Global critical chart for local Calabi–Yau threefolds

The theory of dg-category can be regarded as a non-commutative version of derived algebraic geometry. Indeed, we can regard a derived scheme as a non-commutative space by considering its derived category of perfect complexes. As in usual algebraic geometry, Calabi–Yau structure plays an important role in dg-category theory. In the study of Calabi–Yau dg-category, Keller [Kel11] introduced the Calabi–Yau completion of a dg-category and its deformed version. More precisely, for a given small dg-category ${\mathcal{A}}$, an integer $n$ and a negative cyclic class $c\in\operatorname{HC}^{-}_{n-1}({\mathcal{A}})$, he defined a new small dg-category

$$\Pi_{n+1}({\mathcal{A}},c)\in\operatorname{dgcat}$$

called the $(n+1)$-deformed Calabi–Yau completion of ${\mathcal{A}}$. It carries a natural left $(n+1)$-Calabi–Yau structure in the sense of Brav–Dyckerhoff’s paper [BD19].

It is shown in [IQ] that the undeformed Calabi–Yau completion is a non-commutative generalization of the total space of the canonical bundle on a smooth variety. In this paper, we investigate the geometric meaning of the deformed Calabi–Yau completion.

Let $Y$ be a quasi-projective smooth variety of dimension $n$ over the complex number field. Take a cohomology class $c\in\operatorname{H}^{1}(Y,\omega_{Y})$. Let $X$ be the total space of the $\omega_{Y}$-torsor corresponding to $c$.

The above theorem can be applied in the following situation. Let $C$ be a quasi-projective smooth projective curve and $N$ be a vector bundle over $Z$ of rank two such that there exists an isomorphism $\det(N)\cong\omega_{C}$. Assume that we are given a short exact sequence of locally free sheaves on $C$

$$0\to L_{1}\to N\to L_{2}\to 0$$

such that $L_{1}$ and $L_{2}$ are line bundles. Then $\operatorname{Tot}_{C}(N)$ is the total space of an $\omega_{\operatorname{Tot}_{C}(L_{2})}$-torsor on $\operatorname{Tot}_{C}(L_{2})$. Therefore Theorem 1.1 implies that $\operatorname{Perf}(\operatorname{Tot}_{C}(N))$ is Morita equivalent to a certain deformed $3$-Calabi–Yau completion of $\operatorname{Perf}(\operatorname{Tot}_{C}(L_{2}))$. Combining this result and a theorem due to Bozec–Calaque–Scherotzke [BCS, Corollary 6.19], we obtain the following Corollary:

Corollary 1.2 has a striking application in cohomological Donaldson–Thomas (CoDT) theory. To explain this, we briefly review CoDT theory here.

Donaldson–Thomas (DT) theory studies DT invariants of a Calabi–Yau threefold $X$ which virtually count semistable sheaves on $X$. More precisely, for a homology class $\gamma\in H_{*}(X)$, we can define the Donaldson–Thomas invariant

$$\mathrm{DT}_{\gamma}(X)\in\mathbb{Q}$$

which virtually counts semistable sheaves with the Chern character $\gamma$. Assume that all semistable sheaves on $X$ with the Chern character $\gamma$ is stable and we are given an orientation (in the sense of [Joy15, Definition 2.57]) of the moduli stack ${\mathfrak{M}}_{X,\gamma}$ of stable sheaves on $X$ with the Chern character $\gamma$. Then it follows from [BBD⁺15, Theorem 6.9] that there exists a perverse sheaf

$$\varphi_{M_{X,\gamma}}\in\operatorname{Perv}(M_{X,\gamma})$$

on the good moduli space of stable sheaves on $X$ with the Chern character $\gamma$ such that we have an equality

$$\mathrm{DT}_{\gamma}(X)=\sum_{i}(-1)^{i}\dim\operatorname{H}^{i}(M_{X,\gamma},\varphi_{M_{X,\gamma}}).$$

For general $\gamma$ with a given orientation of the moduli stack ${\mathfrak{M}}_{X,\gamma}$ of semistable sheaves on $X$ with the Chern character $\gamma$, it also follows from [BBD⁺15, Theorem 6.9] that there exists a perverse sheaf

$$\varphi_{{\mathfrak{M}}_{X,\gamma}}\in\operatorname{Perv}({\mathfrak{M}}_{X,\gamma})$$

which recovers the Donaldson–Thomas invariant $\mathrm{DT}_{\gamma}$. CoDT theory aims to study the perverse sheaf $\varphi_{{\mathfrak{M}}_{X,\gamma}}$. At the time of this writing, almost nothing is known in CoDT theory for Calabi–Yau threefolds.

