Non-degeneracy of Cohomological Traces for General Landau–Ginzburg Models

Abstract

We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau–Ginzburg models associated to a pair (X, W), where X is a non-compact complex manifold with trivial canonical line bundle and W is a complex-valued holomorphic function defined on X, assuming only that the critical locus of W is compact (but may not consist of isolated points). These results can be viewed as certain “deformed” versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of W, while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the $\mathbb {Z}_2$-graded category of topological D-branes of such models.

A. Linear Categories and Supercategories with Involutive Shift Functor

In this Appendix, we collect some facts regarding linear categories with involutive shift functors and Serre functors. We are particularly interested in the case of $\mu$-Calabi–Yau categories in the sense of [1]. For simplicity, we assume shift functors and Serre functors to be automorphisms (rather than autoequivalences), since this case suffices for the purpose of the present paper.

1.1 A.1 The $\mathbb {Z}_2$-graded category $\textrm{Mod}_R^s$ and its shift functor

Let R be a unital commutative ring and $\textrm{Mod}_R$ be the category of R-modules. Recall that an R-supermodule is a $\mathbb {Z}_2$-graded R-module, i.e. an R-module M endowed with a direct sum decomposition $M=M^{\hat{0}}\oplus M^{\hat{1}}$ into two submodules $M^{\hat{0}}$ and $M^{\hat{1}}$. Let $\textrm{Mod}_R^{\mathbb {Z}_2}$ denote the ordinary category of $\mathbb {Z}_2$-graded R-modules, whose set of morphisms from an R-supermodule M to an R-supermodule N is the (ungraded) R-module:

$$\begin{aligned} \textrm{Hom}(M,N){\mathop {=}\limits ^{\mathrm{def.}}}\textrm{Hom}_R(M^{\hat{0}},N^{\hat{0}})\oplus \textrm{Hom}_R(M^{\hat{1}},N^{\hat{1}}). \end{aligned}$$

Let $\textrm{Mod}_R^s$ be the category whose objects are the R-supermodules and whose set of morphisms from an object M to an object N is the inner Hom R-supermodule  $\underline{\textrm{Hom}}(M,N)$, whose homogeneous components are defined through:

$$\begin{aligned}{} & {} \underline{\textrm{Hom}}^{\hat{0}}(M,N){\mathop {=}\limits ^{\mathrm{def.}}}\textrm{Hom}_R(M^{\hat{0}},N^{\hat{0}})\oplus \textrm{Hom}_R(M^{\hat{1}},N^{\hat{1}}),\\{} & {} \underline{\textrm{Hom}}^{\hat{1}}(M,N){\mathop {=}\limits ^{\mathrm{def.}}}\textrm{Hom}_R(M^{\hat{0}},N^{\hat{1}})\oplus \textrm{Hom}_R(M^{\hat{1}},N^{\hat{0}})~~ \end{aligned}$$

and whose composition of morphisms is induced from $\textrm{Mod}_R$. We have $\underline{\textrm{Hom}}^{\hat{0}}(M,N)=\textrm{Hom}(M,N)$.

Definition A.1

The parity change functor ${\Pi }$ of $\textrm{Mod}_R^s$ is the automorphism of $\textrm{Mod}_R^s$ defined as follows:

• 1. For any R-supermodule $M=M^{\hat{0}}\oplus M^{\hat{1}}$, the R-supermodule ${\Pi }(M)$ has homogeneous components:

  $$\begin{aligned} {\Pi }(M)^{\hat{0}}{\mathop {=}\limits ^{\mathrm{def.}}}M^{\hat{1}},~~{\Pi }(M)^{\hat{1}}{\mathop {=}\limits ^{\mathrm{def.}}}M^{\hat{0}}. \end{aligned}$$

