An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary

We establish an index theorem for Toeplitz operators on odd-dimensional spin manifolds with boundary. It may be thought of as an odd-dimensional analogue of the Atiyah–Patodi–Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of η type associated to K1 representatives on even-dimensional manifolds, which should be of independent interests. For example, it gives an intrinsic interpretation of the so called Wess–Zumino term in the WZW theory in physics.