Basic Theory of the Generalized Donaldson-Thomas Invariants of Joyce-Song

This is a basic and quick introduction of the theory of generalized Donaldson-Thomas invariants of strict Calabi-Yau $3$-folds due to Joyce-Song in [JS11]¹. We work over $\mathbb C$ to defined consider the generalized DT-invariants. But most of the theory before that we will work over any algebraically closed field $\mathbb K$ of characteristic zero.

0. Review of Donaldson-Thomas Invariants of Calabi-Yau $3$-folds

Definition 0.1. A Calabi-Yau manifold is a smooth projective variety $X$ over $\mathbb K$ such that $K_X\cong\mathscr O_X$. A strict Calabi-Yau manifold is a Calabi-Yau manifold $X$ such that $H^i(X,\mathscr O_X)=0$ for any $0<i<\dim X$.

Then we fix a Calabi-Yau $3$-fold $X$ with a very ample line bundle $\mathscr O_X(1)$ and a Gieseker stability condition $(\tau,G,\leq)$. Consider the moduli stack of semistable sheaves $\mathcal M^{\alpha\text{-ss}}(\tau)$ for some $\alpha\in K^{\text{num}}(\text{coh}(X))$. Then it is a proper Artin stack with a projective good moduli space $ M^{\alpha\text{-ss}}(\tau)$. They contain open locus of stable objects $\mathcal M^{\alpha\text{-st}}(\tau)\subset\mathcal M^{\alpha\text{-ss}}(\tau)$ and $M^{\alpha\text{-st}}(\tau)\subset M^{\alpha\text{-ss}}(\tau)$ which need not be proper.

We can show that there is a perfect obstruction theory of $\mathcal M^{\alpha\text{-ss}}(\tau)$ (and hence $\mathcal M^{\alpha\text{-st}}(\tau)$) comes from their derived enhancement. This is a general fact:

Theorem 0.2. Let $X$ be a smooth projective variety of dimension $n$. Then the natural derived moduli stack $\mathbb R\underline{Coh}(X)$ has cotangent complex \[\mathbb L_{\mathbb R\underline{Coh}(X)}\cong(\mathbb R\mathscr Hom_{\pi}(\mathbb F,\mathbb F)[1])^{\vee}\cong\mathbb R\mathscr Hom_{\pi}(\mathbb F,\mathbb F\otimes\omega_X)[n-1]\] where $\pi:\mathbb R\underline{Coh}(X)\times X\to\mathbb R\underline{Coh}(X)$ and universal sheaf $\mathbb F$ which induce an obstruction theory $\mathbb L_{\mathbb R\underline{Coh}(X)}|_{\underline{Coh}(X)}\to\mathbb L_{\underline{Coh}(X)}$.

Hence by this fact, we know that for the Calabi-Yau $3$-fold $X$, $\mathcal M^{\alpha\text{-ss}}(\tau)$ (and hence $\mathcal M^{\alpha\text{-st}}(\tau)$) has a perfect obstruction theory. But by the theory of Behrend–Fantechi or general virtual pullbacks (see virtual-blog for the introduction), the virtual classes can not construct over a general Artin stack which is not Deligne-Mumfold. However, the good moduli space $\mathcal M^{\alpha\text{-st}}(\tau)\to M^{\alpha\text{-st}}(\tau)$ is actually a $B\mathbb G_m$-gerbe and we can descend the obstruction theory of $\mathcal M^{\alpha\text{-st}}(\tau)$ to $M^{\alpha\text{-st}}(\tau)$, up to some twist (but the $\mathbb R\mathscr Hom(\mathbb F,\mathbb F)$ does not change after the twist). Hence by the theory of Behrend–Fantechi, we get a virtual class \[[M^{\alpha\text{-st}}(\tau)]^{\text{vir}}\in\text{CH}_0(M^{\alpha\text{-st}}(\tau)).\]

Definition 0.3. For a Calabi-Yau $3$-fold $X$ such that $M^{\alpha\text{-ss}}(\tau)=M^{\alpha\text{-st}}(\tau)$, the Donnaldson-Thomas invariants is defined by \[\text{DT}^{\alpha}(\tau):=\int_{[M^{\alpha\text{-st}}(\tau)]^{\text{vir}}}1\in\mathbb Z.\]

The goal of generalized Donaldson-Thomas invariants is to define the DT invariant for any $\alpha$ even $M^{\alpha\text{-ss}}(\tau)\neq M^{\alpha\text{-st}}(\tau)$. The idea comes from another description of DT invariants of Calabi-Yau $3$-fold $X$ such that $M^{\alpha\text{-ss}}(\tau)=M^{\alpha\text{-st}}(\tau)$.

This description follows from a function, we call it the Behrend function:

Definition 0.4. Let $X$ a finite type $\mathbb C$-scheme. Suppose $X\subset M$ is an embedding of $X$ as a closed subscheme of a smooth scheme $M$. Let $C_XM$ be the normal cone with projection $\pi: C_XM\to X$. Define the distinguished cycle is \[\mathfrak c_{X/M}=\sum_{C’}(-1)^{\dim\pi(C’)}\mathrm{mult}(C’)\pi(C’)\in Z_*(X)\] where the sum is over all irreducible components $C’$ of $C_XM$. One can show that this determined by $X$ uniquely, so we let $\mathfrak c_{X}=\mathfrak c_{X/M}$.

Write $\text{CF}_{\mathbb Z}(X)$ for the group of $\mathbb Z$-valued constructible functions on $X$. The local Euler obstruction is a group isomorphism $\text{Eu}:Z_*(X)\to\text{CF}_{\mathbb Z}(X)$ defined as: if $V$ is a prime cycle on $X$, then \[\text{Eu}:V\mapsto\left(x\mapsto\int_{\mu^{-1}(x)}c(\tilde{T})\cap s(\mu^{-1}(x),\tilde{V})\right)\] where $\mu:\tilde{V}\to V$ be the Nash blowup of $V$ with dual of universal quotient bundle $\tilde{T}$. Finally, we define the Behrend function as \[\nu_X:=\text{Eu}(\mathfrak c_{X})\in\text{CF}(X).\]

Small Remark. For the Nash blowup we refer to see Example 4.2.9 in 2022-12-12-blogs and the refs in it.

Then we have the following fact:

Theorem 0.5. (Behrend) Let $M$ be a proper $\mathbb K$-scheme with a symmetric perfect obstruction theory, then \[\int_{[M]^{\text{vir}}}1=\chi(X,\nu_X):=\sum_{k\in\mathbb Z}k\chi(\nu_X^{-1}(k)).\]

Remark 0.6. Here although we do not prove this theorem, we can give some idea of the reason why this result is true. Here we let $\mathbb K=\mathbb C$. Here we focus on the case that consider the Calabi-Yau $3$-fold $X$ and $M:=M^{\alpha\text{-ss}}(\tau)=M^{\alpha\text{-st}}(\tau)$. By the works of derived symplectic geometry in [BBJ19]², the moduli space $M$ is known to be locally written as critical loci of some function. That is for each point $p\in M$, there is an open neighborhood $p\in U$ such that $U\cong\mathrm{Crit}(w)$ where $Y$ is a smooth scheme and $w$ is a regular function on it. So we need to consider the geometry of critical locus.

Let $U$ be a complex manifold with a holomorphic function $f:U\to\mathbb C$. Then we can show that \[\nu_{\mathrm{Crit}(f)}(x)=(-1)^{\dim U}(1-\text{MF}_f(x)),\quad x\in\mathrm{Crit}(f).\] Then as the virtual locus is just the perturb the section locally to give a nice zero locus, here Behrend function is also given by some perturbing, that is, the Milnor fibres. Of course, as a generalization of Milnor fibres, we can using vanishing cycle to give another description. As in [JS11]¹ Theorem 4.9, we have \[\chi_{U_0}(\phi_f(\underline{\mathbb Q}[n-1]))(x)=\nu_{\mathrm{Crit}(f)}(x),\quad x\in\mathrm{Crit}(f)\] where $\phi_f$ be the vanishing-cycle functor.