• 2. For any morphism $f\in \underline{\textrm{Hom}}(M,N)$ of $\textrm{Mod}_R^s$, the morphism ${\Pi }(f)\in \underline{\textrm{Hom}}({\Pi }(M),{\Pi }(N))$ has homogeneous components:

  $$\begin{aligned}{} & {} {\Pi }(f)^{\hat{0}}{\mathop {=}\limits ^{\mathrm{def.}}}f^{\hat{0}}\in \textrm{Hom}_R(M^{\hat{0}},N^{\hat{0}})\oplus \textrm{Hom}_R(M^{\hat{1}},N^{\hat{1}})=\underline{\textrm{Hom}}^{\hat{0}}({\Pi }(M),{\Pi }(N)),\\{} & {} {\Pi }(f)^{\hat{1}}{\mathop {=}\limits ^{\mathrm{def.}}}f^{\hat{1}}\in \textrm{Hom}_R(M^{\hat{0}},N^{\hat{1}})\oplus \textrm{Hom}_R(M^{\hat{1}},N^{\hat{0}})=\underline{\textrm{Hom}}^{\hat{1}}({\Pi }(M),{\Pi }(N)). \end{aligned}$$

It is clear that ${\Pi }$ is involutive, i.e. we have ${\Pi }^2=\textrm{id}_{\textrm{Mod}_R^s}$, where $\textrm{id}_{\textrm{Mod}_R^s}$ denotes the identity functor of $\textrm{Mod}_R^s$. For any R-supermodules M and N, we have:

$$\begin{aligned} \underline{\textrm{Hom}}(M,{\Pi }(N))\simeq {\Pi }\underline{\textrm{Hom}}(M,N)=\underline{\textrm{Hom}}({\Pi }(M),N), \end{aligned}$$

where the second equality results from the first upon replacing M and N with ${\Pi }(M)$ and ${\Pi }(N)$ respectively.

1.2 A.2 Shift functors on linear supercategories

Let $\mathcal {T}$ be an R-linear supercategory, i.e. a category enriched over $\textrm{Mod}_R^{\mathbb {Z}_2}$. A linear functor $F:\mathcal {T}\rightarrow \mathcal {T}$ is called even if the following condition holds for any two objects a and b of $\mathcal {T}$:

$$\begin{aligned} F(\textrm{Hom}_\mathcal {T}^\kappa (a,b))\subset \textrm{Hom}_\mathcal {T}^\kappa (a,b),~~\forall \kappa \in \mathbb {Z}_2. \end{aligned}$$

The even subcategory $\mathcal {T}$ is the subcategory obtained from $\mathcal {T}$ taking the same objects but keeping only those morphisms which have degree ${\hat{0}}\in \mathbb {Z}_2$ (without changing the composition of morphisms). We denote this subcategory by $\textrm{Ev}(\mathcal {T})$ or by $\mathcal {T}^{\hat{0}}$.

Definition A.2

A shift functor for $\mathcal {T}$ is an even automorphism $\varSigma$ of $\mathcal {T}$ which satisfies the following properties:

• 1. We have $\varSigma ^2=\textrm{id}_\mathcal {T}$.

• 2. For any two objects a and b of $\mathcal {T}$, there exist isomorphisms of $\mathbb {Z}_2$-graded R-modules:

$$\begin{aligned} \textrm{Hom}_\mathcal {T}(a,\varSigma (b)){\mathop {\longrightarrow }\limits ^{\rho _{ab}}} {\Pi }\textrm{Hom}_\mathcal {T}(a,b) \end{aligned}$$

(A.1)

which are natural in both a and b. More precisely, there exists an isomorphism:

$$\begin{aligned} \rho :\textrm{Hom}_\mathcal {T}\circ (\textrm{id}_\mathcal {T}\times \varSigma ){\mathop {\rightarrow }\limits ^{\sim }} {\Pi }\circ \textrm{Hom}_{\mathcal {T}}~~ \end{aligned}$$

in the category of functors from $\mathcal {T}\times \mathcal {T}$ to $\mathcal {T}$ and natural transformations between such.

In this case, the pair $(\mathcal {T},\varSigma )$ is called an R-linear supercategory with shift.

