Equivariant analytic torsions in de Rham theory
Let \(F\) denote a Hermitian flat vector bundle on a compact Riemannian manifold \(X\). Let \(G\) denote a compact Lie group with an action on \(X\) by isometries that lifts to an action on \(F\) respecting both the metric and the connection. NEWLINENEWLINENEWLINEUnder these circumstances, the authors discuss and compare two definitions of equivariant analytic torsion in de Rham theory: the equivariant version of Ray-Singer torsion [\textit{D. B. Ray} and \textit{I. M. Singer}, Adv. Math. 7, 145-210 (1971; Zbl 0239.58014)] and their infinitesimal equivariant analytic torsion which is a renormalized version of the torsion suggested by \textit{J. Lott} [J. Funct. Anal. 125, No. 2, 181-236 (2000; Zbl 0824.58044)] and imitates the construction of Chern analytic torsion forms in de Rham theory [\textit{J.-M. Bismut} and \textit{J. Lott}, J. Am. Math. Soc. 8, No. 2, 291-363 (1995; Zbl 0837.58028)]. The difference between these two torsions is then expressed as the integral of local quantities. One of these is the apparently new \(V\)-invariant of odd-dimensional \(G\)-manifolds. Moreover, the difference formula sheds new light on formulas due to \textit{U. Bunke} [Geom. Funct. Anal. 9, No. 1, 67-89 (1999; Zbl 0929.57013), and Am. J. Math. 122, 377-401 (2000)] relating the torsion forms to the equivariant Euler characteristic of the manifold \(X\).
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