2.1 Preliminaries

Throughout the present article, we fix a rank ${\mathsf{r}}\geq 2$ . Only in the very end of Section 4 will we restrict to ${\mathsf{r}} = 2$ . We need ${\mathsf{r}} = 2$ for the analysis of Quot schemes in Section 5.

Let $C$ and $C^{\prime}$ be smooth, non-rational projective curves, and let

\begin{align*}{L_\delta }: = {{\mathcal{O}}_C}(1)\boxtimes{{\mathcal{O}}_{C^{\prime}}}( \delta),{\rm{\;\;\;\;}}\delta \in {\mathbb{Q}_{ \gt 0}}\end{align*}

be an ample $\mathbb{Q}$ -line bundle on the product $C \times C^{\prime}$ . For the extremal values of $\delta $ , we introduce the following notation:

(1) \begin{align}\delta {}&= +\ {\rm{if}}\ \delta \gg 1;\nonumber\\\delta {}&= -\ {\rm{if}}\ \delta \ll 1.\end{align}

Throughout this article, we will be using the identification

\begin{align*}{H^2}\!\left( {C \times C^{\prime},\mathbb{Z}} \right) \cong \mathbb{Z} \oplus {H^1}\!\left( {C,\mathbb{Z}} \right) \otimes {H^1}\!\left( {C^{\prime},\mathbb{Z}} \right) \oplus \mathbb{Z},\end{align*}

as provided by the Kunneth’s decomposition theorem. We define

\begin{align*}{{\rm{\Gamma }}_C}: = {\rm{Jac}}(C)[{\mathsf{r}}]\end{align*}

to be a group of ${\mathsf{r}}$ -torsion lines bundles on $C$ .

Definition 2.1. Let $\mathsf{d}$ and $\mathsf{a}$ be integers such that $0 \leqslant \mathsf{d},\mathsf{a} \lt \mathsf{r}$ . We define

\begin{align*}{M^\delta }\!\left( {C \times C^{\prime},{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right)\end{align*}

to be a moduli space of Higgs sheaves $\left( {F,\phi } \right),$ with a fixed determinant and traceless Higgs field $\phi \in \mathrm{Hom}\!\left( {F,F \otimes {\omega _{C \times C^{\prime}}}} \right)$ on $C \times C^{\prime}$ , which are Gieseker-stable with respect to ${L_\delta }$ . The class of $F$ is given as follows:

\begin{align*}{\rm{rk}}(F) & = {\mathsf{r}};\\{{\rm{c}}_1}(F) & = \left( {{\mathsf{d}},0,{\mathsf{a}}} \right);\\{\rm{\Delta }}(F) & : = {{\rm{c}}_1}{(F)^2} - 2{\mathsf{r}\rm{ch}}_2(F) = 2{{\mathsf{w}}}.\end{align*}

Throughout this article, we assume

\begin{align*}{\rm{gcd}}\!\left( {{\mathsf{r}},{\mathsf{d}},{\mathsf{a}}} \right) = 1,\end{align*}

which implies that there are no strictly semistable Higgs sheaves.

2.2 Vafa–Witten wall-crossing

Conjecturally, Vafa–Witten invariants are independent of stabilities for surfaces with ${p_g}\!\left( S \right) \gt 1$ . Let us present some evidences of this: By [Reference Manschot and MooreMM21], (physically derived) formulas for Vafa–Witten invariants for a surface $S$ with ${b_1}\!\left( S \right) = 0$ and ${p_g}\!\left( S \right) \gt 1$ are independent of stabilities (see also the discussion in [Reference Tanaka and ThomasTT20, Section 1.6]). By [Reference Dijkgraaf, Park and SchroersDPS98], the same holds for Donaldson invariants for a surface with ${b_1}\!\left( S \right) \ne 0$ and ${p_g}\!\left( S \right) \gt 1$ . It is therefore reasonable to expect that Vafa–Witten invariants (with even insertions) for a surface with ${b_1}\!\left( S \right) \ne 0$ and ${p_g}\!\left( S \right) \gt 1$ are also independent of stabilities. Products of non-rational curves are among such surfaces. We now derive a proof for this claim.

Definition 2.4. Following [Reference Tanaka and ThomasTT20], we define Vafa–Witten invariants associated to a moduli space ${M^\delta }\!\left( {C \times C^{\prime},\mathsf{d},\mathsf{a},\mathsf{w}} \right)$ by the $\mathbb{C}_t^*$ -localisation,

\begin{align*} {}&\mathsf{VW}^{{\mathsf{a}}}_{{\mathsf{d}},{\mathsf{w}}}(C\times C^{\prime}):= \int_{[M^\delta(C\times C^{\prime},{\mathsf{d}},{\mathsf{a}}, {\mathsf{w}})^{{\mathbb{C}}_t^*}]^{\mathrm{vir}}} \frac{1 }{e(N^{\mathrm{vir}})} \in {\mathbb{Q}}. \end{align*}

By Claim 2.3, they are independent of $\delta $ . For short, we will write

\begin{align*}\int_{[M^\delta(C\times C^{\prime},{\mathsf{d}},\,{\mathsf{a}}, \mathsf{w} )]^{\mathrm{vir}}}1 := \int_{[M^\delta(C\times C^{\prime},\,{\mathsf{d}},{\mathsf{a}}, \mathsf{w} )^{{\mathbb{C}}_t^*}]^{\mathrm{vir}}} \frac{1 }{e(N^{\mathrm{vir}})},\end{align*}

the same notation of which applies to all integrals that require $\mathbb{C}_t^{\rm{*}}$ -localisations.

2.3 Quasisection invariants

The importance of quasisections was already observed in [Reference OkounkovOko19, Section 7]. Here we apply them in the context of a relative moduli space of Higgs bundles.

Let ${K_{C \times C^{\prime}}}$ be the total space of the canonical bundle ${\omega _{C \times C^{\prime}}}$ on $C \times C^{\prime}$ . The variety ${K_{C \times C^{\prime}}}$ admits projections both to $C$ and to $C^{\prime}$ :

\begin{align*}{\pi _C}:{K_{C \times C^{\prime}}} \to C,{\rm{\;\;\;\;}}{\pi _{C^{\prime}}}:{K_{C \times C^{\prime}}} \to C^{\prime}.\end{align*}

We will consider various moduli spaces (that is, Quot schemes and moduli spaces of Higgs sheaves) relative to these projections.