Remark A.1

Let $(\mathcal {T},\varSigma )$ be an R-linear supercategory with shift. The replacement in (A.1) of a and b by $\varSigma (a)$ and $\varSigma (b)$ respectively gives isomorphisms:

$$\begin{aligned}{} & {} \textrm{Hom}_\mathcal {T}(\varSigma (a),b)= \\{} & {} \textrm{Hom}_\mathcal {T}(\varSigma (a),\varSigma ^2(b))\xrightarrow {\rho _{\varSigma (a)\varSigma (b)}}{\Pi }\textrm{Hom}_\mathcal {T}(\varSigma (a),\varSigma (b)) {\mathop {\longrightarrow }\limits ^{\varSigma }} {\Pi }\textrm{Hom}_\mathcal {T}(a,b), \end{aligned}$$

where we used the relation $\varSigma ^2=\textrm{id}_\mathcal {T}$. We thus have a composite isomorphism:

$$\begin{aligned} \textrm{Hom}_\mathcal {T}(\varSigma (a),b) \xrightarrow {\varSigma \circ \rho _{\varSigma (a)\varSigma (b)}}{\Pi }\textrm{Hom}_\mathcal {T}(a,b), \end{aligned}$$

which is natural in both a and b.

Definition A.3

Let $(\mathcal {T}_1,\varSigma _1)$ and $(\mathcal {T}_2,\varSigma _2)$ be two R-linear supercategories with shifts. A morphism of R-linear supercategories with shifts from $(\mathcal {T}_1,\varSigma _1)$ to $(\mathcal {T}_2,\varSigma _2)$ is a linear functor $F:\mathcal {T}_1\rightarrow \mathcal {T}_2$ such that $F\circ \varSigma _1=\varSigma _2\circ F$.

With this definition of morphisms, R-linear supercategories with shifts form a category denoted ${\textrm{RSCat}}^s$.

1.3 A.3 R-linear categories with involution

Let $\mathcal {C}$ be an R-linear category, i.e. a category enriched over $\textrm{Mod}_R$.

Definition A.4

An involution of $\mathcal {C}$ is a linear automorphism $\varSigma$ of $\mathcal {C}$ such that $\varSigma ^2=\textrm{id}_\mathcal {C}$. In this case, the pair $(\mathcal {C},\varSigma )$ is called an R-linear category with involution.

Definition A.5

Let $(\mathcal {C}_1,\varSigma _1)$ and $(\mathcal {C}_2,\varSigma _2)$ be two R-linear categories with involution. A morphism of R-linear categories with involution from $(\mathcal {C}_1,\varSigma _1)$ to $(\mathcal {C}_2,\varSigma _2)$ is a linear functor $F:\mathcal {C}_1\rightarrow \mathcal {C}_2$ such that $F\circ \varSigma _1=\varSigma _2\circ F$.

With this definition, R-linear categories with involution form a category denoted ${\textrm{RICat}}$.

1.4 A.4 Supercompletion of an R-linear category with involution

An R-linear category with involution can be completed to a $\mathbb {Z}_2$-graded category as follows.

Definition A.6

Let $(\mathcal {C},\varSigma )$ be an R-linear category with involution. The supercompletion of $\mathcal {C}$ along $\varSigma$ is the R-linear $\mathbb {Z}_2$-graded category $\textrm{Gr}_\varSigma (\mathcal {C})$ defined as follows:

• 1. The objects of $\textrm{Gr}_\varSigma (\mathcal {C})$ coincide with those of $\mathcal {C}$.

• 2. For any objects a, b of $\mathcal {C}$ and any $\kappa \in \mathbb {Z}_2$, the R-module of morphisms from a to b in $\mathcal {C}$ has the $\mathbb {Z}_2$-grading given by the decomposition $\textrm{Hom}_{\textrm{Gr}_\varSigma (\mathcal {C})}(a,b)=\textrm{Hom}_{\textrm{Gr}_\varSigma (\mathcal {C})}^{\hat{0}}(a,b)\oplus \textrm{Hom}^{\hat{1}}_{\textrm{Gr}_\varSigma (\mathcal {C})}(a,b)$, where:

  $$\begin{aligned} \textrm{Hom}_{\textrm{Gr}_\varSigma (\mathcal {C})}^\kappa (a,b) {\mathop {=}\limits ^{\mathrm{def.}}}\textrm{Hom}_{\mathcal {C}}(a,\varSigma ^\kappa (b)),~\forall \kappa \in \mathbb {Z}_2. \end{aligned}$$