We can also define CoDT invariants for quivers with potentials whose dimension virtually count semistable representations of their Jacobi algebras. In contrast to the Calabi–Yau threefold case, CoDT theory for quivers with potentials is well-developed (see e.g. [KS11, DM20]) and has a striking applications in representation theory (see e.g. [Dav18]). The main obstruction of CoDT theory for Calabi–Yau threefolds is that the moduli stack ${\mathfrak{M}}_{X,\gamma}$ does not necessarily have a global critical locus description, whereas the moduli stack of representations of a Jacobi algebra does.

Now let us return back to the statement of Corollary 1.2. It states that the moduli stack of semistable sheaves on a local curve, i.e., Calabi–Yau threefold of the form

$$X=\operatorname{Tot}_{C}(N)$$

where $C$ is a smooth projective curve has a global critical locus description. This fact and several observations imply that the argument of Davison–Meinhardt’s paper [DM20] works for local curves and it is possible to establish a foundation of CoDT theory for local curves. When we have $N={\mathcal{O}}_{C}\oplus\omega_{C}$, the CoDT theory for $\operatorname{Tot}_{C}(N)$ is closely related to non-abelian Hodge theory (see [Kin, §5]). Therefore our result also have an application to non-abelian Hodge theory. This point of view will be investigated by the first author and Naoki Koseki [KK].

The main point of the proof of Theorem 1.1 is to work with $k$-linear presentable stable $\infty$-categories rather than small dg-categories (e.g. we work with the stable $\infty$-category of quasi-coherent sheaves rather than the dg-category of perfect complexes). We introduce a definition of the deformed Calabi–Yau completion of $k$-linear stable $\infty$-categories as the module category over a certain monad. An advantage of this definition is that we can use the Barr–Beck–Lurie theorem to construct an equivalence between other $\infty$-categories. For example, one can use the Barr–Beck–Lurie theorem to show that our definition of the deformed Calabi–Yau completion is compatible with Keller’s original definition. Theorem 1.1 is also a simple application of the Barr–Beck–Lurie theorem.

1. (1)

   In [Hit19, §3.5], Hitchin studies the critical locus of a linear function on the Hitchin base in (a certain open subset of) the moduli space of stable $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles. He shows that points in the critical locus corresponds to a one-dimensional sheaves on a certain non-compact Calabi–Yau threefold.

   Corollary 1.2 can be regarded as a $\mathrm{GL}(n,\mathbb{C})$-version of this result. Our result is stated as an equivalence of $-1$-shifted symplectic derived Artin stacks, which is crucial for applications in Donaldson–Thomas theory.

2. (2)

   In [MS21, Theorem 4.5], Maulik and Shen describe the moduli space of stable Higgs bundles on $C$ as the critical locus of a certain linear function of the Hitchin base for $L$-twisted Higgs bundles where $\deg L=2g-1$.

   Corollary 1.2 recovers this result as follows: Let $N$ be the rank two vector bundle given by the non-trivial extension

   $$0\to K_{C}\to N\to{\mathcal{O}}_{C}\to 0.$$

   Then using Corollary 1.2, one can show that the critical locus of Maulik–Shen’s function is isomorphic to the moduli space of one-dimensional stable sheaves on $\operatorname{Tot}_{C}(N)$. Further, one can see that the closed immersion $\operatorname{Tot}_{C}(K_{C})\hookrightarrow\operatorname{Tot}_{C}(N)$ induces an isomorphism of the moduli spaces of one-dimensional sheaves. Combining theses isomorphisms, we obtain the desired claim.

The paper is organized as follows:

In Section 2 we recall some basic definitions and results on $k$-linear stable $\infty$-categories.

In Section 3 we introduce the deformed Calabi–Yau completion of a $k$-linear stable $\infty$-categories.

In Section 4 we prove Theorem 1.1.

In Section 5 we apply Theorem 1.1 to prove that the moduli stack of coherent sheaves on a local curve is described as a global critical locus.

In this section, we recall some basic concepts in $\infty$-category theory. Standard references are [HTT] and [HA].

An $\infty$-category ${\mathcal{C}}$ with a zero object is called stable if it admits finite limits and colimits and a square in ${\mathcal{C}}$ is a pushout square if and only if it is a pullback square. The homotopy category of a stable $\infty$-category carries a natural triangulated structure.

An $\infty$-category ${\mathcal{C}}$ is called presentable if it admits all small colimits and it satisfies a certain set-theoretic assumption called accessibility (see [HTT, §5.5] for the detail). It is shown in [HTT, Corollary 5.5.2.4] that a presentable $\infty$-category admits all small limits. The adjoint functor theorem [HTT, Corollary 5.5.2.9] states that a functor between presentable $\infty$-categories $F\colon{\mathcal{C}}\to{\mathcal{D}}$ has a right adjoint precisely when it preserves small colimits, and has a left adjoint precisely when it is accessible and it preserves small limits. For presentable $\infty$-categories ${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$, the $\infty$-category $\operatorname{LFun}({\mathcal{C}}_{1},{\mathcal{C}}_{2})$ is again presentable (see [HTT, Proposition 5.5.3.8]).