However, when $M^{\alpha\text{-ss}}(\tau)\neq M^{\alpha\text{-st}}(\tau)$, then locally $M^{\alpha\text{-ss}}(\tau)$ is not a critical locus. But it is for moduli stack $\mathcal M^{\alpha\text{-ss}}(\tau)$! So we need to consider the Behrend function on Artin stacks. But it is not obvious how to take its weighted Euler characteristics. For example, let us take $\alpha$ with Hilbert polynomial $P = (2, 0, 0, 0)$ so that the only semistable sheaf is $\mathscr O_X^{\oplus 2}$. Then $\mathcal M^{\alpha\text{-ss}}(\tau)\cong B\text{GL}_2(\mathbb C)$. Their euler number can not defined $1/(\chi(\text{GL}_2(\mathbb C)))$ or its $\mathbb C^*$-rigidification $1/(\chi(\text{PGL}_2(\mathbb C)))$ as $\chi(\text{GL}_2(\mathbb C))=\chi(\text{PGL}_2(\mathbb C))=0$.

An issue caused here is that, for a strictly semistable sheaf $E$, the dimension of the maximal torus of $\text{Aut}(E)$ is greater than one. A crucial idea in Joyce is that, by taking the logarithm of the moduli stack $\mathcal M^{\alpha\text{-ss}}(\tau)$, we can regard it as a virtual stack whose stabilizer groups have one-dimensional maximal torus. Then by getting rid of the one-dimensional maximal torus and integrating over the Behrend function, the desired DT invariant (called generalized DT invariant) is defined.

1. Constructible Functions and Stack Functions

Here are some basic definitions we need in the whole theory.

1.1. Constructible Functions

Definition 1.1. Let $\mathbb{K}$ be an algebraically closed field of characteristic zero, and $\mathfrak{F}$ an Artin $\mathbb{K}$-stack. We call $C \subseteq \mathfrak{F}(\mathbb{K})$ constructible if $C = \bigcup_{i \in I} \mathfrak{F}_i(\mathbb{K})$, where ${\mathfrak{F}_i : i \in I}$ is a finite collection of finite type Artin $\mathbb{K}$-substacks $\mathfrak{F}_i$ of $\mathfrak{F}$. We call $S \subseteq \mathfrak{F}(\mathbb{K})$ locally constructible if $S \cap C$ is constructible for all constructible $C \subseteq \mathfrak{F}(\mathbb{K})$.

A function $f : \mathfrak{F}(\mathbb{K}) \to \mathbb{Q}$ is called constructible if $f(\mathfrak{F}(\mathbb{K}))$ is finite and $f^{-1}(c)$ is a constructible set in $\mathfrak{F}(\mathbb{K})$ for each $c \in f(\mathfrak{F}(\mathbb{K})) \setminus {0}$. A function $f : \mathfrak{F}(\mathbb{K}) \to \mathbb{Q}$ is called locally constructible if $f \cdot \delta_C$ is constructible for all constructible $C \subseteq \mathfrak{F}(\mathbb{K})$, where $\delta_C$ is the characteristic function of $C$. Write $\text{CF}(\mathfrak{F})$ and $\text{LCF}(\mathfrak{F})$ for the $\mathbb{Q}$-vector spaces of $\mathbb{Q}$-valued constructible and locally constructible functions on $\mathfrak{F}$.、

Definition 1.2. Let $\mathbb{K}$ have characteristic zero, and $\mathfrak{F}$ be an Artin $\mathbb{K}$-stack with affine geometric stabilizers and $C \subseteq \mathfrak{F}(\mathbb{K})$ be constructible. Then we define the naïve Euler characteristic $\chi^{\text{na}}(C)$ of $C$ to be some $\chi(Y)$ for some algebraic scheme $Y$ on $\mathbb K$ with $C\cong Y(\mathbb K)$ by some Weil cohomology theory (we will omit this). It is called naïve as it takes no account of stabilizer groups. For $f \in \text{CF}(\mathfrak{F})$, define $\chi^{\text{na}}(\mathfrak{F}, f)$ in $\mathbb{Q}$ by \[\chi^{\text{na}}(\mathfrak{F}, f) = \sum_{c \in f (\mathfrak{F}(\mathbb{K})) \setminus {0}} c \chi^{\text{na}} (f^{-1}(c)).\]

Let $\mathfrak{F}, \mathfrak{G}$ be Artin $\mathbb{K}$-stacks with affine geometric stabilizers, and $\phi : \mathfrak{F} \to \mathfrak{G}$ a 1-morphism. For $f \in \text{CF}(\mathfrak{F})$, define $\text{CF}^na(\phi)f : \mathfrak{G}(\mathbb{K}) \to \mathbb{Q}$ by \[ \text{CF}^{\text{na}}(\phi)f(y) = \chi^{\text{na}}(\mathfrak{F}, f \cdot \delta_{\phi^{-1}(y)}) \quad \text{for } y \in \mathfrak{G}(\mathbb{K}), \] where $\delta_{\phi^{-1}(y)}$ is the characteristic function of $\phi^{-1}({y}) \subseteq \mathfrak{G}(\mathbb{K})$ on $\mathfrak{G}(\mathbb{K})$. Then $\text{CF}^{\text{na}}(\phi) : \text{CF}(\mathfrak{F}) \to \text{CF}(\mathfrak{G})$ is a $\mathbb{Q}$-linear map called the naïve pushforward.

Now suppose $\phi$ is representable. Then for any $x \in \mathfrak{F}(\mathbb{K})$ we have an injective morphism $\phi_* : \text{Iso}_{\mathfrak{F}}(x) \to \text{Iso}_{\mathfrak{G}}(\phi_*(x))$ of affine algebraic $\mathbb{K}$-groups. The image $\phi_*(\text{Iso}_{\mathfrak{F}}(x))$ is an affine algebraic $\mathbb{K}$-group closed in $\text{Iso}_{\mathfrak{G}}(\phi_*(x))$, so the quotient $\text{Iso}_{\mathfrak{G}}(\phi_*(x))/(\phi_*(\text{Iso}_{\mathfrak{F}}(x)))$ exists as a quasiprojective $\mathbb{K}$-variety. Define a function $m_\phi : \mathfrak{F}(\mathbb{K}) \to \mathbb{Z}$ by $m_\phi(x) = \chi(\text{Iso}_{\mathfrak{G}}(\phi_*(x))/(\phi_*(\text{Iso}_{\mathfrak{F}}(x))) )$ for $x \in \mathfrak{F}(\mathbb{K})$. For $f \in \text{CF}(\mathfrak{F})$, define $\text{CF}^{stk}(\phi)f : \mathfrak{G}(\mathbb{K}) \to \mathbb{Q}$ by \[ \text{CF}^{stk}(\phi)f(y) = \chi^{\text{na}}(\mathfrak{F}, m_\phi \cdot f \cdot \delta_{\phi^{-1}(y)}) \quad \text{for } y \in \mathfrak{G}(\mathbb{K}). \]

An alternative definition is \[ \text{CF}^{stk}(\phi)f(y) = \chi(\mathfrak{F} \times_{\phi, \mathfrak{G}, y}\text{Spec}\, \mathbb{K}, \pi_{\mathfrak{F}}^{*}(f)) \quad \text{for } y \in \mathfrak{G}(\mathbb{K}), \] where $\mathfrak{F} \times_{\phi, \mathfrak{G}, y}\text{Spec}\, \mathbb{K}$ is a $\mathbb{K}$-scheme (or algebraic space) as $\phi$ is representable, and $\chi(\cdots)$ is the Euler characteristic of this $\mathbb{K}$-scheme weighted by $\pi_{\mathfrak{F}}^{*}(f)$. These two definitions are equivalent as the projection $\pi_1 : \mathfrak{F} \times_{\phi, \mathfrak{G}, y}\text{Spec}\, \mathbb{K} \to \mathfrak{F}$ induces a map on $\mathbb{K}$-points $(\pi_1)_* : (\mathfrak{F} \times_{\phi, \mathfrak{G}, y}\text{Spec}\, \mathbb{K})(\mathbb{K}) \to \phi_{*}^{-1}(y) \subset \mathfrak{F}(\mathbb{K})$, and the fibre of $(\pi_1)_*$ over $x \in \phi_{*}^{-1}(y)$ is $(\text{Iso}_{\mathfrak{G}}(\phi_*(x))/(\phi_*(\text{Iso}_{\mathfrak{F}}(x))))(\mathbb{K})$, with Euler characteristic $m_\phi(x)$. Then $\text{CF}^{stk}(\phi) : \text{CF}(\mathfrak{F}) \to \text{CF}(\mathfrak{G})$ is a $\mathbb{Q}$-linear map called the stack pushforward. If $\mathfrak{F}, \mathfrak{G}$ are $\mathbb{K}$-schemes then $\text{CF}^na(\phi), \text{CF}^{stk}(\phi)$ coincide, and we write them both as $\text{CF}(\phi) : \text{CF}(\mathfrak{F}) \to \text{CF}(\mathfrak{G})$.

