On Determinant Line Bundles

Determinant line bundles entered differential geometry in a remarkable paper of Quillen [Q]. He attached a holomorphic line bundle L to a particular family of Cauchy-Riemann operators over a Riemann surface, constructed a Hermitian metric on L, and calculated its curvature. At about the same time Atiyah and Singer [AS2] made the connection between determinant line bundles and anomalies in physics. Somewhat later, Witten [W1] gave a formula for “global anomalies ” in terms of η-invariants. He suggested that it could be interpreted as the holonomy of a connection on the determinant line bundle. These ideas have been developed by workers in both mathematics and physics. Our goal here is to survey some of this work. We consider arbitrary families of Dirac operators D on a smooth compact manifold X. The associated Laplacian has discrete spectrum, which leads to a patching construction for the determinant line bundle L. The determinant det D is a section of L. Quillen uses the analytic torsion of Ray and Singer [RS1] to define a metric on L. An extension of these ideas produces a unitary connection whose curvature and holonomy can be computed explicitly. The holonomy formula reproduces Witten’s global anomaly. Section 1 represents joint work with Jean-Michel Bismut, whose proof of the index theorem for families [B] is a crucial ingredient in the curvature formula (1.30). These basic themes allow many variations, two of which we play out in §2 and §3. Suppose X is a complex