Curvature on determinant bundles and first Chern forms

✍ Scribed by Sylvie Paycha; Steven Rosenberg

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 290 KB
• Volume: 45
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(01)00079-1

✦ Synopsis

The Quillen-Bismut-Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut-Freed connection on the determinant bundle. In finite dimensions, these forms agree (up to sign), but in infinite dimensions there is a correction term, which we express in terms of Wodzicki residues.

We illustrate these results with a string theory computation. There is a natural super vector bundle over the manifold of smooth almost complex structures on a Riemannian surface. The Bismut-Freed superconnection is identified with classical Teichmüller theory connections, and its curvature and regularized first Chern form are computed.