Let $\theta : \mathfrak{F} \to \mathfrak{G}$ be a finite type 1-morphism. If $C \subseteq \mathfrak{G}(\mathbb{K})$ is constructible then so is $\theta_{、*}^{-1}(C) \subseteq \mathfrak{F}(\mathbb{K})$. It follows that if $f \in \text{CF}(\mathfrak{G})$ then $f \circ \theta_*$ lies in $\text{CF}(\mathfrak{F})$. Define the pullback $\theta^{*} : \text{CF}(\mathfrak{G}) \to \text{CF}(\mathfrak{F})$ by $\theta^{*}(f) = f \circ \theta_*$. It is a linear map.

There are several functorial and commutative properties of these pullback pushforward we will omit these. We refer Theorem 2.4 in [JS11]¹.

1.2. Stack Functions

Definition 1.3. Let $\mathbb{K}$ be an algebraically closed field, and $\mathfrak{F}$ be an Artin $\mathbb{K}$-stack with affine geometric stabilizers. Consider pairs $(\mathfrak{R}, \rho)$, where $\mathfrak{R}$ is a finite type Artin $\mathbb{K}$-stack with affine geometric stabilizers and $\rho : \mathfrak{R} \to \mathfrak{F}$ is a 1-morphism. We call two pairs $(\mathfrak{R}, \rho)$, $(\mathfrak{R}’ , \rho’)$ equivalent if there exists a 1-isomorphism $\iota : \mathfrak{R} \to \mathfrak{R}’$ such that $\rho’ \circ \iota$ and $\rho$ are 2-isomorphic 1-morphisms $\mathfrak{R} \to \mathfrak{F}$. Write $[(\mathfrak{R}, \rho)]$ for the equivalence class of $(\mathfrak{R}, \rho)$. If $(\mathfrak{R}, \rho)$ is such a pair and $\mathfrak{S}$ is a closed $\mathbb{K}$-substack of $\mathfrak{R}$ then $(\mathfrak{S}, \rho|_{\mathfrak{S}})$, $(\mathfrak{R}\setminus \mathfrak{S}, \rho|_{\mathfrak{R}\setminus \mathfrak{S}})$ are pairs of the same kind.

Define $\underline{\mathrm{SF}}(\mathfrak{F})$ to be the $\mathbb{Q}$-vector space generated by equivalence classes $[(\mathfrak{R}, \rho)]$ as above, with for each closed $\mathbb{K}$-substack $\mathfrak{S}$ of $\mathfrak{R}$ a relation

\[ [(\mathfrak{R}, \rho)] = [(\mathfrak{S}, \rho|_{\mathfrak{S}})] + [(\mathfrak{R}\setminus \mathfrak{S}, \rho|_{\mathfrak{R}\setminus \mathfrak{S}})]. \]

Define $\mathrm{SF}(\mathfrak{F})$ to be the $\mathbb{Q}$-vector space generated by $[(\mathfrak{R}, \rho)]$ with $\rho$ representable, with the same relations above. Then $\mathrm{SF}(\mathfrak{F}) \subseteq \underline{\mathrm{SF}}(\mathfrak{F})$.

Small Remark: Elements of $\mathrm{SF}(\mathfrak{F})$ will be called stack functions. We write stack functions either as letters $f, g, \ldots$, or explicitly as sums $\sum_{i=1}^m c_i[(\mathfrak{R}_i, \rho_i)]$. If $[(\mathfrak{R}, \rho)]$ is a generator of $\mathrm{SF}(\mathfrak{F})$ and $\mathfrak{R}^{\mathrm{red}}$ is the reduced substack of $\mathfrak{R}$ then $\mathfrak{R}^{\mathrm{red}}$ is a closed substack of $\mathfrak{R}$ and the complement $\mathfrak{R} \setminus \mathfrak{R}^{\mathrm{red}}$ is empty. Hence

\[ [(\mathfrak{R}, \rho)] = [(\mathfrak{R}^{\mathrm{red}}, \rho|_{\mathfrak{R}^{\mathrm{red}}})]. \]

Thus, the relations (2.5) destroy all information on nilpotence in the stack structure of $\mathfrak{R}$.

We now can relate $\mathrm{CF}(\mathfrak{F})$ and $\mathrm{SF}(\mathfrak{F})$:

Definition 1.4. Let $\mathfrak{F}$ be an Artin $\mathbb{K}$-stack with affine geometric stabilizers, and $C \subseteq \mathfrak{F}(\mathbb{K})$ be constructible. Then $C = \prod_{i=1}^n \mathfrak{R}_i(\mathbb{K})$, for $\mathfrak{R}_1, \ldots, \mathfrak{R}_n$ finite type $\mathbb{K}$-substacks of $\mathfrak{F}$. Let $\rho_i : \mathfrak{R}_i \to \mathfrak{F}$ be the inclusion 1-morphism. Then $[(\mathfrak{R}_i, \rho_i)] \in \mathrm{SF}(\mathfrak{F})$. Define $\delta_C = \sum_{i=1}^n [(\mathfrak{R}_i, \rho_i)] \in \mathrm{SF}(\mathfrak{F})$. We think of this stack function as the analogue of the characteristic function $\delta_C \in \mathrm{CF}(\mathfrak{F})$ of $C$. When $\mathbb{K}$ has characteristic zero, define a $\mathbb{Q}$-linear map $\iota_{\mathfrak{F}} : \mathrm{CF}(\mathfrak{F}) \to \mathrm{SF}(\mathfrak{F})$ by $\iota_{\mathfrak{F}}(f) = \sum_{0 \neq c \in f (\mathfrak{F}(\mathbb{K}))} c \cdot \delta_{f^{-1}(c)}$. Define $\mathbb{Q}$-linear $\pi_{\mathfrak{F}}^{\mathrm{stk}} : \mathrm{SF}(\mathfrak{F}) \to \mathrm{CF}(\mathfrak{F})$ by

\[ \pi_{\mathfrak{F}}^{\mathrm{stk}} (\sum_{i=1}^n c_i [(\mathfrak{R}_i, \rho_i)]) = \sum_{i=1}^n c_i \mathrm{CF}_{\rho_i}^{\mathrm{stk}} (\rho_i) \mathbf{1}_{\mathfrak{R}_i}, \]

where $\mathbf{1}_{\mathfrak{R}_i}$ is the function in $\mathrm{CF}(\mathfrak{R}_i)$. Then we can show that $\pi_{\mathfrak{F}}^{\mathrm{stk}} \circ \iota_{\mathfrak{F}}$ is the identity on $\mathrm{CF}(\mathfrak{F})$. Thus, $\iota_{\mathfrak{F}}$ is injective and $\pi_{\mathfrak{F}}^{\mathrm{stk}}$ is surjective. In general $\iota_{\mathfrak{F}}$ is far from surjective, and $\underline{\mathrm{SF}}, \mathrm{SF}(\mathfrak{F})$ are much larger than $\mathrm{CF}(\mathfrak{F})$.

Some operators:

Definition 1.5. Define multiplication ‘.’ on $\underline{\mathrm{SF}}(\mathfrak{F})$ by

\[ [(\mathfrak{R}, \rho)] \cdot [(\mathfrak{S}, \sigma)] = [(\mathfrak{R} \times_{\rho, \mathfrak{F}, \sigma} \mathfrak{S}, \rho \circ \pi_{\mathfrak{R}})]. \]

This extends to a $\mathbb{Q}$-bilinear product $\underline{\mathrm{SF}}(\mathfrak{F}) \times \underline{\mathrm{SF}}(\mathfrak{F}) \to\underline{\mathrm{SF}}(\mathfrak{F})$ which is commutative and associative, and $\mathrm{SF}(\mathfrak{F})$ is closed under ‘.’. Let $\phi : \mathfrak{F} \to \mathfrak{G}$ be a $1$-morphism of Artin $\mathbb{K}$-stacks with affine geometric stabilizers. Define the pushforward $\phi^{*} : \underline{\mathrm{SF}}(\mathfrak{F}) \to \underline{\mathrm{SF}}(\mathfrak{G})$ by

\[ \phi^{*} : \sum_{i=1}^m c_i[(\mathfrak{R}_i, \rho_i)] \mapsto \sum_{i=1}^m c_i[(\mathfrak{R}_i, \phi \circ \rho_i)]. \]

