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.