The point of Proposition <ref> is that we understand $\varinjlim \hom(\mathfrak{m}^r, \O_{Z_1}) / \O_{Z_1}$. Furthermore, the quotient $\varinjlim \hom(\mathfrak{m}^r, T_{1}(\O_Z) ) / T_{1}(\O_Z)$ is 1-dimensional and is described similarly to the original work of Pandharipande-Thomas [129] (extended to the 4-fold case in [34] and subsequently studied further in [107, 40, 134]).

For Example <ref>, we have $Z = Z_1$ and $\varinjlim \hom(\mathfrak{m}^r, \O_{Z}) / \O_{Z} \cong T_0(\O_W) = \mathcal{O}_W$ is depicted in Figure <ref>. For Example <ref>, we also have $Z = Z_1$ but $T_0(\O_W) \neq \O_W$. In particular, the non-zero weight spaces of $\varinjlim \hom(\mathfrak{m}^r, \O_{Z}) / \O_{Z} \cong T_0(\O_W)$ correspond to the boxes at weights $(i,0,j,1)$ with $0 \leq i,j \leq 1$ in Figure <ref>(right).

In Example <ref>, we always have $\varinjlim \hom(\mathfrak{m}^r, \O_{Z_1}) / \O_{Z_1} \cong T_0(\O_W) \neq 0$ and $\varinjlim \hom(\mathfrak{m}^r, T_{1}(\O_Z) ) / T_1(\O_Z) = 0$. We now discuss an example for which the former direct limit is zero and the latter is non-zero. Consider the solid partition of Figure <ref>(left), which is defined by the ideal $(x_1^2 x_2 x_3^2,x_2x_3^3,x_3^4,x_4)$. The corresponding scheme is scheme-theoretically supported in $Z(x_4)$. Then $\varinjlim \hom(\mathfrak{m}^r, T_{1}(\O_Z) ) / T_1(\O_Z)$ is very similar to the type of modules studied in <cit.>: its weight spaces are depicted in Figure <ref>(right), where the only boxes in $\Z_{\geq 0}^4$ are the the ones at weights $(0,0,2,0)$ and $(1,0,2,0)$, and the two “legs” extend to $-\infty$ along the $x_1,x_2$ axes as indicated.

The solid partition and direct limit of Example <ref>.

The following corollary is as an equivariant analog of Proposition <ref>

Let $Z = Z_{\boldsymbol{\lambda}} \subset \C^4$ be a pure 2-dimensional $(\C^*)^4$-fixed subscheme. Then the $(\C^*)^4$-fixed $\PT_0$ pairs on $\C^4$ with support $Z$ are in bijective correspondence with the closed points of
[${\curly Quot}_W(T_0(\O_W) ,m)$ is the Quot scheme of length $m$ 0-dimensional quotients of $T_0(\O_W)$.]

\bigcup_{m \geq 0} {\curly Quot}_W(T_0(\O_W) ,m)^{(\C^*)^4},

where $T_0(\O_W) \subset \O_W$ is the torsion subsheaf of

W = (Z_{\lambda_{12}} \cup Z_{\lambda_{13}} \cup Z_{\lambda_{23}}) \cap (Z_{\lambda_{14}} \cup Z_{\lambda_{24}} \cup Z_{\lambda_{34}}).

More precisely, a $(\C^*)^4$-equivariant quotient $[T_0(\O_W) \twoheadrightarrow \zeta]$ with kernel $Q \subset T_0(W)$ corresponds to a $(\C^*)^4$-equivariant $\PT_0$ pair $(F,s)$ via the short exact sequences

0 [r]     Ø_Z @=[d] [r]     F [r] @^(->[d]     Q @^(->[d] [r]     0

0 [r]     Ø_Z [r]     Ø_Z_λ_12 ∪Z_λ_13 ∪Z_λ_23 ⊕Ø_Z_λ_14 ∪Z_λ_24 ∪Z_λ_34 [r]     Ø_W [r]     0,

where the second square is Cartesian.

Fix plane partitions $\boldsymbol{\mu}$ with corresponding 2-dimensional subscheme $Z = Z_{\boldsymbol{\mu}} \subset \C^4$ (Definition <ref>) and let $M:= \varinjlim \hom(\mathfrak{m}^r,\O_{Z})$, where $\mathfrak{m} \subset \O_{Z}$ denotes the ideal sheaf of the origin. We refer to the finitely generated $(\C^*)^4$-invariant submodules $B \subset M / \O_Z$ as $\PT_0$ box configurations with asymptotics $\boldsymbol{\mu}$. We denote the collection of $\PT_0$ box configurations with asymptotics $\boldsymbol{\mu}$ by $\mathcal{B}^{(0)}_{\boldsymbol{\mu}}$. We say that $B \in \mathcal{B}^{(0)}_{\boldsymbol{\mu}}$ has no $\PT_0$ moduli when there exists no $B' \in \mathcal{B}^{(0)}_{\boldsymbol{\mu}}$ with $B' \neq B$ and

\dim_{\C}(B_w) = \dim_{\C}(B'_w), \quad \forall w \in \Z^4,

where $B_w, B'_w$ denote the weight spaces of $B,B'$ for the character $w \in \Z^4$ respectively. We say that the choice of partitions $\boldsymbol{\mu}$ has no $\PT_0$ moduli when $\mathcal{B}^{(0)}_{\boldsymbol{\mu}}$ contains no elements with $\PT_0$ moduli. We refer to $|B| = \sum_w \dim_{\C}(B_w)$ as the size of $B$.

Fix plane partitions $\boldsymbol{\mu}$ with corresponding 2-dimensional subscheme $Z = Z_{\boldsymbol{\mu}} \subset \C^4$ (Definition <ref>). Evidently, the choice of partitions $\boldsymbol{\mu}$ has no $\PT_0$ moduli if and only if[Similar to the previous definition, $(M / \O_Z)_w$ denotes the weight space of $M / \O_Z$ for the character $w \in \Z^4$.]

\dim_{\C} (M / \O_Z)_w \leq 1, \quad \forall w \in \Z^4.

When this is the case, the elements of $\mathcal{B}^{(0)}_{\boldsymbol{\mu}}$ are particularly easy to describe. They are the finite subsets $B \subset \mathcal{R} := \{w \in \Z^4 \, : \, \dim_{\C} (M / \O_Z)_w = 1 \}$ such that

if $w = (w_1,w_2,w_3,w_4) \in   \mathcal{R}$ and one of $(w_1-1,w_2,w_3,w_4)$, $(w_1,w_2-1,w_3,w_4)$, $(w_1,w_2,w_3-1,w_4)$, $(w_1,w_2,w_3,w_4-1)$ lies in $B$, then $w \in B$.[Informally, the elements of $\mathcal{B}^{(0)}_{\boldsymbol{\mu}}$ are obtained by piling finitely many boxes in the region $ \mathcal{R}$ subject to the condition that gravity pulls in direction $(1,1,1,1)$.]

Proposition <ref> is helpful for finding choices of plane partitions $\boldsymbol{\mu}$ with no $\PT_0$ moduli. For finite partitions $\boldsymbol{\lambda}$, there is a minimal plane partition $\mu_1^{\boldsymbol{\lambda}}$ compatible with $\{\lambda_{ab}\}_{b \neq 1}$, i.e., satisfying (<ref>).[Note that the scheme $Z_{\mu_1^{\boldsymbol{\lambda}}} \subset \C^3$ has no embedded 0-dimensional components.] Defining $\mu_a^{\boldsymbol{\lambda}}$ analogously for all $a$, we deduce that the choice of partitions

\begin{equation} \label{eqn:lambdamu}
\boldsymbol{\mu}^{\boldsymbol{\lambda}} := (\mu_1^{\boldsymbol{\lambda}},\mu_2^{\boldsymbol{\lambda}},\mu_3^{\boldsymbol{\lambda}},\mu_4^{\boldsymbol{\lambda}})
\end{equation}

has no $\PT_0$ moduli. In other words, $(\C^*)^4$-fixed $\PT_0$ pairs on $\C^4$, which are also $\PT_1$ pairs, have no moduli.

In Example <ref>(ii), we have $\O_{W_0} = \C$ (with weight $(0,0,0,0)$). In the second case of Example <ref>(iii) we have $\O_{W_0} = \C \cdot t_4$. In these cases there are no $\PT_0$ moduli.

For Examples <ref>, <ref>, <ref> we have drawn a blue box for each element of the region $\mathcal{R} \subset \mathbb{Z}^4$ defined in Remark <ref> — see Figure <ref>. Recall that these boxes have no boxes stacked on top of them in the direction of the 4th axis not drawn in the diagram (in these examples: directions $x_4$, $x_2$, $x_4$ respectively). These blue boxes indicate the “container” inside of which we stack $\PT_0$ box configurations with gravity pulling into the $(1,1,1,1)$ direction, i.e., according to the rule of Remark <ref>. The boxes of the $\PT_0$ box configurations are drawn in red. From left to right, we have depicted a $\PT_0$ box configuration of size 4, 1, 9 respectively. Note that $\mathcal{R}$ has finite size in the first two examples and infinite size in the third example.

Examples of $\PT_0$ box configurations.

The following corollary gives many examples of choices of plane partitions with no $\PT_0$ moduli.

Let $\boldsymbol{\lambda}$ be finite partitions and let $\boldsymbol{\mu}$ be compatible plane partitions, i.e., satisfying (<ref>). Let $a,b,c,d \in \{1,2,3,4\}$ be mutually distinct, suppose $\mu_a$, $\mu_b$ have no further restrictions, and $\mu_c$, $\mu_d$ are minimal, i.e., $\mu_c = \mu_c^{\boldsymbol{\lambda}}$, $\mu_d = \mu_d^{\boldsymbol{\lambda}}$.[I.e., the scheme $Z_{\boldsymbol{\mu}}$ has embedded 1-dimensional components in at most two directions.] Then $\boldsymbol{\mu}$ has no $\PT_0$ moduli.

We use Proposition <ref>. Consider the weight spaces

\Big( \varinjlim \hom(\mathfrak{m}^r, T_{1}(\O_Z) ) / T_{1}(\O_Z) \Big)_w, \quad \Big( \varinjlim \hom(\mathfrak{m}^r, \O_{Z_1}) / \O_{Z_1} \Big)_w

These weight spaces are at most 1-dimensional and they are not 1-dimensional for the same $w$.

(i) In general, when there are $\geq 3$ embedded legs, there will be $\PT_0$ moduli (as in the curve case). For example, let $I_Z = (x_1x_2x_3, x_1x_3^2, x_2 x_3^2, x_4)$. This corresponds to $\lambda_{12} = 1$, $\mu_1 = t_3 + \frac{1}{1-t_2}$, $\mu_2 = t_3 + \frac{1}{1-t_1}$, $\mu_3 = 1$, and $\lambda_{ab} = \mu_4 = \varnothing$ otherwise. Then $(\varinjlim \hom(\mathfrak{m}^r, \O_{Z}) / \O_{Z})_{(0,0,1,0)} = \C^2$ and there are $\PT_0$ moduli.

(ii) However, there are examples with $\geq 3$ embedded legs for which there are no $\PT_0$ moduli such as $I_Z = (x_2x_3, x_1 x_3^2, x_1x_4, x_3x_4, x_4^2)$. This corresponds to $\lambda_{12} = 1$, $\mu_1 = t_3 + \frac{1}{1-t_2}$, $\mu_2 = t_4 + \frac{1}{1-t_1}$, $\mu_3 = 1$, and $\lambda_{ab} = \mu_4 = \varnothing$ otherwise.

$\PT_1$ case. By Proposition <ref>,  the $(\C^*)^4$-fixed $\PT_1$ pairs on $\C^4$ with support $Z$ correspond to the finitely generated $(\C^*)^4$-invariant submodules of

\begin{equation*}
\varinjlim \hom(\mathfrak{I}^r, \O_{Z}) / \O_Z,
\end{equation*}

where $\mathfrak{I} \subset \O_{Z}$ denotes the ideal sheaf of the union of coordinate axes contained in $Z$. Recall that $Z = Z_{\boldsymbol{\lambda}}$ for some finite partitions $\boldsymbol{\lambda}$ (Lemma <ref>) and

Z_{\boldsymbol{\lambda}} = \bigcup_{1 \leq a<b \leq 4} Z_{\lambda_{ab}}.

Let $Z = Z_{\boldsymbol{\lambda}} \subset \C^4$ and denote by $\mathfrak{I} \subset \O_Z$ the ideal sheaf of the union of the coordinate axes contained in $Z$. Then

\varinjlim \hom(\mathfrak{I}^r,\O_Z) = \bigoplus_{1 \leq a<b \leq 4} \O_{Z_{\lambda_{ab}}}[x_a^{-1}, x_b^{-1}].

From the restrictions $ \O_Z \to \O_{Z_{\lambda_{ab}}}$, we obtain a short exact sequence

\begin{equation*}
0 \to \O_Z \to \bigoplus_{1 \leq a<b \leq 4} \O_{Z_{\lambda_{ab}}} \to C \to 0,
\end{equation*}

where the cokernel $C$ is supported on the union of coordinate axes contained in $Z$. Then

\varinjlim \hom(\mathfrak{I}^r,C) = 0

and therefore

\varinjlim \hom(\mathfrak{I}^r,\O_Z) = \bigoplus_{1 \leq a<b \leq 4} \varinjlim \hom(\mathfrak{I}^r,\O_{Z_{\lambda_{ab}}}).

Finally, we have

\begin{equation*}
\varinjlim \hom(\mathfrak{I}^r,\O_{Z_{\lambda_{ab}}}) \cong \O_{Z_{\lambda_{ab}}}[x_a^{-1},x_b^{-1}]. \qedhere
\end{equation*}

Consider the solid partition corresponding to the scheme $Z$ defined by the ideal $(x_3^2,x_4)$. In Figure <ref> we have drawn the weight spaces of $\varinjlim \hom(\mathfrak{I}^r, \O_{Z}) / \O_Z$, where we recall that the blue colour indicates that there are no boxes stacked on top into the positive $x_4$-direction. The double layer of boxes fill all quadrants of the $x_1x_2$-plane except for the first quadrant (since we take the quotient by $\O_Z$).

The direct limit of Example <ref>.

Similar to the $\PT_0$ case, we define:

Fix finite partitions $\boldsymbol{\lambda}$ with corresponding 2-dimensional subscheme $Z = Z_{\boldsymbol{\lambda}} \subset \C^4$ (Definition <ref>). Let $M:= \varinjlim \hom(\mathfrak{I}^r, \O_{Z})$, where $\mathfrak{I} \subset \O_{Z}$ denotes the ideal sheaf of the union of the coordinate axes contained in $Z$. We refer to finitely generated $(\C^*)^4$-invariant submodules $B \subset M / \O_Z$ as $\PT_1$ box configurations with asymptotics $\boldsymbol{\lambda}$. We denote the collection of $\PT_1$ box configurations with asymptotics $\boldsymbol{\lambda}$ by $\mathcal{B}^{(1)}_{\boldsymbol{\lambda}}$. We say that $B \in \mathcal{B}^{(1)}_{\boldsymbol{\lambda}}$ has no $\PT_1$ moduli when there exists no $B' \in \mathcal{B}^{(1)}_{\boldsymbol{\lambda}}$ such that $B' \neq B$ and

\dim_{\C}(B_w) = \dim_{\C}(B'_w), \quad \forall w \in \Z^4.

We say that the choice of partitions $\boldsymbol{\lambda}$ has no $\PT_1$ moduli when $\mathcal{B}^{(1)}_{\boldsymbol{\lambda}}$ contains no elements with $\PT_1$ moduli. We refer to $|B| = \sum_w \dim_{\C}(B_w)$ as the size of $B$. Unlike the $\PT_0$ case, $|B|$ may be infinite.

Fix finite partitions $\boldsymbol{\lambda}$ with corresponding 2-dimensional subscheme $Z = Z_{\boldsymbol{\lambda}} \subset \C^4$ (Definition <ref>).
Then the choice of partitions has no $\PT_1$ moduli if and only if

\dim_{\C} (M / \O_Z)_w \leq 1, \quad \forall w \in \Z^4.

When this is the case, the elements of $\mathcal{B}^{(1)}_{\boldsymbol{\lambda}}$ are again easy to describe. They are the (not necessarily finite!) subsets $B \subset \mathcal{R} := \{w \in \Z^4 \, : \, \dim_{\C} (M / \O_Z)_w = 1 \}$ such that

if $w = (w_1,w_2,w_3,w_4) \in   \mathcal{R}$ and one of $(w_1-1,w_2,w_3,w_4)$, $(w_1,w_2-1,w_3,w_4)$, $(w_1,w_2,w_3-1,w_4)$, $(w_1,w_2,w_3,w_4-1)$ lies in $B$, then $w \in B$. Moreover, $B$ is required to support a finitely generated module.

For Example <ref> we have drawn a blue box for each element of the region $\mathcal{R} \subset \mathbb{Z}^4$ defined in Remark <ref> — see Figure <ref>. Similar to Example <ref>, these blue boxes indicate the “container” inside of which we stack $\PT_1$ box configurations with gravity pulling into the $(1,1,1,1)$ direction, i.e., according to the rule of Remark <ref>. In Figure <ref>, we have drawn in red an example of a $\PT_1$ box configuration. In this case, there are no finite $\PT_1$ box configurations. Note that placing a red box at $(-1,-1,1,0)$ forces us to put red boxes at $(i,-1,1,0)$ and $(-1,j,1,0)$ for all $i,j \geq 0$. Similarly, placing a red box at $(1,-1,0,0)$ forces us to put a red box at $(i,-1,0,0)$ for all $i \geq 2$.

Example of a $\PT_1$ box configuration.

The following is an immediate consequence of Proposition <ref>.

A collection of finite partitions $\boldsymbol{\lambda}$ has no $\PT_1$ moduli if and only if $\lambda_{ab} = \varnothing$ unless $(a,b) = (1,2)$ or $(3,4)$ (up to relabelling).

By Example <ref>(iii) it can happen that $\boldsymbol{\lambda}$ has $\PT_1$ moduli but that $\boldsymbol{\mu}^{\boldsymbol{\lambda}}$ (the minimal plane partitions compatible with $\boldsymbol{\lambda}$) has no $\PT_0$ moduli. This simply means that $(\C^*)^4$-equivariant $\PT_1$ pairs on supported on $Z_{\boldsymbol{\lambda}}$ with 1-dimensional cokernel may have moduli but those with 0-dimensional cokernel do not.

We will see shortly that Corollaries <ref> and <ref> together imply that for toric Calabi-Yau 4-folds of the form $X = \mathrm{Tot}(L_1 \oplus L_2) \to S$, where $S$ is a smooth projective toric surface and $L_1, L_2$ line bundles on $S$ satisfying $L_1 \otimes L_2 \cong K_S$, the $T_X \cong (\C^*)^4$ fixed loci of the $\DT,\PT_0,\PT_1$ moduli spaces are always 0-dimensional (Remark <ref>).

Consider the solid partition corresponding to $S \cong \C^2$ defined by the ideal $(x_3,x_4)$. Then $\varinjlim \hom(\mathfrak{I}^r, \O_{S}) / \O_S = \C[x_1^{\pm 1},x_2^{\pm 1}] / \C[x_1,x_2]$. In Figure <ref>(left) we depict an example of a $\PT_1$ box configuration. Since $S$ is smooth, by Proposition <ref>, $\PT_1$ pairs $(F,s)$ with support $S$ and non-zero cokernel $Q$ correspond to nestings $Z \subset C \subset S$ where $Z$ is 0-dimensional and $C$ is an effective divisor. Furthermore, the cokernel $Q$ satisfies $Q \cong \jmath_*I_{Z/C}(C)$, and $(F,s)$ is $(\C^*)^4$-equivariant if and only if $Z \subset C \subset S$ is a $(\C^*)^2$-invariant nesting. In Figure <ref>(right), we have drawn in red the weight spaces corresponding to $I_{Z/C}(C) \subset \O_C(C)$ and in green the weight spaces corresponding to $\O_Z(C)$.

$\PT_1$ pair of Example <ref>.

§.§ Fixed loci — Global

We turn our attention to the characterization of $T_X \cong (\C^*)^4$ fixed $\PT_q$ stable pairs on an arbitrary smooth quasi-projective toric 4-fold $X$. Given a $T_X$-fixed $\PT_q$ stable pair $(F,s)$ on $X$, we can consider the restrictions

(F^{(\alpha)}, s^{(\alpha)}) := (F,s)|_{U_\alpha}, \quad \alpha \in V(X).

Since $U_\alpha \cong \C^4$, we can use the previous section to describe $(F^{(\alpha)}, s^{(\alpha)})$ combinatorially. Denote the scheme theoretic support of $(F,s)$ by $Z$ and recall the closed subschemes

Z \supset Z_0 \supset Z_1

defined by the torsion filtration of $\O_Z$. Recall that for $q=0$, we have $Z = Z_0$ and for $q=1$ we have $Z = Z_1$. We define $Z^{(\alpha)} := Z \cap U_\alpha$, $Z^{(\alpha)}_0 := Z_0 \cap U_\alpha$, and $Z^{(\alpha)}_1 := Z_1 \cap U_\alpha$. Then there exist collections of finite partitions and compatible (<ref>) plane partitions

\boldsymbol{\lambda}^{(\alpha)} = \{\lambda_{ab}^{(\alpha)}\}_{1 \leq a < b \leq 4 \atop \alpha \in V(X)}, \quad \boldsymbol{\mu}^{(\alpha)} = \{\mu_{a}^{(\alpha)} \}_{1 \leq a \leq 4 \atop \alpha \in V(X)}

such that (Definition <ref>)

Z_0^{(\alpha)} = Z_{\boldsymbol{\mu}^{(\alpha)} }, \quad Z_1^{(\alpha)} = Z_{\boldsymbol{\lambda}^{(\alpha)}}.