If $\phi$ is representable then $\phi^{*}$ maps $\mathrm{SF}(\mathfrak{F}) \to \mathrm{SF}(\mathfrak{G})$. For $\phi$ of finite type, define pullbacks $\phi^{*} :\underline{\mathrm{SF}}(\mathfrak{G}) \to\underline{\mathrm{SF}}(\mathfrak{F})$, $\phi^{*} : \mathrm{SF}(\mathfrak{G}) \to \mathrm{SF}(\mathfrak{F})$ by

\[ \phi^{*} : \sum_{i=1}^m c_i[(\mathfrak{R}_i, \rho_i)] \mapsto \sum_{i=1}^m c_i[(\mathfrak{R}_i \times_{\rho_i, \mathfrak{G}, \phi} \mathfrak{F}, \pi_{\mathfrak{F}})]. \]

The tensor product $\otimes :\underline{\mathrm{SF}}(\mathfrak{F}) \times\underline{\mathrm{SF}}(\mathfrak{G}) \to\underline{\mathrm{SF}}(\mathfrak{F} \times \mathfrak{G})$ or $\mathrm{SF}(\mathfrak{F}) \times \mathrm{SF}(\mathfrak{G}) \to \mathrm{SF}(\mathfrak{F} \times \mathfrak{G})$ is

\[ (\sum_{i=1}^m c_i[(\mathfrak{R}_i, \rho_i)]) \otimes (\sum_{j=1}^n d_j[(\mathfrak{G}_j, \sigma_j)]) = \sum_{i,j} c_i d_j[(\mathfrak{R}_i \times \mathfrak{G}_j, \rho_i \times \sigma_j)]. \]

There are several functorial and commutative properties of these pullback pushforward we will omit these. We refer Theorem 2.8, 2.9 in [JS11]¹.

1.3. Projections of Virtual Ranks

As we have seen, aiming to define the generalized DT-invariants we need to find a logarithm of the moduli stack $\mathcal M^{\alpha\text{-ss}}(\tau)$, we can regard it as a virtual stack whose stabilizer groups have one-dimensional maximal torus. Here we will introduce a projection which will maps the stack functions to the stack functions with some fixed virtual rank of stabilizer groups.

In general, a projection $\Pi^{\mu}$ of stack functions is associated to a weight function:

Definition 1.6. A weight function $\mu$ is a map

\[ \mu : { \mathbb{K} \text{-groups } \mathbb{G}_m^{k} \times K, \, k \geq 0, \, K \text{ finite abelian}, \, \text{up to isomorphism} } \longrightarrow \mathbb{Q}. \]

In our idea, if $\mathfrak{R}$ has abelian stabilizer groups, then $\Pi^\mu([(\mathfrak{R}, \rho)])$ simply weights each point $r$ of $\mathfrak{R}$ by $\mu(\text{Isos}_\mathfrak{R}(r))$. But if $\mathfrak{R}$ has nonabelian stabilizer groups, then $\Pi^\mu([(\mathfrak{R}, \rho)])$ replaces each point $r$ with stabilizer group $G$ by a $\mathbb{Q}$-linear combination of points with stabilizer groups $C_G({t})$ for $t \in T^G$, where $T^G$ is the maximal torus and the $\mathbb{Q}$-coefficients depend on the values of $\mu$ on subgroups of $T^G$:

Definition 1.7. For any Artin $\mathbb{K}$-stack $\mathfrak{F}$ with affine geometric stabilizers, we will define linear maps $\Pi^\mu : \mathrm{SF}(\mathfrak{F}) \to \mathrm{SF}(\mathfrak{F})$ and $\Pi^\mu : \mathrm{SF}(\mathfrak{F}) \to \mathrm{SF}(\mathfrak{F})$. Now $\mathrm{SF}(\mathfrak{F})$ is generated by $[(\mathfrak{R}, \rho)]$ with $\mathfrak{R}$ 1-isomorphic to a quotient $[X/G]$, for $X$ a quasiprojective $\mathbb{K}$-variety and $G$ a special algebraic $\mathbb{K}$-group, with maximal torus $T^G$.

Let $\mathcal{S}(T^G)$ be the set of subsets of $T^G$ defined by Boolean operations upon closed $\mathbb{K}$-subgroups $L$ of $T^G$. Given a weight function $\mu$ as above, define a measure $d\mu : \mathcal{S}(T^G) \to \mathbb{Q}$ to be additive upon disjoint unions of sets in $\mathcal{S}(T^G)$, and to satisfy $d\mu(L) = \mu(L)$ for all algebraic $\mathbb{K}$-subgroups $L$ of $T^G$. Define

\[ \Pi^\mu ([(\mathfrak{R}, \rho)]) = \int_{t \in T^G}\frac{|{w \in W(G, T^G) : w \cdot t = t}|}{|W(G,T^G)|}\left[ ([X^{(t)} / C_G((t))], \rho \circ l^{(t)}) \right]d\mu. \]

Here $X^{(t)}$ is the subvariety of $X$ fixed by $t$, and $l^{(t)} : [X^{(t)} / C_G((t))] \to [X/G]$ is the obvious 1-morphism of Artin stacks.

We can show that $\Pi^\mu ([(\mathfrak{R}, \rho)])$ is indeoendent of $X,G,T^G$ and $\mathfrak R\cong[X/G]$.

Now $\Pi^{\mu}$ commute with pushforward and $\Pi^{\mu}\Pi^{\gamma}=\Pi^{\mu\gamma}$.

For our special case, we define $\Pi^{\text{vir}}_n:=\Pi^{\mu_n}$ where $\mu_n([H])=1$ if rank of $H$ is $1$ and $0$ otherwise.

1.4. Stack Function Spaces Twisted by Euler Characteristics

Let $\mathcal{Q}(G,T^{G})$ is a set of closed subgroups $Q$ of $T^G$ such that $Q=T^G\cap C(C_G(Q))$. We call $G$ is very special if $Q,C_G(Q)$ are special for $Q\in\mathcal{Q}(G,T^{G})$. One can show that $\text{CL}_k(\mathbb K)$ is very special.

Definition 1.8. Let $\mathfrak{F}$ be an Artin $\mathbb{K}$-stack with affine geometric stabilizers. Consider pairs $(\mathfrak{R},\rho)$, where $\mathfrak{R}$ is a finite type Artin $\mathbb{K}$-stack with affine geometric stabilizers and $\rho:\mathfrak{R}\rightarrow\mathfrak{F}$ is a 1-morphism, with equivalence of pairs as in Definition 2.5. Define $\underline{\widehat{\mathrm{SF}}}(\mathfrak{F},\chi,\mathbb{Q})$ to be the $\mathbb{Q}$-vector space generated by equivalence classes $[(\mathfrak{R},\rho)]$ as above, with the following relations:

• (i) Given $[(\mathfrak{R},\rho)]$ as above and $\mathfrak{S}$ a closed $\mathbb{K}$-substack of $\mathfrak{R}$ we have $[(\mathfrak{R},\rho)]=[(\mathfrak{S},\rho|_{\mathfrak{S}})]+[(\mathfrak{R} \setminus\mathfrak{S},\rho|_{\mathfrak{R}\setminus\mathfrak{S}})]$, as in (2.5).

• (ii) Let $\mathfrak{R}$ be a finite type Artin $\mathbb{K}$-stack with affine geometric stabilizers, $U$ a quasiprojective $\mathbb{K}$-variety, $\pi_{\mathfrak{R}}:\mathfrak{R}\times U\rightarrow\mathfrak{R}$ the natural projection, and $\rho:\mathfrak{R}\rightarrow\mathfrak{F}$ a 1-morphism. Then $[(\mathfrak{R}\times U,\rho\circ\pi_{\mathfrak{R}})]=\chi([U])[(\mathfrak{R},\rho)]$. Here $\chi(U)\in\mathbb{Z}$ is the Euler characteristic of $U$. It is a motivic invariant of $\mathbb{K}$-schemes, that is, $\chi(U)=\chi(V)+\chi(U\setminus V)$ for $V\subset U$ closed.

• (iii) Given $[(\mathfrak{R},\rho)]$ as above and a 1-isomorphism $\mathfrak{R}\cong[X/G]$ for $X$ a quasiprojective $\mathbb{K}$-variety and $G$ a very special algebraic $\mathbb{K}$-group acting on $X$ with maximal torus $T^{G}$, we have \[[(\mathfrak{R},\rho)]=\sum_{Q\in\mathcal{Q}(G,T^{G})}F(G,T^{G},Q)\bigl{[}([X/Q], \rho\circ t^{\mathbb{Q}})\bigr{]},\] where $t^{\mathbb{Q}}:[X/Q]\rightarrow\mathfrak{R}\cong[X/G]$ is the natural projection 1-morphism.