• 3. Given three objects a, b, c of $\mathcal {C}$, the R-bilinear composition of morphisms $\circ :\textrm{Hom}_{\textrm{Gr}_\varSigma (\mathcal {C})}(b,c)\times \textrm{Hom}_{\textrm{Gr}_\varSigma (\mathcal {C})}(a,b)\rightarrow \textrm{Hom}_{\textrm{Gr}_\varSigma (\mathcal {C})}(a,c)$ of $\textrm{Gr}_\varSigma (\mathcal {C})$ is uniquely determined by the condition:

  $$\begin{aligned} g\circ _{\textrm{Gr}_\varSigma (\mathcal {C})} f{\mathop {=}\limits ^{\mathrm{def.}}}\varSigma ^\kappa (g)\circ f\in \textrm{Hom}_{\mathcal {C}}(a,\varSigma ^{\kappa +\nu }(c))=\textrm{Hom}_{\textrm{Gr}_\varSigma (\mathcal {C})}^{\kappa +\nu }(a,c), \end{aligned}$$

  for $f\in \textrm{Hom}_{\textrm{Gr}_\varSigma (\mathcal {C})}^{\kappa }(a,b)= \textrm{Hom}_{\mathcal {C}}(a,\varSigma ^\kappa (b))$ and $g\in \textrm{Hom}_{\textrm{Gr}_\varSigma (\mathcal {C})}^{\nu }(b,c)= \textrm{Hom}_{\mathcal {C}}(b,\varSigma ^\nu (c))$ (where $\kappa ,\nu \in \mathbb {Z}_2$).

The proof of the following statements is elementary and left to the reader:

Proposition A.7

Let $(\mathcal {C},\varSigma )$ be an R-linear category with involution. Consider the functor $\textrm{Gr}(\varSigma ):\textrm{Gr}_\varSigma (\mathcal {C})\rightarrow \textrm{Gr}_\varSigma (\mathcal {C})$ defined as follows:

• 1. For any object a of $\mathcal {C}$, let $\textrm{Gr}(\varSigma )(a){\mathop {=}\limits ^{\mathrm{def.}}}\varSigma (a)$.

• 2. For any morphism $f=u\oplus v\in \textrm{Hom}_{\textrm{Gr}(\mathcal {C})}(a,b)=\textrm{Hom}_{\mathcal {C}}(a,b)\oplus \textrm{Hom}_{\mathcal {C}}(a,\varSigma (b))$ (where $u\in \textrm{Hom}_{\mathcal {C}}(a,b)$ and $v\in \textrm{Hom}_{\mathcal {C}}(a,\varSigma (b))$), let:

  $$\begin{aligned} \textrm{Gr}(\varSigma )(f){\mathop {=}\limits ^{\mathrm{def.}}}\varSigma (u)\oplus \varSigma (v) \in \textrm{Hom}_{\textrm{Gr}_\varSigma (\mathcal {C})}(\varSigma (a),\varSigma (b))=\textrm{Hom}_{\mathcal {C}}(\varSigma (a),\varSigma (b))\oplus \textrm{Hom}_{\mathcal {C}}(\varSigma (a),b). \end{aligned}$$

Then $\textrm{Gr}(\varSigma )$ is a shift functor for the supercompletion $\textrm{Gr}_\varSigma (\mathcal {C})$.

Proposition A.8

Let $F:(\mathcal {C}_1,\varSigma _1)\rightarrow (\mathcal {C}_2,\varSigma _2)$ be a morphism of R-linear categories with involution. Consider the functor $\textrm{Gr}(F):\textrm{Gr}_{\varSigma _1}(\mathcal {C}_1)\rightarrow \textrm{Gr}_{\varSigma _2}(\mathcal {C}_2)$ defined through:

• 1. For any object a of $\mathcal {C}_1$, set $\textrm{Gr}(F)(a){\mathop {=}\limits ^{\mathrm{def.}}}F(a)$.

