 
Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 354, Number 1, Pages 107122
S 00029947(01)02863X
Article electronically published on August 20, 2001
APS BOUNDARY CONDITIONS, ETA INVARIANTS AND
ADIABATIC LIMITS
XIANZHE DAI
Abstract. We prove an adiabatic limit formula for the eta invariant of a
manifold with boundary. The eta invariant is defined using the AtiyahPatodi
Singer boundary condition and the underlying manifold is fibered over a man
ifold with boundary. Our result extends the work of BismutCheeger to man
ifolds with boundary.
1. Introduction
The invariant, introduced by AtiyahPatodiSinger in their seminal work [APS],
is the correction term (from the boundary) for the index formula on a manifold with
boundary. 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 the invariant is initiated by E. Witten [W], who relates the adiabatic limit of
the invariant to the holonomy of determinant line bundle, the socalled "global