The partitions $\boldsymbol{\lambda}^{(\alpha)}, \boldsymbol{\mu}^{(\alpha)}$ satisfy gluing conditions. For two charts $U_\alpha,U_\beta$ intersecting in $\C^* \times \C^3$, let us label the coordinates in each chart such that the first coordinate corresponds to $\C^*$. Then

\begin{equation} \label{eqn:glucon1}
\mu_{1}^{(\alpha)}  = \mu_{1}^{(\beta)}, \quad \lambda^{(\alpha)}_{1b} = \lambda^{(\beta)}_{1b}, \quad \forall b = 2,3,4.
\end{equation}

By the compatibility (<ref>), the equations on plane partitions imply the ones on partitions.
For two charts $U_\alpha,U_\beta$ intersecting in $(\C^*)^2 \times \C^2$, let us label the coordinates in each chart such that the first two coordinates correspond to $(\C^*)^2$. Then

\begin{equation} \label{eqn:glucon2}
\lambda^{(\alpha)}_{12} = \lambda^{(\beta)}_{12}.
\end{equation}

For $q \in \{-1,0\}$, the gluing conditions (<ref>), (<ref>) completely determine when a collection of $(\C^*)^4$-equivariant $\PT_q$ pairs $\{(F^{(\alpha)}, s^{(\alpha)})\}_{\alpha \in V(X)}$ on the charts $U_\alpha$ glues to a $T_X$-equivariant $\PT_q$ pair on $X$. For $q=1$, one additionally requires that the $\PT_1$ box configurations glue.

In Section <ref>, we discussed the moduli spaces

\begin{equation} \label{eq:toricsecmoduli}
\PTqvX, \quad q \in \{-1,0,1\}, \quad v = (0,0,\gamma,\beta,n -  \gamma \cdot \td_2(X)) \in H_c^*(X,\Q).
\end{equation}

Let $X$ be a toric Calabi-Yau 4-fold and denote the (3-dimensional) subtorus preserving the Calabi-Yau volume form by $T \leq T_X$. The following theorem provides a nice collection of cases where Oh-Thomas localization for 0-dimensional reduced $T$-fixed loci as in Proposition <ref> applies.

Let $\curly P:= \PTqvX$.

$\mathrm{(1)}$ For $q=-1$, we have $\curly P^{T_X} = \curly P^{T}$ and it is 0-dimensional and reduced.

$\mathrm{(2)}$ For $q=0$ and $\PT_0 = \PT_1$, i.e. $F$ is pure for all $(F,s) \in \curly P^{T_X}$, we have $\curly P^{T_X} = \curly P^{T}$ and it is 0-dimensional and reduced.

$\mathrm{(3)}$ For $q=0$, suppose for any $(F,s) \in \curly P^{T_X}$ and any $\alpha \in V(X)$ the scheme theoretic support of $F|_{U_\alpha}$ has at most two embedded 1-dimensional components and its pure part is Cohen-Macaulay. Then $\curly P^{T_X} = \curly P^{T}$ and it is 0-dimensional and reduced.

$\DT$ case. Clearly, we have

\curP_v^{(-1)}(X)^{T_X} \subset \curP_v^{(-1)}(X)^T.

By comparing $T$-fixed ideals and $(\C^*)^4$-fixed ideals on each chart $U_\alpha \cong \C^4$, it is easy to see that this is an equality at the level of closed points. The argument is written out in <cit.> for $\gamma=\beta = 0$, but it works for 2-dimensional subschemes as well. Next, we claim that for any $Z \in \curP_v^{(-1)}(X)^{T_X}$, we have $\Hom_X(I_Z, \O_Z)^T = 0$, which establishes (1).

In fact, we claim the stronger statement

\Hom_{U_\alpha}(I_{Z_{\alpha}}, \O_{Z_{\alpha}})^T = 0

on each $U_\alpha \cong \C^4$. Since $I_{Z_{\alpha}}$ is a monomial ideal, $\Hom_{U_\alpha}(I_{Z_{\alpha}}, \O_{Z_{\alpha}})$ has an explicit combinatorial basis in terms of so-called Haiman arrows [114]. From this description, it is clear that there are no non-zero $T$-fixed Haiman arrows. This is worked out in <cit.> for $\gamma=\beta = 0$, but the argument holds for 2-dimensional subschemes as well (but it is crucial that $Z_\alpha$ has codimension $\geq 2$).

$\PT_0$ case.
Since the fixed locus $\curly P^T$ is proper and $\curly P^{T_X} \subset \curly P^T$, in order to show that $\curly P^{T} = \curly P^{T_X}$ is 0-dimensional reduced, it suffices to show that $\Hom_X(I^\mdot, F)^T = 0$ for any $I^\mdot = [\O_X \to F] \in \curly P^{T_X}$.

We start by following <cit.>. Take $I^\mdot = [\O_X \to F] \in \curly P^{T_X}$, denote its cokernel by $Q$, and its scheme theoretic support by $Z$. We write $I^\mdot_\alpha := I^\mdot|_{U_\alpha}$, $F_\alpha := F|_{U_\alpha}$, $Q_\alpha := Q|_{U_\alpha}$, and $Z_\alpha = Z|_{U_\alpha}$. Consider the restriction morphism

\Hom_X(I^\mdot, F) \to \bigoplus_{\alpha \in V(X)} \Hom_{U_\alpha}(I_\alpha^\mdot, F_\alpha).

From the exact triangle $I_{Z_\alpha} \to I^\mdot_\alpha \to Q_\alpha[-1]$ we obtain the left exact sequence

0 \to \Ext^1(Q_\alpha, F_\alpha) \to \Hom(I_\alpha^\mdot, F_\alpha) \to \Hom(I_{Z_{\alpha}}, F_\alpha),

which remains left exact after taking $T$-fixed part. Suppose $\Hom(I_{Z_\alpha},F_\alpha)^T = 0$ for all $\alpha \in V(X)$. Then $\Ext^1(Q_\alpha, F_\alpha)^T \cong \Hom(I_\alpha^\mdot, F_\alpha)^T$. Moreover, since $Q_\alpha$ is 0-dimensional, we deduce (by [129] paragraph below equation (3.2)) that the restriction morphism induces an isomorphism

\begin{equation*}
\Hom_X(I^\mdot, F)^T \cong \bigoplus_{\alpha \in V(X)} \Hom_{U_\alpha}(I_\alpha^\mdot, F_\alpha)^T.
\end{equation*}

Next, we define $M_\alpha:=\varinjlim \hom(\mathfrak{m}_\alpha^r, \O_{Z_\alpha})$, where $\mathfrak{m}_\alpha \subset \O_{Z_\alpha}$ is the ideal sheaf of the origin. In fact, it is easier to work with a sufficient approximation $M_\alpha^r:=\hom(\mathfrak{m}_\alpha^r, \O_{Z_\alpha})$, with $r \gg 0$ such that $F_\alpha \subset M_\alpha^r$ (so $M_\alpha^r$ is coherent).
We then obtain an exact sequence

\Hom(Q_\alpha,M_\alpha^r) \to \Hom(Q_\alpha,M_\alpha^r / F_\alpha ) \to \Ext^1(Q_\alpha,F_\alpha) \to \Ext^1(Q_\alpha,M_\alpha^r).

Summarizing: in order to prove that $\curly P^T$ is 0-dimensional reduced, it suffices to show that for any $I^\mdot = [\O_X \to F] \in \curly P^{T_X}$ we have $\Hom(I_{Z_{\alpha}}, F_\alpha)^T = \Ext^1(Q_\alpha, F_\alpha)^T = 0$ for all $\alpha \in V(X)$. In turn, for this it suffices to show

\begin{align*}
\Hom(I_{Z_{\alpha}}, F_\alpha)^T &= 0, \\
\Ext^1(Q_\alpha,M^r_\alpha)^T &= 0, \\
\Hom(Q_\alpha,M_\alpha^r / F_\alpha )^T &=0,
\end{align*}

for all $I^\mdot = [\O_X \to F] \in \curly P^{T_X}$ and $\alpha \in V(X)$.

We now prove (2). We first study $\Hom(I_{Z_{\alpha}}, F_\alpha)^{T}$. Since we assume $\PT_0 = \PT_1$, $Z_\alpha$ is pure 2-dimensional.
We first observe that, by the $\DT$ case, we have $\Hom(I_{Z_{\alpha}}, \O_{Z_\alpha})^{T} = 0$. We now consider $\Hom(I_{Z_{\alpha}}, Q_\alpha)^{T}$. Take a homogeneous element of $\Hom(I_{Z_{\alpha}}, Q_\alpha)^{T}$. By Proposition <ref>, it will map a homogeneous element $g$ of $I_{Z_{\alpha}}$ to a weight space contained in $T_0(\O_W)$. Then $g$ maps to zero by an argument similar to the $\DT$ case.

It suffices to show that $\Ext^1(Q_\alpha, F_\alpha)^T = 0$. Writing $Z_\alpha = Z_{\boldsymbol{\lambda}}$, we set $S_1 := Z_{\lambda_{12}} \cup Z_{\lambda_{13}} \cup Z_{\lambda_{23}}$ and $S_2 := Z_{\lambda_{14}} \cup Z_{\lambda_{24}} \cup Z_{\lambda_{34}}$. Then Corollary <ref> gives a short exact sequence

\cdots \rightarrow \Hom(Q_\alpha, \O_{S_1 \cap S_2} / Q_\alpha) \rightarrow \Ext^1(Q_\alpha, F_\alpha) \rightarrow \Ext^1(Q_\alpha,\O_{S_1} \oplus \O_{S_2}) \rightarrow \cdots,

where the scheme theoretic support of $Q_\alpha$ is contained in $S_1 \cap S_2$.
Since $Q_\alpha$ is 0-dimensional and $\O_{S_i}$ is pure 2-dimensional and Cohen-Macaulay (Proposition <ref>), we have $\Ext^1(Q_\alpha, \O_{S_i}) = 0$. Indeed, since $Q_\alpha$ is 0-dimensional $\Ext^1(Q_\alpha, \O_{S_i})$ is finite-dimensional and dual to $\Ext^3(\O_{S_i},Q_\alpha)$. At the stalk at 0, the module $\O_{S_i,0}$ has depth 2 (Cohen-Macaulay property) and hence projective dimension 2 (Auslander-Buchsbaum), so

\Ext_{\O_{\C^4,0}}^3(\O_{S_i,0},Q_{\alpha,0}) = 0.

Furthermore, similar to the $\DT$ case, one can show

\Hom(Q_\alpha, \O_{S_1 \cap S_2} / Q_\alpha)^T = 0.

Next, we prove (3). By assumption, the support of $T_1(\O_{Z_\alpha})$ has at most two irreducible components for all $\alpha \in V(X)$. We first study $\Hom(I_{Z_{\alpha}}, F_\alpha)^{T}$ and note that, by the $\DT$ case, $\Hom(I_{Z_{\alpha}}, \O_{Z_\alpha})^{T} = 0$. We consider $\Hom(I_{Z_{\alpha}}, Q_\alpha)^{T}$. Take a homogeneous element of $\Hom(I_{Z_{\alpha}}, Q_\alpha)^{T}$. It maps a homogeneous element $g$ of $I_{Z_{\alpha}}$ to a weight space of $Q_\alpha$ supported on
$$\varinjlim \hom(\mathfrak{m}^r_\alpha, T_1(\O_{Z_\alpha})) / T_1(\O_{Z_\alpha})$$
by Proposition <ref>. It can then be seen that $g$ maps to zero by a reasoning similar to <cit.>. Thus $\Hom(I_{Z_{\alpha}}, F_\alpha)^{T} = 0$.

By assumption, the pure part $Z_1 \subset Z_\alpha$ is Cohen-Macaulay. Propositions <ref>, <ref> give a short exact sequence

0 \to \varinjlim \hom(\mathfrak{m}_\alpha^r, T_{1}(\O_{Z_\alpha}) ) \to M_\alpha \to \O_{Z_1} \to 0.

Furthermore, by assumption, the support of $T_{1}(\O_{Z_\alpha})$ has at most two irreducible components. In particular, $(F_\alpha,s_\alpha)$ has no $\PT_0$ moduli (Corollary <ref>). Without loss of generality, we take the reduced support of $T_{1}(\O_{Z_\alpha})$ to be contained in the union of the $x_1$- and $x_2$-axis. We first show $\Ext^1(Q_\alpha,M^r_\alpha)^T = 0$.
There exists a $(\C^*)^3$-equivariant 0-dimensional $\C[x_2,x_3,x_4]$-module $N_1$ and $(\C^*)^3$-equivariant 0-dimensional $\C[x_1,x_3,x_4]$-module $N_2$ such that

\varinjlim \hom(\mathfrak{m}^r_\alpha, T_{1}(\O_{Z_\alpha}) ) = M_1 \oplus M_2, \quad M_i  = N_i[x_i,x_i^{-1}], \quad i=1,2.

Using the truncated versions $M_\alpha^r, M_1^r, M_2^r$ of $M_\alpha, M_1, M_2$, we have a short exact sequence

\begin{equation} \label{eqn:truncses}
0 \to M_1^r \oplus M_2^r \to M_\alpha^r \to \O_{Z_1} \to 0.
\end{equation}

Note that

M_i^{r+N} = M_i^r \otimes t_i^{-N},

for all $N > 0$. Since $\Ext^1(Q_\alpha,M_i^r)$ are finite $T$-representations ($Q_\alpha$ is 0-dimensional), we can take $N$ sufficiently large such that

\Ext^1(Q_\alpha,M_i^{r+N}) = \Ext^1(Q_\alpha,M_i^r) \otimes t_i^{-N}

has no $T$-fixed part (this trick is also used in <cit.>). Furthermore, since $Q_\alpha$ is 0-dimensional and $Z_1$ is 2-dimensional Cohen-Macaulay, we have $\Ext^1(Q_\alpha, \O_{Z_1}) = 0$ (as we saw above). Consequently, for $r \gg 0$, (<ref>) implies $\Ext^1(Q_\alpha,M^r_\alpha)^T = 0$.

Finally, we show that $\Hom(Q_\alpha,M_\alpha^r / F_\alpha )^T = 0$. Consider

\Hom(Q_\alpha,M_\alpha^r / F_\alpha ) \cong  \Hom(Q_\alpha, (\hom(\mathfrak{m}^r_\alpha, T_{1}(\O_{Z_\alpha}) ) \cap F_\alpha) / T_{1}(\O_{Z_\alpha})).

The vanishing of the $T$-fixed part on the right hand side follows by an argument which is almost the same as <cit.>.

We expect that there are many more cases where $\curly P^{T_X} = \curly P^{T}$ is 0-dimensional and reduced, but the combinatorics involved in the tangent space analysis becomes increasingly complicated:

Let $X$ be a toric Calabi-Yau 4-fold and $\curly P := \PTqvX$ for $q \in \{-1,0,1\}$. If $\curly P^{T_X}$ is 0-dimensional, then $\curly P^{T_X} = \curly P^{T}$ and it is reduced.

Evidence for the conjecture is provided by the cases proved in Theorem <ref>. Further indications that this conjecture is true come from the fact that in all examples we calculated where $\curly P^{T_X}$ is 0-dimensional (including, e.g., for $q=1$ the examples in Section <ref>), we never found a $T$-fixed term in $T_{\curly P}^{\vir}|_{I\udot}$. This does not imply that $\Hom_X(I\udot, F)^{T}$ is zero, because it could cancel against a $T$-fixed term of the same dimension in the obstructions. See also <cit.> for an analogous result in the 3-fold case.

Suppose $X = \mathrm{Tot}(L_1 \oplus L_2) \to S$, where $S$ is a smooth projective toric surface and $L_1, L_2$ line bundles on $S$ satisfying $L_1 \otimes L_2 \cong K_S$. Let $\curly P := \PTqvX$ for $q \in \{-1,0,1\}$ and $v = (0,0,\gamma,\beta,n - \gamma \cdot \td_2(X)) \in H_c^*(X,\Q)$. By Corollaries <ref>, <ref>, and the gluing description at the beginning of this section, it is clear that $\curly P^{T_X}$ is always 0-dimensional. For $q \in \{-1,0\}$, we have $\curly P^{T_X} = \curly P^{T}$ is reduced by Theorem <ref>(1) and (3). For $q=1$ and assuming Conjecture <ref>, we have $\curly P^{T_X} = \curly P^{T}$ is reduced.

§ VERTEX FORMALISM

In this section, we investigate the calculation of $K$-theoretic $\PT_q$ invariants of toric Calabi-Yau 4-folds for $q \in \{-1,0,1\}$. We also formulate a $K$-theoretic $\DT$–$\PT_0$ vertex correspondence, which we verify in some examples.

§.§ Redistribution

Let $X$ be a toric Calabi-Yau 4-fold. Let $\curly P:= \PTqvX$ with $q \in \{-1,0,1\}$ with $v$ as in (<ref>). We now present a method to determine the virtual tangent representation[Under this isomorphism, $[\O_{\pt}]$ corresponds to $1$.]

R\Hom_X(I^\mdot,I^\mdot)_0[1] \in K_0^{T_X}(\pt) \cong \Z[t_1^{\pm 1},t_2^{\pm 1},t_3^{\pm 1},t_4^{\pm 1}],

for any $T_X$-fixed $I^\mdot \in \curly P$. Note that the spaces $\Ext_X^i(I\udot,I\udot)_0$ are finite-dimensional (Section <ref>). We observe that the rank of this complex is even because $N_{S_f/X}$ is split for all $f \in F(X)$ (Lemma <ref>). There exists a smooth projective toric 4-fold $\overline{X}$ and a $(\C^*)^4$-equivariant good compactification $j : X \hookrightarrow \overline{X}$. We then have

\[
\vd := n - \frac{1}{2} (j_* \gamma)^2 \in \Z.
\]

We write out the case $q=-1$ and comment on the differences for $q \in \{0,1\}$ in Remarks <ref>, <ref>.
Then $I^\mdot  = I_Z$ for some $T_X$-fixed subscheme $Z \subset X$. The $q=-1$ was recently also developed by Nekrasov-Piazzalunga [122] (see Section <ref>).

Consider the open affine invariant cover $\{U_\alpha\}_{\alpha \in V(X)}$ of $X$. We write $U_{\alpha\beta} = U_\alpha \cap U_{\beta}$, $U_{\alpha\beta\gamma} = U_\alpha \cap U_\beta \cap U_\gamma$, etc. and we choose a total order on the indices $\alpha$. We also write $Z_\alpha = Z \cap U_\alpha$,  $I_\alpha = I_{Z_{\alpha}}$, $Z_{\alpha\beta} = Z \cap U_{\alpha\beta}$,  $I_{\alpha\beta} = I_{Z_{\alpha\beta}}$, $Z_{\alpha\beta\gamma} = Z \cap U_{\alpha\beta\gamma}$,  $I_{\alpha\beta\gamma} = I_{Z_{\alpha\beta\gamma}}$ etc. Using the local-to-global spectral sequence and Čech cohomology, as in <cit.>, we obtain

\begin{align*}
R \mathrm{Hom}_X(I_Z,I_Z)_0[1] = &\sum_{\alpha \in V(X)} R\mathrm{Hom}_{U_\alpha}(I_\alpha,I_\alpha)_0[1] - \sum_{\alpha<\beta \in V(X)} R\mathrm{Hom}_{U_{\alpha\beta}}(I_{\alpha\beta},I_{\alpha\beta})_0[1] \\
&+ \sum_{\alpha<\beta<\gamma \in V(X)} R\mathrm{Hom}_{U_{\alpha\beta\gamma}}(I_{\alpha\beta\gamma},I_{\alpha\beta\gamma})_0[1] - \cdots.
\end{align*}

We always label edges $\alpha\beta \in E(X)$ so that $\alpha < \beta$. The previous equation should be read as an identity in the additive group $\mathbb{Q}(\!(t_1,t_1^{-1}, \ldots, t_4,t_4^{-1})\!)$: the individual summands on the right hand side are generally not finite but, when adding them, there are infinitely many cancellations resulting in an element in $\mathbb{Q}[t_1,t_1^{-1}, \ldots, t_4,t_4^{-1}] \subset \mathbb{Q}(\!(t_1,t_1^{-1}, \ldots, t_4,t_4^{-1})\!)$.

Since $Z$ is $T_X$-invariant, 2-dimensional, and proper, the only contributions in the above sum are:

* $\alpha \in V(X)$, which corresponds to $U_\alpha \cong \C^4$,

* $\alpha < \beta \in V(X)$ such that $\alpha\beta \in E(X)$, which corresponds to $U_{\alpha\beta} \cong \C^* \times \C^3$,

* $\alpha < \beta \in V(X)$ such that $\alpha\beta \notin E(X)$ but $\alpha, \beta \in f$ for some $f \in F(X)$, which corresponds to $U_{\alpha\beta} \cong (\C^*)^2 \times \C^2$,

* $\alpha_1 < \cdots < \alpha_N \in V(X)$ with $N \geq 3$ such that $\alpha_1, \ldots, \alpha_N \in f$ for some $f \in F(X)$, which corresponds to $U_{\alpha_1 \cdots \alpha_N} \cong (\C^*)^2 \times \C^2$.

In order to deduce these statements, it is useful to recall the description of fans of toric Calabi-Yau 4-folds. Let $N = \mathbb{Z}^4$ and $N_{\R} = N \otimes_{\Z} \R$. Consider the hyperplane $\{x_4 = 1\} \subset N_{\R}$ and any collection of (not necessarily regular) rational tetrahedra in $\{x_4=1\}$ such that any two are disjoint or intersect in a common vertex, edge, or facet. Then we can consider the induced fan $\Delta$ of strongly convex rational polyhedral cones in $N_{\R}$ spanned by the origin and the tetrahedra in $\{x_4=1\}$. Fans obtained in this way are precisely the fans of toric Calabi-Yau 4-folds.