• 2. For any morphism $f\!=\!u\oplus v\in \! \textrm{Hom}_{\textrm{Gr}_{\varSigma _1}\!(\mathcal {C}_1)}\!(a,b)$, where $u\in \!\textrm{Hom}_{\mathcal {C}_1}\!(a,b)$ and $v\in \!\textrm{Hom}_{\mathcal {C}_1}(a,\varSigma _1(b))$, set:

  $$\begin{aligned} \textrm{Gr}(F)(f)\!{\mathop {=}\limits ^{\mathrm{def.}}}\!F(u) \oplus F(v) \in \! \textrm{Hom}_{\textrm{Gr}_{\varSigma _2}(\mathcal {C}_2)}(\!F(a),\!F(b))\!=\!\textrm{Hom}_{\mathcal {C}_2}(\!F(a),\!F(b))\oplus \textrm{Hom}_{\mathcal {C}_2}(\!F(a),\!\varSigma _2(\!F(b))), \end{aligned}$$

  where we used the relation $F\circ \varSigma _1=\varSigma _2\circ F$.

Then $\textrm{Gr}(F):(\textrm{Gr}_{\varSigma _1}(\mathcal {C}_1),\textrm{Gr}(\varSigma _1))\rightarrow (\textrm{Gr}_{\varSigma _2}(\mathcal {C}_2),\textrm{Gr}(\varSigma _2))$ is a morphism of R-linear supercategories with shift.

Proposition A.9

$\textrm{Gr}$ is a functor from $\textrm{RICat}$ to $\textrm{RSCat}^s$.

1.5 A.5 The even subcategory of an R-linear supercategory with shift

A quasi-inverse of the supercompletion functor $\textrm{Gr}$ can be constructed as follows, where the proof of the various statements is left to the reader.

Proposition A.10

Let $(\mathcal {T},\varSigma )$ be an R-linear supercategory with shift. Then $\varSigma$ is an involution of the even subcategory $\mathcal {T}^{\hat{0}}$.

Proposition A.11

Given a morphism of R-linear supercategories with shifts $F:(\mathcal {T}_1,\varSigma _1)\rightarrow (\mathcal {T}_2,\varSigma _2)$, consider the functor $\textrm{Ev}(f):\textrm{Ev}(\mathcal {T}_1)=\mathcal {T}_1^{\hat{0}}\rightarrow \textrm{Ev}(\mathcal {T}_2)=\mathcal {T}_2^{\hat{0}}$ obtained by restricting F to the subcategory $\mathcal {T}_1^{\hat{0}}$ of $\mathcal {T}_1$. Then $\textrm{Ev}(f)$ is a morphism in $\textrm{RICat}$ from $(\mathcal {T}_1^{\hat{0}},\varSigma _1)$ to $(\mathcal {T}_2^{\hat{0}},\varSigma _2)$.

Proposition A.12

$\textrm{Ev}$ is a functor from $\textrm{RSCat}^s$ to $\textrm{RICat}$.

Finally, one easily proves the following:

Theorem A.13

The functors $\textrm{Gr}$ and $\textrm{Ev}$ are mutually quasi-inverse equivalences between $\textrm{RICat}$ and $\textrm{RSCat}^s$.

This shows, in particular, that R-linear supercategories with shift can be reconstructed from their even part, which is an R-linear category with involution.

1.6 A.6 Calabi–Yau supercategories with shift

In this subsection we consider the case $R=\mathbb {C}$. Recall the following notion used in [1]:

Definition A.14

A Calabi–Yau supercategory of parity $\mu \in \mathbb {Z}_2$ is a pair $(\mathcal {T},\textrm{tr})$, where:

• A. $\mathcal {T}$ is a $\mathbb {Z}_2$-graded and $\mathbb {C}$-linear Hom-finite category.