Here $F(G,T^{G},Q)\in\mathbb{Q}$ are a system of rational coefficients with a complicated definition which we will not repeat.

Similarly, define $\widehat{\mathrm{SF}}(\mathfrak{F},\chi,\mathbb{Q})$ to be the $\mathbb{Q}$-vector space generated by $[(\mathfrak{R},\rho)]$ with $\rho$ representable, and relations (i)-(iii) as above. Then $\widehat{\mathrm{SF}}(\mathfrak{F},\chi,\mathbb{Q})\subset\underline{\widehat{\mathrm{SF}}}( \mathfrak{F},\chi,\mathbb{Q})$. Define projections $\Pi_{\mathfrak{F}}^{\chi,\mathbb{Q}}:\underline{\widehat{\mathrm{SF}}}(\mathfrak{F}) \to\underline{\widehat{\mathrm{SF}}}(\mathfrak{F},\chi,\mathbb{Q})$ and $\widehat{\mathrm{SF}}(\mathfrak{F})\rightarrow\widehat{\mathrm{SF}}(\mathfrak{F}, \chi,\mathbb{Q})$ by $\Pi_{\mathfrak{F}}^{\chi,\mathbb{Q}}:\sum_{i\in I}c_{i}[(\mathfrak{R}_{i},\rho_{i })]\mapsto\sum_{i\in I}c_{i}[(\mathfrak{R}_{i},\rho_{i})]$.

Define multiplication ‘.’, pushforwards $\phi_{*}$, pullbacks $\phi^{*}$, and tensor products $\otimes$ on the spaces $\widehat{\mathrm{SF}},\underline{\widehat{\mathrm{SF}}}(*,\chi,\mathbb{Q})$ as in Definition 2.7, and projections $\Pi_{n}^{vi}$.

An important structure result:

Proposition 1.9. $\widehat{\mathrm{SF}},\underline{\widehat{\mathrm{SF}}}(\mathfrak{F},\chi, \mathbb{Q})$ are spanned over $\mathbb{Q}$ by elements $[(U \times [\operatorname{Spec} \mathbb{K}/T], \rho)]$, for $U$ a quasiprojective $\mathbb{K}$-variety and $T$ an algebraic $\mathbb{K}$-group isomorphic to $\mathbb{G}_m^k \times K$ for $k \geq 0$ and $K$ finite abelian.

Suppose $\sum_{i \in I} c_i [(U_i \times [\operatorname{Spec} \mathbb{K}/T_i], \rho_i)] = 0$ in $\underline{\widehat{\mathrm{SF}}}(\mathfrak{F}, \chi, \mathbb{Q})$ or $\widehat{\mathrm{SF}}(\mathfrak{F}, \chi, \mathbb{Q})$, where $I$ is finite set, $c_i \in \mathbb{Q}$, $U_i$ is a quasiprojective $\mathbb{K}$-variety, and $T_i$ is an algebraic $\mathbb{K}$-group isomorphic to $\mathbb{G}_m^{k_i} \times K_i$ for $k_i \geq 0$ and $K_i$ finite abelian, with $T_i \neq T_j$ for $i \neq j$. Then $c_j [(U_j \times [\operatorname{Spec} \mathbb{K}/T_j], \rho_j)] = 0$ for all $j \in I$.

Note that in this representation, the operators $\Pi_n^{\mathrm{vi}}$ are easy to define: we have $\Pi_n^{\mathrm{vi}} ([(U \times [\operatorname{Spec} \mathbb{K}/T], \rho)]) = [(U \times [\operatorname{Spec} \mathbb{K}/T], \rho)]$ if $\text{dim } T = n$ and $\Pi_n^{\mathrm{vi}} ([(U \times [\operatorname{Spec} \mathbb{K}/T], \rho)]) =0$ otherwise.

This proposition says that a general element $[(\mathfrak{R}, \rho)]$ of $\widehat{\mathrm{SF}},\underline{\widehat{\mathrm{SF}}}(\mathfrak{F}, \chi, \mathbb{Q})$, whose stabilizer groups $\mathrm{Iso}_{\mathfrak{R}}(x)$ for $x \in \mathfrak{R}(\mathbb{K})$ are arbitrary affine algebraic $\mathbb{K}$-groups, may be written as a $\mathbb{Q}$-linear combination of elements $[(U \times [\operatorname{Spec} \mathbb{K}/T], \rho)]$ whose stabilizer groups $T$ are of the form $\mathbb{G}_m^k \times K$ for $k \geq 0$ and $K$ finite abelian. That is, by working in $\widehat{\mathrm{SF}},\underline{\widehat{\mathrm{SF}}}(\mathfrak{F}, \chi, \mathbb{Q})$, we can treat all stabilizer groups as if they are abelian. Furthermore, although $\widehat{\mathrm{SF}},\underline{\widehat{\mathrm{SF}}}(\mathfrak{F}, \chi, \mathbb{Q})$ forget information about nonabelian stabilizer groups, they do remember the difference between abelian stabilizer groups of the form $\mathbb{G}_m^k \times K$ for finite $K$.

2. Ringle-Hall Algebra and Its Applications

Here we will focus on and assume our abelian category $\mathcal A$ and $K(\mathcal A)$ which is a quotient of G-group $K_0(\mathcal A)$ with some expected properties (see Assumption 3.2 in [JS11]¹). Note that these properties holds for $\mathrm{Coh}(X)$ of smooth projective variety $X$ with $K(\mathrm{Coh}(X))=K^{\text{num}}(\mathrm{Coh}(X))$; the caetgory of reps of quiver with relations.

We will use the following notation:

• Define the ‘positive cone’ $C(\mathcal A)$ in $K(\mathcal A)$ to be \[ C(\mathcal A) = {[E] \in K(\mathcal A) : 0 \not\equiv E \in\mathcal A} \subset K(\mathcal A). \]

• Write $\mathfrak{M}_{\mathcal A}$ for the moduli stack of objects in $\mathcal A$. It is an Artin $\mathbb{K}$-stack, locally of finite type. Elements of $\mathfrak{M}_{\mathcal A}(\mathbb{K})$ correspond to isomorphism classes $[E]$ of objects $E$ in $\mathcal A$, and the stabilizer group $\text{Iso}_{\mathfrak{M}_{\mathcal A}}([E])$ in $\mathfrak{M}_{\mathcal A}$ is isomorphic as an algebraic $\mathbb{K}$-group to the automorphism group $\text{Aut}(E)$.

• For $\alpha \in C(A)$, write $\mathfrak{M}_{\mathcal A}^\alpha$ for the substack of objects $E \in {\mathcal A}$ in class $\alpha$ in $K(\mathcal A)$. It is an open and closed $\mathbb{K}$-substack of $\mathfrak{M}_{\mathcal A}$.

• Write $\mathfrak{Exact}_{\mathcal A}$ for the moduli stack of short exact sequences $0 \to E_1 \to E_2 \to E_3 \to 0$ in $A$. It is an Artin $\mathbb{K}$-stack, locally of finite type.

• For $j = 1, 2, 3$ write $\pi_j : \mathfrak{Exact}_{\mathcal A} \to \mathfrak{M}_{\mathcal A}$ for the 1-morphism of Artin stacks projecting $0 \to E_1 \to E_2 \to E_3 \to 0$ to $E_j$. Then $\pi_2$ is representable, and $\pi_1 \times \pi_3 : \mathfrak{Exact}_{\mathcal A} \to \mathfrak{M}_{\mathcal A} \times \mathfrak{M}_{\mathcal A}$ is of finite type.

Definition 2.1. Define bilinear operations $*$ on the stack function spaces $\underline{\text{SF}}, \text{SF}(\mathfrak{M}_{\mathcal A})$ and $\underline{\widehat{\text{SF}}}, \widehat{\text{SF}}(\mathfrak{M}_{\mathcal A}, \chi, \mathbb{Q})$ by

\[ f * g = (\pi_2)_*((\pi_1 \times \pi_3)^*(f \otimes g)), \]

using pushforwards, pullbacks and tensor products we defined. They are well-defined as $\pi_2$ is representable, and $\pi_1 \times \pi_3$ is of finite type. We can sbhow that this $*$ is associative, and makes $\underline{\text{SF}}, \text{SF}(\mathfrak{M}_{\mathcal A})$ and $\underline{\widehat{\text{SF}}}, \widehat{\text{SF}}(\mathfrak{M}_{\mathcal A}, \chi, \mathbb{Q})$ into noncommutative $\mathbb{Q}$-algebras, with identity $\bar{\delta}_{[0]}$, where $[0] \in \mathfrak{M}_{\mathcal A}$ is the zero object. We call them Ringel-Hall algebras. The natural inclusions and projections $\Pi_{\mathfrak{M}_{\mathcal A}}^{\chi,\mathbb{Q}}$ between these spaces are algebra morphisms.

