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Let $\Sigma$ be a closed genus $g$ surface. Assume that $\mathcal{C}$ is a smooth and proper dg- (or $A_\infty$-) category which admits a faithful action $$ MCG(\Sigma) \to Auteq(\mathcal{C})$$ by the mapping class group by auto-equivalences.

  1. What good properties about $\mathcal{C}$ does that imply? (or : why can representation theory tell me about the category or its auto-equivalences)
  2. Are there any known cases where such an action arises which do not come from symplectic topology/mirror symmetry? The cases I know of so far come from Seidel-Thomas spherical twists or Ivan's Floer cohomology and pencils of quadrics paper. I assume there must be some examples from geometric representation theory or categorification and alike...
  3. What happens if I replace the mapping class group with the Torelli group $\mathcal{I}_g$ (or more generally, level $k$ in the Johnson filtration)?
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    $\begingroup$ I suspect it's probably a little tricky to make precise what "faithful" ought to mean here. There are at least two definitions you might think of, namely 1) each nontrivial element of the group acts nontrivially and 2) the action does not factor through an action of any quotient group. And these are not equivalent. $\endgroup$ – Qiaochu Yuan Mar 3 '17 at 5:45
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    $\begingroup$ Anyway, you can cook up all sorts of silly examples if you want. Suppose you have any group $G$ whatsoever acting faithfully on any set $X$ whatsoever; this induces an action of $G$ on the category of functors $X \to C$, or equivalently the product $C^X$ of $X$ copies of $C$, for any kind of category $C$ whatsoever, which ought to be faithful in any reasonable sense as soon as $C$ has at least two isomorphism classes of objects. And since $C$ is pretty arbitrary I can't think what good properties could possibly always hold. think you need more hypotheses. $\endgroup$ – Qiaochu Yuan Mar 3 '17 at 5:47
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    $\begingroup$ 1) Can you give an example of the difference? 2) would adding smooth and proper help (or do I need more assumptions to make it interesting)? $\endgroup$ – Nati Mar 3 '17 at 14:39
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    $\begingroup$ 1) Let me work in the setting of linear categories here. You can show that $k$-linear actions of a group $G$ on the category of $k$-vector spaces are classified by $H^2(G, k^{\times})$. This is despite the fact that every $k$-linear autoequivalence of $k$-vector spaces is naturally isomorphic to the identity, so no such action can be faithful in the first sense. It's faithful in the second sense iff the corresponding cohomology class is not pulled back from any nontrivial quotient of $G$, which e.g. is guaranteed if it's nonzero and $G$ is simple. $\endgroup$ – Qiaochu Yuan Mar 4 '17 at 0:00
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[This is an elaboration of parts of Mark Penney's answer]

A natural source of categorical actions of the mapping class group is the category assigned by any 4d TFT to a surface. Such categories are often written as Fukaya categories - eg Donaldson theory and Seiberg-Witten theory will attach Fukaya categories of moduli of G-bundles or symmetric products of the curve - but there are other examples, coming e.g. from factorization homology constructions as Mark wrote.

Here's what seems like the most natural example: the MCG acts on moduli spaces of G local systems ("character varieties" - or really character derived stacks) for G a complex reductive group say. So it acts on any category of sheaves on these moduli spaces. One natural answer is to look at quasicoherent sheaves - i.e. to any $\Sigma$ we look at $QC(Loc_G\Sigma)$. This is neither smooth nor proper as dg category, but it's faithful and interesting.. It's also the output of factorization homology, when the input is the braided (in fact symmetric) monoidal category $Rep(G)$ of representations of $G$. A more interesting answer coming from [Betti version of] geometric Langlands is to replace $QC$ by $IndCoh$ (possibly with fixed singular support). This is supposed to indeed be the value of an interesting 4d TFT on $\Sigma$ - namely the Kapustin-Witten B-twist of $\mathcal N=4$ super Yang-Mills. [An even more interesting (but conjectural) answer is the A-side of geometric Langlands.]