• B. $\textrm{tr}=(\textrm{tr}_a)_{a\in \textrm{Ob}\mathcal {T}}$ is a family of $\mathbb {C}$-linear maps $\textrm{tr}_a:\textrm{End}_\mathcal {T}(a)\rightarrow \mathbb {C}$ of $\mathbb {Z}_2$-degree $\mu$

such that the following conditions are satisfied:

• 1. For any two objects $a,b\in \textrm{Ob}\mathcal {T}$, the $\mathbb {C}$-bilinear pairing $\langle \cdot , \cdot \rangle _{a,b}:\textrm{Hom}_\mathcal {T}(a,b)\times \textrm{Hom}_\mathcal {T}(b,a)\rightarrow \mathbb {C}$ defined through:

  $$\begin{aligned} \langle t_1,t_2\rangle _{a,b}{\mathop {=}\limits ^{\mathrm{def.}}}\textrm{tr}_b (t_1\circ t_2),~~\forall t_1\in \textrm{Hom}_\mathcal {T}(a,b),~\forall t_2\in \textrm{Hom}_\mathcal {T}(b,a) \end{aligned}$$

  is non-degenerate.

• 2. For any two objects $a,b\in \textrm{Ob}\mathcal {T}$ and any $\mathbb {Z}_2$-homogeneous elements $t_1\in \textrm{Hom}_\mathcal {T}(a,b)$ and $t_2\in \textrm{Hom}_\mathcal {T}(b,a)$, we have:

  $$\begin{aligned} \langle t_1,t_2\rangle _{a,b}=(-1)^{\mathrm{deg\,}t_1\,\mathrm{deg\,}t_2}\langle t_2,t_1\rangle _{b,a}. \end{aligned}$$

  (A.2)

We are interested in the case of Calabi–Yau supercategories which admit a shift functor compatible with the traces $\textrm{tr}_a$.

Definition A.15

A Calabi–Yau supercategory of parity $\mu \in \mathbb {Z}_2$ with compatible shift functor is a triplet $(\mathcal {T},\textrm{tr},\varSigma )$ such that:

• 1. $(\mathcal {T},\textrm{tr})$ is a Calabi–Yau supercategory of parity $\mu$.

• 2. $\varSigma$ is a parity change functor on the $\mathbb {Z}_2$-graded $\mathbb {C}$-linear category $\mathcal {T}$.

• 3. We have:

  $$\begin{aligned} \textrm{tr}_{\varSigma (a)}(\varSigma (t))=\textrm{tr}_\varSigma (t),~~\forall a\in \textrm{Ob}\mathcal {T},~\forall t\in \textrm{End}_\mathcal {T}(a). \end{aligned}$$

1.7 A.7 Serre functors

Recall the notion of Serre functor introduced by Bondal and Kapranov [15]⁶:

Definition A.16

A Serre functor on a Hom-finite $\mathbb {C}$-linear category $\mathcal {C}$ is a linear autoequivalence S of $\mathcal {C}$ such that for any two objects a, b of $\mathcal {C}$, there exists a linear isomorphism:

$$\begin{aligned} \textrm{Hom}_{\mathcal {C}}(a, S(b))\simeq \textrm{Hom}_{\mathcal {C}}(b,a)^\vee \end{aligned}$$

which is natural in both a and b. More precisely, there exists an isomorphism of functors:

$$\begin{aligned} \textrm{Hom}_{\mathcal {C}}(\textrm{id}_\mathcal {C}\times S)\simeq \textrm{D}\circ \textrm{Hom}_{\mathcal {C}}\circ \tau , \end{aligned}$$

where $\tau :\mathcal {T}\times \mathcal {T}\rightarrow \mathcal {T}\times \mathcal {T}$ is the transposition functor and $\textrm{D}:\textrm{vect}_\mathbb {C}\rightarrow \textrm{vect}_\mathbb {C}$ is the dualization functor on the category $\textrm{vect}_\mathbb {C}$ of finite-dimensional vector spaces.

One has the following equivalent description:

Proposition A.17

Let $\mathcal {C}$ be a Hom-finite $\mathbb {C}$-linear category and let S be a linear automorphism of $\mathcal {C}$. Then the following statements are equivalent:

• (a) S is a Serre functor for $\mathcal {C}$.