An object $x$ in an $\infty$-category ${\mathcal{C}}$ is called compact if the functor

$$\operatorname{Map}_{{\mathcal{C}}}(x,-)\colon{\mathcal{C}}\to{\mathcal{S}}$$

preserves small filtered colimits. An $\infty$-category ${\mathcal{C}}$ is compactly generated if it admits small colimits and it is generated under colimits by a set of compact objects. Compactly generated $\infty$-categories are presentable.

We let $\mathop{{\mathcal{P}}\mathrm{r}^{L}}\subset\mathop{{\mathcal{C}}\mathrm{at}_{\infty}}$ (resp. $\mathop{{\mathcal{P}}\mathrm{r}^{R}}\subset\mathop{{\mathcal{C}}\mathrm{at}_{\infty}}$) denote the subcategory consisting of presentable $\infty$-categories and left (resp. right) adjoint functors. We have a natural equivalence $(\mathop{{\mathcal{P}}\mathrm{r}^{L}})^{\mathrm{op}}\simeq\mathop{{\mathcal{P}}\mathrm{r}^{R}}$. We let $\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\omega}}\subset\mathop{{\mathcal{P}}\mathrm{r}^{L}}$ (resp. $\mathop{{\mathcal{P}}\mathrm{r}^{R}_{\omega}}\subset\mathop{{\mathcal{P}}\mathrm{r}^{R}}$) denote the subcategory consisting of compactly generated $\infty$-categories and left adjoint functors that preserve the compact objects (resp. right adjoint functors which preserves small filtered colimits). The above equivalence restricts to $(\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\omega}})^{\operatorname{op}}\simeq\mathop{{\mathcal{P}}\mathrm{r}^{R}_{\omega}}$. We let $\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\mathrm{st}}}\subset\mathop{{\mathcal{P}}\mathrm{r}^{L}}$ denote the full subcategory spanned by presentable stable $\infty$-categories. We define $\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\mathrm{st},\omega}}\coloneqq\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\omega}}\cap\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\mathrm{st}}}$.

In [HA, §4.8], Lurie defines a natural closed symmetric monoidal structure on $\mathop{{\mathcal{P}}\mathrm{r}^{L}}$ with $\operatorname{LFun}$ being the internal hom, i.e., the tensor product is characterized by $\operatorname{LFun}({\mathcal{C}}\otimes{\mathcal{D}},{\mathcal{E}})\simeq\operatorname{LFun}({\mathcal{C}},\operatorname{LFun}({\mathcal{D}},{\mathcal{E}}))$. The unit is the $\infty$-category ${\mathcal{S}}$ of spaces.

It is shown in [HA, §4.8.2] that $\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\mathrm{st}}}$ inherits a symmetric monoidal structure from that of $\mathop{{\mathcal{P}}\mathrm{r}^{L}}$. The unit is the $\infty$-category $\operatorname{Sp}$ of spectra. A compactly generated stable $\infty$-category is dualizable in $\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\mathrm{st}}}$ (see [SAG, Proposition D.7.2.3]).

Recall that we have fixed an algebraically closed field $k$ of characteristic zero. We let $\operatorname{Mod}_{k}$ denote the derived $\infty$-category of differential graded $k$-modules. The $\infty$-category $\operatorname{Mod}_{k}$ is stable and compactly generated, and its symmetric monoidal structure defines a commutative algebra object $\operatorname{Mod}_{k}\in\operatorname{CAlg}(\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\mathrm{st},\omega}})$. We define $\mathop{\mathrm{LinCat}_{k}^{\mathrm{St}}}\coloneqq\operatorname{Mod}_{\operatorname{Mod}_{k}}(\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\mathrm{st}}})$ and call its object a $k$-linear stable $\infty$-category and its morphism a $k$-linear functor. The $\infty$-category $\mathop{\mathrm{LinCat}_{k}^{\mathrm{St}}}$ inherits a symmetric monoidal closed structure from $\mathop{{\mathcal{P}}\mathrm{r}^{L}}$. Explicitly, the relative tensor product is the colimit of the simplicial object ${\mathcal{C}}_{1}\otimes\operatorname{Mod}_{k}^{\otimes\bullet}\otimes{\mathcal{C}}_{2}$ in $\mathop{{\mathcal{P}}\mathrm{r}^{L}}$, and the internal hom, which is denoted by $\operatorname{LFun}_{k}({\mathcal{C}}_{1},{\mathcal{C}}_{2})$, is the limit of the cosimplicial object $\operatorname{LFun}({\mathcal{C}}_{1}\otimes\operatorname{Mod}_{k}^{\otimes\bullet},{\mathcal{C}}_{2})\simeq\operatorname{LFun}({\mathcal{C}}_{1},\operatorname{LFun}(\operatorname{Mod}_{k}^{\otimes\bullet},{\mathcal{C}}_{2}))$. The space of objects (the maximal sub $\infty$-groupoid) of $\operatorname{LFun}_{k}({\mathcal{C}}_{1},{\mathcal{C}}_{2})$ is $\operatorname{Map}_{\mathop{\mathrm{LinCat}_{k}^{\mathrm{St}}}}({\mathcal{C}}_{1},{\mathcal{C}}_{2})$.

