# Analytic Torsion in the Derived Category

I recently learned about analytic torsion and about the amazing Cheeger-Muller theorem identifying analytic and Reidemeister torsion for compact Riemannian manifolds.

Now analytic torsion is defined in terms of the eigenvalues of the Laplacian, acting on the de Rham complex; essentially, one takes the alternating product of the zeta-regularized determinants of the Laplacian, suitably normalized. This is all reasonably natural.

What bothers me about this definition is that it depends so strongly on properties of the de Rham complex, which I see as just a (particularly nice) choice of acyclic resolution for the constant sheaves $\underline{\mathbb{R}}$ or $\underline{\mathbb{C}}$. So my question is:

Is it possible to recover the analytic torsion from the derived category of sheaves over (a compact Riemannian manifold) $M$?

I would also be happy with any other "intrinsic" characterization of analytic torsion.

EDIT (3/15/2011): This is profilesdroxford54's suggested recipe as I understand it: (1) Take the determinant of any perfect complex resolving the constant sheaf, with its natural integral structure coming from the inclusion of the constant sheaf $\underline{\mathbb{Z}}$ in $\underline{\mathbb{R}}$; (2) Take the determinant of the de Rham complex with the integral structure given by zeta regularization; (3) The difference in the two integral structures is the analytic torsion (though I do not immediately see why there is a natural map between the two determinant lines, which is essential). Of course this recipe does not free us from the de Rham complex, but it suggests a way to do so, which I'll flesh out in three more focused questions.

(1) Is there a good theory of the derived category of sheaves on a compact Riemannian manifold, with some extra structure (i.e. self-dual, as in the case of the de Rham complex over compact Riemannian manifolds, or equivalently, with an inner product)? This seems to me like a very natural thing to study, so I am not as pessimistic as profilesdroxford54.

(2) In the framework of such a derived category (which must contain lots of infinite-dimensional objects, e.g. the de Rham complex) is there a good theory of the determinant, and does it agree with the zeta-regularized determinant? Someone recently mentioned some determinantal theory in the infinite-dimensional derived setting to me, but I know next to nothing about this sort of thing; references are welcome.

(3) Why is there a canonical map (up to sign?) between this new determinant line and the usual one (which is necessary to compare integral structure)?

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What does depend on the "derived category" is the notion of determinant line. Thus take any local system $\mathcal E$ of $R$-modules on $M$ (though one should probably work with constructible sheaves of some kind), then compute cohomology any way you want -- de Rham, etc. So long as the the cohomology is a perfect complex of $R$ modules, it has a determinant -- a free rank one projective $R$-module.

So far this has nothing to do with the Riemannanian structure. Now suppose there is a second structure -- say $M$ is Riemannian and $\mathcal E$ is a local system of inner product spaces, where $R$ is a subring of the reals. Then you can view the DeRham cx of $\mathcal E$ as a perfect complex of (infinite dimensional) inner-product spaces, and the zeta normalization gives a way of defining the determinant of this as a one dimensional inner product space.

The torsion is the relationship between these two structures. EG if $R={\mathbf Z}$ then it is the real number given by the length of a generator of the line.

To give this part of the construction a more "derived category" feel, one would need a notion of quasi-isomorphism "preserving inner products". I am not sure anyone has really thought hard about this.

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Yeah, this is essentially where I got to in my thinking on the subject. There are two issues I'm looking for clarification on: (1) My impression is that there are (high-tech) ways of taking determinants of infinite-dimensional complexes in the derived category setting, and I was wondering if analytic torsion matched up w/ any of them, and (2) Whether there is a way to replace the de Rham cpx with any other resolution of $\underline{\mathbb{R}}$ satisfying some conditions, perhaps in the sense you allude to in your last paragraph. –  Daniel Litt Mar 14 '11 at 21:01
I'm not an expert on analytic torsion, but the combinatorial (Reidemeister) torsion's definition requires some sort of finiteness condition on the chain complex. This is avoided in the analytic setting by knowing that you have a trace class operator for which you can take the determinant (but this too is a sort of finiteness condition!). Somehow I think this has to built into whatever complex of sheaves you choose. Analogously, no one knows how to define Reidemeister torsion directly from the singular chains on a space---one must pass to a finite chain complex to do it. –  John Klein Mar 14 '11 at 21:46
In response to both comments. To my knowledge you always need the complex to be perfect i.e. locally quasi-iso to a bounded complex of projectives. This is the construction explained by Deligne. To every object in the derived "category of perfect complexes" it associates a rank one projective, turning triangles into tensor products. The method is to choose a bounded complex of f.g. projectives quasi-iso to the given perfect complex, takes its determinant, and then check the answer does not depend on choices. The key point is that an acyclic cx of f.g. projectives has trivial determinant. –  user10849 Mar 14 '11 at 23:44
"The key point is that an acyclic cx of f.g. projectives has trivial determinant." I don't understand why that should be the case. For example the acyclic short complex $\cdot 0 \to \Bbb R \to \Bbb R\to 0$ where we use the identity map of $\Bbb R$ has determinant $\pm 1$. –  John Klein Mar 14 '11 at 23:57
Secondly, I do think "the answer depends on the choices" if one is allowed to vary within all perfect complexes quasi-isomorphic to the given one. The torsion depends in a serious way on "equivalence classes" of finiteness structures. For example, if $X$ is a space homotopy equivalent to a CW complex $Y$, and $L$ is a locally constant sheaf on $X$ such that $H_*(X;L)$ is trivial, then the Reidemeister torsion of $X$ {\it relative to the given finiteness structure} is defined---but it depends in a serious way on what $Y$ and what homotopy equivalence to $X$ you are choosing! –  John Klein Mar 14 '11 at 23:59