• (b) For any object a of $\mathcal {C}$, there exists a linear map $\textrm{tr}_a:\textrm{Hom}_{\mathcal {C}}(a,S(a))\rightarrow \mathbb {C}$ such that the following conditions are satisfied for any objects a and b of $\mathcal {C}$:

  • $\textrm{tr}_a(g\circ f)=\textrm{tr}_b(S(f)\circ g),~\forall f\in \textrm{Hom}_\mathcal {C}(a,b),~\forall g\in \textrm{Hom}_{\mathcal {C}}(b,S(a))$.

  • The bilinear map $\langle \cdot , \cdot \rangle _{a,b}^S:\textrm{Hom}_\mathcal {C}(a,b)\times \textrm{Hom}_{\mathcal {C}}(b,S(a))\rightarrow \mathbb {C}$ defined through:

    $$\begin{aligned} \langle f,g\rangle _{a,b}^S{\mathop {=}\limits ^{\mathrm{def.}}}\textrm{tr}_b(S(f)\circ g),~\forall f\in \textrm{Hom}_\mathcal {C}(a,b),~\forall g\in \textrm{Hom}_{\mathcal {C}}(b,S(a)) \end{aligned}$$

    is non-degenerate.

1.8 A.8 Calabi–Yau categories with involution

Given a $\mathbb {C}$-linear category $\mathcal {T}$ with involution $\varSigma$ and an element $\mu \in \mathbb {Z}_2$, we define:

$$\begin{aligned} \varSigma ^\mu {\mathop {=}\limits ^{\mathrm{def.}}} \left\{ \begin{array}{ll} \textrm{id}_{\mathcal {T}^{\hat{0}}~~} &{} \text{ if } \mu ={\hat{0}} ~~ , \\ \varSigma ~~ &{} \text{ if } \mu ={\hat{1}} ~~. \end{array} \right. \end{aligned}$$

Definition A.18

Let $\mu \in \mathbb {Z}_2$. Then a $\mathbb {C}$-linear category with involution $(\mathcal {C},\varSigma )$ is called $\mu$-Calabi–Yau if $\varSigma ^\mu$ is a Serre functor for $\mathcal {C}$.

For any $\mu \in \mathbb {Z}_2$, let $\textrm{ICYCat}(\mu )$ denote the full subcategory of ${\mathbb {C}} {\textrm{ICat}}$ consisting of all $\mu$-Calabi–Yau categories with involution and $\mathrm {SCYCat^s}(\mu )$ denote the full subcategory of ${\mathbb {C}} {\mathrm {SCat^s}}$ consisting of all Calabi–Yau supercategories of signature $\mu$. The proof of the following statement is immediate:

Proposition A.19

The restrictions of the functors $\textrm{Gr}$ and $\textrm{Ev}$ give mutually quasi-inverse equivalences between the categories $\textrm{ICYCat}(\mu )$ and $\mathrm {SCYCat^s}$.

1.9 A.9 Shift functors on $\mathbb {Z}_2$-graded dg-categories

Definition A.20

Let R be a unital commutative ring and $\mathcal {A}$ be $\mathbb {Z}_2$-graded and R-linear dg-category. A differential shift functor on $\mathcal {A}$ is a shift functor $\varSigma$ on the underlying R-linear supercategory of $\mathcal {A}$ which satisfies the following condition for any objects a and b of $\mathcal {A}$:

$$\begin{aligned} \varSigma _{a,b}\circ \textrm{d}_{a,b}=\textrm{d}_{\varSigma (a),\varSigma (b)}\circ \varSigma _{a,b}~: \textrm{Hom}_{\mathcal {A}}(a,b)\rightarrow \textrm{Hom}_{\mathcal {A}}(\varSigma (a),\varSigma (b)), \end{aligned}$$

(A.3)

where $\textrm{d}_{a,b}$ and $\textrm{d}_{\varSigma (a),\varSigma (b)}$ are the odd differentials on the R-supermodules $\textrm{Hom}_{\mathcal {A}}(a,b)$ and respectively $\textrm{Hom}_{\mathcal {A}}(\varSigma (a),\varSigma (b))$.

A differential shift functor $\varSigma$ on $\mathcal {A}$ induces a shift functor on the total cohomology category $\textrm{H}(\mathcal {A})$, which we again denote by $\varSigma$.