By Lemma <ref>, $I_\alpha \subset \O_{U_\alpha} \cong  \O_{\C^4}$ corresponds to a solid partition $\pi^{(\alpha)} \subset \Z_{\geq 0}^4$ with asymptotic finite partitions and (compatible) plane partitions

\boldsymbol{\lambda}^{(\alpha)} =  \{\lambda_{ab}^{(\alpha)}\}_{1 \leq a<b \leq 4}, \quad \boldsymbol{\mu}^{(\alpha)} = \{ \mu_{a}^{(\alpha)}\}_{a=1}^{4}.

Suppose we choose coordinate functions $(x_1,x_2,x_3,x_4)$ on $U_\alpha \cong \C^4$ such that $t:=(t_1,t_2,t_3,t_4) \in T_X \cong (\C^*)^4$ acts by

\begin{equation} \label{eqn:stndact}
t \cdot (x_1,x_2,x_3,x_4) = (t_1x_1,t_2x_2,t_3x_3,t_4x_4).
\end{equation}

Then we denote the $T_X$-character of $\O_{Z_\alpha}$ by[For any $M \in K_0^{(\C^*)^4}(\C^4)$, we denote by $\tr_M$ the virtual $(\C^*)^4$-representation of $M$. It is viewed as a rational function in $t_1,t_2,t_3,t_4$ and the dimensions of the weight spaces of $M$ are obtained by expanding this rational function in ascending powers of $t_1,t_2,t_3,t_4$. For example, for $M = \O_Z$ and $Z = Z(x_1,x_2) \cup Z(x_3,x_4)$, we have $\tr_{\O_Z} = (1-t_1)^{-1}(1-t_2)^{-1}+(1-t_3)^{-1}(1-t_4)^{-1} -1$.]
\begin{equation} \label{Zalpha}
Z_\alpha := \tr_{\O_{Z_\alpha}} = \sum_{(i_1,i_2,i_3,i_4) \in \pi^{(\alpha)}} t_1^{i_1} t_2^{i_2} t_3^{i_3} t_4^{i_4}.
\end{equation}

Let $\alpha\beta \in E(X)$. Then $U_{\alpha\beta} \cong \C^* \times \C^3$, and we label coordinates $(x_1,x_2,x_3,x_4)$ such that $x_1$ corresponds to the first $\C^*$ factor (and $T_X \cong (\C^*)^4$ acts by the standard torus action (<ref>)).
Let $\alpha_1, \ldots, \alpha_N \in V(X)$ such that $\alpha_1, \ldots, \alpha_N \in f$ for some $f \in F(X)$, where for $N=2$, we assume $\alpha_1\alpha_2 \not\in E(X)$.
Then $U_{\alpha_1 \cdots \alpha_N} \cong (\C^*)^2 \times \C^2$ (Remark <ref>).
We label coordinates $(x_1,x_2,x_3,x_4)$ on $U_{\alpha_1 \cdots \alpha_N}$ such that $(x_1,x_2)$ corresponds to the $(\C^*)^2$ factor (and $T_X \cong (\C^*)^4$ acts by the standard torus action (<ref>)).
Similar to (<ref>), we define

Z_{\alpha\beta} := \tr_{\O_{Z_{\alpha\beta}}}, \quad Z_{\alpha_1\cdots\alpha_N} := \tr_{\O_{Z_{\alpha_1\cdots\alpha_N}}}.

Using the Taylor resolution [151]

0 \to F_s \to \cdots \to F_1 \to F_0 \to I_\alpha \to 0,

a calculation similar to <cit.> gives

\begin{align} \label{prevertex}
\tr_{R \mathrm{Hom}_{U_\alpha}(I_\alpha,I_\alpha)_0[1]} =  Z_\alpha + \frac{\overline{Z}_{\alpha}}{t_1t_2t_3t_4} - \frac{P_{1234}}{t_1t_2t_3t_4} Z_\alpha \overline{Z}_\alpha,
\end{align}

where $\overline{(\cdot)}$ denotes the dual representation, which amounts to replacing each $t_i$ by $t_i^{-1}$, and

\begin{align*}
P_{1234} :=(1-t_1)(1-t_2)(1-t_3)(1-t_4).
\end{align*}

The rational function (<ref>) in $t_1,t_2,t_3,t_4$ should be expanded in ascending powers of $t_i$.

Furthermore, we have

\begin{align} \label{preedge}
\tr_{R \mathrm{Hom}_{U_{\alpha\beta}}(I_{\alpha\beta},I_{\alpha\beta})_0[1]} =  \delta(t_1)\Bigg(Z_{\alpha\beta} - \frac{\overline{Z}_{\alpha\beta}}{t_2t_3t_4} + \frac{P_{234}}{t_2t_3t_4} Z_{\alpha\beta} \overline{Z}_{\alpha\beta}\Bigg),
\end{align}

\begin{align*}
\quad\delta(t_i) := \sum_{i \in \Z} t_i.
\end{align*}

The rational function (<ref>) in $t_2,t_3,t_4$ between brackets should be expanded in ascending powers of $t_2, t_3, t_4$.

Finally, we have

\begin{align*}
\tr_{R \mathrm{Hom}_{U_{\alpha_1\cdots\alpha_N}}(I_{\alpha_1\cdots\alpha_N},I_{\alpha_1\cdots\alpha_N})_0[1]} =  \delta(t_1)\delta(t_2)\Bigg(Z_{\alpha_1\cdots\alpha_N} + \frac{\overline{Z}_{\alpha_1\cdots\alpha_N}}{t_3t_4} - \frac{P_{34}}{t_3t_4} Z_{\alpha_1\cdots\alpha_N} \overline{Z}_{\alpha_1\cdots\alpha_N}\Bigg),
\end{align*}

\begin{align*}
\quad P_{ij}:=(1-t_i)(1-t_j).
\end{align*}

Note that

Z_{\alpha_1\cdots\alpha_N} + \frac{\overline{Z}_{\alpha_1\cdots\alpha_N}}{t_3t_4} - \frac{P_{34}}{t_3t_4} Z_{\alpha_1\cdots\alpha_N} \overline{Z}_{\alpha_1\cdots\alpha_N}

is a Laurent polynomial in $t_3, t_4$. However, (<ref>) and the term between brackets in (<ref>) are in general infinite Laurent series (in ascending powers of the $t_i$). We therefore redistribute the terms.
This means we want to attach Laurent polynomials to all elements of $V(X)$, $E(X)$, $F(X)$ which sum up to the $T_X$-representation $R\Hom_X(I_Z,I_Z)_0[1]$.

In this section, we denote the vertices neighbouring a vertex $\alpha$ by $\beta_1,\beta_2,\beta_3,\beta_4$.[Since $X$ is non-compact, $\alpha$ may lie on a non-compact edge in which case there are $<4$ neighbouring vertices.] Then we label the coordinates $(x_1,x_2,x_3,x_4)$ on $U_\alpha \cong \C^4$ such that

L_{\alpha\beta_i} = Z(x_j,x_k,x_l),

for all $i,j,k,l \in \{1,2,3,4\}$ distinct (and $T_X \cong (\C^*)^4$ acts by the standard torus action (<ref>)).

In the following four definitions, we fix $Z \subset X$ a proper 2-dimensional $T_X$-fixed subscheme. Recall that, for any $\alpha$, $Z_\alpha$ determines partitions in various dimensions (Lemma <ref>)

\{\pi^{(\alpha)}\}_{\alpha \in V(X)}, \quad \{\boldsymbol{\mu}^{(\alpha)}\}_{\alpha \in V(X)}, \quad \{\boldsymbol{\lambda}^{(\alpha)}\}_{\alpha \in V(X)}.

By the inclusion-exclusion principle, there exist unique Laurent polynomials $W_{\alpha\beta_i}$, $W_\alpha$ such that $W_{\alpha\beta_i}$ does not depend on $t_i$ and

\begin{align*}
Z_\alpha =  \sum_{1 \leq i<j \leq 4} \frac{Z_{\alpha\beta_i\beta_j}}{(1-t_i)(1-t_j)} + \sum_{i=1}^{4} \frac{W_{\alpha\beta_i}}{1-t_i} + W_\alpha.
\end{align*}

Recall that $Z_{\alpha\beta_i\beta_j}$ are also Laurent polynomials. Then $|Z_\alpha| := \rk(W_\alpha)$ is called the renormalized volume of $Z_\alpha$. We also write

|\pi^{(\alpha)}| := \rk(W_\alpha).

Furthermore, we have

\begin{align*}
Z_{\alpha\beta_i} =  \sum_{a \in \{j,k,l\}} \frac{Z_{\alpha\beta_i\beta_a}}{1-t_a} + W_{\alpha\beta_i}.
\end{align*}

Then $|Z_{\alpha\beta_i}| := \rk(W_{\alpha\beta_i})$ is called the renormalized volume of $Z_{\alpha\beta_i}$. Moreover

|\mu_i^{(\alpha)}| := \rk(W_{\alpha\beta_i})

coincides with the renormalized volume of plane partitions defined in <cit.>.

We define the vertex term

\begin{align*}
\mathsf{V}_{\alpha} &:= \tr_{R \mathrm{Hom}_{U_\alpha}(I_\alpha,I_\alpha)_0[1]} + \sum_{i=1}^{4} \frac{\mathsf{A}_{\alpha\beta_i}}{1-t_i} + \sum_{1 \leq i<j \leq 4} \frac{\mathsf{A}_{\alpha\beta_i\beta_j}}{(1-t_i)(1-t_j)},
\end{align*}

\begin{align*}
\mathsf{A}_{\alpha\beta_i} &:= - Z_{\alpha\beta_i} + \frac{\overline{Z}_{\alpha\beta_i}}{t_{j} t_{k} t_{l}} - \frac{P_{jkl}}{t_{j}t_{k}t_{l}}  Z_{\alpha\beta_i}   \overline{Z}_{\alpha\beta_i}, \\
\mathsf{A}_{\alpha\beta_i\beta_{j}} &:= Z_{\alpha\beta_i\beta_j} + \frac{\overline{Z}_{\alpha\beta_i\beta_j}}{t_{k} t_{l}} - \frac{P_{kl}}{t_{k}t_{l}}  Z_{\alpha\beta_i\beta_j}   \overline{Z}_{\alpha\beta_i\beta_j},
\end{align*}

and $\tr_{R \mathrm{Hom}_{U_\alpha}(I_\alpha,I_\alpha)_0[1]}$ is given by (<ref>).
Here $i,j,k,l \in \{1,2,3,4\}$ are mutually distinct.
As before, the expressions in this definition should be expanded in ascending powers of $t_i$ (though we will see shortly that $\mathsf{V}_\alpha$ is a Laurent polynomial in the $t_i$).
When we want to stress the dependence of these expressions on $Z_\alpha, Z_{\alpha\beta}, Z_{\alpha\beta\gamma}$, we write $\mathsf{V}_{Z_\alpha}, \mathsf{A}_{Z_{\alpha\beta}}, \mathsf{A}_{Z_{\alpha\beta\gamma}}$. Referring to their corresponding ideals, we also denote these expressions by $\mathsf{V}_{I_\alpha}, \mathsf{A}_{I_{\alpha\beta}}, \mathsf{A}_{I_{\alpha\beta\gamma}}$ respectively.

Recall that the normal bundle to $L_{\alpha\beta_i} \cong \PP^1$ is of the form (Section <ref>)

\begin{align*}
&N_{L_{\alpha\beta_i}/X} \cong \O(m_{\alpha\beta_i}) \oplus \O(m_{\alpha\beta_i}') \oplus \O(m_{\alpha\beta_i}''), \\
&m_{\alpha\beta_i} + m'_{\alpha\beta_i} + m''_{\alpha\beta_i} = -2.
\end{align*}

We define the edge term

\begin{align*}
\mathsf{E}_{\alpha\beta_i} &:= \frac{t_i^{-1}  \mathsf{B}_{\alpha\beta_i}(t_j,t_k,t_l) }{1-t_i^{-1}} -  \frac{\mathsf{B}_{\alpha\beta_i}(t_i^{-m_{\alpha\beta_i}}t_j,t_i^{-m_{\alpha\beta_i}'} t_k,t_i^{-m_{\alpha\beta_i}''}t_l) }{1-t_i^{-1}},
\end{align*}

\begin{align*}
\mathsf{B}_{\alpha\beta_i} &:= \mathsf{A}_{\alpha\beta_i}  + \sum_{a \in \{j,k,l\}} \frac{\mathsf{A}_{\alpha\beta_i\beta_a} }{1-t_a},
\end{align*}

where $i,j,k,l \in \{1,2,3,4\}$ are mutually distinct.

The expressions in this definition should be expanded in ascending powers of $t_i^{-1}$ (!), $t_j, t_k, t_l$ (though we will see shortly that $\mathsf{E}_{\alpha\beta_i}$ is a Laurent polynomial in $t_1,t_2,t_3,t_4$).
When we want to stress the dependence of these expressions on $Z_\alpha, Z_{\alpha\beta}$, we write $\mathsf{E}_{Z_{\alpha\beta}}, \mathsf{B}_{Z_{\alpha\beta}}$. Referring to their corresponding ideals, we also denote these expressions by $\mathsf{E}_{I_{\alpha\beta}}, \mathsf{B}_{I_{\alpha\beta}}$ respectively.

One can also consider the expressions of Definitions <ref>, <ref> for $Z \subset X$ of dimension $\leq 1$. Then they reduce to the redistributions studied in [34] (in turn based on [108]).

Fix $f \in F(X)$. Denote by $Z_f \subset X$ the unique $T_X$-fixed 2-dimensional closed subscheme set-theoretically supported on $S_f$ such that

Z_f \cap U_{\alpha_1\alpha_2\alpha_3} = Z \cap U_{\alpha_1\alpha_2\alpha_3}

for any $\alpha_1, \alpha_2, \alpha_3 \in f$ distinct. Then $Z_f$ is determined by a single finite partition. Let $I_f := I_{Z_f} \subset \O_X$ be the corresponding ideal, then we define the face term by

\mathsf{F}_f := \tr_{R\Hom_X(I_{f},I_{f})_0[1]}.

Note that, since $Z_f$ is proper, $\mathsf{F}_f$ is a Laurent polynomial in the $t_i$.
When we want to stress the dependence of $\mathsf{F}_f $ on $Z_f$ we write $\mathsf{F}_{Z_f}$. Referring to the corresponding ideal, we also denote this expression by $\mathsf{F}_{I_f}$.

The main result of this section is the following:

Let $X$ be a toric Calabi-Yau 4-fold and $Z \subset X$ a 2-dimensional $T_X$-fixed closed subscheme with proper support. Then

\begin{equation} \label{eq:redisteq}
R \mathrm{Hom}(I_Z,I_Z)_0[1] = \sum_{\alpha \in V(X)} \mathsf{V}_{\alpha} + \sum_{\alpha\beta \in E(X)} \mathsf{E}_{\alpha\beta} + \sum_{f \in F(X)} \mathsf{F}_{f},
\end{equation}

where $\mathsf{V}_{\alpha}, \mathsf{E}_{\alpha\beta}, \mathsf{F}_{f}$ are Laurent polynomials.

We first show the equality (<ref>) and then establish polynomiality of $\mathsf{V}_{\alpha}$, $\mathsf{E}_{\alpha\beta}$, $\mathsf{F}_{f}$. Note that

\begin{align*}
&\sum_{\alpha \in V(X)} \tr_{R \mathrm{Hom}_{U_\alpha}(I_\alpha,I_\alpha)_0[1]} - \sum_{\alpha\beta \in E(X)} \tr_{R \mathrm{Hom}_{U_{\alpha\beta}}(I_{\alpha\beta},I_{\alpha\beta})_0[1]} \\
&= \sum_{\alpha \in V(X)} \Bigg( -R\mathrm{Hom}_{U_\alpha}(I_\alpha,I_\alpha)_0 + \sum_{i=1}^{4} \frac{\mathsf{A}_{\alpha\beta_i}}{1-t_i} \Bigg) \\
&+ \sum_{\alpha\beta \in E(X)} \Bigg( \frac{t_1^{-1}  \mathsf{A}_{\alpha\beta}(t_2,t_3,t_4) }{1-t_1^{-1}} -  \frac{\mathsf{A}_{\alpha\beta}(t_1^{-m_{\alpha\beta}}t_2 ,t_1^{-m'_{\alpha\beta}}t_3 ,t_1^{-m''_{\alpha\beta}}t_4 ) }{1-t_1^{-1}} \Bigg),
\end{align*}

where the coordinates of summand $\alpha\beta$ are labelled such that $L_{\alpha\beta} \cap U_\alpha = Z(x_2,x_3,x_4)$ and $L_{\alpha\beta} \cap U_\beta = Z(x_2',x_3',x_4')$.

By Remark <ref>, in order to calculate $R \mathrm{Hom}(I_Z,I_Z)_0[1]$, we are left with the contributions of (1) $\alpha < \beta \in V(X)$ such that $\alpha\beta \notin E(X)$ and $\alpha, \beta \in f$ for some $f \in F(X)$; and (2) $\alpha_1 < \cdots <\alpha_N \in V(X)$, with $N \geq 3$, such that $\alpha_1, \ldots, \alpha_N \in f$ for some $f \in F(X)$. For any face $f \in F(X)$, we now label the vertices of $f$ as $\alpha_{1} < \cdots  < \alpha_{v_f}$, where $v_f = e(S_f)$ is the topological Euler characteristic of $S_f$. On $U_{\alpha_i}$, we use coordinates $(x_1^{(i)},x_2^{(i)},x_3^{(i)},x_4^{(i)})$ such that $(t_1^{(i)},t_2^{(i)},t_3^{(i)},t_4^{(i)}) \in (\mathbb{C}^*)^4 \cong T_X$ acts by the standard torus action. We label the coordinates such that for any  distinct $\alpha_i, \alpha_j,\alpha_k$, the first two coordinates of $U_{\alpha_i \alpha_j \alpha_k} \cong (\C^*)^2 \times \C^2$ are $x_1^{(i)}, x_2^{(i)}$, then

\begin{align*}
\tr_{R \mathrm{Hom}_{U_{\alpha_i \alpha_j \alpha_k}}(I_{\alpha_{i}\alpha_{j}\alpha_{k}},I_{\alpha_{i}\alpha_{j}\alpha_{k}})_0[1]} &= \delta(t_1^{(i)})\delta(t_2^{(i)}) \mathsf{A}_{f}(t_3^{(i)},t_4^{(i)}) \\
&= \delta(t_1^{(j)})\delta(t_2^{(j)}) \mathsf{A}_{f}(t_3^{(j)},t_4^{(j)})\\
&= \delta(t_1^{(k)})\delta(t_2^{(k)}) \mathsf{A}_{f}(t_3^{(k)},t_4^{(k)}),
\end{align*}

\[
\mathsf{A}_f(t_3,t_4) := Z_{\lambda_f} + \frac{\overline{Z}_{\lambda_f}}{t_3t_4} - \frac{P_{34}}{t_3 t_4} Z_{\lambda_f} \overline{Z}_{\lambda_f} \in \mathbb{Z}[t_3^{\pm 1}, t_4^{\pm 1}]
\]

and $\lambda_f$ is the finite partition determining the scheme $Z_f$ in Definition <ref> and $Z_{\lambda_f} := \sum_{(i,j) \in \lambda_f} t_3^{i} t_4^{j}$. More generally, for any $\alpha_{i_1} < \cdots < \alpha_{i_l}$ with $l \geq 2$, where for $l=2$ we assume $\alpha_{i_1}, \alpha_{i_2}$ are not adjacent, we have

\tr_{R \mathrm{Hom}_{U_{\alpha_{i_1} \ldots \alpha_{i_l}}} (I_{\alpha_{i_1} \ldots \alpha_{i_l}},I_{\alpha_{i_1} \ldots \alpha_{i_l}})_0[1]} = \delta(t_1^{(1)}) \delta(t_2^{(1)}) \mathsf{A}_f(t_3^{(1)}, t_4^{(1)}).

We define $N_f := -(-v_f + \sum_{i=2}^{v_f} (-1)^{i} \binom{v_f}{i}) =1$ and

\begin{align}
\begin{split} \label{def:Fprime}
&\mathsf{F}_f'  := N_f \cdot \delta(t_1^{(1)}) \delta(t_2^{(1)}) \mathsf{A}_f(t_3^{(1)}, t_4^{(1)}) - \sum_{i} \frac{\mathsf{A}_f(t_3^{(i)}, t_4^{(i)})}{(1-t_1^{(i)})(1-t_2^{(i)})} \\
&- \sum_{i} \Bigg( \frac{(t_1^{(i)})^{-1} \mathsf{A}_{f}(t_3^{(i)},t_4^{(i)})}{(1 - (t_1^{(i)})^{-1})(1-t_2^{(i)})} - \frac{\mathsf{A}_{f} ( (t_1^{(i)})^{-m_{\alpha_i \alpha_{i+1}}'} t_3^{(i)}, (t_1^{(i)})^{-m_{\alpha_i \alpha_{i+1}}''} t^{(i)}_4) }{(1 - (t_1^{(i)})^{-1})(1- (t_1^{(i)})^{-m_{\alpha_i\alpha_{i+1}}}  t_2^{(i)} )}\Bigg).
\end{split}
\end{align}

Then, by Definitions <ref> and <ref>, we have

\begin{align*}
R \mathrm{Hom}(I_Z,I_Z)_0[1] = &\sum_{\alpha \in V(X)} \Bigg( -R\mathrm{Hom}_{U_\alpha}(I_\alpha,I_\alpha)_0 + \sum_{i=1}^{4} \frac{\mathsf{A}_{\alpha\beta_i}}{1-t_i} \Bigg) \\
&+ \sum_{\alpha\beta \in E(X)} \Bigg( \frac{t_1^{-1}  \mathsf{A}_{\alpha\beta}(t_2,t_3,t_4) }{1-t_1^{-1}} -  \frac{\mathsf{A}_{\alpha\beta}(t_1^{-m_{\alpha\beta}}t_2 ,t_1^{-m'_{\alpha\beta}}t_3 ,t_1^{-m''_{\alpha\beta}}t_4 ) }{1-t_1^{-1}} \Bigg) \\
&+ \sum_{f \in F(X)} N_f \cdot \delta(t_1^{(1)}) \delta(t_2^{(1)}) \mathsf{A}_f(t_3^{(1)}, t_4^{(1)}) \\
= &\sum_{\alpha \in V(X)} \mathsf{V}_{\alpha} + \sum_{\alpha\beta \in E(X)} \mathsf{E}_{\alpha\beta} + \sum_{f \in F(X)} \mathsf{F}_{f}'.
\end{align*}

In order to complete the proof, we need to show $\mathsf{F}_f' = \mathsf{F}_f$ (Definition <ref>), and $\mathsf{V}_{\alpha}, \mathsf{E}_{\alpha\beta}, \mathsf{F}_{f}$ are Laurent polynomials.

