2000/09/18 by Sylvie Paycha, Paycha, Sylvie, Steven Rosenberg +1
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.DG #math.MP
paper · pdf · doi:10.48550/arxiv.math/0009172
arxiv created 2000/09/18 · arxiv updated 2009/11/30
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 Teichmuller theory connections, and its curvature and regularized first Chern form are computed.