The half-torsion of a BGG sequence
by
Thomas Branson
University of Iowa
Coauthors: A. Rod Gover

Let D be a conformally invariant differential operator with suitable ellipticity properties. A choice of a Riemannian metric on a compact manifold assigns a (pure point) spectrum to D. This spectrum, and therefore spectral invariants like the functional determinant det(D), are not conformal invariants: they move as we move within a conformal class of metrics. However they lead to nice max/min problems which tend to pick out distinguished metrics within a conformal class, since one can write a nice (Polyakov) formula for the quotient of such determinants in conformally related metrics; this has been an important theme in string theory.

Though the Laplacians of the de Rham complex are not conformally invariant, a certain combination of their determinants admits a nice Polyakov formula. In even dimension n=2m this is \prod_k=0^m-1(det\Delta₀)^(-1)k(m-k). This half-torsion has considerably less invariance than the Reidemeister-Ray-Singer torsion; it may be viewed as a functional determinant for the Maxwell operator that ``feels'' effect of that operator's gauge (as well as conformal) invariance.

We show that the proper setting for the above principle is that of generalized Bernstein-Gelfand-Gelfand (gBGG) sequences, and that each of these has a half-torsion with a Polyakov formula, at least in the locally conformally flat case. (The exponents are now not just (-1)^k(m-k), but are determined by the conformal weights of bundles in the gBGG sequence.)

The next most elementary gBGG sequence (after the de Rham complex) is the metric deformation complex, an important construct in quantum gravity. We discuss the possible interpretation of the resulting half-torsion for gravitational theory.

Date received: October 25, 2001