Let us denote

\begin{align*}{\mathsf{v}}=({\mathsf{r}},{\mathsf{d}}) \in H^{\mathrm{ev}}(C,{\mathbb{Z}}).\end{align*}

As in the absolute case, we have a determinant-line-bundle map

\begin{align*}\lambda :{H^{{\rm{ev}}}}\!\left( {C,\mathbb{Z}} \right) \to {\rm{Pic}}\big( {\mathfrak{M}_C^{{\rm{rel}}}(\mathsf{d})} \big),\end{align*}

such that a class $u \in {H^{{\rm{ev}}}}\!\left( {C,\mathbb{Z}} \right)$ that satisfies

\begin{align*}\chi({\mathsf{v}} \cdot u)=\int_C{\mathsf{v}} \cdot u \cdot {\mathrm{td}}_C={\mathsf{r}} \cdot u_2+{\mathsf{d}} \cdot u_1+{\mathsf{r}} \cdot u_1(1-g)=1\end{align*}

gives a trivilisation of the ${\mathbb{C}^{\rm{*}}}$ -gerbe

or, in other words, a universal family on $\mathfrak{M}_{{\rm{rg}},C}^{{\rm{rel}}}(\mathsf{d})$ . For a class $u \in {H^{{\rm{ev}}}}\!\left( {C,\mathbb{Z}} \right)$ such that $\chi \!\left( {{\rm{v}} \cdot u} \right) = 0$ , the line bundle $\lambda \!\left( u \right)$ descends to $\mathfrak{M}_{{\rm{rg}},C}^{{\rm{rel}}}(\mathsf{d})$ .

We define the theta line bundle ${\rm{\Theta }} \in {\rm{Pic}}\Big( {\mathfrak{M}_{{\rm{rg}},C}^{{\rm{rel}}}(\mathsf{d})} \Big)$ as follows:

\begin{align*}\theta & = \left( { - {\mathsf{r}},{\mathsf{d}} - {\mathsf{r}}\!\left( {g - 1} \right)} \right)\!;\\{\rm{\Theta }} & = \lambda \!\left( \theta \right)\!.\end{align*}

We also define the class of ${\rm{SL}}$ -trivialisations of the universal family of $\mathfrak{M}_C^{{\rm{rel}}}(\mathsf{d}):$

\begin{align*}\alpha \in {H^2}\Big( {\mathfrak{M}_C^{{\rm{rel}}}(\mathsf{d}),{\mathbb{Z}_{\mathsf{r}}}} \Big)\!.\end{align*}

Equivalently, $\alpha $ is the Kunneth component of the first Chern class of the universal family modulo ${\rm{r}}$ . The class $\alpha $ is the gerbe class of [Reference Hausel and ThaddeusHT03]. The classes ${\rm{\Theta }}$ and $\alpha $ will be used to define degrees of quasisections.

Definition 2.7. A quasisection of $M_C^{\mathrm{rel}}( \mathsf{d})$ is a section of the projection ${p_{C^{\prime}}}:\mathfrak{M}_{rg,C}^{\mathrm{rel}}( \mathsf{d}) \to C^{\prime}$ ,

\begin{align*}f:C^{\prime} \to \mathfrak{M}_{{\rm{rg}},C}^{{\rm{rel}}}(\mathsf{d}),{\rm{\;\;\;\;}}{p_{C^{\prime}}} \circ f = {\rm{i}}{{\rm{d}}_{C^{\prime}}},\end{align*}

which maps generically to $M_C^{{\rm{rel}}}(\mathsf{d})$ . A quasisection is of degree $\left( {{{\mathsf{w}}},{\mathsf{a}}} \right) \in \mathbb{Z} \oplus {\mathbb{Z}_{\mathsf{r}}}: = \mathbb{Z} \oplus \mathbb{Z}/{\mathsf{r}}\mathbb{Z}$ , if

\begin{align*}{\rm{deg}}\!\left( {{f^{\rm{*}}}{\rm{\Theta }}} \right) = {{\mathsf{w}}},{\rm{\;\;\;\;}}{f^{\rm{*}}}\alpha = {\rm{a}}.\end{align*}

We denote the moduli space of quasisections of $M_C^{{\rm{rel}}}(\mathsf{d})$ of degree $\left( {{{\mathsf{w}}},{\mathsf{a}}} \right)$ by $Q\!\left( {M_C^{{\rm{rel}}}(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \right)$ .

The moduli spaces $Q\!\left( {M_C^{{\rm{rel}}}(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \right)$ inherit $\mathbb{C}_t^{\rm{*}}$ -actions from $\mathfrak{M}_C^{{\rm{rel}}}(\mathsf{d})$ . The properness of quasisections and the existence of a perfect obstruction theory is proven in the same way as in [Reference NesterovNes21, Reference NesterovNes23]; see also [Reference Lee and WenLW23].

Definition 2.8. If $\mathrm{gcd}( {\mathsf{r},\mathsf{d}}) = 1$ , we define

\begin{align*} \mathsf{QM} ^{{\mathsf{a}}}_{{\mathsf{d}}, \mathsf{w} }(C)=\int_{[Q(M^{{\mathrm{rel}}}_{C}({\mathsf{d}}),{\mathsf{a}},\mathsf{w} )]^{\mathrm{vir}}}1 \in {\mathbb{Q}} \end{align*}

to be quasisection invariants associated to a moduli space $Q\!\left( {M_C^{{\rm{rel}}}(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \right)$ .

Note that by the definition of a relative moduli of sheaves, a section

\begin{align*}f:C^{\prime} \to \mathfrak{M}_C^{{\rm{rel}}}(\mathsf{d})\end{align*}

is given by a sheaf on

\begin{align*}{K_{C \times C^{\prime}}}{ \times _{C^{\prime}}}C^{\prime} = {K_{C \times C^{\prime}}}.\end{align*}

Hence by [Reference NesterovNes23, Proposition 5.10], a moduli space of ${L_ + }$ -stable Higgs sheaves on $C \times C^{\prime}$ is naturally a ${{\rm{\Gamma }}_{C^{\prime}}}$ -torsor over the moduli space of quasisections of $M_C^{{\rm{rel}}}(\mathsf{d})$ . On the other hand, the moduli space of ${L_ - }$ -stable Higgs sheaves on $C \times C^{\prime}$ is naturally a ${{\rm{\Gamma }}_C}$ -torsor over the moduli space of quasisections of $M_{C^{\prime}}^{{\rm{rel}}}(\mathsf{d})$ . Moreover, the corresponding obstruction theories match. This is summarised in the following proposition:

Proposition 2.9. If $\mathrm{gcd}( {\mathsf{r},\mathsf{d}}) = 1$ , we have

\begin{align*}Q\Big( {M_{C^{\prime}}^{{\rm{rel}}}(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}}\Big) & \cong \left[ {{{{M^ - }\!\left( {C \times C^{\prime},{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right)}}{{{{\rm{\Gamma }}_C}}}} \right]\\Q\Big( {M_C^{{\rm{rel}}}(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \Big) & \cong \left[ {{M^ + }\!\left( {C \times C^{\prime},{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right)\!/{{\rm{\Gamma }}_{C^{\prime}}}} \right],\end{align*}

such that the naturally defined obstruction theories on both sides match.

