Analytic Torsion

Let  be a compact -dimensional oriented Riemannian manifold without boundary, let  be a group representation of  by orthogonal matrices, and let  be the associated vector bundle. Suppose further that the Laplacian  is strictly negative on  where  is the linear space of  differential k-forms on  with values in . In this context, the analytic torsion  is defined as the positive real root of

where the -function is defined by

for  the collection of eigenvalues of , the restriction of  to the collection  of  bundle sections of the sheaf .

Intrinsic to the above computation is that  is a real manifold. However, there is a collection of literature on analytic torsion for complex manifolds, the construction of which is nearly identical to the construction given above. Analytic torsion on complex manifolds is sometimes called del bar torsion.

REFERENCES:

Ray, D. B. and Singer, I. M. "-Torsion and the Laplacian on Riemannian Manifolds." Adv. Math. 7, 145-210, 1971.

Ray, D. B. and Singer, I. M. "Analytic Torsion for Complex Manifolds." Ann. Math., Second Series, 98, 154-177, 1973.