Consider a vertex $\alpha \in V(X)$ with neighbouring vertices $\beta_1,\beta_2,\beta_3, \beta_4 \in V(X)$ (possibly fewer if $\alpha$ lies on a non-compact edge). Recall that $Z_{\alpha\beta_i\beta_j}$ are Laurent polynomials. Consider the Laurent polynomials $W_{\alpha\beta_i}$, $W_\alpha$ of Definition <ref>, which satisfy

\begin{align}
\begin{split} \label{eq:Ztosubst}
Z_\alpha &=  \sum_{1 \leq i<j \leq 4} \frac{Z_{\alpha\beta_i\beta_j}}{(1-t_i)(1-t_j)} + \sum_{i=1}^{4} \frac{W_{\alpha\beta_i}}{1-t_i} + W_\alpha, \\
Z_{\alpha\beta_i} &=  \sum_{a \in \{j,k,l\}} \frac{Z_{\alpha\beta_i\beta_a}}{1-t_a} + W_{\alpha\beta_i}.
\end{split}
\end{align}

Substituting into $\mathsf{V}_{\alpha}$ gives an expression in terms of $W_{\alpha\beta_i}$, $W_\alpha$ (and their duals). An explicit (lengthy) calculation shows that all denominators cancel and $\mathsf{V}_{\alpha}$ is a Laurent polynomial.
Similarly, substituting (<ref>) into $\mathsf{B}_{\alpha\beta_i}$, all denominators cancel and we find that $\mathsf{B}_{\alpha\beta_i}$ is a Laurent polynomial. Since the numerator and denominator of

\begin{align*}
\mathsf{E}_{\alpha\beta_i} = \frac{t_i^{-1}  \mathsf{B}_{\alpha\beta_i}(t_j,t_k,t_l) }{1-t_i^{-1}} -  \frac{\mathsf{B}_{\alpha\beta_i}(t_j t_i^{-m_{\alpha\beta_i}},t_k t_i^{-m_{\alpha\beta_i}'},t_l t_i^{-m_{\alpha\beta_i}''}) }{1-t_i^{-1}}
\end{align*}

are zero for $t_i^{-1} = 1$, we deduce that $\mathsf{E}_{\alpha\beta_i} $ is a Laurent polynomial in $t_1, t_2, t_3, t_4$ as well.

Recall the definition of $Z_f \subset X$ of Definition <ref>, and let $Z_{f,\alpha} = Z_f \cap U_\alpha$ and $Z_{f,\alpha\beta} = Z_{f} \cap U_{\alpha\beta}$. Then

\begin{align*}
\mathsf{F}_{Z_f} := R\Hom_X(I_{Z_f},I_{Z_f})_0[1] = \sum_{\alpha \in V(X)} \mathsf{V}_{Z_{f,\alpha}} + \sum_{\alpha\beta \in E(X)} \mathsf{E}_{Z_{f,\alpha\beta}} + \mathsf{F}_{Z_f}'.
\end{align*}

It is easy to calculate $\mathsf{V}_{Z_{f,\alpha}} = \mathsf{E}_{Z_{f,\alpha\beta}} = 0$ for all $\alpha \in V(X)$ and $\alpha\beta \in E(X)$. Hence $\mathsf{F}_{Z_f}' = \mathsf{F}_{Z_f} = R\Hom_X(I_{Z_f},I_{Z_f})_0[1]$ and it is a Laurent polynomial because $Z_f \subset X$ is proper.

The proof of Theorem <ref> leads to a nice expression for $\mathsf{F}_f$. For a $T_X$-fixed 2-dimensional subscheme $Z \subset X$ with proper support, denote by $\lambda_f$ the finite partition corresponding to the subscheme $Z_f$ from Definition <ref>. Let

\[
\mathsf{A}_f(t_3,t_4) := Z_{\lambda_f} + \frac{\overline{Z}_{\lambda_f}}{t_3t_4} - \frac{P_{34}}{t_3 t_4} Z_{\lambda_f} \overline{Z}_{\lambda_f} \in \mathbb{Z}[t_3^{\pm 1}, t_4^{\pm 1}],
\]

where $Z_{\lambda_f} := \sum_{(i,j) \in \lambda_f} t_3^{i} t_4^{j}$. Denote by $\alpha_1< \cdots < \alpha_{e(S_f)}$ the vertices of $f$, where $e(S_f)$ denotes the topological Euler characteristic of $S_f$. As before, for any $\alpha_i \in f$, we use coordinates $(x_1^{(i)}, x_2^{(i)}, x_3^{(i)}, x_4^{(i)})$ on $U_{\alpha_i}$ such that $(t_1^{(i)},t_2^{(i)},t_3^{(i)},t_4^{(i)}) \in (\mathbb{C}^*)^4 \cong T_X$ acts by the standard torus action. Then

\begin{equation} \label{eqn:exprFf}
\mathsf{F}_f = \sum_{i=1}^{e(S_f)} \frac{\mathsf{A}_f(t_3^{(i)},t_4^{(i)})}{(1-t_1^{(i)})(1-t_2^{(i)})}.
\end{equation}

This equality should be interpreted carefully and can be proved as follows.[We thank Schmiermann for pointing out the correct interpretation of (<ref>).] Firstly, writing $t_1 := t_1^{(1)}$, $t_2 := t_2^{(1)}$, we view each $t_1^{(i)}$, $t_2^{(i)}$ as a function of $t_1,t_2$ determined by the coordinate change between $U_{\alpha_{1}}$ and $U_{\alpha_i}$ (<ref>). Secondly, the left hand side of (<ref>) is the expression in (<ref>), which lives in $\mathbb{Q}[t_1^{\pm 1}, \ldots, t_4^{\pm 1}] \subset \mathbb{Q}(\!(t_1,t_1^{-1}, \ldots ,t_4, t_4^{-1})\!)$, whereas the right hand side of (<ref>) (a priori) is an element of $\mathbb{Q}(t_1,\ldots,t_4)$. We consider the field extensions

\[
\mathbb{Q}(t_1, \ldots, t_4) \hookrightarrow \mathbb{Q}(t_1,t_2,t_3)(\!(t_4)\!) \hookrightarrow \cdots \hookrightarrow \mathbb{Q}(\!(t_1)\!)\cdots(\!(t_4)\!),
\]

obtained by expanding denominators in ascending powers of the $t_i$ (e.g. for the first extension we expand $1/(1-f(t_4)) = 1 + f(t_4) + f(t_4)^2 + \ldots$ for any $f(t_4) \in t_4 \Q(t_1,t_2,t_3)[t_4]$).
Note that $\mathbb{Q}(\!(t_1)\!)\cdots(\!(t_4)\!) \subset \mathbb{Q}(\!(t_1,t_1^{-1}, \ldots, t_4,t_4^{-1})\!)$. This embeds the right hand side of (<ref>) into $\mathbb{Q}(\!(t_1,t_1^{-1}, \ldots ,t_4, t_4^{-1})\!)$ and after multiplying left and right hand side of (<ref>) by $\prod_i(1-t_1^{(i)})(1-t_2^{(i)})$, and using $\delta(t_a^{(i)})(1-t_a^{(i)}) = 0$, they are obviously equal.[Although $\mathbb{Q}(\!(t_1,t_1^{-1}, \ldots ,t_4, t_4^{-1})\!)$ is not a ring, it is a $\mathbb{Q}[t_1^{\pm 1}, \ldots, t_4^{\pm 1}]$-module.] This shows that the right hand side of (<ref>) is in fact an element $\mathbb{Q}[t_1^{\pm 1}, \ldots, t_4^{\pm 1}] \subset \mathbb{Q}(t_1,\ldots,t_4)$ and it is equal to the left hand side of (<ref>).

We use the following notation, borrowed from Nekrasov-Piazzalunga [121], for the symmetrized $T$-equivariant $K$-theoretic Euler class. As before, $T \leq T_X$ denotes the Calabi-Yau torus (<ref>).

Consider the action of $T \times \C^*$ on $\pt$, where $T = Z(t_1t_2t_3t_4- 1) \leq T_X$ as before, and we denote the equivariant parameter of the second torus by $y$. We define

[E] := \widehat{\Lambda}\udot E^\vee,

for any class $E \in K_0^{T \times \C^*}(\mathrm{pt},\Z[\tfrac{1}{2}]) \cong \Z[t_1^{\pm \frac{1}{2}},t_2^{\pm \frac{1}{2}},t_3^{\pm \frac{1}{2}},t_4^{\pm \frac{1}{2}},y^{\pm \frac{1}{2}}] / (t_1t_2t_3t_4-1)$. In particular

[L] = L^{\frac{1}{2}} - L^{-\frac{1}{2}},

for the class of a $T \times \C^*$-equivariant line bundle $L$.

Next, we incorporate the $K$-theoretic insertion (Sections <ref>, <ref>)

\widehat{\Lambda}\udot (R\pi_{\curly P*} \mathcal{O}_{\cZ} \otimes y^{-1})

into the vertex formalism. At a fixed point $Z \in \curP_v^{(-1)}(X)^T$, we have

\widehat{\Lambda}\udot ( R\pi_{\curly P *} \mathcal{O}_{\cZ} \otimes y^{-1})|_Z = \widehat{\Lambda}\udot (R\Gamma(X,\O_Z) \otimes y^{-1}).

Using Čech cohomology, we deduce

\begin{align*}
R\Gamma(X,\O_Z) = &\sum_{\alpha \in V(X)} \Gamma(U_\alpha,\O_{Z_\alpha}) - \sum_{\alpha < \beta \in V(X)} \Gamma(U_{\alpha\beta}, \O_{Z_{\alpha\beta}}) \\
&+ \sum_{\alpha<\beta<\gamma \in V(X)} \Gamma(U_{\alpha\beta\gamma},\O_{Z_{\alpha\beta\gamma}}) - \cdots.
\end{align*}

These terms can be redistributed similar to Definitions <ref>, <ref>, <ref>

\begin{align*}
\widetilde{\mathsf{V}}_{\alpha} := &\mathsf{V}_{\alpha} - y \overline{Z}_\alpha + \sum_{i=1}^{4} \frac{y \overline{Z}_{\alpha\beta_i}}{1-t_i^{-1}} - \sum_{1 \leq i<j \leq 4} \frac{y \overline{Z}_{\alpha\beta_i\beta_j}}{(1-t_i^{-1})(1-t_j^{-1})} \\
&\quad \ - y^{-1} Z_\alpha + \sum_{i=1}^{4} \frac{y^{-1} Z_{\alpha\beta_i}}{1-t_i} - \sum_{1 \leq i<j \leq 4} \frac{y^{-1} Z_{\alpha\beta_i\beta_j}}{(1-t_i)(1-t_j)} \\
\widetilde{\mathsf{E}}_{\alpha\beta_i} := &\mathsf{E}_{\alpha\beta_i} + \frac{t_i}{1-t_i} \Bigg\{ y \overline{Z}_{\alpha\beta_i} - \sum_{a \in \{j,k,l\}} \frac{y \overline{Z}_{\alpha\beta_i\beta_a}}{1-t_a^{-1}} \Bigg\} \\
&\quad\quad -\frac{1}{1-t_i} \Bigg\{ y \overline{Z}_{\alpha\beta_i} - \sum_{a \in \{j,k,l\}} \frac{y \overline{Z}_{\alpha\beta_i\beta_a}}{1-t_a^{-1}} \Bigg\}\Bigg|_{(t_j t_i^{-m_{\alpha\beta_i}},t_k t_i^{-m_{\alpha\beta_i}'},t_l t_i^{-m_{\alpha\beta_i}''})} \\
&\quad\quad + \frac{t_i^{-1}}{1-t_i^{-1}} \Bigg\{ y^{-1} Z_{\alpha\beta_i} - \sum_{a \in \{j,k,l\}} \frac{y^{-1} Z_{\alpha\beta_i\beta_a}}{1-t_a} \Bigg\} \\
&\quad\quad- \frac{1}{1-t_i^{-1}} \Bigg\{ y^{-1} Z_{\alpha\beta_i} - \sum_{a \in \{j,k,l\}} \frac{y^{-1} Z_{\alpha\beta_i\beta_a}}{1-t_a} \Bigg\}\Bigg|_{(t_j t_i^{-m_{\alpha\beta_i}},t_k t_i^{-m_{\alpha\beta_i}'},t_l t_i^{-m_{\alpha\beta_i}''})} \\
\widetilde{\mathsf{F}}_f  := &\mathsf{F}_f - y R\Gamma(X, \O_{Z_f})^\vee - y^{-1} R\Gamma(X, \O_{Z_f}).
\end{align*}

Using the notation of Remark <ref>, we have

\widetilde{\mathsf{F}}_f = \mathsf{F}_f - \sum_{i=1}^{e(S_f)} \frac{y \overline{Z}_{\lambda_f}(t_3^{(i)},t_4^{(i)})}{(1-(t_1^{(i)})^{-1})(1-(t_2^{(i)})^{-1})} - \sum_{i=1}^{e(S_f)} \frac{y^{-1} Z_{\lambda_f}(t_3^{(i)},t_4^{(i)})}{(1-t_1^{(i)})(1-t_2^{(i)})},

where the two sums are elements of $\Q[t_1^{\pm 1}, \ldots, t_4^{\pm 1}, y^{\pm 1}] \subset \Q(t_1,\cdots,t_4)[y^{\pm 1}]$.

Recall that, in the toric case, $\vd \in \Z$ (Lemma <ref>) and we define invariants by the virtual localization formula (Definition <ref>). Therefore, combining Proposition <ref> and Theorems <ref>, <ref>(i), we obtain:

Let $X$ be a toric Calabi-Yau 4-fold and fix $v = (0,0,\gamma,\beta,n - \gamma \cdot \td_2(X)) \in H_c^*(X,\Q)$. Let $\curly P := \curly P_v^{(-1)}(X)^T$ and denote by $\vd$ the virtual dimension and $\rk$ the rank of the tautological complex. Then

\[
\langle\!\langle \O_X \rangle\!\rangle_{X,v}^{\DT} = \sum_{Z \in \curly P^T} \pm \sqrt{(-1)^{\vd - \rk}\Big[-\sum_{\alpha \in V(X)} \widetilde{\mathsf{V}}_{Z_\alpha} - \sum_{\alpha\beta \in E(X)} \widetilde{\mathsf{E}}_{Z_{\alpha\beta}} - \sum_{f \in F(X)} \widetilde{\mathsf{F}}_{Z_f} \Big]},
\]

where the right hand side depends on a choice of $\pm \sqrt{\cdot}$ for each fixed point.

For notational convenience, we considered the insertion

\widehat{\Lambda}\udot ( R\pi_{\curly P*} \O_{\cZ} \otimes y^{-1}).

The insertion $\widehat{\Lambda}\udot ( R\pi_{\curly P *} (\O_{\cZ} \otimes L)  \otimes y^{-1})$, for an arbitrary $T_X$-equivariant line bundle $L$ on $X$, can be easily incorporated as follows. Define $L_\alpha = L|_{U_\alpha}$, $L_{\alpha\beta} = L|_{U_{\alpha\beta}}$ etc. Then, in the definition of $\mathsf{V}_\alpha$, we replace $y$ by $L_\alpha^* y$ and we denote the resulting expression by $\mathsf{V}_\alpha(L_\alpha)$ (and similarly for the edge and face).

We will now show that the vertex, edge, and face terms in Proposition <ref> contain no positive $T$-fixed terms, which justifies writing[It follows from (<ref>) in Section <ref> that $\rk(\widetilde{\mathsf{V}}_{Z_\alpha})$ is even. Similarly, it can be seen that $\rk(\widetilde{\mathsf{E}}_{Z_{\alpha\beta}})$ is even. It is also clear from Definition <ref> and Lemma <ref> that $\rk(\widetilde{\mathsf{F}}_{Z_f})$ is even.]

\begin{align*}
\langle\!\langle \O_X \rangle\!\rangle_{X,v}^{\DT} = \sum_{Z \in \curly P^T} \pm &\prod_{\alpha \in V(X)} \sqrt{(-1)^{\frac{\rk(\widetilde{\mathsf{V}}_{Z_\alpha})}{2}} [-\widetilde{\mathsf{V}}_{Z_\alpha}]} \cdot \prod_{\alpha\beta \in E(X)} \sqrt{(-1)^{\frac{\rk \widetilde{\mathsf{E}}_{Z_{\alpha\beta}}}{2}}  [-\widetilde{\mathsf{E}}_{Z_{\alpha\beta}}]} \cdot \\
&\prod_{f \in F(X)} \sqrt{(-1)^{\frac{\rk \widetilde{\mathsf{F}}_{Z_f}}{2}} [-\widetilde{\mathsf{F}}_{Z_f}]}.
\end{align*}

$\mathsf{V}_\alpha$, $\mathsf{E}_{\alpha\beta}$, $\mathsf{F}_f$ have no positive $T$-fixed terms.

For $\mathsf{E}_{\alpha\beta}$, $\mathsf{F}_f$ we in fact show that they have no $T$-fixed terms at all. First note that $\mathsf{A}_f(t_3^{(i)}, t_4^{(i)})$ in (<ref>) has no terms of the form $(t_3^{(i)} t_4^{(i)})^n$ because tangent spaces to Hilbert schemes of points of $\C^2$ at monomial ideals have no such terms (by a simple version of the argument with Haiman arrows in the proof of Theorem <ref>(i)). From this it follows that $\mathsf{F}_f$ has no $T$-fixed terms.

Next, we note that $\mathsf{B}_{\alpha\beta_1}$ in Definition <ref>, which only depends on variables $t_2,t_3,t_4$, is precisely minus the vertex term of Donaldson-Thomas theory on toric 3-folds <cit.>. This is known to have no terms of the form $(t_2t_3t_4)^n$ <cit.>. Therefore $\mathsf{E}_{\alpha\beta_1}$ has no $T$-fixed terms.

Finally, we turn to the vertex term. We fix a $(\C^*)^4$-invariant 2-dimensional subscheme $Z_{\alpha_0} \subset U_{\alpha_0} = \C^4$ and we want to show that $\mathsf{V}_{Z_{\alpha_0}}$ has no positive $T$-fixed term (we use a distinguished label $\alpha_0$). We pick a toric Calabi-Yau 4-fold $X$ which is a partial toric compactification of $U_{\alpha_0}$ in the following sense: $\alpha_0$ has four neighbouring vertices $\beta_1, \ldots, \beta_4$ and each $\alpha_0,\beta_i,\beta_j$, with $i \neq j$, are vertices of a proper smooth toric surface in $X$.[In fact, it suffices to add ten copies of $\C^4$ in such a way that $X$ contains precisely six $\PP^2$'s which compactify the six coordinate 2-planes in $U_{\alpha_0} = \C^4$.] We consider the minimal proper $T_X$-fixed 2-dimensional subscheme $Z \subset X$ such that $Z \cap U_{\alpha_0} = Z_{\alpha_0}$. It is then easy to see that for each $\alpha \neq \alpha_0$, we have $\mathsf{V}_{Z_\alpha} = 0$. More precisely, in Definitions <ref>, <ref>: putting $Z_{\alpha\beta_i\beta_j} = 0$ for $(i,j) \neq (1,4), (2,4), (3,4)$, $W_{\alpha\beta_i} = 0$ for $i \neq 4$, and $W_\alpha = 0$ gives $\mathsf{V}_{Z_\alpha} = 0$ by explicit calculation. Therefore

\begin{equation} \label{eqn:completionarg}
R\Hom_X(I_Z,I_Z)_0[1] = \mathsf{V}_{\alpha_0} + \sum_{\alpha\beta \in E(X)} \mathsf{E}_{\alpha\beta} + \sum_{f \in F(X)} \mathsf{F}_f.
\end{equation}

The left hand side of (<ref>) has no positive $T$-fixed terms since $\Ext_X^3(I_Z,I_Z)_0^* \cong \Ext_X^1(I_Z,I_Z)_0$ and $\Ext_X^1(I_Z,I_Z)_0 \cong \Hom_X(I_Z,\O_Z)$ (<cit.>), which has no $T$-fixed term by Theorem <ref>(i). Since the edge and face terms have no $T$-fixed terms, the result follows.

It is interesting to provide an explicit “half in $K_0^T(\pt)$” of the vertex, edge, and face terms and hence the term under the square root symbol in Proposition <ref>. Using the spectral construction, we give such expressions when $X = \mathrm{Tot}(K_Y)$ (Section <ref>). In general, such an expression is exhibited in [122]. However, it should be noted that even after such a “halving”, one can still make any choice of signs assigned to the $T$-fixed points (Definition <ref>). As we will see, in order to get interesting answers, one typically has to assign non-trivial signs even after “halving”.