We use Proposition 2.9 as a justification for the following definition of invariants in the case of ${\rm{gcd}}({{\rm{r}},{\rm{d}}}) \ne 1:$

Definition 2.10. If $\mathrm{gcd}( {\mathsf{r},\mathsf{d}}) \ne 1$ , we define

\begin{align*}\mathsf{QM} ^{{\mathsf{a}}}_{{\mathsf{d}}, \mathsf{w} }( C):= \int_{[M^+(C\times C^{\prime},{\mathsf{d}},{\mathsf{a}},\mathsf{w} )/\Gamma_{C^{\prime}}]^{\mathrm{vir}}}1 \in {\mathbb{Q}}.\end{align*}

If ${\rm{gcd}}( {{\mathsf{r}},{\mathsf{a}}}) \ne 1$ , we define

\begin{align*}\mathsf{QM} ^{{\mathsf{d}}}_{{\mathsf{a}}, \mathsf{w} }( C^{\prime}):= \int_{[M^-(C\times C^{\prime},{\mathsf{d}},{\mathsf{a}},\mathsf{w} )/\Gamma_{C}]^{\mathrm{vir}}}1 \in {\mathbb{Q}}.\end{align*}

Using Claim 2.3 and Proposition 2.9, we obtain a curious correspondence between quasisection invariants of $M_C^{{\rm{rel}}}(\mathsf{d})$ and $M_{C^{\prime}}^{{\rm{rel}}}( {\mathsf{a}})$ .

Corollary 2.12. If $\mathsf{r}$ is prime, we have

\begin{align*}{{\mathsf{r}}^{2g\!\left( {C^{\prime}} \right)}}{\mathsf{QM}}_{{\mathsf{d}},{{\mathsf{w}}}}^{\mathsf{a}}(C) = {\mathsf{VW}}_{{\mathsf{d}},{{\mathsf{w}}}}^{\mathsf{a}}\!\left( {C \times C^{\prime}} \right) = {{\mathsf{r}}^{2g(C)}}{\mathsf{QM}}_{{\mathsf{a}},{{\mathsf{w}}}}^{\mathsf{d}}\!\left( {C^{\prime}} \right).\end{align*}

3.1 Group actions

For the rest of this article we assume that $C^{\prime}$ is an elliptic curve,

\begin{align*}C^{\prime} = E.\end{align*}

Since ${\pi _E}:{K_{C \times E}} \to E$ is a trivial fibration, the moduli space of quasisections to $M_C^{{\rm{rel}}}(\mathsf{d})$ is canonically isomorphic to a moduli space of quasimaps from $E$ to an absolute moduli space of Higgs bundles ${M_C}(\mathsf{d})$ on $C$ :

\begin{align*}Q\Big( {M_C^{{\rm{rel}}}(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \Big) \cong {Q_E}\!\left( {{M_C}(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \right).\end{align*}

In fact, our primary interest is in quasimaps up to translations of $E$ ; that is, in the quotient

\begin{align*}\left[ {{Q_E}\!\left( {M(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \right)\!/E} \right],\end{align*}

where $E$ acts ${\rm{on}}\;{Q_E}\!\left( {M(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \right)$ by precomposition with a translation. A similar action exists on the level of moduli spaces ${M^\delta }\!\left( {C \times E,{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right)$ , which we now explain.

The group

\begin{align*}E \times {\rm{Jac}}(E)\end{align*}

naturally acts on sheaves. Here, $E$ acts by pulling back a sheaf with respect to a translation ${\tau _p}$ by a point $p \in E$ , while ${\rm{Jac}}(E)$ acts by tensoring a sheaf with a line bundle $L$ . These operations commute. Overall,

\begin{align*}F \mapsto \tau _p^{\rm{*}}F \otimes L.\end{align*}

Let

\begin{align*}{\rm{\Phi }}\!\left( {\mathcal{L}} \right) \subset E \times {\rm{Jac}}(E)\end{align*}

be the subgroup that fixes the determinant line bundle ${\mathcal{L}}$ of sheaves in a moduli space ${M^\delta }\!\left( {C \times E,{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right)$ . We define

(2) \begin{align}{{\rm{\Phi }}_{\mathsf{a}}} = {({\rm{id}},{\rm{r}})^{ - 1}}{\rm{\Phi }}\!\left( {\mathcal{L}} \right).\end{align}

The group ${{\rm{\Phi }}_{\mathsf{a}}}$ preserves rank ${\mathsf{r}}$ sheaves with determinant ${\mathcal{L}}$ . The action of ${{\rm{\Phi }}_{\mathsf{a}}}$ on sheaves therefore restricts to an action on ${M^\delta }\!\left( {C \times E,{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right)$ .

By our assumption on the classes in Definition 2.1, the line bundle ${\mathcal{L}}$ is of the form $L \boxtimes L^{\prime}$ . Hence, ${\rm{\Phi }}\!\left( {\mathcal{L}} \right)$ and therefore ${{\rm{\Phi }}_{\mathsf{a}}}$ depend only on the degree of ${\mathsf{a}}$ . The group ${{\rm{\Phi }}_{\mathsf{a}}}$ also acts on $\mathfrak{M}_E^{{\rm{rel}}}( {\mathsf{a}})$ . In the case of

\begin{align*}Q\!\left( {M_E^{{\rm{rel}}}( {\mathsf{a}}),{\mathsf{d}},{{\mathsf{w}}}} \right) \cong \left[ {{M^ - }\!\left( {C \times E,{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right)/{{\rm{\Gamma }}_C}} \right],\end{align*}

the action of ${{\rm{\Phi }}_{\mathsf{a}}}$ on $Q\!\left( {M_E^{{\rm{rel}}}( {\mathsf{a}}),{\mathsf{d}},{{\mathsf{w}}}} \right)$ can be seen as identification of maps by the automorphisms of the target. The importance of this action is due to the next two lemmas:

Lemma 3.1. There is a canonical identification

\begin{align*}\left[ {Q_E}\!\left( {{M_C}(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \right)/E\right] \cong \left[{M^ + }\!\left( {C \times E,{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right)/{{\rm{\Phi }}_{\mathsf{a}}} \right]\end{align*}

such that the naturally defined obstruction theories on both sides match.

