On Determinant Line Bundles

Abstract

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 detD 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 manifold and the family of Dirac operators (or Cauchy-Riemann operators) varies holomorphically. Then L carries a natural complex structure, and under appropriate restrictions on the geometry the canonical connection is compatible with the holomorphic structure. The proper geometric hypothesis, that the total space swept out by X be Kahler, at least locally in the parameter space, also ensures that the operators vary holomorphically. Many special cases of this result can be found in the literature; the version we prove is due to Bismut, Gillet, and Soule [BGS]. One novelty here is the observation that detD has a natural square root if the dimension ofX is congruent to 2 modulo 8. On topological grounds one can argue the existence of L1/2 using Rohlin’s theorem, which is linked to real K-theory. However, one needs the differential geometry to see that detD also admits a square root. There is an extension of the holonomy theorem to L1/2. In §4 we study Riemann surfaces. This is the case originally considered by Quillen. Faltings [Fa] considered determinants on Riemann surfaces in an arithmetic context. These determinants also form the