We let $\mathop{\mathrm{LinCat}_{k}^{\mathrm{St},\omega}}\coloneqq\mathop{\mathrm{LinCat}_{k}^{\mathrm{St}}}\times_{\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\mathrm{st}}}}\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\mathrm{st},\omega}}\subset\mathop{\mathrm{LinCat}_{k}^{\mathrm{St}}}$ denote the subcategory consisting of compactly generated $k$-linear stable $\infty$-categories and $k$-linear functors preserving compact objects. This category is equivalent to $\operatorname{Mod}_{\operatorname{Mod}_{k}}(\mathop{{\mathcal{P}}\mathrm{r}^{L}_{\mathrm{st},\omega}})$.

The following statement is a consequence of [GR17, Capter 1, 9.3.6 and 3.5.3]:

Here we recall the relation between small dg-categories and $k$-linear stable $\infty$-categories.

A small dg-category is a small category enriched over the symmetric monoidal category of complexes of $k$-vector spaces. In other words, a small dg-category ${\mathcal{A}}$ consists of a set of objects $\operatorname{Obj}({\mathcal{A}})$, a dg-vector space of morphisms ${\mathcal{A}}(a,a^{\prime})$ for each $a,a^{\prime}\in\operatorname{Obj}({\mathcal{A}})$, and the identity and the composition maps satisfying the associativity relation. A dg-functor $F\colon{\mathcal{A}}_{1}\to{\mathcal{A}}_{2}$ between small dg-categories is given by a map of sets $F\colon\operatorname{Obj}({\mathcal{A}}_{1})\to\operatorname{Obj}({\mathcal{A}}_{2})$ and maps of dg-vector space ${\mathcal{A}}_{1}(a,a^{\prime})\to{\mathcal{A}}_{2}(F(a),F(a^{\prime}))$ preserving the identity and the composition maps.

For small dg-categories ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$, the tensor product ${\mathcal{A}}_{1}\otimes_{k}{\mathcal{A}}_{2}$ is defined as follows:

• •

  The set of objects $\mathrm{Obj}({\mathcal{A}}_{1}\otimes_{k}{\mathcal{A}}_{2})$ is the product $\mathrm{Obj}({\mathcal{A}}_{1})\times\mathrm{Obj}({\mathcal{A}}_{2})$.

• •

  For a pair of objects $(x,y),(x^{\prime},y^{\prime})\in\mathrm{Obj}({\mathcal{A}}_{1}\otimes_{k}{\mathcal{A}}_{2})$, the morphism dg-vector space is defined by the tensor product

  $$\mathrm{Mor}((x,y),(x^{\prime},y^{\prime}))=\mathrm{Mor}(x,y)\otimes_{k}\mathrm{Mor}(x^{\prime},y^{\prime}).$$

• •

  The composition is defined in the natural manner.

We can also define the internal mapping space $\operatorname{Fun}_{\operatorname{dg}}({\mathcal{A}}_{1},{\mathcal{A}}_{2})$ as follows:

• •

  The set of objects $\operatorname{Fun}_{\operatorname{dg}}({\mathcal{A}}_{1},{\mathcal{A}}_{2})$ is the set of dg-functors.

• •

  For a pair of objects $F,G\in\mathrm{Obj}(\operatorname{Fun}_{\operatorname{dg}}({\mathcal{A}}_{1},{\mathcal{A}}_{2}))$, the morphism dg-vector space $\mathrm{Mor}(F,G)$ is defined as the dg-vector space of natural transformations.

• •

  The composition is defined in the natural manner.