Proof. There exists a natural map

(3) \begin{align}{M^ + }\!\left( {C \times E,{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right) \to {Q_E}\!\left( {{M_C}(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \right)\end{align}

that is a ${{\rm{\Gamma }}_E}$ -torsor. There also exists a natural projection

(4) \begin{align}{{\rm{\Phi }}_{\mathsf{a}}} \to E\end{align}

that is also a ${{\rm{\Gamma }}_E}$ -torsor. The map (3) is equivariant with respect to (4) and the corresponding actions of ${{\rm{\Phi }}_{\mathsf{a}}}$ and $E$ on the source and the target. It is not difficult to check that we obtain the desired identification after taking quotients. The rest follows from the same arguments as in [Reference NesterovNes23, Section 5.5].

The action of ${{\rm{\Phi }}_{\mathsf{a}}}$ can be exchanged for an insertion. We are interested in $\mu $ -insertions, which are defined as follows:

\begin{align*}\mu :{H^{\rm{*}}}\!\left( {C \times E,\mathbb{Q}} \right) & \to H_{{\mathbb{C}^{\rm{*}}}}^{{\rm{*}}}\left( {{M^\delta }\!\left( {C \times E,{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right),\mathbb{Q}} \right);\\\beta & \mapsto {\pi _{M{\rm{*}}}}\left( {{\rm{\Delta }}\!\left( {\mathcal{F}} \right)\!/2{\rm{r}} \cdot \pi _{X \times E}^{\rm{*}}\beta } \right),\end{align*}

where ${\mathcal{F}}$ is the universal sheaf on ${M^\delta }\!\left( {C \times E,{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right)$ . Consider now the class

\begin{align*}{B_{{\mathsf{w}}}}: = \frac{{\mathbb{1} \boxtimes [ {{\rm{pt}}}]}}{{{\rm{rw}}}} \in {H^{\rm{*}}}\!\left( {C \times E,\mathbb{Q}} \right).\end{align*}

Lemma 3.2. We have

\begin{align*}{\mathsf{VW}}^{{\mathsf{a}}, \bullet}_{{\mathsf{d}},\mathsf{w} }(C\times E):=\int_{[M^\delta(C\times E,{\mathsf{d}},{\mathsf{a}},\mathsf{w} )/\Phi_{{\mathsf{a}}}]^{\mathrm{vir}}}1=\int_{[M^\delta(C\times E,{\mathsf{d}},{\mathsf{a}},\mathsf{w} )]^{\mathrm{vir}}}\mu(B_\mathsf{w} ).\end{align*}

by Lemma 3.2 and Claim 2.3, invariants associated to a moduli space $\left[ {{M^\delta }\!\left( {C \times E,{\mathsf{d}},{\mathsf{a}},{{\mathsf{w}}}} \right)\!/{{\rm{\Phi }}_{\mathsf{a}}}} \right]$ are independent of $\delta $ .

Definition 3.3. If $\rm{gcd}( {\mathsf{r},\mathsf{d}}) = 1$ , we define

\begin{align*}\mathsf{QM} ^{{\mathsf{a}}, \bullet}_{{\mathsf{d}}, \mathsf{w} }(C)= \int_{[Q_E(M({\mathsf{d}}),{\mathsf{a}},\mathsf{w} )/E]^{\mathrm{vir}}}1\in {\mathbb{Q}} t\end{align*}

to be invariants associated with quotient moduli spaces $\left[ {{Q_E}\!\left( {M(\mathsf{d}),{\mathsf{a}},{{\mathsf{w}}}} \right)/E} \right]$ . If ${\rm{gcd}}\!\left( {{\mathsf{r}},{\mathsf{a}}} \right) = 1$ , we also define

\begin{align*}\mathsf{QM} ^{{\mathsf{d}}, \bullet}_{{\mathsf{a}},\mathsf{w} }( E) = \int_{[Q(M^{\mathrm{rel}}_E({\mathsf{d}}), {\mathsf{a}},\mathsf{w} )/\Phi_{{\mathsf{a}}}]^{\mathrm{vir}}} 1\in {\mathbb{Q}} t\end{align*}

to be invariants associated with the quotient moduli space $\left[ {Q\!\left( {M_E^{{\rm{rel}}}( {\mathsf{a}}),{\mathsf{d}},{{\mathsf{w}}}} \right)/{{\rm{\Phi }}_{\mathsf{a}}}} \right]$ .

We use Lemma 3.1 as a justification for the following definition of invariants in the case of ${\rm{gcd}}\!\left( {{\mathsf{r}},{\mathsf{d}}} \right) \ne 1$ and ${\rm{gcd}}\!\left( {{\mathsf{r}},{\mathsf{a}}} \right)$ .

Definition 3.4. If $\rm{gcd}( {\mathsf{r},\mathsf{d}}) \ne 1$ , we define

\begin{align*}\mathsf{QM} ^{{\mathsf{a}}, \bullet}_{{\mathsf{d}}, \mathsf{w} }(C) = \int_{[M^+(C\times E,{\mathsf{d}},{\mathsf{a}},\mathsf{w} )/\Phi_{{\mathsf{a}}}]^{\mathrm{vir}}}1 \in {\mathbb{Q}} t.\end{align*}

If $\mathrm{gcd}\!\left( {{\mathsf{r}},{\mathsf{a}}} \right) \ne 1$ , we define

\begin{align*}\mathsf{QM} ^{{\mathsf{d}}, \bullet}_{{\mathsf{a}}, \mathsf{w} }(E)=\int_{[M^-(C\times E,{\mathsf{d}},{\mathsf{a}},\mathsf{w} )/\Phi_{{\mathsf{a}}}]^{\mathrm{vir}}}1 \in {\mathbb{Q}} t.\end{align*}

Using Claim 2.3 and Lemma 3.2, we obtain the following result:

Corollary 3.7. If $\mathsf{r}$ is prime, we have

\begin{align*}{\mathsf{QM}}_{{\mathsf{d}},{{\mathsf{w}}}}^{{\mathsf{a}}, \bullet }(C) = {\mathsf{VW}}_{{\mathsf{d}},{{\mathsf{w}}}}^{{\mathsf{a}}, \bullet }\!\left( {C \times E} \right) = {{\mathsf{r}}^{2g(C)}}{\mathsf{QM}}_{{\mathsf{a}},{{\mathsf{w}}}}^{{\mathsf{d}}, \bullet }(E).\end{align*}

