The analytic torsion 
is a positive real invariant associated with a compact oriented Riemannian
manifold 
and an orthogonal representation 
, defined as follows. 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 the positive real number defined by
 |
where the 
-function
is defined by
 |
for 
the collection of eigenvalues of 
, the restriction of 
to the space 
of 
bundle sections of
the sheaf 
.
The preceding construction is for a real Riemannian manifold. Ray and Singer (1973) defined an analogous analytic torsion for complex
manifolds, often called 
-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.