We let $\operatorname{Mod}_{k}^{\operatorname{dg}}$ the dg-category of complexes of $k$-vector spaces. For a small dg-category ${\mathcal{A}}$, we define

$$\operatorname{RMod}^{\operatorname{dg}}_{{\mathcal{A}}}\coloneqq\operatorname{Fun}_{\operatorname{dg}}({\mathcal{A}}^{\operatorname{op}},\operatorname{Mod}_{k}^{\operatorname{dg}})$$

and an object in $\operatorname{RMod}^{\operatorname{dg}}_{{\mathcal{A}}}$ is called right ${\mathcal{A}}$-module. We let $\operatorname{RMod}_{{\mathcal{A}}}^{\operatorname{ord}}$ the underlying ordinary category of $\operatorname{RMod}^{\operatorname{dg}}_{{\mathcal{A}}}$. The category $\operatorname{RMod}_{{\mathcal{A}}}^{\operatorname{ord}}$ carries a combinatorial model structure induced by the projective model structure on $\operatorname{Mod}_{k}^{\operatorname{ord}}$. The weak equivalences of right ${\mathcal{A}}$-modules with respect to this model structure are the objectwise quasi-isomorphisms. We define the derived $\infty$-category $\operatorname{RMod}_{{\mathcal{A}}}$ by the underlying $\infty$-category of the model category $\operatorname{RMod}_{{\mathcal{A}}}^{\operatorname{ord}}$, i.e., the $\infty$-categorical localization

$$\mathcal{N}(\operatorname{RMod}_{{\mathcal{A}}}^{\operatorname{ord}})[W^{-1}]$$

where $\mathcal{N}$ denotes the ordinary nerve functor and $W$ is the class of quasi-isomorphisms.

Let ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ be small dg-categories. A dg-functor $F\colon{\mathcal{A}}_{1}\to{\mathcal{A}}_{2}$ is called Morita equivalence if the restriction functor

$$F^{*}\colon\operatorname{RMod}_{{\mathcal{A}}_{2}}\to\operatorname{RMod}_{{\mathcal{A}}_{1}}$$

defines an equivalence of $\infty$-categories.

For small dg-categories ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$, we define the dg-category ${}_{{\mathcal{A}}_{1}}\mathrm{BMod}_{{\mathcal{A}}_{2}}^{\operatorname{dg}}$ of ${\mathcal{A}}_{1}$-${\mathcal{A}}_{2}$-bimodules by

$${}_{{\mathcal{A}}_{1}}\mathrm{BMod}_{{\mathcal{A}}_{2}}^{\operatorname{dg}}\coloneqq\operatorname{RMod}_{{\mathcal{A}}_{1}^{\operatorname{op}}\otimes_{k}{\mathcal{A}}_{2}}^{\operatorname{dg}}.$$

For an ${\mathcal{A}}_{1}$-${\mathcal{A}}_{2}$-bimodule $M$ and objects $a_{1}\in{\mathcal{A}}_{1}$ and $a_{2}\in{\mathcal{A}}_{2}$, we let ${}_{a_{1}}M_{a_{2}}$ the value of the functor $M$ at $a_{1}\otimes a_{2}\in\operatorname{Obj}({\mathcal{A}}_{1}^{\operatorname{op}}\otimes_{k}{\mathcal{A}}_{2})$.

Let ${\mathcal{A}}_{i}$ ($i=1,2,3$) be a small dg-category and $M$ (resp. $N$) be an ${\mathcal{A}}_{1}$-${\mathcal{A}}_{2}$-bimodule (resp. ${\mathcal{A}}_{2}$-${\mathcal{A}}_{3}$-bimodule). We define the ${\mathcal{A}}_{1}$-${\mathcal{A}}_{3}$-bimodule $M\otimes_{{\mathcal{A}}_{2}}N$ in a natural way such that for $a_{1}\in\operatorname{Obj}({\mathcal{A}}_{1})$ and $a_{3}\in\operatorname{Obj}({\mathcal{A}}_{3})$, the following equality holds:

$${}_{a_{1}}(M\otimes_{{\mathcal{A}}_{2}}N)_{a_{3}}=\int^{a_{2}\in\operatorname{Obj}({\mathcal{A}}_{2})}({}_{a_{1}}M_{a_{2}})\otimes_{k}({}_{a_{2}}N_{a_{3}}).$$

Let $\operatorname{dgcat}_{k}^{\operatorname{ord}}$ be the ordinary category of small dg-categories whose morphisms are dg-functors. In [Tab05] Tabuada constructed a model structure called Morita model structure on $\operatorname{dgcat}_{k}^{\operatorname{ord}}$ whose weak equivalences are Morita equivalences. We let $\operatorname{dgcat}_{k}^{\mathrm{M}}$ denote the underlying $\infty$-category of this model category.

Consider the assignment

$${\mathcal{A}}\mapsto\operatorname{RMod}_{{\mathcal{A}}}$$

for small dg-category ${\mathcal{A}}$. This assignment can be made $\infty$-functorial, and it is shown in [Coh, Corollary 5.7] that it induces an equivalence of $\infty$-categories

(2.1)

$$\operatorname{dgcat}_{k}^{\mathrm{M}}\simeq\mathop{\mathrm{LinCat}_{k}^{\mathrm{St},\omega}}$$

Under this equivalence, an ${\mathcal{A}}_{1}$-${\mathcal{A}}_{2}$-bimodule $X:{\mathcal{A}}_{1}\to\operatorname{RMod}^{\operatorname{dg}}_{{\mathcal{A}}_{2}}$ corresponds to a $k$-linear functor $\operatorname{RMod}_{{\mathcal{A}}_{1}}\to\operatorname{RMod}_{{\mathcal{A}}_{2}}$ via left Kan extension. The tensor product of bimodules corresponds to the composition of $k$-linear functors.