3.2 Moduli spaces of Higgs sheaves and sheaves

From now on, we will assume that ${\rm{gcd}}\!\left( {{\mathsf{r}},{\mathsf{a}}} \right) = 1$ , unless stated otherwise. A moduli space of rank ${\mathsf{r}}$ and degree ${\mathsf{a}}$ stable Higgs ${\rm{G}}{{\rm{L}}_{\rm{r}}}$ -bundles on $E$ , denoted by $M_E^{{\rm{GL}}}( {\mathsf{a}})$ , is isomorphic to ${K_E}$ via the determinant-trace map,

\begin{align*}\left( {{\rm{det}},{\rm{tr}}} \right):M_E^{{\rm{GL}}}( {\mathsf{a}}) \mathop\to\limits^{\cong\,} {K_E}.\end{align*}

Hence, a moduli space of stable Higgs sheaves on $E$ with a fixed determinant and a traceless Higgs field is a point,

\begin{align*}{M_E}( {\mathsf{a}}) = \left\{ {\left( {G,0} \right)} \right\} = {\rm{pt}},\end{align*}

where $G$ is the unique stable sheaf with the given determinant. This is also holds relatively for the projection ${\pi _C}:{K_{C \times E}} \to C$ ,

\begin{align*}M_E^{{\rm{rel}}}( {\mathsf{a}}) = \left\{ {\left( {G,0} \right)} \right\} \times C = C.\end{align*}

Using Corollary 2.12, we obtain an immediate consequence for quasimaps of degree ${{\mathsf{w}}} = 0$ , which confirms a part of [Reference NesterovNes23, Conjecture B].

Corollary 3.9. If $\mathsf{r}$ is prime and $\mathsf{a} \ne 0$ , then

\begin{align*}{\mathsf{QM}}_{0,0}^{\mathsf{a}}(C) = {{\mathsf{r}}^{2g - 2}}.\end{align*}

Let us now consider quasimaps of degree $\;{{\mathsf{w}}} \ne 0$ . Since the unique Higgs sheaf in $M_E^{{\rm{rel}}}( {\mathsf{a}})$ has a zero Higgs field, a quasisection to $M_E^{{\rm{rel}}}( {\mathsf{a}})$ must factor through the moduli stack of Higgs sheaves with zero Higgs fields. The latter is just a moduli stack of sheaves on $E$ ,

(5) \begin{align}\mathfrak{N}_E^{{\rm{rel}}}( {\mathsf{a}}) \hookrightarrow \mathfrak{M}_E^{{\rm{rel}}}( {\mathsf{a}}).\end{align}

Since $\mathfrak{N}_E^{{\rm{rel}}}( {\mathsf{a}})$ is a relative moduli space of sheaves associated to a trivial fibration $C \times E \to C$ , it trivialises canonically,

(6) \begin{align}\mathfrak{N}_E^{{\rm{rel}}}( {\mathsf{a}}) = {\mathfrak{N}_E}( {\mathsf{a}}) \times C.\end{align}

The same applies for rigidified stacks.

The obstruction theory of $\mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})$ is constructed as follows: Let

\begin{align*}{\mathcal{G}} \in {\rm{Coh}}\!\left( {{K_{C \times E}}\,{ \times _C}\,\mathfrak{M}_E^{{\rm{rel}}}( {\mathsf{a}})} \right){\rm{\;\;\;\;and\;\;\;\;}}\pi :{K_{C \,\times\, E}}\,{ \times _C}\,\mathfrak{M}_E^{{\rm{rel}}}( {\mathsf{a}}) \to \mathfrak{M}_E^{{\rm{rel}}}( {\mathsf{a}})\end{align*}

be the universal $C$ -relative 1-dimensional sheaf and the canonical projection. The complex $R{\mathcal{H}}{om_\pi }\!\left( {{\mathcal{G}},{\mathcal{G}}} \right)$ descends to $\mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})$ . The obstruction theory for Higgs sheaves on a surface is constructed in [Reference Tanaka and ThomasTT20, Section 6], and the construction applies to Higgs sheaves on a curve. The spectral-curve construction identifies $C$ -relative 1-dimensional sheaves on ${K_{C \times E}}$ with $C$ -relative Higgs sheaves on $C \times E$ with Higgs fields valued in ${\omega _{C \times E}}$ . Hence, the $C$ -relative obstruction theory of $\mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})$ is given by the complex

\begin{align*}\mathbb{T}_{\mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})}^{{\rm{vir}}} = R{\mathcal{H}}o{m_\pi }{({\mathcal{G}},{\mathcal{G}})_0}{\rm{\;}}\left[ 1 \right],\end{align*}

where $R{\mathcal{H}}o{m_\pi }{({\mathcal{G}},{\mathcal{G}})_0}$ is defined to be the cone

\begin{align*}{\rm{Cone}}\!\left( {R{\mathcal{H}}o{m_\pi }\!\left( {{\mathcal{G}},{\mathcal{G}}} \right) \to \left( {{H^{{\rm{*}} - 1}}\left( {{\omega _E}} \right) \oplus {H^{\rm{*}}}\!\left( {{{\mathcal{O}}_E}} \right)} \right) \otimes {\omega _C}} \right)\left[ { - 1} \right].\end{align*}

Note that $\mathbb{T}_{\mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})}^{{\rm{vir}}}$ does not restrict to the virtual tangent complex of the stack $\mathfrak{N}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})$ .

3.3 Chern characters

For the purposes of wall-crossing, one needs to make a choice for a universal family on the rigidified stack $\mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})$ . As explained in [Reference NesterovNes23, Section 3], this amounts to choosing ${\mathsf{u}} \in {H^{{\rm{ev}}}}\!\left( {E,\mathbb{Z}} \right)$ such that $\chi \!\left( {{\mathsf{u}} \cdot {\mathsf{v}}} \right) = 1.$ For a choice of such class ${\mathsf{u}} = \left( {{{\mathsf{u}}_1},{{\mathsf{u}}_2}} \right)$ , the sheaf $F$ on $C \times E$ associated to a quasisection $f:C \to \mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})$ of degree ${{\mathsf{w}}}$ has the following Chern character,

\begin{align*}{\mathrm{ch}}(F)=({\mathsf{v}}, \check{\mathsf{w}} ) \in H^{\mathrm{ev}}(E,{\mathbb{Z}})\oplus H^{\mathrm{ev}}(E,{\mathbb{Z}})(-1), \end{align*}

where $\check{\mathsf{w}} $ is defined by the following system of equations:

