The analytic torsion 
is a positive real invariant associated with a compact oriented Riemannian
manifold 
and an orthogonal group representation 
, defined as follows. Let 
be a compact 
-dimensional oriented Riemannian manifold without boundary, let 
be a group
representation of the fundamental group 
by orthogonal
matrices, and let 
be the associated vector bundle. Suppose further
that the Laplacian 
is nonpositive on 
where 
is the vector 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 initially defined, for sufficiently large 
, by
 |
for 
the collection of eigenvalues of 
, the restriction of 
to the space 
of 
bundle sections of
the sheaf 
. Only nonzero eigenvalues are included. The series
has a meromorphic analytic
continuation that is regular at 
, and 
is the derivative
of this continuation.
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.