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.
-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.