(7) \begin{align}\begin{split} & \check{\mathsf{w}} _1\cdot {\mathsf{u}}_2+ \check{\mathsf{w}} _2\cdot {\mathsf{u}}_1=0, \\ & \check{\mathsf{w}} _1 \cdot {\mathsf{a}} -\check{\mathsf{w}} _2 \cdot {\mathsf{r}} =\mathsf{w} . \end{split}\end{align}

For example, if $( {{\mathsf{r}},{\mathsf{a}}}) = \left( {{\rm{r}},1} \right)$ , then ${\mathsf{u}} = \left( {1,0} \right)$ clearly satisfies

\begin{align*}\chi \!\left( {{\mathsf{v}} \cdot {\mathsf{u}}} \right) = 1.\end{align*}

Using (7), we deduce that in this case,

\begin{align*} \check{\mathsf{w}} =(\mathsf{w}, 0).\end{align*}

4.1 $\boldsymbol\epsilon $ -stable quasisections

We will use Corollary 3.7 to compute genus 1 quasimap invariants of moduli spaces of Higgs bundles on $C$ . If ${\rm{gcd}}( {{\mathsf{r}},{\mathsf{a}}}) = 1$ , then

\begin{align*}{M_E}( {\mathsf{a}}) = \left\{ {\left( {G,0} \right)} \right\}\!;\end{align*}

hence, there are no sections of nonzero degree, and there is a unique section of degree zero. The quasimap wall-crossing for ${{\mathsf{w}}} \gt 0$ is therefore particularly simple here, as it gives equality of invariants associated to $\epsilon = {0^ + }$ and to the wall-crossing invariants. However, there are two complications:

• • the action of ${{\rm{\Phi }}_{\mathsf{a}}}$ on $\mathfrak{M}_E^{{\rm{rel}}}( {\mathsf{a}})$ and

• • $C$ -relative setup,

which obscure otherwise-simple computations.

Let us start with defining $\epsilon $ -stable quasisections. From now on, we simplify the notation in the following way:

\begin{align*}Q( {{\mathsf{a}},{{\mathsf{w}}}})& : = Q\left( {M_E^{{\rm{rel}}}( {\mathsf{a}}),{{\mathsf{w}}}} \right),\\Q{({\mathsf{a}},{{\mathsf{w}}})^ \bullet } & : = \left[ {{{Q\left( {M_E^{{\rm{rel}}}( {\mathsf{a}}),{{\mathsf{w}}}} \right)}}{{{{\rm{\Phi }}_{\mathsf{a}}}}}}\right];\end{align*}

the same applies to other related spaces.

Definition 4.1. A marked bubbling of $C$ is a pair $\left( {C^{\prime},\mathbf{p},\iota } \right)$ , where $\left( {C^{\prime},\mathbf{p}} \right)$ is a connected, marked nodal curve of genus equal to $g(C)$ and where

\begin{align*}\iota :C \hookrightarrow C^{\prime}\end{align*}

is a closed immersion. In other words, $C^{\prime}$ is an isotrivial, semistable degeneration of $C$ .

Definition 4.2. Given $\epsilon \in {\mathbb{Q}_{ \gt 0}}$ , we define $Q_k^\epsilon ({\mathsf{a},\mathsf{w}})$ to be the moduli space of quasimaps of degree $\mathsf{w}$ ,

\begin{align*}f:\left( {C^{\prime},{\bf{p}}} \right) \to \mathfrak{N}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}}) = {\mathfrak{N}_{{\rm{rg}},E}}( {\mathsf{a}}) \times C,\end{align*}

such that

• • ${f_{\mathfrak{N}( {\mathsf{a}})}}:\left( {C^{\prime},{\bf{p}}} \right) \to {\mathfrak{N}_{{\rm{rg}},E}}( {\mathsf{a}})$ is $\epsilon $ -stable [Reference NesterovNes23, Definition 3.5];

• • $\left( {C^{\prime},{\bf{p}}} \right)$ is a marked bubbling of $C$ with $k$ markings;

• • $\left[ {{f_C} \circ \iota :C \to C} \right] = {\rm{i}}{{\rm{d}}_C}$ .

Since quasisections to $\mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})$ factor through $\mathfrak{N}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})$ , a moduli space $Q_k^\epsilon ( {{\mathsf{a}},{{\mathsf{w}}}})$ should be viewed as a moduli space of $\epsilon $ -stable quasisections to $\mathfrak{M}_E^{{\rm{rel}}}( {\mathsf{a}})$ . The fact that these moduli spaces are proper follows from the arguments of [Reference NesterovNes21, Reference NesterovNes23]. Recall the embedding (5), which also holds for rigidified stacks,

\begin{align*}\mathfrak{N}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}}) \hookrightarrow \mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}});\end{align*}

we thus endow $Q_k^\epsilon ( {{\mathsf{a}},{{\mathsf{w}}}})$ with the obstruction theory given by the complex

(8) \begin{align}\mathbb{T}_{Q_k^\epsilon ( {{\mathsf{a}},{{\mathsf{w}}}})}^{{\rm{vir}}}: = {\pi _{\rm{*}}}{f^{\rm{*}}}\mathbb{T}_{\mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})}^{{\rm{vir}}}.\end{align}

Its perfectness is proven in the same vein as in [Reference NesterovNes21, Reference NesterovNes23].

Let us indicate what moduli spaces $Q_k^\epsilon ( {{\mathsf{a}},{{\mathsf{w}}}})$ are for the extremal values of $\epsilon $ . Using the same notation as in (1), if $\epsilon = - $ , we get

\begin{align*}Q_0^ - ( {{\mathsf{a}},{{\mathsf{w}}}}) \cong Q( {{\mathsf{a}},{{\mathsf{w}}}}).\end{align*}

If $\epsilon = + $ , then

\begin{align*}Q_0^ + ( {{\mathsf{a}},{{\mathsf{w}}}}) & = {\rm{pt\;\;\;\;if}}\ {\mathsf{w}} = 0;\\[4pt]Q_0^ + ( {{\mathsf{a}},{{\mathsf{w}}}}) & = \emptyset {\rm{\;\;\;\;\;\;\;if}}\ {\mathsf{w}} \ne 0.\end{align*}

We now discuss the wall-crossing between invariants associated to different values of $\epsilon $ .

