  In previous work, we introduced eta invariants for even dimensional
manifolds. It plays the same role as the eta invariant of Atiyah-Patodi-Singer,
which is for odd dimensional manifolds. It is associated to $K^1$
representatives on even dimensional manifolds and is closely related to the so
called WZW theory in physics. In fact, it is an intrinsic interpretation of the
Wess-Zumino term without passing to the bounding 3-manifold. Spectrally the eta
invariant is defined on a finite cylinder, rather than on the manifold itself.
Thus it is an interesting question to find an intrinsic spectral interpretation
of this new invariant. We address this issue here using adiabatic limit
technique. The general formulation relates the (mod $\mathbb Z$ reduction of)
eta invariant for even dimensional manifolds with the holonomy of the
determinant line bundle of a natural family of Dirac type operators. In this
sense our result might be thought of as an even dimensional analogue of
Witten's holonomy theorem proved by Bismut-Freed and Cheeger independently.