Let ${\mathcal{C}}$ be a $k$-linear stable $\infty$-category. The functor category $\operatorname{LFun}_{k}({\mathcal{C}},{\mathcal{C}})$ carries a natural monoidal structure whose monoidal product is given by the composition. We let

$$\operatorname{Mnd}_{k}({\mathcal{C}})\coloneqq\operatorname{Alg}(\operatorname{LFun}_{k}({\mathcal{C}},{\mathcal{C}}))$$

be the category of ($k$-linear) monads acting on ${\mathcal{C}}$. Note that the $\infty$-category ${\mathcal{C}}$ is left-tensored over $\operatorname{LFun}_{k}({\mathcal{C}},{\mathcal{C}})$. Therefore for a monad $T\in\operatorname{Mnd}_{k}({\mathcal{C}})$, we can define the $\infty$-category $\operatorname{LMod}_{T}({\mathcal{C}})$ of left $T$-modules in ${\mathcal{C}}$. It follows from [HA, Corollary 4.2.3.7(1)] that the $\infty$-category $\operatorname{LMod}_{T}({\mathcal{C}})$ is presentable. One can see that it is also naturally a $k$-linear stable $\infty$-category.

A morphism of monads $T_{1}\to T_{2}$ induces a forgetful functor

$$\operatorname{LMod}_{T_{2}}({\mathcal{C}})\to\operatorname{LMod}_{T_{1}}({\mathcal{C}}).$$

It preserves limits and colimits by [HA, Corollary 4.2.3.7(2)]. One can also show that it has a natural $k$-linear structure. Therefore, the adjoint functor theorem and Proposition 2.1 imply that we have a $k$-linear adjunction

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We collect some useful lemmas on monads and its modules:

Now we recall Lurie’s $\infty$-categorical version of the Barr–Beck theorem [HA, Theorem 4.7.3.5]. Let $U\colon{\mathcal{D}}\to{\mathcal{C}}$ be a $k$-linear functor between $k$-linear stable $\infty$-categories which admits a left adjoint functor $U^{L}$. Then the endomorphism monad $T$ of $U$ ([HA, Definition 4.7.3.2]) exists, and its underlying functor is $U\circ U^{L}$. One can show that $U$ naturally lifts to a $k$-linear functor $\bar{U}\colon{\mathcal{D}}\to\operatorname{LMod}_{T}({\mathcal{C}})$ such that the following diagram commutes:

We say that $U$ is monadic if $\bar{U}$ is an equivalence of $\infty$-categories.

Let ${\mathcal{C}}$ be a compactly generated (hence dualizable) $k$-linear stable $\infty$-category. Consider the following composition

$$\operatorname{Mod}_{k}\xrightarrow[]{\operatorname{coev}_{{\mathcal{C}}}}{\mathcal{C}}\otimes{\mathcal{C}}^{\vee}\simeq{\mathcal{C}}^{\vee}\otimes{\mathcal{C}}\xrightarrow[]{\operatorname{ev}_{{\mathcal{C}}}}\operatorname{Mod}_{k}$$

where $\operatorname{coev}_{{\mathcal{C}}}$ denotes the coevaluation functor and $\operatorname{ev}_{{\mathcal{C}}}$ denotes the evaluation functor. We define the Hochschild homology complex of ${\mathcal{C}}$, denoted by $\operatorname{HH}({\mathcal{C}})$, to be the image of the one-dimensional vector space under the above composition. It follows from [HSS17, Theorem 2.14] that the Hochschild homology complex $\operatorname{HH}({\mathcal{C}})$ carries a natural $S^{1}$-action hence a mixed differential $\delta\colon\operatorname{HH}({\mathcal{C}})\to\operatorname{HH}({\mathcal{C}})[-1]$. We define the negative cyclic complex $\operatorname{HC}^{-}({\mathcal{C}})$ to be the homotopy $S^{1}$-fixed point of $\operatorname{HH}({\mathcal{C}})$. It is shown in [HSS17, Theorem 2.14] that if we are given a $k$-linear functor between $k$-linear stable $\infty$-categories $F\colon{\mathcal{C}}\to{\mathcal{D}}$ which preserves the compact objects, we have a natural map $\operatorname{HH}(F)\colon\operatorname{HH}({\mathcal{C}})\to\operatorname{HH}({\mathcal{D}})$ of $S^{1}$-equivariant complexes.