Definition 4.3. Let $GQ({\mathsf{a},\mathsf{w}})$ be a moduli space of prestable quasimaps of degree $\mathsf{w}$ ,

\begin{align*}f:{\mathbb{P}^1} \to \mathfrak{N}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}}) = {\mathfrak{N}_{{\rm{rg}},E}}( {\mathsf{a}}) \times C,\end{align*}

such that $\infty \in {\mathbb{P}^1}$ is mapped to the stable locus. Consider a $\mathbb{C}_z^{\rm{*}}$ -action on the source ${\mathbb{P}^1}$ with weight $1$ at $0 \in {\mathbb{P}^1}$ . It thus induces a $\mathbb{C}_z^{\rm{*}}$ -action on $GQ( {{\mathsf{a}},{{\mathsf{w}}}})$ . We define

\begin{align*}{W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}) \subset GQ( {{\mathsf{a}},{{\mathsf{w}}}})\end{align*}

to be the $\mathbb{C}_z^{\rm{*}}$ -fixed locus.

As in the case of $Q_k^\epsilon ( {{\mathsf{a}},{{\mathsf{w}}}})$ , we endow $GQ( {{\mathsf{a}},{{\mathsf{w}}}})$ with the obstruction theory given by the complex

\begin{align*}\mathbb{T}_{GQ( {{\mathsf{a}},{{\mathsf{w}}}})}^{{\rm{vir}}}: = {\pi _{\rm{*}}}{f^{\rm{*}}}\mathbb{T}_{\mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})}^{{\rm{vir}}},\end{align*}

using the embedding $\mathfrak{N}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}}) \hookrightarrow \mathfrak{M}_{{\rm{rg}},E}^{{\rm{rel}}}( {\mathsf{a}})$ . The space ${W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ inherits the obstruction theory defined by the fixed part of the obstruction theory of $GQ( {{\mathsf{a}},{{\mathsf{w}}}})$ , as well as the virtual normal bundle ${N^{{\rm{vir}}}}$ defined by the moving part of the obstruction theory.

4.2 Moduli spaces of flags

As before, there exists a canonical identification of moduli spaces

(9) \begin{align}{W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}) = W( {{\mathsf{a}},{{\mathsf{w}}}}) \times C,\end{align}

where $W( {{\mathsf{a}},{{\mathsf{w}}}})$ is the analogous space defined via quasimaps to ${\mathfrak{N}_E}( {\mathsf{a}})$ . By [Reference OberdieckObe21, Section 4], the space $W( {{\mathsf{a}},{{\mathsf{w}}}})$ admits a description in terms of moduli spaces of flags, which we now recall. In what follows, by $G$ we denote the unique stable sheaf of degree ${\rm{a}}$ with a fixed determinant supported on the zero section in ${K_E}$ . We also define

\begin{align*}{\bf{z}}: = {\rm{weight}}\ 1\ {\rm{representation\ of}}\ \mathbb{C}_z^{\rm{*}}\ {\rm{on}}\ \mathbb{C}.\end{align*}

Let

\begin{align*}{\rm{Fl}}( {\mathsf{a}}) = \left\{ {{F_1} \subseteq {F_2} \subseteq \ldots \subset {F_{r - 1}} \subseteq {F_r} = G} \right\}\end{align*}

be a moduli space of flags such that consecutive terms are allowed to be equal. To each ${F_ \bullet }$ and a choice of an integer $k \in \mathbb{Z}$ , we can associated a ${\mathbb{C}^{\rm{*}}}$ -equivariant, torsion-free sheaf ${\mathcal{F}}$ on ${K_E} \times {\mathbb{A}^1}$ ,

(10) \begin{align}{\mathcal{F}} = {F_1}{{\bf{z}}^{k + 1}} \oplus {F_2}{{\bf{z}}^{k + 2}} \oplus \ldots {F_{r - 1}}{{\bf{z}}^{k + r - 1}} \oplus G{{\bf{z}}^{k + r}} \oplus G{{\bf{z}}^{k + r + 1}} \ldots \end{align}

In fact, for torsion-free sheaves $G$ , such association is an equivalence between ${\mathbb{C}^{\rm{*}}}$ -fixed torsion-free sheaves on ${K_E} \times {\mathbb{A}^1}$ and weighted flags (up to a choice of $k$ ). Moreover, each ${\mathbb{C}^{\rm{*}}}$ -fixed sheaf in $W( {{\mathsf{a}},{{\mathsf{w}}}})$ is canonically ${\mathbb{C}^{\rm{*}}}$ -equivariant. Let us denote by

\begin{align*}{\rm{Fl}}( {{\mathsf{a}},{{\mathsf{w}}}}) \subset {\rm{Fl}}( {\mathsf{a}})\end{align*}

the locus of flags that correspond to sheaves in $W( {{\mathsf{a}},{{\mathsf{w}}}})$ . By construction, we have

(11) \begin{align}W( {{\mathsf{a}},{{\mathsf{w}}}}) \cong {\rm{Fl}}( {{\mathsf{a}},{{\mathsf{w}}}}).\end{align}

Analogously, let ${\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ be the relative moduli space of flags of ${(\pi _E^{\rm{*}}G)_{|C \times E}}$ on the relative surface ${\pi _C}:{K_{C \times E}} \to C$ . Viewing quotients of $G$ as quotients of a sheaf on $E$ , on ${K_E}$ or $C$ -relatively on ${K_{C \times E}}$ is equivalent. Hence, identification (11) also holds relatively:

(12) \begin{align}{W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}) \cong {\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}) \cong {\rm{Fl}}( {{\mathsf{a}},{{\mathsf{w}}}}) \times C.\end{align}

Let

\begin{align*}{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}) \subset {\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})\end{align*}

be the connected component corresponding to Quot schemes; that is, flags with $r = 2$ . By

\begin{align*}{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}{({\mathsf{a}},{{\mathsf{w}}})^c} \subset {\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}),\end{align*}

we denote its complement. We define

\begin{align*}{{\mathcal{Q}}^{{\rm{rel}}}}: = {{\mathcal{F}}_2}/{{\mathcal{F}}_1}{\rm{\;\;\;\;and\;\;\;\;}}{{\mathcal{K}}^{{\rm{rel}}}}: = {{\mathcal{F}}_1}\end{align*}

to be the universal quotient and the universal kernel of ${\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ , respectively. Quot schemes of stable sheaves on smooth curves are smooth; hence, so are ${\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ by (12).