Let $\boldsymbol{\mu} = \{\mu_{a} \}_{a = 1}^{4}$ be a collection of plane partitions corresponding to subschemes of dimension $\leq 1$ of $\C^3$ (and not all finite). We denote by $\mathcal{P}_{\boldsymbol{\mu}}$ the collection of solid partitions $\pi$ such that $Z_\pi$ has the property that its maximal subscheme without embedded points is $Z_{\boldsymbol{\mu}}$ (Definition <ref>). We define the $(K$-theoretic) $\DT$ topological vertex by

\mathsf{V}^{\DT}_{\mu_1\mu_2\mu_3\mu_4}(q) := \sum_{\pi \in \mathcal{P}_{\boldsymbol{\mu}}} \pm  \sqrt{(-1)^{\frac{\rk(\widetilde{\mathsf{V}}_{Z_\pi})}{2}} [-\widetilde{\mathsf{V}}_{Z_\pi}]} \, q^{|\pi|},

where $|\pi|$ denotes renormalized volume (Definition <ref>) and $\pm\sqrt{\cdot}$ is a choice of square root with sign for each $\pi \in \mathcal{P}_{\boldsymbol{\mu}}$. In particular, $\mathsf{V}^{\DT}_{\mu_1\mu_2\mu_3\mu_4}(q)$ depends on a choice of sign attached to each $\pi \in \mathcal{P}_{\boldsymbol{\mu}}$.

The previous definition extends the $\DT$ topological vertex for curves studied in [34, 35, 115] and for points in [121]. It also matches the DT topological vertex in [122]

Let $I^\mdot \in \curP_v^{(0)}(X)^{T_X}$. Its scheme theoretic support $Z \subset X$ is a $T_X$-fixed proper 2-dimensional subscheme with no embedded 0-dimensional components.
Then $\mathsf{V}_\alpha$, $\mathsf{E}_{\alpha\beta}$, $\mathsf{F}_f$, $\widetilde{\mathsf{V}}_\alpha$, $\widetilde{\mathsf{E}}_{\alpha\beta}$, $\widetilde{\mathsf{F}}_f$ can be determined in exactly the same way as in the $\DT$ case with the following replacement. Instead of (<ref>) we now take

Z_\alpha := \tr_{\O_{Z_\alpha}} + \tr_{B_\alpha}

where $\tr_{B_\alpha}$ is the $T$-representation of the $\PT_0$ box configuration corresponding to $I^\mdot|_{U_\alpha}$ (Definition <ref>). Since the local description of $I^\mdot|_{U_{\alpha_1\cdots \alpha_l}}$ for $l \geq 2$ is identical to the $\DT$ case, no modifications are needed for the edge and face terms.
Under the assumption $\curP_v^{(0)}(X)^{T_X} = \curP_v^{(0)}(X)^{T}$ is 0-dimensional reduced (e.g. in the cases of Theorem <ref>(ii),(iii)), we conclude that Proposition <ref> holds in the $\PT_0$ case.

Let $\boldsymbol{\mu}$ be a choice of plane partitions with no $\PT_0$ moduli (Definition <ref>). Recall that $\mathcal{B}_{\boldsymbol{\mu}}^{(0)}$ denotes the collection of $\PT_0$ box configurations with asymptotics $\boldsymbol{\mu}$ (Definition <ref>). We denote by $\pi^{\boldsymbol{\mu} }$ the solid partition corresponding to $Z_{\boldsymbol{\mu}}$, i.e., the 2-dimensional subscheme without embedded 0-dimensional components determined by $\boldsymbol{\mu}$ (Definition <ref>). Each element $B \in \mathcal{B}^{(0)}_{\boldsymbol{\mu}}$ then determines a $\PT_0$ pair $I_B\udot$ on $\C^4$. We define the ($K$-theoretic) $\PT_0$ topological vertex by

\mathsf{V}^{\PT_0}_{\mu_1\mu_2\mu_3\mu_4}(q) := \sum_{B \in \mathcal{B}^{(0)}_{\boldsymbol{\mu}}} \pm  \sqrt{(-1)^{\frac{\rk(\widetilde{\mathsf{V}}_{I_B\udot})}{2}} [-\widetilde{\mathsf{V}}_{I_B\udot}]} \, q^{|B|+|\pi^{\boldsymbol{\mu} }|}.

This definition is subject to the following restriction. We must assume that for all $B \in \mathcal{B}^{(0)}_{\boldsymbol{\mu}}$, the vertex term $\mathsf{V}_{I_B\udot}$ has no positive $T$-fixed part (so we do not divide by zero). We expect that this is always the case when $\boldsymbol{\mu}$ is a choice of plane partitions with no $\PT_0$ moduli. It is certainly the case when $\boldsymbol{\mu}$ is chosen such that $\PT_0 = \PT_1$; or such that there are embedded curves in at most 2 directions and the underlying pure 2-dimensional subscheme is Cohen-Macaulay. Then the proof of Lemma <ref> goes through.[The only modification is that this time the analog of the left hand side of (<ref>) has no positive $T$-fixed part by the proof of Theorem <ref>(ii), (iii).]

The previous definition extends the $\PT$ topological vertex for curves studied in [34, 35, 107, 134].

Let $I^\mdot \in \curP_v^{(1)}(X)^{T_X}$. Its scheme theoretic support $Z \subset X$ is a $T_X$-fixed proper pure 2-dimensional subscheme.
Then $\mathsf{V}_\alpha$, $\mathsf{E}_{\alpha\beta}$, $\mathsf{F}_f$, $\widetilde{\mathsf{V}}_\alpha$, $\widetilde{\mathsf{E}}_{\alpha\beta}$, $\widetilde{\mathsf{F}}_f$ can be determined in exactly the same way as in the $\DT$ case with the following replacements. Instead of (<ref>) we now take

Z_\alpha := \tr_{\O_{Z_\alpha}} + \tr_{B_\alpha}

where $\tr_{B_\alpha}$ is the $T$-representation of the $\PT_1$ box configuration corresponding to $I^\mdot|_{U_\alpha}$ (Definition <ref>). In contrast to the $\PT_0$ case, the $T$-representation $\tr_{B_\alpha}$ can now be infinite. Moreover, the edge terms need to be modified as well. Let $\alpha\beta \in E(X)$. Suppose we label our coordinates such that $U_{\alpha\beta} = \C^* \times \C^3$. From Proposition <ref>, it follows that there exists a $(\C^*)^3$-equivariant $\PT$ pair $J^\mdot$ on $\C^3$ such that

I^\mdot|_{U_{\alpha\beta}} \cong J^\mdot [x_1^{-1},x_1].

Therefore it determines a box configuration $B_{\alpha\beta}$ as defined in <cit.>. Then instead of $Z_{\alpha\beta} = \tr_{\O_{Z_{\alpha\beta}}}$, we now take

Z_{\alpha\beta} := \tr_{\O_{Z_{\alpha\beta}}} + \tr_{B_{\alpha\beta}}.

Recall that a box configuration $B_{\alpha\beta}$ may also have moduli (labelled by $\PP^1$'s, <cit.>).
The face term needs no modification. With the above replacements in mind, we have now defined $\mathsf{V}_\alpha$, $\mathsf{E}_{\alpha\beta}$, $\mathsf{F}_f$, $\widetilde{\mathsf{V}}_\alpha$, $\widetilde{\mathsf{E}}_{\alpha\beta}$, $\widetilde{\mathsf{F}}_f$ and they are again Laurent polynomials.[It is also instructive to note that $\mathsf{B}_{\alpha\beta}$ in Definition <ref> is identical to minus the PT vertex for toric 3-folds studied in [129].]
Under the assumption $\curP_v^{(1)}(X)^{T_X} = \curP_v^{(1)}(X)^{T}$ is 0-dimensional reduced, we conclude that Proposition <ref> holds in the $\PT_1$ case.

§.§ Vertex dtpt0 correspondence

The easiest $\DT$ topological vertex is for $\boldsymbol{\mu}$ all empty. In this case, the $\DT$ topological vertex is a sum over all finite solid partitions viewed as the fixed points of the Hilbert schemes

\bigcup_{n \geq 0} \Hilb^n(\C^4)^{T_X} = \bigcup_{n \geq 0} \Hilb^n(\C^4)^{T}.

In [94], it is shown that $\Hilb^n(\C^4)$ is globally the the zero locus of an isotropic section of an orthogonal bundle on the smooth non-commutative Hilbert scheme $\mathrm{ncHilb}^n(\C^4)$. Moreover, from this description, one can construct an orientation on $\Hilb^n(\C^4)$ such that the following holds.

There exist orientations on $\Hilb^n(\C^4)$ for which the corresponding signs, induced by the Oh-Thomas localization formula, give

\begin{align*}
\sum_{n=0}^{\infty} \langle \! \langle \O_{\mathbb{C}^4} \rangle \! \rangle_{\C^4,0,0,n}^{\DT} q^n = &\mathsf{V}^{\DT}_{\varnothing\varnothing\varnothing\varnothing}(q) = \Exp\Bigg( \frac{[t_1t_2][t_1t_3][t_2t_3][y]}{[t_1][t_2][t_3][t_4][y^{\frac{1}{2}}q][y^{\frac{1}{2}}q^{-1}]}\Bigg).
\end{align*}

The second equality was conjectured in physics by Nekrasov [119] and conjecturally extended to punctual Quot schemes[In physics parlance: “the coloured case”] of $\O_{\C^4}^{\oplus r}$ by Nekrasov-Piazzalunga [121], who also proposed the formula for the signs (which we discuss in Section <ref>). We refer to the second equality as the Magnificent Four formula. Let us explain the notation on the right hand side. For any formal variable $x$, we set

[x] := x^{\frac{1}{2}} - x^{-\frac{1}{2}}.

Furthermore, for any formal power series $f(p_1, \ldots, p_s; q)$ in $\Q(p_1, \ldots, p_r)[\![q]\!]$ with $f(p_1, \ldots, p_s,0) = 0$,
its plethystic exponential is defined by

\begin{align*}
\Exp(f(p_1, \ldots, p_s;q)) &:= \exp\Big( \sum_{n=1}^{\infty} \frac{1}{n} f(p_1^n, \ldots, p_s^n;q^n) \Big).
\end{align*}

In the theorem, before taking the plethystic exponential, the Exponent is written as a rational function in $q$ and expanded as a formal power series in $q$. For $\mathsf{V}^{\DT}_{\varnothing\varnothing\varnothing\varnothing}(q)$, we always fix the signs as in Theorem <ref>.

We now formulate the main conjecture of this section.

For any plane partitions $\boldsymbol{\mu} = \{\mu_a\}_{a=1}^{4}$ with no $\PT_0$ moduli (Definition <ref>), there exists a choice of $\pm \sqrt{\cdot}$ assigned to the $(\C^*)^4$-fixed points such that[Note that the expectation at the end of Remark <ref> is part of the conjecture: for $\boldsymbol{\mu}$ with no $\PT_0$ moduli and for all $B \in \mathcal{B}_{\boldsymbol{\mu}}^{(0)}$, the $\PT_0$ vertex $\mathsf{V}_{I_B\udot}$ has no positive $T$-fixed term.]

\frac{\mathsf{V}^{\DT}_{\mu_1\mu_2\mu_3\mu_4}(q)}{\mathsf{V}^{\DT}_{\varnothing\varnothing\varnothing\varnothing}(q)} = \mathsf{V}^{\PT_0}_{\mu_1\mu_2\mu_3\mu_4}(q).

For $\boldsymbol{\mu}$ all finite (the curve case), the right hand side should be viewed as the $\PT$ vertex [35, 107]. In this case, the above was first conjectured in [35]. Before we provide computational evidence for this conjecture, we discuss some consequences.

Let $\boldsymbol{\lambda} = \{\lambda_{ab}\}_{1 \leq a < b \leq 4}$ be finite partitions and $\boldsymbol{\mu}^{\boldsymbol{\lambda}} = \{\boldsymbol{\mu}^{\boldsymbol{\lambda}}_a\}_{a=1}^{4}$ be the minimal plane partitions compatible with $\boldsymbol{\lambda}$ (<ref>). Then there are no $\PT_0$ moduli (Corollary <ref>). Furthermore, by Proposition <ref>, any $\PT_0$ pair is a $\PT_1$ pair with cokernel $Q$ of length

\ell(Q) \leq \ell(\ext_{\C^4}^3(\O_{Z_{\boldsymbol{\lambda}}}, \O_{\C^4} )).

\mathsf{V}^{\PT_0}_{\mu_1^{\boldsymbol{\lambda}}\mu_2^{\boldsymbol{\lambda}}\mu_3^{\boldsymbol{\lambda}}\mu_4^{\boldsymbol{\lambda}}}(q)

is a Laurent polynomial in $q$ with at most $\ell(\ext_{\C^4}^3(\O_{Z_{\boldsymbol{\lambda}}}, \O_{\C^4} ))+1$ terms. Conjecture <ref> implies that $\mathsf{V}^{\DT}_{\mu_1^{\boldsymbol{\lambda}}\mu_2^{\boldsymbol{\lambda}}\mu_3^{\boldsymbol{\lambda}}\mu_4^{\boldsymbol{\lambda}}}(q)/\mathsf{V}^{\DT}_{\varnothing\varnothing\varnothing\varnothing}(q)$ is equal to this Laurent polynomial.

In the following examples, $Z_{\boldsymbol{\lambda}} $ is not Cohen-Macaulay and we used Section <ref> to determine the $\PT_0$ topological vertex.

* For $\lambda_{12} = \lambda_{34} = 1$, and $\lambda_{ab} = \varnothing$ otherwise, we have

\sfV^{\PT_0}_{\mu_1^{\boldsymbol{\lambda}}\mu_2^{\boldsymbol{\lambda}}\mu_3^{\boldsymbol{\lambda}}\mu_4^{\boldsymbol{\lambda}}}(q) = \frac{[t_1t_3][t_2t_3]}{[y]} q^{-1}  + [t_1t_2].

* For $\lambda_{12} = 1+t_4$, $\lambda_{13} = \lambda_{34} = 1$, and $\lambda_{ab} = \varnothing$ otherwise, we have

$$ \sfV^{\PT_0}_{\mu_1^{\boldsymbol{\lambda}}\mu_2^{\boldsymbol{\lambda}}\mu_3^{\boldsymbol{\lambda}}\mu_4^{\boldsymbol{\lambda}}}(q) = \frac{[t_1t_3][t_2t_3][t_1 t_2^2 t_3^2]}{[t_1t_2t_3 y]} q^{-1} + [t_1t_2][t_2t_3].

* For $\lambda_{12} = 1+t_3$, $\lambda_{34} = 1$, and $\lambda_{ab} = \varnothing$ otherwise, we have

\sfV^{\PT_0}_{\mu_1^{\boldsymbol{\lambda}}\mu_2^{\boldsymbol{\lambda}}\mu_3^{\boldsymbol{\lambda}}\mu_4^{\boldsymbol{\lambda}}}(q) = \frac{[t_1t_3][t_2t_3][t_1t_3^2][t_2t_3^2]}{[y][y t_3^{-1}]} q^{-2} + \frac{[t_1t_2][t_1t_3][t_2t_3][t_3^2]}{[y][t_3]} q^{-1} + [t_1t_2][t_1t_2t_3^{-1}].

The fact that $y$ can appear in the denominator is caused by the redistribution. The negative powers in $q$ come from the renormalized volumes (Definition <ref>).

Using the vertex formalism of Section <ref>, we verified Conjecture <ref> in several cases. In the following proposition, the “bar” means that we normalized the generating series by its leading term (which is the same for $\DT/\PT_0$), so the generating series starts with 1.

There exist choices of $\pm \sqrt{\cdot}$ such that

\overline{\mathsf{V}}^{\DT}_{\mu_1\mu_2\mu_3\mu_4}(q) = \overline{\mathsf{V}}^{\PT_0}_{\mu_1\mu_2\mu_3\mu_4}(q) \, \mathsf{V}^{\DT}_{\varnothing\varnothing\varnothing\varnothing}(q)

holds modulo $q^N$ in the following cases:

$\mathrm{(1)}$ $\lambda_{12} = 1$, $\lambda_{ab} = \varnothing$ otherwise, $\boldsymbol{\mu} = \boldsymbol{\mu}^{\boldsymbol{\lambda}}$, and $N=4$.

$\mathrm{(2)}$ $\lambda_{12} = 1+t_3$, $\lambda_{ab} = \varnothing$ otherwise, $\boldsymbol{\mu} = \boldsymbol{\mu}^{\boldsymbol{\lambda}}$, and $N=3$.

$\mathrm{(3)}$ $\lambda_{12} = 1+t_3+t_4$, $\lambda_{ab} = \varnothing$ otherwise, $\boldsymbol{\mu} = \boldsymbol{\mu}^{\boldsymbol{\lambda}}$, and $N=3$.

$\mathrm{(4)}$ $\lambda_{12} = \lambda_{13} = 1$, $\lambda_{ab} = \varnothing$ otherwise, $\boldsymbol{\mu} = \boldsymbol{\mu}^{\boldsymbol{\lambda}}$, and $N=3$.

$\mathrm{(5)}$ $\lambda_{12} = \lambda_{34} = 1$, $\lambda_{ab} = \varnothing$ otherwise,  $\boldsymbol{\mu} = \boldsymbol{\mu}^{\boldsymbol{\lambda}}$, and $N=3$.

$\mathrm{(6)}$ $\lambda_{12} = \lambda_{13} = \lambda_{23} = 1$, $\lambda_{ab} = \varnothing$ otherwise, $\boldsymbol{\mu} = \boldsymbol{\mu}^{\boldsymbol{\lambda}}$, and $N=3$.

$\mathrm{(7)}$ $\lambda_{ab} = 1$ for all $a<b$, $\boldsymbol{\mu} = \boldsymbol{\mu}^{\boldsymbol{\lambda}}$, and $N=3$.

$\mathrm{(8)}$ $\lambda_{12} = \lambda_{34} =\lambda_{23}  = 1$, $\lambda_{ab} = \varnothing$ otherwise, $\boldsymbol{\mu} = \boldsymbol{\mu}^{\boldsymbol{\lambda}}$, and $N=3$.

$\mathrm{(9)}$ $\lambda_{12} = 1+t_4$, $\lambda_{13} = \lambda_{34} = 1, \lambda_{ab} = \varnothing$ otherwise,  $\boldsymbol{\mu} = \boldsymbol{\mu}^{\boldsymbol{\lambda}}$, and $N=3$.

$\mathrm{(10)}$ $\lambda_{12} = 1+t_3$, $\lambda_{34} = 1, \lambda_{ab} = \varnothing$ otherwise,  $\boldsymbol{\mu} = \boldsymbol{\mu}^{\boldsymbol{\lambda}}$, and $N=3$.

$\mathrm{(11)}$ $\mu_1 = \frac{1}{1-t_2}+t_3$, $\mu_2 = \frac{1}{1-t_1}$, $\mu_a = \varnothing$ otherwise, and $N=4$.

$\mathrm{(12)}$ $\mu_1 = \frac{1}{1-t_2}+t_3+t_2 t_3$, $\mu_2 = \frac{1}{1-t_1}$, $\mu_a = \varnothing$ otherwise, and $N=3$.

$\mathrm{(13)}$ $\mu_1 = \frac{1}{1-t_2}+ t_3+t_3^2$, $\mu_2 = \frac{1}{1-t_1}$, $\mu_a = \varnothing$ otherwise, and $N=3$.

$\mathrm{(14)}$ $\mu_1 = \frac{1}{1-t_2}$, $\mu_2 = \frac{1}{1-t_1}$, $\mu_3 = 1$, $\mu_4 = \varnothing$, and $N=3$.

$\mathrm{(15)}$ $\mu_1 = \frac{1}{1-t_2} + t_3$, $\mu_2 = \frac{1}{1-t_1} + t_4$, $\mu_3=1$, $\mu_4 = \varnothing$, and $N=2$.

In this list, the number of fixed points on the $\DT$ side can be rather large. In case (1), for 3 embedded points we have 19 torus fixed points. In case (11), for 3 embedded points we have 33 torus fixed points.

Since face and edge terms coincide for $\DT$ and $\PT_0$ theory, Proposition <ref>, Remark <ref>, and Conjecture <ref> imply the global $T$-equivariant $K$-theoretic $\DT$–$\PT_0$ correspondence (Conjecture <ref>):

Let $X$ be a toric Calabi-Yau 4-fold and let $L$ be a $T_X$-equivariant line bundle on $X$. Let $\gamma \in H_c^4(X,\Q)$ and $\beta \in H_c^6(X,\Q)$ such that $\curP_{v}^{(0)}(X)^{T_X} = \curP_{v}^{(0)}(X)^{T}$ is empty or 0-dimensional and reduced for $v = (0,0,\gamma,\beta,n-\gamma \cdot \td_2(X))$ and all $n \in \Z$. Suppose Conjecture <ref> holds. Then there are choices of $\pm \sqrt{\cdot}$ such that

\frac{\sum_{n} \langle \! \langle L  \rangle \! \rangle_{X,\gamma,\beta,n}^{\DT} q^n}{\sum_{n} \langle \! \langle L  \rangle \! \rangle_{X,0,0,n}^{\DT} q^n} = \sum_{n} \langle \! \langle L  \rangle \! \rangle_{X,\gamma,\beta,n}^{\PT_0} q^n.

