for
manifolds with boundary has since been established in full generality by Melrose
Piazza in [31], [32].
Around the same time, Mazzeo and Melrose took on the analytic aspect of the