For later use, we recall some basic facts about negative cyclic homology. Firstly, by a generality of mixed complexes (see [Lod98, §2.5.13]) we obtain the Connes exact sequence

$$\cdots\to\operatorname{HC}^{-}_{n+2}({\mathcal{C}})\to\operatorname{HC}^{-}_{n}({\mathcal{C}})\xrightarrow[]{\pi}\operatorname{HH}_{n}({\mathcal{C}})\xrightarrow[]{\tilde{\delta}}\operatorname{HC}^{-}_{n+1}({\mathcal{C}})\to\cdots$$

where $\pi$ is the map forgetting the $S^{1}$-fixed point structure and $\tilde{\delta}$ is a natural lift of the mixed differential. Now take ${\mathcal{C}}=\operatorname{Qcoh}(X)$ for some $d$-dimensional smooth variety $X$. It is shown in [BD19, Lemma 5.10] that for $i>d$ the negative cyclic homology group $\operatorname{HC}^{-}_{i}(\operatorname{Qcoh}(X))$ vanishes. Therefore we have natural isomorphisms

(2.3)

$$\operatorname{HC}^{-}_{d}(\operatorname{Qcoh}(X))\cong\operatorname{HH}_{d}(\operatorname{Qcoh}(X))\cong H^{0}(X,\omega_{X})$$

and a natural short exact sequence

(2.4)

$$0\to\operatorname{HC}^{-}_{d-1}(\operatorname{Qcoh}(X))\to\operatorname{HH}_{d-1}(\operatorname{Qcoh}(X))\xrightarrow[]{\delta}\operatorname{HH}_{d}(\operatorname{Qcoh}(X)).$$

We now discuss Hochschild homology for a special class of $k$-linear stable $\infty$-categories called smooth. A $k$-linear stable $\infty$-category ${\mathcal{C}}$ is called smooth if the evaluation map

$$\operatorname{ev}_{{\mathcal{C}}}\colon{\mathcal{C}}^{\vee}\otimes{\mathcal{C}}\to\operatorname{Mod}_{k}$$

has a left adjoint $\operatorname{ev}_{{\mathcal{C}}}^{L}$. In this case, we define the inverse dualizing functor $\operatorname{Id}_{{\mathcal{C}}}^{!}\colon{\mathcal{C}}\to{\mathcal{C}}$ to be the image of the one-dimensional vector space under the functor

$$\operatorname{Mod}_{k}\xrightarrow[]{\operatorname{ev}_{{\mathcal{C}}}^{L}}{{\mathcal{C}}}^{\vee}\otimes{\mathcal{C}}\simeq\operatorname{LFun}_{k}({\mathcal{C}},{\mathcal{C}}).$$

By the definition of the inverse dualizing functor, we have a natural equivalence

$$\operatorname{HH}({\mathcal{C}})\simeq\operatorname{Map}_{\operatorname{LFun}_{k}}(\operatorname{Id}_{{\mathcal{C}}}^{!},\operatorname{Id}_{{\mathcal{C}}}).$$

Let $F\colon{\mathcal{C}}\to{\mathcal{D}}$ be a $k$-linear functor between smooth $k$-linear stable $\infty$-categories with a $k$-linear right adjoint $F^{R}\colon{\mathcal{D}}\to{\mathcal{C}}$. We can define a natural transformation

(2.5)

$$\eta\colon\operatorname{Id}_{{\mathcal{D}}}^{!}\to F\circ\operatorname{Id}_{{\mathcal{C}}}^{!}\circ F^{R}$$

as follows: Firstly note that we have a natural transformation

$$\alpha\colon\operatorname{ev}_{{\mathcal{C}}}\to\operatorname{ev}_{{\mathcal{D}}}\circ((F^{R})^{\vee}\otimes F)$$

where $\operatorname{ev}_{{\mathcal{C}}}\colon{\mathcal{C}}^{\vee}\otimes{\mathcal{C}}\to\operatorname{Mod}_{k}$ and $\operatorname{ev}_{{\mathcal{D}}}\colon{\mathcal{D}}^{\vee}\otimes{\mathcal{D}}\to\operatorname{Mod}_{k}$ are evaluation functors. Then consider the following composition

$\displaystyle\operatorname{ev}_{{\mathcal{D}}}^{L}\to\operatorname{ev}_{{\mathcal{D}}}^{L}\circ\operatorname{ev}_{{\mathcal{C}}}\circ\operatorname{ev}_{{\mathcal{C}}}^{L}\xrightarrow{\operatorname{ev}_{{\mathcal{D}}}^{L}\alpha\operatorname{ev}_{{\mathcal{C}}}^{L}}\operatorname{ev}_{{\mathcal{D}}}^{L}\circ\operatorname{ev}_{{\mathcal{D}}}\circ((F^{R})^{\vee}\otimes F)\circ\operatorname{ev}_{{\mathcal{C}}}^{L}\to((F^{R})^{\vee}\otimes F)\circ\operatorname{ev}_{{\mathcal{C}}}^{L}.$