Definition 2.2. Suppose $[(\mathfrak{R}, \rho)]$ is a generator of $\text{SF}(\mathfrak{M}_{\mathcal A})$. Let $r \in \mathfrak{R}(\mathbb{K})$ with $\rho_*(r) = [E] \in \mathfrak{M}_{\mathcal A}(\mathbb{K})$. Then $\rho$ induces an injective morphism of stabilizer $\mathbb{K}$-groups $\rho_* : \text{Iso}_{\mathfrak{R}}(r) \rightarrow \text{Iso}_{\mathfrak{M}_A}([E]) \cong \text{Aut}(E)$ as $\rho$ is representable this is injective. Hence induces an isomorphism of $\text{Iso}_{\mathfrak{R}}(r)$ with a $\mathbb{K}$-subgroup of $\text{Aut}(E)$. Now $\text{Aut}(E) = \text{End}(E)^{\times}$ is the $\mathbb{K}$-group of invertible elements in a finite-dimensional $\mathbb{K}$-algebra $\text{End}(E) = \text{Hom}(E, E)$. We say that $[(\mathfrak{R}, \rho)]$ has algebra stabilizers if whenever $r \in \mathfrak{R}(\mathbb{K})$ with $\rho_*(r) = [E]$, the $\mathbb{K}$-subgroup $\rho_*(\text{Iso}_{\mathfrak{R}}(r))$ in $\text{Aut}(E)$ is the $\mathbb{K}$-group $A^{\times}$ of invertible elements in a $\mathbb{K}$-subalgebra $A$ in $\text{End}(E)$.

Write $\text{SF}_{\text{al}}(\mathfrak{M}_{\mathcal A}), \widehat{\text{SF}}_{\text{al}}(\mathfrak{M}_{\mathcal A}, \chi, \mathbb{Q})$ for the subspaces spanned over $\mathbb{Q}$ by $[(\mathfrak{R}, \rho)]$ with algebra stabilizers. These are subalgebras of Ringel-Hall algebras and one can show that they are closed under $\Pi^{\text{vir}}_{n}$. We define $\text{SF}_{\text{al}}^{\text{ind}}(\mathfrak{M}_{\mathcal A}), \widehat{\text{SF}}_{\text{al}}^{\text{ind}}(\mathfrak{M}_{\mathcal A}, \chi, \mathbb{Q})$ to be the subspaces consist of $f$ with $\Pi_1^i(f) = f$.

We can show that $\text{SF}_{\text{al}}^{\text{ind}}(\mathfrak{M}_{\mathcal A}), \widehat{\text{SF}}_{\text{al}}^{\text{ind}}(\mathfrak{M}_{\mathcal A}, \chi, \mathbb{Q})$ are closed under the Lie bracket $[f, g] = f * g - g * f$. Thus, these are Lie subalgebras. The projection $\Pi_{\mathfrak{M}_{\mathcal A}}^{\chi,\mathbb Q}:\text{SF}_{\text{al}}^{\text{ind}}(\mathfrak{M}_{\mathcal A}) \to\widehat{\text{SF}}_{\text{al}}^{\text{ind}}(\mathfrak{M}_{\mathcal A}, \chi, \mathbb{Q})$ is a Lie algebra morphism.

So our $\text{SF}_{\text{al}}^{\text{ind}}(\mathfrak{M}_{\mathcal A}), \widehat{\text{SF}}_{\text{al}}^{\text{ind}}(\mathfrak{M}_{\mathcal A}, \chi, \mathbb{Q})$ are stack functions with virtual rank $1$! Using the local structure result before, we can show:

Proposition 2.3. $\widehat{\text{SF}}_{\text{al}}(\mathfrak{M}_{\mathcal A}, \chi, \mathbb{Q})$ is spanned over $\mathbb{Q}$ by elements of the form $[(U \times [\text{Spec } \mathbb{K}/\mathbb{G}_{m}^{k}], \rho)]$ with algebra stabilizers, for $U$ a quasiprojective $\mathbb{K}$-variety and $k \geq 0$. Also $\widehat{\text{SF}}_{\text{al}}^{\text{ind}}(\mathfrak{M}_{\mathcal A}, \chi, \mathbb{Q})$ is spanned over $\mathbb{Q}$ by $[(U \times [\text{Spec} \mathbb{K}/\mathbb{G}_{m}^{k}], \rho)]$ with algebra stabilizers, for $U$ a quasiprojective $\mathbb{K}$-variety.

Finally we will taking the logarithm of the moduli stack of semistable objects and we can regard it as a virtual stack whose stabilizer groups have one-dimensional maximal torus, that is, lies in $\text{SF}_{\text{al}}^{\text{ind}}(\mathfrak{M}_{\mathcal A})$.

Now let $\mathbb K, A, K(A)$ be abelian categories with K-group, and $(\tau, T, \leq)$ be a permissible (that is, $A$ is $\tau$-Artinian and the moduli stack of $\alpha$-semistable objects is of finite type for $\alpha\in C(A)$) weak stability condition on $A$. Let $\mathfrak M^{\alpha\text{-ss}}_{A}(\tau),\mathfrak M^{\alpha\text{-st}}_{A}(\tau)$ be the moduli stack of semistable, stable objects.

Definition 2.4. Define stack functions $\delta_{\text{ss}}^{\alpha}(\tau) = \delta_{\mathfrak{M}^{\alpha\text{-ss}}(\tau)}$ in $\text{SF}_{\text{al}}(\mathfrak{M}_A)$ for $\alpha \in C(A)$. We define elements $\bar{\epsilon}^{\alpha}(\tau)$ in $\text{SF}_{\text{al}}(\mathfrak{M}_A)$ by

\[ \bar{\epsilon}^{\alpha}(\tau) = \sum_{n \geq 1, \, \alpha_1, \ldots, \alpha_n \in C(A): \, \alpha_1 + \cdots + \alpha_n = \alpha, \, \tau(\alpha_i) = \tau(\alpha), \, \text{all } i} \frac{(-1)^{n-1}}{n} \delta_{\text{ss}}^{\alpha_1}(\tau) * \delta_{\text{ss}}^{\alpha_2}(\tau) * \cdots * \delta_{\text{ss}}^{\alpha_n}(\tau), \]

where $*$ is the Ringel-Hall multiplication in $\text{SF}_{\text{al}}(\mathfrak{M}_A)$. Then we can prove that

\[ \delta_{\text{ss}}^{\alpha}(\tau) = \sum_{n \geq 1, \, \alpha_1, \ldots, \alpha_n \in C(A): \, \alpha_1 + \cdots + \alpha_n = \alpha, \, \tau(\alpha_i) = \tau(\alpha), \, \text{all } i} \frac{1}{n!} \bar{\epsilon}^{\alpha_1}(\tau) * \bar{\epsilon}^{\alpha_2}(\tau) * \cdots * \bar{\epsilon}^{\alpha_n}(\tau). \]

There are only finitely many nonzero terms above, because as the family of $\tau$-semistable sheaves in class $\alpha$ is bounded, there are only finitely ways to write $\alpha = \alpha_1 + \cdots + \alpha_n$ with $\tau$-semistable sheaves in class $\alpha_i$ for all $i$.

Small Remark. Here is a way to interpret these informally in terms of log and exp: working in a completed version, so that appropriate classes of infinite sums make sense, for fixed $t \in T$ we have

\[ \sum_{\alpha \in C(A): \tau(\alpha) = t} \tilde{\epsilon}^\alpha(\tau) = \log \left[ \bar{\delta}_0 + \sum_{\alpha \in C(A): \tau(\alpha) = t} \bar{\delta}_\text{ss}^\alpha(\tau) \right], \]

and

\[ \bar{\delta}_0 + \sum_{\alpha \in C(A): \tau(\alpha) = t} \bar{\delta}_\text{ss}^\alpha(\tau) = \exp \left[ \sum_{\alpha \in C(A): \tau(\alpha) = t} \tilde{\epsilon}^\alpha(\tau) \right], \]

where $\bar{\delta}_0$ is the identity 1 in $\widetilde{\text{SF}}_{\text{al}}(\mathfrak{M}_A)$. For $\alpha \in C(A)$ and $t = \tau(\alpha)$, using the power series $\log(1 + x) = \sum_{n \geq 1} \frac{(-1)^{n-1}}{n} x^n$ and $\exp(x) = 1 + \sum_{n \geq 1} \frac{1}{n!} x^n$ we see that (3.4)-(3.5) are the restrictions of (3.6)-(3.7) to $\mathfrak{M}_A^\alpha$. This makes clear why (3.4) and (3.5) are inverse, since log and exp are inverse. Thus, knowing the $\tilde{\epsilon}^\alpha(\tau)$ is equivalent to knowing the $\bar{\delta}_\text{ss}^\alpha(\tau)$.