The character varieties have a symplectic form preserved by $MCG(\Sigma)$, so one can try to quantize them and get more interesting categorical actions. That's what is done e.g. in my work with Brochier and Jordan that Mark referenced, using factorization homology.

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    $\begingroup$ Thanks David. I have a (possibly stupid) question: is there a connection between the Fukaya category of the $SU(2)$-representation variety studied by Ivan and the picture you described? $\endgroup$ – Nati Mar 5 '17 at 3:19
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    $\begingroup$ The Fukaya categories of compact group representation varieties are part of Donaldson theory ($\mathcal N=2$ super Yang-Mills) - the story I'm describing is "double" that ($\mathcal N=4$), so eg the A-side of geometric Langlands has to do with Fukaya categories of moduli of Higgs bundles, i.e. roughly the cotangents of the representation varieties you're discussing. $\endgroup$ – David Ben-Zvi Mar 5 '17 at 3:51
  • $\begingroup$ Thanks! Do you know of a place where this is elaborated upon? (and is accessible to someone with background in Floer theory but really no knowledge about geometric Langlands...) $\endgroup$ – Nati May 21 '17 at 17:54
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As a (partial) answer to question 2, factorization homology provides examples of objects carrying actions of the mapping class groups. I will follow the notation of the paper Factorization homology for topological manifolds by Ayala-Francis avaiable at https://arxiv.org/abs/1206.5522, so see there for more details.

We start with a symmetric monoidal $\infty$-category $\mathcal{C}$ such that the underlying category is presentable and the product distributes over colimits. Then the short story is that factorization homology takes as input a $\mathrm{Disk}_2^\mathrm{or}$-algebra $A$ in $\mathcal{C}$ and produces a symmetric monoidal functor $\int_A: \mathrm{Mfld}_2^\mathrm{or} \to \mathcal{C}$. In particular, for each surface $\Sigma$ the object $\int_A \Sigma \in \mathcal{C}$ has an action of $\mathrm{Aut}(\Sigma) = \mathrm{MCG}(\Sigma)$.

A $\mathrm{Disk}_2^\mathrm{or}$-algebra is defined in the following way: Let $\mathrm{Mfld}_2^\mathrm{or}$ be the symmetric monoidal $\infty$-category having as objects oriented $2$-manifolds (admitting finite good covers) and morphism spaces orientation preserving embeddings. The symmetric monoidal structure is given by disjoint union. Define $\mathrm{Disk}_2^\mathrm{or}$ to be the full subcategory of $\mathrm{Mfld}_2^\mathrm{or}$ on those objects whose underlying manifold is a finite disjoint union of $\mathbb{R}^2$. Then a $\mathrm{Disk}_2^\mathrm{or}$-algebra in $\mathcal{C}$ is simply a symmetric monoidal functor $A: \mathrm{Disk}_2^\mathrm{or} \to \mathcal{C}$. Note that this is equivalent to being an algebra over the framed little 2-disks operad of Salvatore-Wahl.

I can't say much about dg- or $A_\infty$-categories, so instead let's consider a simpler setting. Let $\mathrm{Rex}$ be the symmetric monoidal category of finitely cocomplete $\mathbb{C}$-linear categories and right exact functors between them with the Deligne-Kelly tensor product $\boxtimes$. Then the symmetric monoidal $\infty$-category $\mathcal{C}$ obtained by localising $\mathrm{Rex}$ at the categorical equivalences satisfies the properties we need. $\mathrm{Disk}_2^\mathrm{or}$-algebras in $\mathcal{C}$ correspond to what are known as balanced tensor categories.

Examples of balanced tensor categories include the categories of representations of the quantum groups $U_q\mathfrak{g}$, or more generally of ribbon Hopf algebras. Ben--Zvi-Brochier-Jordan have a project studying factorization homology of balanced tensor categories with particular focus on quantum group representations. I suggest you have a look there for interesting examples of linear categories carrying mapping class group actions.

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