By evaluating this natural transformation at $k$, we obtain the desired map (2.5). It is shown in [BD, Proposition 4.4] that the map $\operatorname{HH}(F)$ is given by the following composition

$$\operatorname{Hom}(\operatorname{Id}_{{\mathcal{C}}}^{!},\operatorname{Id}_{{\mathcal{C}}})\to\operatorname{Hom}(F\circ\operatorname{Id}_{{\mathcal{C}}}^{!}\circ F^{R},F\circ F^{R})\to\operatorname{Hom}(\operatorname{Id}_{{\mathcal{D}}}^{!},\operatorname{Id}_{{\mathcal{D}}})$$

where the latter map is defined using $\eta$ and the counit map.

Let ${\mathcal{C}}$ be a $k$-linear stable $\infty$-category and $F\colon{\mathcal{C}}\to{\mathcal{C}}$ be a $k$-linear endofunctor. As is shown in Lemma-Definition 2.4, we can construct the free monad $TF\in\operatorname{Alg}(\operatorname{LFun}_{k}({\mathcal{C}},{\mathcal{C}}))$ generated by $F$. We define the tensor algebra $\infty$-category $T_{{\mathcal{C}}}(F)\in\mathop{\mathrm{LinCat}_{k}^{\mathrm{St}}}$ by

$$T_{{\mathcal{C}}}(F)\coloneqq\operatorname{LMod}_{TF}({\mathcal{C}}).$$

Let us see that our definition of tensor algebra $\infty$-category is compatible with Keller’s definition of tensor dg-category [Kel11, §4.1]. Let ${\mathcal{A}}$ be a small dg-category and $X$ be a ${\mathcal{A}}$-${\mathcal{A}}$-bimodule. Consider the following ${\mathcal{A}}$-${\mathcal{A}}$-bimodule

$$T_{{\mathcal{A}}}^{\operatorname{dg}}(X)\coloneqq{\mathcal{A}}\oplus X\oplus(X\otimes_{{\mathcal{A}}}X)\oplus\cdots.$$

For objects $a_{i}\in{\mathcal{A}}$ (i = 1, 2, 3), we have a natural morphism of dg-vector spaces

$$({}_{a_{3}}T_{{\mathcal{A}}}^{\operatorname{dg}}(X)_{a_{2}})\otimes_{k}({}_{a_{2}}T_{{\mathcal{A}}}^{\operatorname{dg}}(X)_{a_{1}})\to{}_{a_{3}}T_{{\mathcal{A}}}^{\operatorname{dg}}(X)_{a_{1}}.$$

Therefore $T_{{\mathcal{A}}}(X)$ carries a dg-category structure, whose set of objects are $\operatorname{Obj}({\mathcal{A}})$ and the dg-vector space of morphisms is defined by

$$T_{{\mathcal{A}}}(X)^{\operatorname{dg}}(a_{1},a_{2})={}_{a_{2}}T_{{\mathcal{A}}}^{\operatorname{dg}}(X)_{a_{1}}.$$

The dg-category $T_{{\mathcal{A}}}(X)$ is called tensor dg-category.

Now we introduce the notion of Calabi–Yau completion as a special case of tensor algebra $k$-linear $\infty$-category.

It follows from Proposition 3.1 that our definition of Calabi–Yau completion is compatible with Keller’s definition in [Kel11, §4.1].

Let ${\mathcal{C}}$ be a smooth $k$-linear stable $\infty$-category and take a Hochschild homology class $c\colon k[n-1]\to\operatorname{HH}({\mathcal{C}})\simeq\operatorname{Map}(\operatorname{Id}_{{\mathcal{C}}}^{!},\operatorname{Id}_{{\mathcal{C}}})$. We let $i_{c}^{\#}$ be the morphism of monads $T\operatorname{Id}_{{\mathcal{C}}}^{!}[n-1]\to\operatorname{Id}_{{\mathcal{C}}}$ induced by $c$. It defines a $k$-linear functor $\Pi_{n}({\mathcal{C}})\to{\mathcal{C}}$ which we denote by $i_{c}^{*}$. We define the deformed $(n+1)$-Calabi–Yau completion $\Pi_{n+1}({\mathcal{C}},c)$ of ${\mathcal{C}}$ associated with $c$ by the following pushout square in $\mathop{\mathrm{LinCat}_{k}^{\mathrm{St},\omega}}$:

(3.1)

It follows from [Kel11, Proposition 5.5] that our definition of the deformed Calabi–Yau completion is compatible with Keller’s definition in [Kel11, §5.1].

We recall the notion of left Calabi–Yau structure and its relative version introduced in [BD19].