§.§ Cohomological limits

Let $X$ be a toric Calabi-Yau 4-fold and $L$ a $T_X$-equivariant line bundle on $X$. Let $\curly P := \curly P_v^{(q)}(X)$ for any $v$ as in (<ref>) and $q \in \{-1,0,1\}$. Besides the $K$-theoretic invariants $\langle\!\langle L \rangle\!\rangle_{X,v}^{\PT_q}$,
we are also interested in the following “tautological” and “cohomological” invariants

\begin{align*}
\langle L \rangle_{X,v}^{\PT_q} \in \frac{\Q(s_1,s_2,s_3,s_4,m)}{(s_1+s_2+s_3+s_4)}, \quad \int_{[\curly P]^{\vir}} 1 \in \frac{\Q(s_1,s_2,s_3,s_4)}{(s_1+s_2+s_3+s_4)},
\end{align*}

where the former were defined in Section <ref>, and we recall that $c_1^T(t_i) = s_i$ and $c_1^{\C^*}(y) = m$ are the cohomological equivariant parameters.
Similar to the $K$-theoretic case, these invariants can be calculated using the vertex, edge, face terms $\widetilde{\mathsf{V}}_{\alpha}$, $\widetilde{\mathsf{E}}_{\alpha\beta}$, $\widetilde{\mathsf{F}}_{f}$ of Section <ref>. The only differences are the following:

* tautological invariants: replace

[t_1^{i_1} t_2^{i_2} t_3^{i_3} t_4^{i_4} y^{i_5}] := t_1^{\frac{i_1}{2}} t_2^{\frac{i_2}{2}} t_3^{\frac{i_3}{2}} t_4^{\frac{i_4}{2}} y^{\frac{i_5}{2}} - t_1^{-\frac{i_1}{2}} t_2^{-\frac{i_2}{2}} t_3^{-\frac{i_3}{2}} t_4^{-\frac{i_4}{2}} y^{-\frac{i_5}{2}}$$

$$e(t_1^{i_1} t_2^{i_2} t_3^{i_3} t_4^{i_4} y^{i_5}) := i_1 s_1+ i_2 s_2+ i_3 s_3+i_4 s_4+ i_5 m,$$

* cohomological invariants: the same as the previous case but omit all terms involving $y$ from the vertex, edge, and face terms of Section <ref>, in other words, use $\mathsf{V}_{\alpha}$, $\mathsf{E}_{\alpha\beta}$, $\mathsf{F}_{f}$ instead of $\widetilde{\mathsf{V}}_{\alpha}$, $\widetilde{\mathsf{E}}_{\alpha\beta}$, $\widetilde{\mathsf{F}}_{f}$.

For the vertex formalism in the curve case we have <cit.>

\rk(\widetilde{\mathsf{V}}_{\alpha}) = \rk(\widetilde{\mathsf{E}}_{\alpha\beta}) = 0.

This is no longer the case for surfaces. Then $\widetilde{\mathsf{V}}_{\alpha}$, $\widetilde{\mathsf{E}}_{\alpha\beta}$, $\widetilde{\mathsf{F}}_{f}$ can have non-zero rank (e.g. see Example <ref>).
However, we have the following weaker result, which we formulate for the case $q=-1$, but it also holds for $q=0,1$.

Let $Z, Z' \subset \C^4$ be 2-dimensional $(\C^*)^4$-invariant closed subschemes with the same underlying pure 2-dimensional sheaf, i.e., $Z_1 = Z_1'$.  Then

\rk(\widetilde{\mathsf{V}}_{Z}) = \rk(\widetilde{\mathsf{V}}_{Z'}).

Consider the definition of $\widetilde{\mathsf{V}}_Z$ (Definition <ref>) and make the substitutions of Definition <ref> (where we write $Z = Z_\alpha$, $W = W_\alpha$, $W_i := W_{\alpha\beta_i}$, $W_{ij} = Z_{\alpha\beta_i\beta_j}$). Taking $t_1=\cdots=t_4=1$, an explicit calculation gives

\begin{equation} \label{eqn:rkVtilde}
\widetilde{\mathsf{V}}_Z \Big|_{t_1=\cdots=t_4=1} = \Big( W + \overline{W} - y^{-1} W - y \overline{W} - \sum_{i,j,k,l \in \{1,2,3,4\} \atop \textrm{distinct } i<j, k<l} W_{ij} \overline{W}_{kl} \Big)\Big|_{t_1=\cdots=t_4=1},
\end{equation}

which is independent of $W_i$. We see that for fixed 2-dimensional asymptotics $W_{ij}$, adding or removing embedded points or curves keeps the rank of $\widetilde{\mathsf{V}}_Z$ unchanged as claimed.

We formulate the following result for $q=-1$, but it holds analogously for $q=0,1$.

Let $Z, Z' \subset \C^4$ be 2-dimensional $(\C^*)^4$-invariant closed subschemes with the same underlying pure 2-dimensional sheaf, i.e., $Z_1 = Z_1'$. Suppose $\widetilde{V}_{Z'}$ has no $T$-fixed term. Then

\begin{align*}
\lim_{b \mapsto 0} \frac{[-\widetilde{\mathsf{V}}_{Z}]}{[-\widetilde{\mathsf{V}}_{Z'}]} \Big|_{t_i = e^{b s_i}, y = e^{b m}} &= \frac{e(-\widetilde{\mathsf{V}}_{Z})}{e(-\widetilde{\mathsf{V}}_{Z'})}, \\
\lim_{b \mapsto 0 \atop m \to \infty}  \frac{[-\widetilde{\mathsf{V}}_{Z}] / (-m^2)^{|Z|}}{[-\widetilde{\mathsf{V}}_{Z'}]/ (-m^2)^{|Z'|}} \Big|_{t_i = e^{b s_i}, y = e^{b m}} &= \frac{e(-\mathsf{V}_{Z})}{e(-\mathsf{V}_{Z'})},
\end{align*}

where $|Z|$, $|Z'|$ denote the renormalized volumes of $Z,Z'$.

Using multi-index notation $\eta = (t_1,t_2,t_3,t_4,y)$, for any rank 0 finite complex $E = \sum_v \eta^v - \sum_w \eta^w$, we have

[E] = \frac{\prod_v [\tau^v]}{\prod_w [\tau^w]} \Big|_{t_i = e^{b s_i}, y = e^{b m}} = \frac{\prod_v (v_1 s_1+v_2s_2+v_3s_3+v_4s_4 + v_5 m + O(b))}{\prod_w (w_1s_1+w_2s_2+w_3s_3+w_4s_4 + w_5 m + O(b))},

and therefore

\lim_{b \mapsto 0} \widehat{\Lambda}\udot E^* = \lim_{b \mapsto 0} [E] = e(E).

The first statement then follows from Lemma <ref>.

Now we turn to the second statement. Let $Z \subset \C^4$ be the 2-dimensional subscheme corresponding to $\pi$ and use the notation of Definition <ref> (with $Z = Z_\alpha$, $W = W_\alpha$). The only term involving $y$ in $[-\widetilde{\mathsf{V}}_{Z}]/(-m^2)^{|Z| }$ is $[y^{-1} W + y \overline{W}]/(-m^2)^{|Z| }$, where $|Z| = \rk(W)$ is the renormalized volume (Definition <ref>). We use multi-index notation $\tau = (t_1,t_2,t_3,t_4)$. Writing $W = \sum_v \tau^v - \sum_w \tau^w$ and taking the limit $b \to 0$, it contributes

\begin{align*}
\frac{[y^{-1} W + y \overline{W}]}{(-m^2)^{|Z| }} &=  (-m^2)^{-(\sum_v 1 -\sum_w 1) } \frac{\prod_v (-1)(v_1s_1+v_2s_2+v_3s_3+v_4s_4 - m)^2}{\prod_w (-1)(w_1 s_1+w_2s_2+w_3s_3+w_4s_4 - m)^2} \\
&=  \frac{\prod_v (\frac{v_1}{m}s_1+\frac{v_2}{m}s_2+\frac{v_3}{m}s_3+\frac{v_4}{m}s_4 -1)^2}{\prod_w (\frac{w_1}{m} s_1+\frac{w_2}{m}s_2+\frac{w_3}{m}s_3+\frac{w_4}{m}s_4 - 1)^2}.
\end{align*}

This term goes to 1 as $m \to \infty$.

For $q=-1,0$, we accordingly define the tautological and cohomological topological vertex

\mathsf{V}^{\PT_q, \textrm{taut}}_{\mu_1\mu_2\mu_3\mu_4}(q), \quad \mathsf{V}^{\PT_q, \textrm{coho}}_{\mu_1\mu_2\mu_3\mu_4}(q).

In the following corollary, the “bar” means that we normalized the generating series by its leading term, so the generating series starts with 1.

Let ${\boldsymbol{\mu}}$ be plane partitions corresponding to subschemes of dimension $\leq 1$ of $\C^3$. Then for $q=-1,0$ we have

\begin{align*}
\lim_{b \mapsto 0} \overline{\mathsf{V}}^{\PT_q, \mathrm{taut}}_{\mu_1\mu_2\mu_3\mu_4}(q) \Big|_{t_i = e^{b s_i}, y = e^{b m}} &= \overline{\mathsf{V}}^{\PT_q}_{\mu_1\mu_2\mu_3\mu_4}(q), \\
\lim_{b \mapsto 0 \atop m \to \infty} \overline{\mathsf{V}}^{\PT_q, \mathrm{coho}}_{\mu_1\mu_2\mu_3\mu_4}(q') \Big|_{t_i = e^{b s_i}, y = e^{b m}, q' = - \frac{q}{m}} &= \overline{\mathsf{V}}^{\PT_q}_{\mu_1\mu_2\mu_3\mu_4}(q).
\end{align*}

In particular, the $K$-theoretic vertex $\DT$–$\PT_0$ correspondence (Conjecture <ref>) implies a tautological and cohomological vertex $\DT$–$\PT_0$ correspondence (for the same choice of signs). Consequently, we also obtain global analogs of Corollary <ref> for tautological and numerical insertions.

§.§ Local 3-folds

For $Y$ a smooth quasi-projective toric 3-fold, we consider the toric Calabi-Yau 4-fold

p : X = \mathrm{Tot}(K_Y) \to Y,

i.e., the total space of the canonical line bundle on $Y$. Denote by $T_Y$ the torus acting on $Y$ and by $\C^*$ the torus acting on the fibres, then $T_X = T_Y \times \C^*$. The torus $T \leq T_X$ preserving the Calabi-Yau volume form is isomorphic to $T_Y$. Let $v = (0,0,\gamma,\beta,n-\gamma \cdot \td_2(X)) \in H_c^*(X,\Q)$ and $\curly P := \curly P_v^{(q)}(X)$. In Section <ref>, assuming  $\curly P^T$ is 0-dimensional and reduced, we deduced

\begin{equation} \label{eqn:invKYinitial}
\langle\!\langle L \rangle\!\rangle_{X,v}^{\PT_q} = \sum_{I\udot = [\O_X \to F] \in \curly P^T} (-1)^{\sigma_{I\udot}} \frac{\widehat{\Lambda}\udot (R\Gamma(X,F \otimes L) \otimes y^{-1})}{\widehat{\Lambda}\udot (R\Gamma(Y,p_*F) - R\Hom_Y(p_* F,p_*F))^\vee},
\end{equation}

for any $T_X$-equivariant line bundle $L$ on $X$ and choice of sign attached to each $T$-fixed point. Recall that in Section <ref>, we focused on the case $L = \O_X(Y)$, but now $L$ is arbitrary.

Let $I\udot = [\O_X \to F] \in \curly P^{T_X}$. In this section, we develop a vertex formalism for the calculation of

R\Gamma(Y, p_* F) - R\Hom_Y(p_* F, p_* F))

and comment on the signs $(-1)^{\sigma_{I\udot}}$.

We start with the following definitions. Note that for $X = \mathrm{Tot}(K_Y)$, we have $V(X) = V(Y)$, $E(X) = E(Y)$, $F(X) = F(Y)$, because we only consider compact edges and faces. For notational simplicity, we consider the case $q = -1$, but the cases $q=0,1$ hold similarly (subject to the modifications of Remarks <ref> and <ref>). Let $Z \subset X$ be a closed subscheme of dimension $\leq 2$ with proper support. For $\alpha_1, \ldots, \alpha_N \in V(X)$, such that $U_{\alpha_1 \cdots \alpha_N} \cong (\C^*)^k \times \C^{4-k}$ with $k=0,1,2$, we always label coordinates on $(\C^*)^k \times \C^{4-k}$ in such a way that $Z \cap U_{\alpha_1 \cdots \alpha_N}$ is set theoretically (but not necessarily scheme theoretically!) supported on $Z(x_4)$. We also choose the coordinates such that $T_X$ acts on $U_{\alpha_1 \cdots \alpha_N}$ by the standard torus action (<ref>). In other words, locally, the fibre coordinate of $X = \mathrm{Tot}(K_Y)$ always corresponds to $x_4$. We use the notation of Section <ref>.

We define the halved vertex term

\begin{align*}
\mathsf{v}_{\alpha} &:= Z_\alpha + \frac{P_{123}}{t_1t_2t_3} Z_\alpha \overline{Z}_{\alpha} + \sum_{i=1}^{3} \frac{\mathsf{a}_{\alpha\beta_i}}{1-t_i} + \sum_{1 \leq i<j \leq 3} \frac{\mathsf{a}_{\alpha\beta_i\beta_j}}{(1-t_i)(1-t_j)},
\end{align*}

where, for any distinct $i,j,k \in \{1,2,3\}$, we set

\begin{align*}
\mathsf{a}_{\alpha\beta_i} &:= - Z_{\alpha\beta_i}  + \frac{P_{jk}}{t_{j}t_{k}}  Z_{\alpha\beta_i}   \overline{Z}_{\alpha\beta_i}, \\
\mathsf{a}_{\alpha\beta_i\beta_{j}} &:= Z_{\alpha\beta_i\beta_j} + \frac{P_{k}}{t_{k}}  Z_{\alpha\beta_i\beta_j}   \overline{Z}_{\alpha\beta_i\beta_j}.
\end{align*}

When we want to stress the dependence of these expressions on $Z_\alpha, Z_{\alpha\beta}, Z_{\alpha\beta\gamma}$, we write $\mathsf{v}_{Z_\alpha}, \mathsf{a}_{Z_{\alpha\beta}}, \mathsf{a}_{Z_{\alpha\beta\gamma}}$.

For any distinct $i,j,k \in \{1,2,3\}$, we define the halved edge term by

\begin{align*}
\mathsf{e}_{\alpha\beta_i} &:= \frac{t_i^{-1}  \mathsf{b}_{\alpha\beta_i}(t_j,t_k,t_4) }{1-t_i^{-1}} -  \frac{\mathsf{b}_{\alpha\beta_i}(t_i^{-m_{\alpha\beta_i}} t_j,t_i^{-m_{\alpha\beta_i}'} t_k,t_i^{-m_{\alpha\beta_i}''}t_4) }{1-t_i^{-1}},
\end{align*}

\begin{align*}
\mathsf{b}_{\alpha\beta_i} &:= \mathsf{a}_{\alpha\beta_i}  + \sum_{a \in \{i,j,k\} \setminus \{i\}} \frac{\mathsf{a}_{\alpha\beta_i\beta_a} }{1-t_a} .
\end{align*}

When we want to indicate the dependence of these expressions on $Z_\alpha, Z_{\alpha\beta}$, we write $\mathsf{e}_{Z_{\alpha\beta}}, \mathsf{b}_{Z_{\alpha\beta}}$.

Fix $f \in F(X) = F(Y)$. Denote by $Z_f \subset X$ the unique $T_X$-fixed 2-dimensional closed subscheme set-theoretically supported on $S_f \subset Y \subset X$ such that

Z_f \cap U_{\alpha_1\alpha_2\alpha_3} = Z \cap U_{\alpha_1\alpha_2\alpha_3}

for any distinct $\alpha_1, \alpha_2, \alpha_3 \in f$.  We define the halved face term by

\mathsf{f}_f := \tr_{R\Gamma(Y,p_* \O_{Z_f})-R\Hom_Y(p_* \O_{Z_f}, p_*\O_{Z_f})}.

When we want to stress the dependence of $\mathsf{f}_f $ on $Z_f$ we write $\mathsf{f}_{Z_f}$.

For $\dim Z = 0$, $\mathsf{v}_{\alpha}$ reduces to the halved vertex term first introduced by Nekrasov-Piazzalunga [119, 121]. For $\dim Z = 1$, $\mathsf{v}_{\alpha}$ and $\mathsf{e}_{\alpha\beta}$ reduce to the halved vertex and edge terms studied in [35]. The following lemma justifies the use of the adjective “halved”.

For all $\alpha \in V(X)$, $\alpha\beta \in E(X)$, $f \in F(X)$, the following equations hold modulo the relation $t_1t_2t_3t_4=1$

\begin{align*}
\mathsf{v}_{\alpha} + \overline{\mathsf{v}}_{\alpha} = \mathsf{V}_\alpha, \quad \mathsf{e}_{\alpha\beta} + \overline{\mathsf{e}}_{\alpha\beta} = \mathsf{E}_{\alpha\beta}, \quad \mathsf{f}_{f} + \overline{\mathsf{f}}_{f} = \mathsf{F}_f.
\end{align*}

The first two equations follow from a straight-forward calculation involving identities such as

\frac{P_{123}}{t_1t_2t_3} + \frac{\overline{P}_{123}}{t_1^{-1}t_2^{-1}t_3^{-1}} = - \frac{P_{1234}}{t_1t_2t_3t_4},

which uses the relation  $t_1t_2t_3t_4=1$. The third identity follows from (<ref>).

The following proposition is proved similarly to Theorem <ref>.

Consider $p  : X = \mathrm{Tot}(K_Y) \to Y$, where $Y$ is a smooth quasi-projective toric 3-fold. Then for any $T_X$-fixed closed subscheme $Z \subset X$ with proper support and dimension $\leq 2$, we have

R\Gamma(Y, p_* \O_Z) - R\Hom_Y(p_* \O_Z, p_* \O_Z) = \sum_{\alpha \in V(X)} \mathsf{v}_{\alpha} + \sum_{\alpha\beta \in E(X)} \mathsf{e}_{\alpha\beta} + \sum_{f \in F(X)} \mathsf{f}_{f},

where $\mathsf{v}_{\alpha}, \mathsf{e}_{\alpha\beta}, \mathsf{f}_{f}$ are Laurent polynomials.

Denote by $\{U_\alpha\}_{\alpha \in V(Y)}$ the open cover of $Y$ by maximal $T_Y$-invariant affine open subsets. Choose a total order on the indices $\alpha \in V(Y)$. Using Čech cohomology and the local-to-global spectral sequence as in the beginning of Section <ref>, we obtain

\begin{align*}
&R\Gamma(Y, p_* \O_Z) - R\Hom_Y(p_* \O_Z,p_* \O_Z) = \\
&\quad\quad\quad\quad \sum_{\alpha \in V(Y)} \Big( R\Gamma(U_\alpha, p_* \O_Z) - R\Hom_{U_{\alpha}}(p_* \O_Z,p_* \O_Z) \Big) \\
&\quad\quad\quad\quad- \sum_{\alpha < \beta \in V(Y)} \Big( R\Gamma(U_{\alpha\beta}, p_* \O_Z) - R\Hom_{U_{\alpha\beta}}(p_* \O_Z,p_* \O_Z) \Big) \\
&\quad\quad\quad\quad+ \sum_{\alpha < \beta < \gamma \in V(Y)} \Big( R\Gamma(U_{\alpha\beta\gamma}, p_* \O_Z) - R\Hom_{U_{\alpha\beta\gamma}}(p_* \O_Z,p_* \O_Z) \Big) + \cdots,
\end{align*}

where we suppress restrictions to $U_\alpha$, $U_{\alpha\beta}$, $U_{\alpha\beta\gamma}$, $\ldots$

There exist finitely many $T_Y$-invariant closed subschemes

Y \supset Z(0) \supset Z(1) \supset \cdots

such that

p_* \O_Z = \bigoplus_{i \geq 0} \O_{Z(i)} \otimes t_4^i.

We denote their restrictions by $Z_\alpha(i) = Z(i) \cap U_\alpha$, $Z_{\alpha\beta}(i) = Z(i) \cap U_{\alpha\beta}$, $Z_{\alpha\beta\gamma}(i) = Z(i) \cap U_{\alpha\beta\gamma}$, etc. Using Taylor resolutions [151], it is easy to see that

\begin{align*}
&R\Gamma(U_\alpha, p_* \O_Z) - R\Hom_{U_{\alpha}}(p_* \O_Z,p_* \O_Z) \\
&= \sum_{i \geq 0} R\Gamma(U_\alpha,\O_{Z_\alpha(i)})  t_4^i - \sum_{i,j \geq 0} R\Hom_{U_\alpha}(\O_{Z_{\alpha}(i)}, \O_{Z_{\alpha}(j)}) t_4^{j-i} \\
&=\sum_{i \geq 0} Z_{\alpha}(i) t_4^i - \sum_{i,j \geq 0} \frac{P_{123}}{t_1t_2t_3} \overline{Z_\alpha(i)} Z_\alpha(j) t_4^{j-i} = Z_\alpha + \frac{P_{123}}{t_1t_2t_3} Z_\alpha \overline{Z}_\alpha.
\end{align*}

Let $\alpha\beta \in E(Y)$. Let $\alpha_1,\ldots, \alpha_N \in V(Y)$ such that $\alpha_1, \ldots, \alpha_N \in f$ for some $f \in F(Y)$, where for $N=2$ we assume $\alpha_1\alpha_2$ is not an edge. Then $U_{\alpha\beta} \cong \C^* \times \C^2$ and $U_{\alpha_1\cdots\alpha_N} \cong (\C^*)^2 \times \C$, and we similarly obtain

\begin{align*}
&R\Gamma(U_{\alpha\beta}, p_* \O_Z) - R\Hom_{U_{\alpha\beta}}(p_* \O_Z,p_* \O_Z) = \delta(t_1)\Big(Z_{\alpha\beta}  - \frac{P_{23}}{t_{2}t_{3}}  Z_{\alpha\beta}   \overline{Z}_{\alpha\beta}\Big), \\
&R\Gamma(U_{\alpha_1\cdots \alpha_N}, p_* \O_Z) - R\Hom_{U_{\alpha_1\cdots \alpha_N}}(p_* \O_Z,p_* \O_Z) = \\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \delta(t_1)\delta(t_2)\Big(  Z_{\alpha_1\cdots \alpha_N} + \frac{P_{3}}{t_{3}}  Z_{\alpha_1\cdots \alpha_N}   \overline{Z}_{\alpha_1\cdots \alpha_N}  \Big).
\end{align*}

For the remainder of the proof, we can follow the proof of Theorem <ref>.

