An Index Theorem for Toeplitz Operators on Odd Dimensional Manifolds with Boundary

2001/03/30 by Xianzhe Dai, Dai, Xianzhe, Weiping Zhang +1
Mathematics · #58Gxx #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #math.DG #msc:58Gxx

paper · pdf · doi:10.48550/arxiv.math/0103230

Completely revised. A gap in the proof fixed. To appear in JFA

openalex publication_date 2001/03/30 · arxiv created 2006/05/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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