Equivariant de Rham torsions. (English) Zbl 1094.58013

Bismut, Jean-Michel; Goette, Sebastian

Ann. Math. (2) 159, No. 1, 53-216 (2004).

The main result of this paper establishes a comparison formula for two natural versions of equivariant analytic torsion in de Rham theory.
The first is the equivariant analytic torsion which has been introduced by J. Lott and M. Rothenberg [J. Differ. Geom. 34, 431–481 (1991; Zbl 0744.57021)]. It generalizes the classical Ray-Singer torsion, and is associated to an equivariant flat vector bundle over a closed manifold. J.-M. Bismut and W. Zhang [Geom. Funct. Anal. 4, 136–212 (1994; Zbl 0830.58030)] have compared this torsion to the equivariant Reidemeister torsion, generalizing the results of J. Lott and M. Rothenberg [loc. cit.] for unitarily flat vector bundles.
The second is the so-called Chern equivariant infinitesimal analytic torsion, a renormalized version of the infinitesimal analytic torsion, motivated by normalizations appearing in [J.-M. Bismut and S. Goette, Families torsion and Morse functions Astérisque 275 (2001; Zbl 1071.58025)].
The main theorem of the paper under review expresses the difference of the equivariant analytic torsion and the Chern equivariant infinitesimal analytic torsion as the sum of two terms. The first one is an integral of a local quantity over the fixed point manifold. This term is ‘predicted’ by the anomaly formulas, and contains contributions from the even dimensional components of the fixed point manifold only. The second summand is a subtler, but still local term, to which only the odd dimensional components of the fixed point manifold contribute.
The essential part of the latter is a new invariant associated to a manifold equipped with the action of a compact Lie group, and is referred to as the \(V\)-invariant.

Reviewer: Stefan Haller (Wien)

MSC:

58J52

Determinants and determinant bundles, analytic torsion

Keywords:

equivariant analytic torsion; Chern equivariant infinitesimal analytic torsion

Citations:

Zbl 0744.57021; Zbl 0830.58030; Zbl 1071.58025