Similar to Remark <ref>, we can obtain a nice expression for $\mathsf{f}_f$ and $f \in F(Y)$. For a 2-dimensional $T_X$-fixed closed subscheme $Z \subset X$ with proper support, denote by $\lambda_f$ the finite partition corresponding to the subscheme $Z_f$ from Definition <ref>. Let

\[
\mathsf{a}_f(t_3,t_4) := Z_{\lambda_f} + \frac{P_{3}}{t_3} Z_{\lambda_f} \overline{Z}_{\lambda_f} \in \mathbb{Z}[t_3^{\pm 1}, t_4^{\pm 1}],
\]

where $Z_{\lambda_f} = \sum_{(i,j) \in \lambda_f} t_3^{i} t_4^{j}$. Then

\begin{equation} \label{eqn:exprff}
\mathsf{f}_f = \sum_{i=1}^{e(S_f)} \frac{\mathsf{a}_f(t_3^{(i)},t_4^{(i)})}{(1-t_1^{(i)})(1-t_2^{(i)})},
\end{equation}

where the right hand side is an element of $\Q[t_1^{\pm 1}, \ldots, t_4^{\pm 1}] \subset \Q(t_1, \ldots, t_4)$.
We observe that, modulo the relation $t_1t_2t_3t_4=1$, it is also clear by direct computation that the sum of (<ref>) and its conjugate equals (<ref>). In fact, (<ref>) provides a “half” in $K$-theory of (<ref>) for any toric Calabi-Yau 4-fold (not necessarily of the form $X = \mathrm{Tot}(K_Y)$).

Next, we incorporate the tautological insertion

\widehat{\Lambda}\udot ( R\pi_{\curly P*}\O_{\cZ} \otimes y^{-1})

into the vertex formalism, which we already carried out  in Section <ref>. We define for all $i,j,k \in \{1,2,3\}$ distinct

\begin{align*}
\widetilde{\mathsf{v}}_{\alpha} := &\mathsf{v}_{\alpha} - y \overline{Z}_\alpha + \sum_{i=1}^{3} \frac{y \overline{Z}_{\alpha\beta_i}}{1-t_i^{-1}} - \sum_{1 \leq i<j \leq 3} \frac{y \overline{Z}_{\alpha\beta_i\beta_j}}{(1-t_i^{-1})(1-t_j^{-1})}, \\
\widetilde{\mathsf{e}}_{\alpha\beta_i} := &\mathsf{e}_{\alpha\beta_i} + \frac{t_i}{1-t_i} \Bigg\{ y \overline{Z}_{\alpha\beta_i} - \sum_{a \in \{j,k\}} \frac{y \overline{Z}_{\alpha\beta_i\beta_a}}{1-t_a^{-1}} \Bigg\} \\
&\quad\quad- \frac{1}{1-t_i} \Bigg\{ y \overline{Z}_{\alpha\beta_i} - \sum_{a \in \{j,k\}} \frac{y \overline{Z}_{\alpha\beta_i\beta_a}}{1-t_a^{-1}} \Bigg\}\Bigg|_{(t_j t_i^{-m_{\alpha\beta_i}},t_k t_i^{-m_{\alpha\beta_i}'},t_4 t_i^{-m_{\alpha\beta_i}''})}, \\
\widetilde{\mathsf{f}}_f  := &\mathsf{f}_f -y R\Gamma(X, \O_{Z_f})^\vee.
\end{align*}

Combining (<ref>) and Proposition <ref>, we obtain the following result.

Let $X = \mathrm{Tot}(K_Y)$ for a smooth quasi-projective toric 3-fold $Y$ and fix $v = (0,0,\gamma,\beta,n - \gamma \cdot \td_2(X)) \in H_c^*(X,\Q)$. Let $\curly P^T := \curly P_v^{(-1)}(X)^T$. Then

\[
\langle\!\langle \O_X \rangle\!\rangle_{X,v}^{\DT} = \sum_{Z \in \curly P^T} \pm \Big[-\sum_{\alpha \in V(Y)} \widetilde{\mathsf{v}}_{Z_\alpha} - \sum_{\alpha\beta \in E(Y)} \widetilde{\mathsf{e}}_{Z_{\alpha\beta}} - \sum_{f \in F(Y)} \widetilde{\mathsf{f}}_{Z_f} \Big],
\]

where the right hand side depends on a choice of $\pm$ for each fixed point.

By Lemmas <ref> and <ref>, $\mathsf{v}_\alpha$, $\mathsf{e}_{\alpha\beta}$, $\mathsf{f}_f$ have no positive $T$-fixed part, so we may also write

\begin{align*}
\langle\!\langle \O_X \rangle\!\rangle_{X,v}^{\DT} = \sum_{Z \in \curly P^T} \pm \prod_{\alpha \in V(Y)} [-\widetilde{\mathsf{v}}_{Z_\alpha}] \cdot \prod_{\alpha\beta \in E(Y)} [-\widetilde{\mathsf{e}}_{Z_{\alpha\beta}}] \cdot \prod_{f \in F(Y)}  [-\widetilde{\mathsf{f}}_{Z_f}].
\end{align*}

We stress that the elements of $\curly P^T$ are set theoretically, but not necessarily scheme theoretically, supported on the zero section $Y \subset X$. The invariants $\langle\!\langle L \rangle\!\rangle_{X,v}^{\DT} $, for other $T_X$-equivariant line bundles $L$ on $X$, can also easily be incorporated into the vertex formalism by Remark <ref>.

Let $\boldsymbol{\mu} = \{\mu_{a} \}_{a = 1}^{4}$ be a collection of plane partitions corresponding to subschemes of dimension $\leq 1$ of $\C^3$ such that $Z_{\boldsymbol{\mu}}$ (Definition <ref>) is set theoretically supported on $Z(x_4)$. In particular $\mu_4 = \varnothing$. Then the $\DT$ topological vertex equals (Definition <ref>)

\begin{equation} \label{eqn:DTvertexKY}
\mathsf{V}^{\DT}_{\mu_1\mu_2\mu_3\varnothing}(q) = \sum_{\pi \in \mathcal{P}_{\boldsymbol{\mu}}} \pm  [-\widetilde{\mathsf{v}}_{\pi}] \, q^{|\pi|},
\end{equation}

where $\pm$ indicates that this expression involves a choice of sign for each $\pi \in \mathcal{P}_{\boldsymbol{\mu}}$.

Just like in Remark <ref> and <ref>, the results of this section so far hold similarly for $q=0,1$.

In the remainder of this section, we take $q=-1$ and comment on the signs in (<ref>). It is tempting to think that the “natural” sign of every solid partition in (<ref>) is $+1$. However, neither the Magnificent Four formula (Theorem <ref>) nor the $\DT$–$\PT_0$ vertex correspondence (Conjecture <ref>) hold for this choice of signs.

Suppose $\boldsymbol{\mu}$ are all empty. Then it was first found in physics by Nekrasov-Piazzalunga [119, 121] and proved mathematically by the second-named author and Rennemo [94], that the Magnificent Four formula holds when we assign the sign

\begin{equation} \label{eqn:sign0dim}
(-1)^{|\pi| + |\{(i,i,i,j) \in \pi  \, : \, i<j \}|}
\end{equation}

to each finite solid partition $\pi$. Therefore, (<ref>) provides the correct sign choice when all $\boldsymbol{\mu}$ are empty.

Suppose $\boldsymbol{\mu}$ are finite plane partitions and $\mu_4 = \varnothing$. For all verifications of the $\DT$–$\PT$ (curve) vertex correspondence performed in [35], it was found that, on the $\DT$ side, the following choice of sign for each $\pi \in \mathcal{P}_{{\boldsymbol{\mu}}}$ works

\begin{equation} \label{eqn:sign1dim}
(-1)^{|\pi|} \prod_{w = (i,i,i,j) \in \pi \atop i< j} (-1)^{1  -  |\{\textrm{legs containing } w\}|}.
\end{equation}

When $ \boldsymbol{\mu}$ are all empty, (<ref>) reduces to (<ref>). Conjecturally, on the $\DT$ side, this provides the sign choice for which the $\DT$–$\PT$ (curve) vertex correspondence holds when all $\boldsymbol{\mu}$ are finite plane partitions and $\mu_4 = \varnothing$.

Now let $\boldsymbol{\mu} = \{\mu_{a} \}_{a = 1}^{4}$ be a collection of plane partitions corresponding to subschemes of dimension $\leq 1$ of $\C^3$ such that $Z_{\boldsymbol{\mu}}$ is set theoretically supported on $Z(x_4)$ (in particular $\mu_4 = \varnothing$). Let $\pi \in \mathcal{P}_{\boldsymbol{\mu}}$ and denote the corresponding subscheme by $Z_\pi \subset \C^4$. Denote by $W:=W_\alpha$ the Laurent polynomial in Definition <ref> (it is in fact a polynomial since $q=-1$). Then $|\pi| = \rk(W)$ is the renormalized volume. We associate to $\pi$ the sign

\begin{equation} \label{eqn:sign2dim}
(-1)^{|\pi| + \mu_\pi}, \quad \mu_\pi := \sum_{(i,i,i,j) \in \pi \atop i < j} \frac{1}{(i!)^3 j!} \frac{\partial^{3i+j} W}{\partial t_1^i \partial t_2^i \partial t_3^i \partial t_4^j} \Bigg|_{t_1=t_2=t_3=t_4=0},
\end{equation}

where $\mu_\pi$ is simply the sum of all coefficients of all terms of the form $t_1^i t_2^i t_3^i t_4^j$ with $(i,i,i,j) \in \pi$ and $i<j$. When all $\boldsymbol{\mu}$ are finite plane partitions and $\mu_4 = \varnothing$, we have $\mu_{\pi} = \sum_{w = (i,i,i,j) \in \pi : i<j} (1  -  |\{\textrm{legs containing } w\}|)$ so (<ref>) is a generalization of (<ref>).

For all the verifications of the $\DT$–$\PT_0$ vertex correspondence in Proposition <ref> for which $Z_{\boldsymbol{\mu}}$ is set theoretically supported on $Z(x_4)$, on the $\DT$ side the correct choice of $\pm \sqrt{\cdot}$ is given by (<ref>) with sign (<ref>). We therefore expect that, when $Z_{\boldsymbol{\mu}}$ is set theoretically supported on $Z(x_4)$, the $\DT$ topological vertex with sign choice for which Conjecture <ref> holds is given by the following formula

\mathsf{V}^{\DT}_{\mu_1\mu_2\mu_3\varnothing}(q) = \sum_{\pi} (-1)^{|\pi| + \mu_\pi} [-\widetilde{\mathsf{v}}_{\pi}] \, q^{|\pi|}.

§.§ Virtual Lefschetz principle – Local

In this section, we study the halved vertex and edge terms $\widetilde{\mathsf{v}}_\alpha$, $\widetilde{\mathsf{e}}_{\alpha\beta}$ of the previous section for the specialization $y = t_4 = (t_1t_2t_3)^{-1}$. For a 2-dimensional scheme $Z$ with torsion filtration $T_0(\O_Z) \subset T_1(Z) \subset \O_Z$ we recall that we denote the corresponding subschemes by $Z \supset Z_0 \supset Z_1$ (Section <ref>).

Let $X = \C^4$, $Y = Z(x_4) \cong \C^3$, $S = Z(x_3,x_4) \cong \C^2$. Let $dS \subset Y$ be the $d$ times thickening of $S$ in $Y$.
Let $I^\mdot$ be a $T_X$-equivariant $\PT_q$ pair on $X$ for $q \in \{-1,0\}$ with scheme theoretic support $Z$. We assume $Z_1 = dS$ and $Z_0 \subset Y$. In particular, only the 0-dimensional embedded components of $Z$ may not lie scheme-theoretically inside $Y$.

* Suppose $Z$ does not lie scheme theoretically in $Y$, which can only happen for $q=-1$. Then $\widetilde{\mathsf{v}}_{I^\mdot} |_{y=t_4}$ has negative $T$-fixed term, in particular

[-\widetilde{\mathsf{v}}_{I^\mdot}] \big|_{y = t_4} = 0.

* Suppose $Z \subset Y$. Denote by $I_Y^\mdot = [ \O_Y \to F ]$ the corresponding $\PT_q$ pair on $Y$. Then there exists a unique 1-dimensional $T_Y$-equivariant pair $J^\mdot = [\O_Y \to G]$ on $Y$ such that

I_Y\udot \cong J\udot(-d S),

where $J^\mdot$ is a 1-dimensional $\DT$ pair on $Y$ for $q=-1$ and a 1-dimensional $\PT$ pair on $Y$ for $q=0$. Moreover

[-\widetilde{\mathsf{v}}_{I^\mdot}] \big|_{y = t_4} = [-\mathsf{V}^{\mathrm{3D}}_{J^\mdot}],

where $\mathsf{V}^{\mathrm{3D}}_{J^\mdot}$ denotes the equivariant vertex term of 3-dimensional $\DT$ theory <cit.> resp. $\PT$ theory <cit.>.

Let $q=-1$ and suppose $Z \subsetneq Y$. Before specializing $y = t_4$, $\widetilde{\mathsf{v}}_{I^\mdot}$ has no positive $T$-fixed term (Lemma <ref>), so we only need to consider terms containing $y$. Using the notation of Definition <ref> (with $Z := Z_\alpha$, $W := W_\alpha$, $W_i := W_{\alpha\beta_i}$, $W_{ij} = Z_{\alpha\beta_i\beta_j}$), these terms are
$ -y \overline{W}.$
Note that $W$ is a Laurent polynomial in $t_1,t_2,t_3,t_4$ but can have terms with negative coefficients.
However, since only the 0-dimensional embedded components of $Z$ may lie outside $Y$, any term in $W$ containing $t_4$ has positive coefficient. Moreover, since $Z \subsetneq Y$, one of these terms is $+t_4$. Therefore, after specializing $y=t_4$, we conclude that $\widetilde{\mathsf{v}}_{I^\mdot}$ has a negative $T$-fixed term and $[-\widetilde{\mathsf{v}}_{I\udot}] |_{y = t_4} = 0$.

Next, we assume $Z \subset Y$. From Proposition <ref>, it is clear that there exists a unique 1-dimensional $(\C^*)^3$-equivariant pair $J^\mdot = [\O_Y \to G]$ on $Y$ such that $I\udot \cong J\udot(-dS)$ (see also the proof of Theorem <ref>). Continuing the notation of Definition <ref>, we have

\begin{align*}
\tr_F &= \frac{1+t_3 + \cdots + t_3^{d-1}}{(1-t_1)(1-t_2)}  + \frac{W_1}{1-t_1} + \frac{W_2}{1-t_2} + W, \\
\tr_G &= \frac{W_1 t_3^{-d}}{1-t_1} + \frac{W_2 t_3^{-d}}{1-t_2}  + W t_3^{-d}.
\end{align*}

An explicit calculation shows

\widetilde{\mathsf{v}}_{I\udot} - \mathsf{V}_{J\udot}^{\mathrm{3D}} = (t_4 - y) \overline{W}.

The statement of the proposition follows by setting $y = t_4$.

The condition that $Z \subset Y$, except for 0-dimensional embedded components, is necessary as shown by the following example.

Let $Z \subset \C^4$ be defined by the ideal

I_Z = (x_3,x_3x_4,x_1x_2x_4,x_4^2),

which is a copy of $\C^2$ with an embedded nodal curve. Then $Z$ has an embedded 1-dimensional component which is not contained in $Y = Z(x_4)$. Moreover, $\widetilde{\mathsf{v}}_{Z}|_{y=t_4}$ has positive $T$-fixed term $+1$ and the specialization $[-\widetilde{\mathsf{v}}_{Z}]|_{y=t_4}$ is not defined.

As an immediate consequence of Proposition <ref>, we obtain a dimensional reduction of the $K$-theoretic $\DT$–$\PT_0$ vertex correspondence of Conjecture <ref> to the 3-dimensional $K$-theoretic $\DT$–$\PT$ (curve) vertex correspondence, which was formulated by Nekrasov-Okounkov <cit.> and which generalizes the cohomological version of Pandharipande-Thomas <cit.>. The 3-dimensional $K$-theoretic $\DT$–$\PT$ vertex correspondence was recently proved in [99]. The 3-dimensional numerical DT/PT vertex correspondence (i.e., for the specialization $t_1t_2t_3=1$) was proved combinatorially by Jenne–Webb–Young [82]. More precisely, we obtain the following.

Fix $\mu_1,\mu_2$ plane partitions and let $\mu_3 = \mu_4 = \varnothing$. Suppose $Z_{\boldsymbol{\mu}}$ (Definition <ref>) is scheme theoretically supported in $\C^3 = Z(x_4) \subset \C^4$. Let $\lambda_1$, $\lambda_2$ be the unique finite partitions with the property that $I_{Z_{\boldsymbol{\mu}}/ \C^3} = I_{Z_{\boldsymbol{\lambda}} / \C^3}(-dS)$, for some $d \geq 0$, where $S = Z(x_3) \subset \C^3$ and $Z_{\boldsymbol{\lambda}} \subset \C^3$ is the Cohen-Macaulay curve determined by $\boldsymbol{\lambda} = (\lambda_1,\lambda_2,\varnothing)$.
Then there exist choices of $\pm \sqrt{\cdot}$ such that

\frac{\mathsf{V}^{\DT}_{\mu_1\mu_2\varnothing\varnothing}(q)}{\mathsf{V}^{\DT}_{\varnothing\varnothing\varnothing\varnothing}(q)} \Big|_{y = t_4} = \mathsf{V}^{\PT_0}_{\mu_1\mu_2\varnothing\varnothing}(q)|_{y = t_4}

is equivalent to 2-leg 3-dimensional $K$-theoretic $\DT$–$\PT$ vertex correspondence

\frac{\mathsf{V}^{\DT}_{\lambda_1\lambda_2\varnothing}(q)}{\mathsf{V}^{\DT}_{\varnothing\varnothing\varnothing}(q)} = \mathsf{V}^{\PT}_{\lambda_1\lambda_2\varnothing}(q).

The final result of this section concerns a $\PT_0$–$\PT_1$ edge correspondence for the specialization $y = t_4$. Recall that the halved edge term $\widetilde{\mathsf{e}}_{I^\mdot}$ in Definition <ref> depends on integers $m,m',m''$, i.e., the weights of the three normal directions (<ref>).

Let $X = \C^* \times \C^3$, $Y = Z(x_4) \cong \C^* \times  \C^2$, $S = Z(x_3,x_4) \cong \C^* \times \C$. Let $dS \subset Y$ be the $d$ times thickening of $S$ in $Y$.
Let $I^\mdot$ be a $T_X$-equivariant $\PT_q$ pair on $X$ for $q \in \{0,1\}$ and denote its scheme theoretic support by $Z$.[Recall that on $X = \C^* \times \C^3$, $T_X$-equivariant $\DT$ and $\PT_0$ pairs are the same.] Assume $Z_1 = dS$ and $m''=0$.

* Suppose $Z$ does not lie scheme theoretically in $Y$, which can only happen for $q=0$, then $\widetilde{\mathsf{e}}_{I^\mdot} |_{y=t_4}$ has negative $T$-fixed term, in particular

[-\widetilde{\mathsf{e}}_{I^\mdot}] \big|_{y = t_4} = 0.

* Suppose $Z \subset Y$. Denote by $I_Y^\mdot = [ \O_Y \to F ]$ the corresponding $\PT_q$ pair on $Y$ with cokernel $Q$. Then there exists a unique 1-dimensional $T_Y$-equivariant $\DT = \PT$ pair[Note that on $Y = \C^* \times \C^2$, 1-dimensional $T_Y$-equivariant $\DT$ and $\PT$ pairs coincide.] $J^\mdot = [\O_Y \to \O_C]$ on $Y$ such that

\begin{align*}
I_Y\udot &\cong I_{C/Y}(-dS), \quad \textrm{for } q=0 \\
\O_C \cdot t_2 t_3  &= \overline{Q(dS)}, \quad\quad \ \ \ \textrm{for } q=1.
\end{align*}

[-\widetilde{\mathsf{e}}_{I^\mdot}] \big|_{y = t_4} = [-\mathsf{E}^{\mathrm{3D}}_{J^\mdot}],

where $\mathsf{E}^{\mathrm{3D}}_{J^\mdot}$ denotes the equivariant edge term of 3-dimensional $\DT$ theory from <cit.> or, equivalently[Recall that the equivariant edge terms of 3-dimensional $\DT$ and $\PT$ theory coincide.], 3-dimensional $\PT$ theory from <cit.>. In particular, the $\PT_0$ and $\PT_1$ edge terms are equal for $y=t_4$.

