Let $M$ be a manifold; $shv(M)$ the (derived) category of sheaves on $M$, and $\mathcal{G}(M)$ the group of autoequivalences of $shv(M)$.

I want to consider the following sort of right-invariant semi-metric (i.e. some things have distance zero) on $\mathcal{G}(M)$:

$$d(\gamma, \eta) = sup_{F} d_{Haus}(ss(\gamma F), ss( \eta F))$$

Here, we take $ss(F)$ in the cosphere bundle so that this Hausdorff distance will be finite unless $ss(F) = \emptyset$; in this case we take $d_{Haus}(S, \emptyset) = \infty$.

Lemma.  Assume $M$ is not a homology sphere.  One has $d(\gamma, 1) = 0$, i.e., $\gamma$ preserves all microsupports, if and only if $\gamma$ is given by shifts and tensoring by rank one local systems.

Pf.  Skyscrapers $\mathbb{Z}_{\mathrm{pt}}$ must be carried to some other sheaf with the same microsupport at the point; computing self-homs it must be an exceptional object.  According to a result of Tatsuki Kuwagaki, if $M$ is not a homology sphere, the only such objects are shifts of the skyscraper. The constant sheaf $\mathbb{Z}_M$ must be carried to some local system; computing hom to the skyscraper it must be rank one.  Compose the given automorphism with a shift and a tensor so that now it fixeds $\mathbb{Z}_M$.  Again computing homs with skyscrapers — or more generally, $\gamma(\mathbb{Z}_U)$ for $U$ any contractible closed set, one has that these are all preserved by $\gamma$; similarly for contractible open sets.  These determine all sheaves.  (Note also the constant sheafs on the open sets are preserved together with compatible trivializations $\mathbb{Z}_M \leftarrow \mathbb{Z}_U$).

Remark.  The hypothesis on not being a homology sphere is most likely necessary: on a homology sphere, perhaps one can perform a spherical twist by $\mathbb{Z}_M$.  (I do not know however that the category has a Serre functor, so perhaps not).

 

Henceforth we restrict attention to the subgroup $\mathcal{G}’$ of autoequivalences which carry $\mathbb{Z}_M \to \mathbb{Z}_M$.  By the above lemma, on this the distance function is a metric.

Here are some natural questions about this metric:

Q1. The Guillermou-Kashiwara-Schapira theorem gives a map from the group of contact isotopies to the connected component  of the identity in $\mathcal{G}'(M)$.  Is it dense here?

Q2.  Is this metric already determined by the constructible sheaves?

Q3. Does this metric have any good properties?

Q4. One could write a similar thing for a metric on e.g. the endomorphisms of the space of distributions on $M$, using the microlocal analytic notion of microsupport.  Do the analysts study this?  What properties does it have?