If $\mathfrak{M}_\text{ss}^\alpha(\tau) = \mathfrak{M}_\text{st}^\alpha(\tau)$ then $\tilde{\epsilon}^\alpha(\tau) = \bar{\delta}_\text{ss}^\alpha(\tau)$. The difference between $\tilde{\epsilon}^\alpha(\tau)$ and $\bar{\delta}_\text{ss}^\alpha(\tau)$ is that $\tilde{\epsilon}^\alpha(\tau)$ ‘counts’ strictly semistable sheaves in a special, complicated way.

Here is an important and highly nontrivial property of the $\tilde{\epsilon}^\alpha(\tau)$, which does not hold for $\delta_{\text{ss}}^{\alpha}(\tau)$, that tell us that taking the logarithm of the moduli stack of semistable objects and we can regard it as a virtual stack whose stabilizer groups have one-dimensional maximal torus:

Theorem 2.5. We have $\bar{\epsilon}^{\alpha}(\tau)\in\text{SF}_{\text{al}}^{\text{ind}}(\mathfrak{M}_A)\subset\text{SF}_{\text{al}}(\mathfrak{M}_A)$.

Using these, we can define the generalized Donaldson-Thomas invariants.

3. Generalized Donaldson-Thomas Invariants

Let $X$ be a strict projective Calabi-Yau $3$-fold over $\mathbb C$. Let $\mathfrak M:=\underline{Coh}(X)$ be the moduli stack of coherent sheaves and $\mathcal M^{\alpha\text{-ss}}(\tau)$ be the stack of semistable objects for some $\alpha\in K^{\text{num}}(\text{coh}(X))$ with a projective good moduli space $ M^{\alpha\text{-ss}}(\tau)$. They contain open locus of stable objects $\mathcal M^{\alpha\text{-st}}(\tau)\subset\mathcal M^{\alpha\text{-ss}}(\tau)$ and $M^{\alpha\text{-st}}(\tau)\subset M^{\alpha\text{-ss}}(\tau)$ which need not be proper.

3.1. Generalized DT-Invariants

We will define a global Behrend function of $\mathfrak M$ first:

Theorem 3.1. Let $X$ be a Calabi-Yau 3-fold over $\mathbb{C}$, and $\mathfrak{M}$ the moduli stack of coherent sheaves on $X$. The Behrend function $\nu_{\mathfrak{M}} : \mathfrak{M}(\mathbb{C}) \to \mathbb{Z}$ is a natural locally constructible function on $\mathfrak{M}$. For all $E_1, E_2 \in \operatorname{coh}(X)$, it satisfies:
\[ \nu_{\mathfrak{M}}(E_1 \oplus E_2) = (-1)^{\bar{X}([E_1],[E_2])} \nu_{\mathfrak{M}}(E_1) \nu_{\mathfrak{M}}(E_2), \] and \[ \int_{[\lambda] \in \mathbb{P}(Ext^1(E_2,E_1))} \nu_{\mathfrak{M}}(F) \, d\chi - \int_{[\bar{\lambda}] \in \mathbb{P}(Ext^1(E_1,E_2))} \nu_{\mathfrak{M}}(\tilde{F}) \, d\chi
= (\dim Ext^1(E_2, E_1) - \dim Ext^1(E_1, E_2)) \nu_{\mathfrak{M}}(E_1 \oplus E_2). \]

Here the correspondence between $[\lambda] \in \mathbb{P}(Ext^1(E_2,E_1))$ and $F \in \operatorname{coh}(X)$ is that $[\lambda] \in \mathbb{P}(Ext^1(E_2,E_1))$ lifts to some $0 \neq \lambda \in Ext^1(E_2,E_1)$, which corresponds to a short exact sequence $0 \to E_1 \to F \to E_2 \to 0$ in $\operatorname{coh}(X)$ in the usual way. The function $[\lambda] \mapsto \nu_{\mathfrak{M}}(F)$ is a constructible function $\mathbb{P}(Ext^1(E_2,E_1)) \to \mathbb{Z}$, and the integrals are integrals of constructible functions using the Euler characteristic as measure.

Here they uses gauge theory and transcendental complex analytic geometry methods.

Definition 3.2. Define a Lie algebra $\tilde{L}(X)$ to be the $\mathbb{Q}$-vector space with basis of symbols $\tilde{\lambda}^{\alpha}$ for $\alpha \in K^{\text{num}}(\text{coh}(X))$, with Lie bracket \[ [\tilde{\lambda}^{\alpha}, \tilde{\lambda}^{\beta}] = (-1)^{\bar{\chi}(\alpha, \beta)} \bar{\chi}(\alpha, \beta) \tilde{\lambda}^{\alpha + \beta}, \]

As $\bar{\chi}$ is antisymmetric, this satisfies the Jacobi identity, and makes $\tilde{L}(X)$ into an infinite-dimensional Lie algebra over $\mathbb{Q}$.

Define a $\mathbb{Q}$-linear map $\tilde{\Psi}^{\chi,\mathbb{Q}}:\widehat{\text{SF}}_{\text{al}}^{ind}(\mathfrak{M}, \chi, \mathbb{Q})\to \tilde{L}(X)$ by \[ \tilde{\Psi}^{\chi,\mathbb{Q}}(f) = \sum_{\alpha \in K^{\text{num}}(\text{coh}(X))} \gamma^{\alpha} \tilde{\lambda}^{\alpha}, \] where $\gamma^{\alpha} \in \mathbb{Q}$ is defined as follows. Write $f|_{\mathfrak{M}^{\alpha}}=\sum_{i=1}^n\delta_i[(U_i\times[\text{Spec}\mathbb C/\mathbb G_m],\rho_i)]$ where $\delta_i\in\mathbb Q$ and $U_i$ is a quasiprojective variety, and set \[ \gamma^{\alpha} = \sum_{i=1}^{n} \delta_i \chi(U_i, \rho_i^* (\nu_{\mathfrak{M}})), \]

where $\rho_i^* (\nu_{\mathfrak{M}})$ is the pullback of the Behrend function $\nu_{\mathfrak{M}}$ to a constructible function on $U_i \times [\text{Spec }\mathbb C / \mathbb{G}_m]$, or equivalently on $U_i$, and $\chi(U_i, \rho_i^* (\nu_{\mathfrak{M}}))$ is the Euler characteristic of $U_i$ weighted by $\rho_i^* (\nu_{\mathfrak{M}})$. Define $\tilde{\Psi} : \text{SF}_{\text{al}}^{ind}(\mathfrak{M}) \rightarrow \tilde{L}(X)$ by $\tilde{\Psi} = \tilde{\Psi} ^{\chi,\mathbb{Q}} \circ{\Pi}_{\mathfrak{M}}^{\chi,\mathbb{Q}}$.

Here is an alternative way to write $\tilde{\Psi}^{\chi,\mathbb{Q}}$, $\tilde{\Psi}$ using constructible functions. Define a $\mathbb{Q}$-linear map $\Pi_{\text{CF}} :\widehat{\text{SF}}_{\text{al}}^{ind}(\mathfrak{M}, \chi, \mathbb{Q}) \rightarrow \text{CF}(\mathfrak{M})$ by \[ \Pi_{\text{CF}} : \sum_{i=1}^{n} \delta_i [(U_i \times [\text{Spec } C / \mathbb{G}_m], \rho_i)] \mapsto \sum_{i=1}^{n} \delta_i \text{CF}^{\text{na}}(\rho_i)(1_{U_i}). \] Then we have \[ \tilde{\Psi}^{\chi,\mathbb{Q}}(f) = \sum_{\alpha \in K^{\text{num}}(\text{coh}(X))} \chi^{\text{na}} (\mathfrak{M}^{\alpha}, (\Pi_{\text{CF}}(f) \cdot \nu_{\mathfrak{M}})|_{\mathfrak{M}^{\alpha}}) \tilde{\lambda}^{\alpha}, \] and \[ \tilde{\Psi}(f) = \sum_{\alpha \in K^{\text{num}}(\text{coh}(X))} \chi^{\text{na}} (\mathfrak{M}^{\alpha}, (\Pi_{\text{CF}} \circ \tilde{\Pi}_{\mathfrak{M}}^{\chi,\mathbb{Q}}(f) \cdot \nu_{\mathfrak{M}})|_{\mathfrak{M}^{\alpha}}) \tilde{\lambda}^{\alpha}. \]

Theorem 3.3. Maps $\tilde{\Psi} : \text{SF}_{\text{al}}^{ind}(\mathfrak{M}) \rightarrow \tilde{L}(X)$ and $\tilde{\Psi}^{\chi,\mathbb{Q}}:\widehat{\text{SF}}_{\text{al}}^{ind}(\mathfrak{M}, \chi, \mathbb{Q})\to \tilde{L}(X)$ are Lie algebra morphisms.