Let $q=0$ and $Z \subsetneq Y$. We use the notation of Definition <ref> with $Z_i := Z_{\alpha\beta_i}$, $W_i := W_{\alpha\beta_i}$, $W_{ij} := Z_{\alpha\beta_i\beta_j}$.
Since $\mathsf{e}_{I^\mdot}$ has no positive $T$-fixed terms (Lemma <ref>), any $T$-fixed term resulting from the specialization $y = t_4$ comes from

\begin{equation} \label{eqn:yWedge}
\frac{t_1}{1-t_1} \Big( y \overline{W}_1 \Big) - \frac{1}{1-t_1} \Big( y \overline{W}_1 |_{(t_2 t_1^{-m}, t_3 t_1^{-m'}, t_4 t_1^{-m''} )} \Big),
\end{equation}

where $m'' = 0$. Note that all terms of $W_1$ are of the form $+t_2^j t_3^k t_4^{l}$ for some $j,k,l \geq 0$ (by our conditions on $Z$). Such a term can only produce a $T$-fixed term under the specialization $y=t_4$ when $j=k=l-1$. Using the fact that $m''=0$, the contribution to (<ref>) of such a term is

\[
\frac{1}{1-t_1} y t_2^{-j} t_3^{-j} t_4^{-j-1} (1 - t_1^{jm +jm'-1}) = y t_2^{-j} t_3^{-j} t_4^{-j-1} (-1-t_1-\cdots -t_1^{-2j}),
\]

where we used that $m''=0$ and $m+m'=-2$ by the Calabi-Yau condition. It follows that such a term produces the $T$-fixed term $-1$ under the specialization $y=t_4$. Since we assume $Z \subsetneq Y$, such terms occur.

Next, we assume $Z \subset Y$. It is easy to see from the the local description of fixed loci (Lemma <ref>) that there exists  a unique 1-dimensional $(\C^*)^3$-equivariant $\DT = \PT$ pair $J^\mdot = [\O_Y \to \O_C]$ on $Y$ such that

\begin{align*}
I_Y\udot &\cong I_{C/Y}(-dS), \quad \textrm{for } q=0 \\
\O_C \cdot t_2 t_3  &= \overline{Q(dS)}, \quad\quad \ \ \ \textrm{for } q=1.
\end{align*}

Let $C := \tr_{\O_C}$.
Then, for $q=0$, from the definitions of the halved edge term and 3-dimensional edge term, we obtain

\begin{align*}
\widetilde{e}_{I\udot} - \mathsf{E}_{J\udot}^{\mathrm{3D}} = &\frac{t_1^{-1}}{1-t_1^{-1}} \Big((1 - yt_1t_2t_3) t_2^{-1}t_3^{-d-1} \overline{C} \Big) - \\
&\frac{1}{1-t_1^{-1}} \Big( \big((1 - yt_1t_2t_3) t_2^{-1}t_3^{-d-1} \overline{C} \big) \Big|_{(t_1^{-1},t_2t_1^{-m},t_3t_1^{-m'})} \Big).
\end{align*}

On the other hand, for $q=1$, we obtain

\begin{align*}
\widetilde{e}_{I\udot} - \mathsf{E}_{J\udot}^{\mathrm{3D}} = &\frac{t_1^{-1}}{1-t_1^{-1}} \Big((1-yt_1t_2t_3) t_3^{-d} C \Big) - \\
&\frac{1}{1-t_1^{-1}} \Big( \big( (1-yt_1t_2t_3) t_3^{-d} C \big) \Big|_{(t_1^{-1},t_2t_1^{-m},t_3t_1^{-m'})} \Big).
\end{align*}

In both cases, we see that $[-\widetilde{e}_{I\udot}]|_{y=t_4}$ is well-defined and equal to $[-\mathsf{E}_{J\udot}^{\mathrm{3D}}]$.

The specialization $y = t_4$ provides a dimensional reduction, which we think of as a local virtual Lefschetz principle as opposed to the global virtual Lefschetz principles studied in Sections <ref> and <ref>.

§.§ Local projective plane

Consider $X = \mathrm{Tot}_{\PP^2}(\O_{\PP^2}(-2) \oplus \O_{\PP^2}(-1))$. Note that

\[
v = (0,0,\gamma,\beta,n - \gamma \cdot \td_2(X)) \in H_c^*(X,\Q)
\]

has the property that $\gamma = d [\PP^2]$ and $\beta = m [\PP^1]$. We therefore write $\gamma = d$ and $\beta = m$. Then, for $q \in \{-1,0,1\}$, the virtual dimension of $\PTqvX$ equals

\[
\vd = n - d^2.
\]

For $q=1$, and fixed $d,m$, there are only finitely many $n$ such that $\vd \geq 0$ <cit.>.

For a fixed $d$, we are interested in the generating series

\begin{align*}
\mathsf{G}^{\PT_q}_d(q,Q,y) := \sum_{m,n} \langle\!\langle \O_X \rangle\!\rangle_{d,m,n}^{\PT_q} Q^m q^n,
\end{align*}

for $q \in \{0,1\}$. By Proposition <ref>, the moduli spaces $\PTqvX$ are projective. Hence the specialization $t_1=t_2=t_3=t_4 = 1$ is well-defined and we will always make this specialization.

We use the description of fixed loci (Section <ref>) and the vertex formalism (Section <ref>) to calculate part of these generating series. We focus mostly on the calculation of $\PT_1$ invariants. Since $X$ is the canonical bundle over a smooth quasi-projective toric 3-fold, we can in fact use Section <ref>, i.e., Proposition <ref>.

Since $X$ is a local surface, Theorem <ref>(3) implies that, for $q=0$, $\PTqvX^{T_X} = \PTqvX^T$ is 0-dimensional and reduced. For $q=1$, the calculations in this section depend on Conjecture <ref>.

We used an implementation into a Maple program to show that there are signs such that the following formula holds:

\begin{align*}
[y]^{-1} \mathsf{G}^{\PT_0}_1 = &\, q Q^{\frac{3}{2}} + \Big( q^3 + (y^{\frac{3}{2}}+y^{-\frac{3}{2}}) q^4 +(y^{2}+y^{-2}) q^5 + (y^{\frac{5}{2}}+y^{-\frac{5}{2}}) q^6 + \cdots \Big)Q^{\frac{5}{2}} + \\
&\Big( 0 \cdot q^4 + 0 \cdot q^5 + \cdots \Big) Q^\frac{7}{2} +  \cdots\,,
\end{align*}

where $\cdots$ means higher order terms in $q$ or $Q$. The term with $Q^{\frac{3}{2}}$ has no embedded curve, i.e., $\PT_0 = \PT_1$. The terms with $Q^{\frac{5}{2}}$ have one embedded $\PP^1$. The terms with $Q^{\frac{7}{2}}$ have two embedded $\PP^1$s. Based on this, it is reasonable to expect that for $m = \frac{5}{2}$ and any $n > 3$, we have

\begin{equation} \label{eqn:localPT0eqn}
[y]^{-1} \langle\!\langle \O_X \rangle\!\rangle_{1,\frac{5}{2},n}^{\PT_0} = y^{\frac{n-1}{2}}+y^{-\frac{n-1}{2}}.
\end{equation}

Equation (<ref>) will be proved using different methods a sequel to this paper. We note that the two zeroes at the end are non-trivial. E.g., the term $0 \cdot q^5 Q^{\frac{7}{2}}$ involves the summation of 15 fixed points which, individually, have non-zero contributions.

Next, we used the vertex formalism to show that there are signs such that:

\begin{align}
\begin{split} \label{eqn:localPT1eqn}
\mathsf{G}^{\PT_1}_1 = &[y]qQ^{\frac{3}{2}}+\Big( 2[y]q+[y^2]q^2+[y^3]q^3 \Big)Q^{\frac{5}{2}} + \\
&\Big(...q+...q^2+...q^3+2[y^4]q^4+[y^5]q^5+[y^6]q^6 \Big) Q^\frac{7}{2} + \cdots,
\end{split}
\end{align}

where $...q^i$ means that we did not determine the coefficient. In a sequel to this paper, we determine $\mathsf{G}^{\PT_1}_1$ up to all orders (essentially by using the nested description of Proposition <ref>). We present the answer. Let $S$ be a smooth projective surface and $\cZ \subset S \times \Hilb^m(S)$ the universal subscheme. Denote the corresponding ideal sheaf by $I_{\cZ}$ and let $\pi : S \times \Hilb^m(S) \to \Hilb^m(S)$ be the projection. For any $L \in \Pic(S)$, consider the complex

\[
T(L) := R\hom_{\pi}(I_{\cZ},I_{\cZ} \otimes L)_0[1],
\]

which is in fact a vector bundle concentrated in degree 0.
Then Carlsson-Okounkov [54] showed

\[
\sum_{m=0}^{\infty} \int_{\Hilb^m(S)} c_{2m}(T(L)) q^m = \Bigg( \prod_{m=1}^{\infty} \frac{1}{1-q^m} \Bigg)^{e(S) + L(L-K_S)},
\]

where $e(S)$ is the topological Euler characteristic of $S$. Denote the coefficient of $q^m$ on the left hand side by $\mathsf{CO}(S,L)_m$. Then in a sequel we show that

\begin{align*}
\langle\!\langle \O_X \rangle\!\rangle_{1,\frac{3}{2}+\delta,n}^{\PT_1} &= \mathsf{CO}_{\chi(\O_{\PP^2}(\delta))-n}(\PP^2,\O(-1)) \cdot [y^n] \\
&= \mathsf{CO}_{\chi(\O_{\PP^2}(\delta))-n}(\PP^2,\O(-1)) [y] (y^{\frac{n-1}{2}} + \cdots + y^{-\frac{n-1}{2}}),
\end{align*}

for all $\delta \geq 0$ and $1 \leq n \leq \chi(\O_{\PP^2}(\delta))$. This formula is consistent with the toric calculations.

The proofs of (<ref>) and (<ref>) in the sequels strongly depend on the fact that, so far, we considered the irreducible class $\gamma = [\PP^2]$. The main strength of the vertex formalism is that it allows us to consider $\gamma = d [\PP^2]$ with $d>1$.

For $d=2$, we have shown that

\begin{align*}
[y]^{-4} \mathsf{G}^{\PT_0}_2 = q^4 Q^4 + \Big( 2(y^{\frac{1}{2}} + y^{-\frac{1}{2}})^3 q^7 + \cdots \Big) Q^5 + \cdots.
\end{align*}

The term $q^7 Q^5$ involves contributions of the zero section $\PP^2 \subset X$ thickened in the direction of $\O(-1)$ as well as in the direction of $\O(-2)$. In the former case, it involves contributions with an embedded curve thickened in the direction of $\O(-1)$ as well as in the direction of $\O(-2)$. Moreover, the signs in Proposition <ref> for fixed points scheme theoretically inside the zero section $\mathrm{Tot}(\O_{\PP^2}(-1)) \subset X$ and those which are not are opposite. Next, we determined the following terms of the $\PT_1$ generating series for $d=2$

\begin{align*}
[y]^{-4} \mathsf{G}^{\PT_1}_2 = &\, q^4 Q^{4}+\Big( 4q^4 +2(y^{\frac{1}{2}}+y^{-\frac{1}{2}})q^5+2(y+y^{-1})q^6 + \\
&2(y^{\frac{1}{2}} + y^{-\frac{1}{2}})(2y  + 1 + 2 y^{-1} ) q^7 \Big)Q^5 + \\
&\Big( ...q^4 +\cdots + 2(y^{\frac{1}{2}}+y^{-\frac{1}{2}})(y+y^{-1})(5y+1+5y^{-1}) q^{9}  +  \\
&(7 y^{3}  +12 y^2 + 15 y + 16 + 15 y^{-1} + 12 y^{-2} + 7 y^{-3}) q^{10}  + \\
&2(y^{\frac{1}{2}} + y^{-\frac{1}{2}})(4 y^{3}  + 3 y^2 + 6 y + 4 + 6 y^{-1} + 3 y^{-2} + 4 y^{-3}) q^{11}   \Big) Q^6 + \cdots.
\end{align*}

The calculation involves thickenings of the zero section $\PP^2$ in the $\O(-1)$ as well as the $\O(-2)$ direction. The computation of the coefficient of $q^9Q^6$ involves 48 fixed points. For $d=3$ we found

\begin{align*}
[y]^{-9} \mathsf{G}^{\PT_1}_3 = &\, \Big(0q^{9} + 3(y^{\frac{1}{2}}+y^{-\frac{1}{2}})q^{10} \Big)Q^{\frac{15}{2}}+ \Big(2q^{9} +18(y^{\frac{1}{2}}+y^{-\frac{1}{2}})q^{10}+ \\
&9(y^{\frac{1}{2}}+y^{-\frac{1}{2}})^2 q^{11}+9(y^{\frac{1}{2}}+y^{-\frac{1}{2}})(y+y^{-1})q^{12}+ \\
&3(3y^2+2y-3+2y^{-1}+3y^{-2}) q^{13}+ \\
&3(y^{\frac{1}{2}}+y^{-\frac{1}{2}})(7y^2+8y+12+8y^{-1}+7y^{-2}) q^{14} \Big)Q^{\frac{17}{2}} + \cdots.
\end{align*}

The calculation of the coefficient of $q^{9} Q^{\frac{17}{2}}$ involves 51 fixed points. For $d=4$, we obtained:

\begin{align*}
[y]^{-16} \mathsf{G}^{\PT_1}_4 = &\, q^{16}  Q^{11}+\Big(8q^{16}+4(y^{\frac{1}{2}}+y^{-\frac{1}{2}}) q^{17}+4(y+y^{-1}) q^{18}+ \\
&4(y^{\frac{3}{2}}+y^{-\frac{3}{2}}) q^{19} +10(y^2+8y+12+8y^{-1}+y^{-2})q^{20} \Big)Q^{12} + \cdots.
\end{align*}

Generally, the poles in the equivariant parameters $t_i$ in the calculations only cancel when the contributions of the fixed points are summed with very specific signs, which we expect to be induced from global canonical orientations. We observe that all terms calculated in this section are invariant under $y \to 1/y$ and, more generally, are consistent with the structure described in Proposition <ref>. This is evidence for the expectation that the sign choices found in this section are induced from global orientations.

We note that all coefficients appearing in the calculations of this section are integer. This is no longer true when one does not include the $K$-theoretic tautological insertion (Definition <ref>). E.g. for $d=1$, $m = \frac{7}{2}$, $n=5$, we have $\chi(\curly P_v^{(1)}(X), \widehat{\O}^{\vir}) = \frac{3}{128}$ (for the correct sign choice).

[1] M. Aganagic, A. Klemm, M. Marinõ, and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425–478.

[6] Y. Bae, M. Kool, and H. Park, Counting surfaces on Calabi-Yau 4-folds I: Foundations, arXiv:2208.09474.

[10] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45–88.

[17] A. Bojko, Orientations for DT invariants on quasi-projective Calabi-Yau 4-folds, Adv. Math. 388 (2021) 107859.
[18] A. Bojko, Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi–Yau 4-folds, arXiv:2102.01056.

[20] D. Borisov and D. Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds, Geom. Topol. 21 (2017) 3231–3311.

[25] C. Brav, V. Bussi, and D. Joyce, A Darboux theorem for derived schemes with shifted symplectic structure, J. Amer. Math. Soc. 32 (2019) no. 2 399–443.

[27] T. Bridgeland, Hall algebras and curve-counting invariants, 24 (2011) 969–998.

[32] Y. Cao, J. Gross, and D. Joyce, Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds, Adv. Math. 368 (2020) 107134.
[33] Y. Cao and M. Kool, Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds, Adv. Math. 338 (2018) 601–648.
[34] Y. Cao and M. Kool, Curve counting and DT/PT correspondence for Calabi-Yau 4-folds, Adv. Math. 375 (2020) 107371.
[35] Y. Cao, M. Kool, and S. Monavari, K-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds, Comm. Math. Phys. 396 (2022) 225–264.
[36] Y. Cao, M. Kool, and S. Monavari, Stable pair invariants of local Calabi-Yau 4-folds, Internat. Math. Res. Not. 2022 (2022) 4753–4798.

[38] Y. Cao and N. C. Leung, Donaldson-Thomas theory for Calabi-Yau 4-folds, arXiv:1407.7659.

[40] Y. Cao and G. Zhao, Quasimaps to quivers with potentials, arXiv:2306.01302.

[54] E. Carlsson and A. Okounkov, Exts and vertex operators, Duke Math. J. 161(9) (2012) 1797–1815.
[55] H.-L. Chang, Y.-H. Kiem, and J. Li, Torus localization and wall crossing for cosection localized virtual cycles, Adv. Math. 308 (2017) 964–986.
[56] J. Choi, S. Katz, and A. Klemm, The refined BPS index from stable pair invariants, Commun. Math. Phys. 328 (2014) 903–954.
[57] D. Cox, J. Little, and H. Schenck, Toric varieties, Grad. Stud. Math., 124, AMS, Providence, RI, 2011.
[59] D. Edidin and W. Graham, Equivariant intersection theory, Invent. Math. 131 (1998) 595–634.

[64] W. Fulton, Introduction to toric varieties,  Ann. of Math. Stud., 131, Princeton University Press, Princeton, NJ, 1993.
[65] A. Gholampour, Y. Jiang, and J. Lo, Stable pairs of 2-dimensional sheaves on 4-folds, arXiv:2110.01342.

[69] T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999) 487–518.

[72] J. Gross, D. Joyce, and Y. Tanaka, Universal structures in $\mathbb{C}$-linear enumerative invariant theories, SIGMA 18 (2022) 068, 61 pages.

[74] A. Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Exp. No. 221, 249–276, Soc. Math. France, Paris, 1995.

[79] L. Illusie, Complexe cotangent et déformations I, II, LNM 239, 283, Springer-Verlag, Berlin-New York, 1971, 1972.
[80] M.-a. Inaba, Toward a definition of moduli of complexes of coherent sheaves on a projective scheme, J. Math. Kyoto Univ. 42 (2002) no. 2 317–329.

[82] H. Jenne, G. Webb, and B. Young, The combinatorial PT–DT correspondence, arXiv:2012.08484.
[83] D. Joyce, Ringel-Hall style vertex algebra and Lie algebra structures on the homology of moduli spaces, in progress (preprint available on the homepage of the author).

[94] M. Kool and J. Rennemo, Proof of a magnificent conjecture, in preparation.

[99] N. Kuhn, H. Liu, and F. Thimm, The 3-fold K-theoretic DT/PT vertex correspondence holds, arXiv:2311.15697 .

[106] M. Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006) no. 1 175–206.
[107] H. Liu, The 4-fold Pandharipande–Thomas vertex, arXiv:2306.12923.
[108] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, I, Compos. Math. 142 (2006) 1263–1285.
[109] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, II, Compos. Math. 142 (2006) 1286–1304.

[114] E. Miller and B. Sturmfels, Combinatorial commutative algebra, GTM 227, Springer-Verlag, 2005.
[115] S. Monavari, Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds, Jour. Geom. Physics 174 (2022) 104466.

[119] N. Nekrasov, Magnificent Four,  Ann. Inst. Henri Poincaré D 7 (2020) no. 4 505–534.
[120] N. Nekrasov and A. Okounkov, Membranes and sheaves, Alg. Geom. 3 (2016) 320–369.
[121] N. Nekrasov and N. Piazzalunga, Magnificent Four with Colors, Comm. Math. Phys. 372 (2019) 573–597.
[122] N. Nekrasov and N. Piazzalunga, Global magni4icence, or: 4G Networks, arXiv:2306.12995.
[123] J. Oh and R. P. Thomas, Counting sheaves on Calabi-Yau 4-folds, I, Duke Math. J. 172 (2023) no. 7 1333–1409.

[125] A. Okounkov, Lectures on K-theoretic computations in enumerative geometry, Geometry of moduli spaces and representation theory, IAS/Park City Math. Ser. 24, AMS Providence RI (2017) 251–380.

[128] R. Pandharipande and R. P. Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009) 407–447.
[129] R. Pandharipande and R. P. Thomas, The 3-fold vertex via stable pairs, Geom. Topol. 13 (2009) 1835–1876.
[130] R. Pandharipande and R. P. Thomas, Stable pairs and BPS invariants, J. Amer. Math. Soc. 1 (2010) 267–297.

[132] T. Pantev, B. Töen, M. Vaquié, and G. Vezzosi, Shifted symplectic structures, Publ. Math. I.H.É.S. 117 (2013) 271–328.
[133] H. Park, Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4-folds, arXiv:2110.03631.
[134] N. Piazzalunga The one-legged K-theoretic vertex of fourfolds from 3d gauge theory, arXiv:2306.12405.
[135] J. Le Potier, Systemes cohérents et structures de niveau, Astérisque 214 (1993).
[136] J. Le Potier, Faisceaux semi-stables et systemes cohérents, Vector bundles in algebraic geometry (Durham, 1993) 208 (1995) 179–239.
[137] A. Preygel, Some remarks on shifted symplectic structures on non-compact mapping spaces, http://www.tolypreygel.com/index-research.html.
[138] F. Qu, Virtual pull-backs in $K$-theory, Ann. Inst. Fourier 68 (2018) no. 4 1609–1641.

[141] A. Ricolfi, The equivariant Atiyah class,
C. R. Math. Acad. Sci. Paris vol. 359 issue 3 (2021) 257–282.
[142] M. Savvas, Cosection localization for $D$-manifolds and $(-2)$-shifted symplectic derived schemes, revisited, arXiv:2309.03066.
[143] R. F. Schmiermann, Murphy's law on a fixed locus of the Quot scheme, arXiv:2307.16272.

[149] H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14–1 (1974) 1–28.

[150] Y. Tanaka and R. P. Thomas, Vafa-Witten invariants for projective surfaces I: stable case, Jour. Alg. Geom. 29 (2020) 603–668.
[151] D. Taylor, Ideals generated by monomials in an R-sequence, PhD thesis, University of Chicago (1966).
[152] R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations J. Differential Geom. 54 (2000) no. 2 367–438.

[154] R. W. Thomason, Une formule de Lefschetz en K-théorie équivariante algébrique, Duke Math. J. 68 (1992) 447–462.

[156] Y. Toda, Curve counting theories via stable objects I. DT/PT correspondence, JAMS 23 (2010) 1119–1157.

<EMAIL_ADDRESS><EMAIL_ADDRESS>hyeonjunpark@kias.re.kr