Definition 3.4. Let $X$ be a strict projective Calabi-Yau 3-fold over $\mathbb{C}$, let $\mathscr O_X(1)$ be a very ample line bundle on $X$, and let $(\tau, G, \leq)$ be Gieseker stability and $(\mu, M, \leq)$ be $\mu$-stability on $\text{coh}(X)$ w.r.t. $\mathscr O_X(1)$. We define generalized Donaldson-Thomas invariants $\bar{\text{DT}}^\alpha(\tau) \in \mathbb{Q}$ and $\bar{\text{DT}}^\alpha(\mu) \in \mathbb{Q}$ for all $\alpha \in C(\text{coh}(X))$ by

\[ \tilde{\Psi}(\bar{\epsilon}^{\alpha}(\tau)) = -\bar{\text{DT}}^\alpha(\tau)\tilde{\lambda}^{\alpha} \quad \text{and} \quad \tilde{\Psi}(\bar{\epsilon}^{\alpha}(\mu)) = -\bar{\text{DT}}^\alpha(\mu)\tilde{\lambda}^{\alpha}. \]

We can also have an alternative expression as above and we will omit this.

These are invariants since the following theorem:

Theorem 3.5. Using the expression of generalized DT invariants by the Joyce-Song stable pair invariants (see section 5.4 in [JS11]¹) and the fact that the Chern character maps $K^{\text{num}}(\text{Coh}(X))$ into a subgroup(a lattice) of $H^{2*}(X,\mathbb Q)$ which force $K^{\text{num}}(\text{Coh}(X))$ depends only on the underlying topological space of $X$ (see section 4.5 in [JS11]¹), we can show that the generalized Donaldson-Thomas invariants $\bar{\text{DT}}^\alpha(\tau) \in \mathbb{Q}$ are unchanged under continuous deformations of complex structures.

3.2. Compare with Usually DT-Invariants

Now we compare this and the ordinary DT-invariants above.

Theorem 3.6. If $\mathcal M^{\alpha\text{-ss}}(\tau)=\mathcal M^{\alpha\text{-st}}(\tau)$, then \[\bar{\text{DT}}^\alpha(\tau)={\text{DT}}^\alpha(\tau):=\int_{[M^{\alpha\text{-st}}(\tau)]^{\text{vir}}}1\in\mathbb Z.\]

Proof. In this case we have $\bar{\epsilon}^{\alpha}(\tau)=\delta^{\alpha}_{\text{ss}}(\tau)=[(\mathcal M^{\alpha\text{-st}}(\tau),\iota)]$. Let $\pi:\mathcal M^{\alpha\text{-st}}(\tau)\to M^{\alpha\text{-st}}(\tau)$ be the good moduli space and then we have \[\bar{\text{DT}}^\alpha(\tau)=-\chi^{\text{na}}(\mathcal M^{\alpha\text{-st}}(\tau),\nu_{\mathcal M^{\alpha\text{-st}}(\tau)})\] as the definition and that $\chi$ is a motivic invariants and $\pi$ is a $B\mathbb G_m$-gerbe. Then \[-\chi^{\text{na}}(\mathcal M^{\alpha\text{-st}}(\tau),\nu_{\mathcal M^{\alpha\text{-st}}(\tau)}) =\chi^{\text{na}}(\mathcal M^{\alpha\text{-st}}(\tau),\pi^*\nu_{M^{\alpha\text{-st}}(\tau)}) =\chi^{\text{na}}(M^{\alpha\text{-st}}(\tau),\nu_{M^{\alpha\text{-st}}(\tau)}).\] Hence \[\bar{\text{DT}}^\alpha(\tau)={\text{DT}}^\alpha(\tau):=\int_{[M^{\alpha\text{-st}}(\tau)]^{\text{vir}}}1 \] by Behrend’s result.

3.3. Where We Use the “Strict”?

There are two places we use $H^1(X,\mathscr O_X)=0$:

• In the proof of Theorem 3.1 about the global stacky Behrend function, we use a fact that $\mathfrak M$ is locally isomorphjic to the moduli stack of vector bundles. The proof of this fact uses the Seidel-Thomas twists by $\mathscr O_X(-n)$. We need $\mathscr O_X(-n)$ is a spherical object, that is, if $H^1(X,\mathscr O_X)=0$.
• In the proof of Theorem 3.5., we use the fact that the Chern character maps $K^{\text{num}}(\text{Coh}(X))$ into a subgroup of $H^{2*}(X,\mathbb Q)$ which force $K^{\text{num}}(\text{Coh}(X))$ depends only on the underlying topological space of $X$. This is right for $H^1(X,\mathscr O_X)=0$.

4. BPS Invariants and the Integrality

Note that by a direct calculation (as section 6.1 in [JS11]¹), we find that given a rigid $\tau$-stable sheaf $E$ in class $\alpha$, the sheaves $mE$ contribute $1/m^2$ to $\bar{\text{DT}}^{m\alpha}(\tau)$ for all $m\geq 1$. Therefore we may guess that the generalized DT-invariants comes from some another invariants of integral value with weights of form $1/m^2$. So we might to define:

Definition 4.1. Let $X$ be a strict projective Calabi-Yau 3-fold over $\mathbb{C}$, let $\mathscr O_X(1)$ be a very ample line bundle on $X$, and let $(\tau, G, \leq)$ be Gieseker stability and $(\mu, M, \leq)$ be $\mu$-stability on $\text{coh}(X)$ w.r.t. $\mathscr O_X(1)$. We define the BPS-invariants as \[\widehat{\text{DT}}^{\alpha}(\tau)=\sum_{m\geq1,m|\alpha}\frac{\mu(m)}{m^2}\bar{\text{DT}}^{\alpha/m}(\tau)\] where $\mu$ is the Mobius function. Hence we have $\bar{\text{DT}}^{\alpha}(\tau)=\sum_{m\geq1,m|\alpha}\frac{1}{m^2}\widehat{\text{DT}}^{\alpha/m}(\tau)$ as we expected.

This $\widehat{\text{DT}}^{\alpha}(\tau)$ is an analogue of invariants $\Omega(\alpha)$ in [KS08]³ which aim to count BPS states. Our coefficients $1/m^2$ in $\bar{\text{DT}}^{\alpha}(\tau)=\sum_{m\geq1,m|\alpha}\frac{1}{m^2}\widehat{\text{DT}}^{\alpha/m}(\tau)$ are related to the dilogarithms $\text{Li}_2(t)=\sum_{m\geq1}t^m/m^2$ in [KS08]³.

Note that if $\mathcal M^{\alpha\text{-ss}}(\tau)=\mathcal M^{\alpha\text{-st}}(\tau)$, then $\mathcal M^{\alpha/m\text{-ss}}(\tau)=\emptyset$ for $m\geq2$. Hence in this case $\bar{\text{DT}}^\alpha(\tau)={\text{DT}}^\alpha(\tau)=\widehat{\text{DT}}^{\alpha}(\tau)$.

Conjecture. Let $X$ be a strict projective Calabi-Yau 3-fold over $\mathbb{C}$, let $\mathscr O_X(1)$ be a very ample line bundle on $X$, and let $(\tau, G, \leq)$ be Gieseker stability and $(\mu, M, \leq)$ be $\mu$-stability on $\text{coh}(X)$ w.r.t. $\mathscr O_X(1)$. Let $(\tau,T,\leq)$ is generic, that is, for all $\alpha,\beta\in C(\text{Coh}(X))$ with $\tau(\alpha)=\tau(\beta)$ we have $\bar{\chi}(\alpha,\beta)=0$. Then $\widehat{\text{DT}}^{\alpha}(\tau)\in\mathbb Z$ for any $\alpha\in C(\text{Coh}(X))$.

This is an analogue of a conjecture in [KS08]³ and have shown by the case of quiver without relations.