4.3 Obstruction theory of flags

By [Reference OberdieckObe21, Section 4], obstruction theories of moduli spaces ${W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ and ${\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ also agree. We now describe the obstruction theory of ${\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ and the associated virtual normal bundle ${N^{{\rm{vir}}}}$ . Let

\begin{align*}{{\mathcal{F}}_1} \subseteq {{\mathcal{F}}_2} \subseteq \ldots \subseteq {{\mathcal{F}}_r} = G\end{align*}

be the universal flag on ${K_{C \times E}}\,{ \times _C}\,{\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ , and let

\begin{align*}\pi :{K_{C \times E}}\,{ \times _C}\,{\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}) \to {\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})\end{align*}

be the natural projection.

Theorem 4.4 The $C$ -relative obstruction theory of ${\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ is given by the complex

\begin{align*}\mathbb{T}_{{\rm{Fl}}{{({\mathsf{a}},{{\mathsf{w}}})}^{{\rm{rel}}}}}^{{\rm{vir}}} = {\rm{Cone}}\!\left( {\mathop {\mathop \oplus \limits_{i = 1} }\limits^{i = r - 1} R{\mathcal{H}}o{m_\pi }\!\left( {{{\mathcal{F}}_i},{{\mathcal{F}}_i}} \right) \to \mathop {\mathop \oplus \limits_{i = 1} }\limits^{i = r - 1} R{\mathcal{H}}o{m_\pi }\!\left( {{{\mathcal{F}}_i},{{\mathcal{F}}_{i + 1}}} \right)} \right).\end{align*}

The $K$ -class of ${N^{{\rm{vir}}}}$ is

\begin{align*} N^{\mathrm{vir}}= &-\sum_{i\geqslant 1}\sum_{k\geqslant 1} R {\mathcal H} om_\pi({\mathcal F}_{i+k}/{\mathcal F}_{i+k-1},{\mathcal F}_i) \mathbf z^{k} \\[3pt] &+ \sum_{i\geqslant 1}\sum_{k\geqslant 1} R {\mathcal H} om_\pi({\mathcal F}_{i+k+1}/{\mathcal F}_{i+k},{\mathcal F}_i)^{\vee}\mathbf z^{-k}. \end{align*}

The next lemmas will be useful for the analysis presented in Section 4.6 and in Section 5.

Lemma 4.5. We have the following identity in the K-group,

\begin{align*}R{\mathcal{H}}o{m_\pi }\!\left( {{{\mathcal{F}}_{j + 1}}/{{\mathcal{F}}_j},{{\mathcal{F}}_i}} \right) = {\mathsf{K}}( {1 - {\omega _C}{\bf{t}}}),\end{align*}

for some $K$ -class ${\mathsf{K}}$ .

Proof. Let us denote ${\mathcal{A}}: = {{\mathcal{F}}_{j + 1}}/{{\mathcal{F}}_j}$ and ${\mathcal{B}}: = {{\mathcal{F}}_i}$ . Both ${\mathcal{A}}$ and ${\mathcal{B}}$ are scheme-theoretically supported on the zero section $C \times E \subset {K_{C \times E}}$ ; they can therefore be extended to the entire ${K_{C \times E}}$ by pulling them back by the projection ${K_{C \times E}} \to C \times E$ . We denote these extensions by $\bar {\mathcal{A}}$ and $\bar {\mathcal{B}}$ .

Consider now the sequence on ${K_{C \times E}}$ ,

\begin{align*}0 \to {\mathcal{O}}\!\left( { - C \times E} \right) \to {{\mathcal{O}}_{{K_{C \times E}}}} \to {{\mathcal{O}}_{C \times E}} \to 0.\end{align*}

We tensor it with $\bar {\mathcal{A}}$ ,

\begin{align*}0 \to \bar {\mathcal{A}}\!\left( { - C \times E} \right) \to \bar {\mathcal{A}} \to {\mathcal{A}} \to 0,\end{align*}

and then we apply $R{\mathcal{H}}o{m_\pi }\!\left( { -, {\mathcal{B}}} \right)$ to obtain the distinguished triangle

(13) \begin{align}R{\mathcal{H}}o{m_\pi }( {{\mathcal{A}},{\mathcal{B}}}) \to R{\mathcal{H}}o{m_\pi }\!\left( {\bar {\mathcal{A}},{\mathcal{B}}} \right) \to R{\mathcal{H}}o{m_\pi }\!\left( {\bar {\mathcal{A}}\!\left( { - C \times E} \right),{\mathcal{B}}} \right) \to .\end{align}

There is a natural $\mathbb{C}_t^{\rm{*}}$ -equivariant identification

(14) \begin{align}{{\mathcal{O}}_{{K_{C \times E}}}}{( - C \times E)_{|C \times E}} \cong \omega _{C \times E}^ \vee {{\bf{t}}^{ - 1}} \cong \omega _C^ \vee {{\bf{t}}^{ - 1}},\end{align}

which gives us that

\begin{align*}R{\mathcal{H}}o{m_\pi }\!\left( {\bar {\mathcal{A}}\!\left( { - C \times E} \right),{\mathcal{B}}} \right) \cong R{\mathcal{H}}o{m_\pi } ( {\bar {\mathcal{A}},{\mathcal{B}}} )\boxtimes {\omega _C}{\bf{t}}.\end{align*}

Passing to the $K$ -group, the distinguished triangle (13) therefore gives us that

\begin{align*}R{\mathcal{H}}o{m_\pi }( {{\mathcal{A}},{\mathcal{B}}}) = R{\mathcal{H}}o{m_\pi }\!\left( {\bar {\mathcal{A}},{\mathcal{B}}} \right)( {1 - {\omega _C}{\bf{t}}}).\end{align*}

This proves the claim.

Lemma 4.6. With respect to the identification (12), the obstruction bundle of a Quot scheme $\mathrm{Quo}{t^{\mathrm{rel}}}({\mathsf{a},\mathsf{w}})$ leads to the following expression:

\begin{align*}{\rm{O}}{{\rm{b}}_{{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})}} \cong {T_{{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})}}\boxtimes {\omega _C}{\bf{t}}.\end{align*}

Proof. Assume $r = 2;$ then

\begin{align*}\mathbb{T}_{{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}\left( {{\mathsf{a}},{{\mathsf{w}}}}\right)}^{{\rm{vir}}} = R{\mathcal{H}}o{m_\pi }\big( {{{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big).\end{align*}

Using the same distinguished triangle (13) and passing to the associated long exact sequence, we obtain

\begin{align*}0 & \to {\mathcal{H}}o{m_\pi }\big( {{{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big) \to {\mathcal{H}}o{m_\pi }\big( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big) \to {\mathcal{H}}o{m_\pi }\big( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}}\!\left( { - C \times E} \right),{{\mathcal{Q}}^{{\rm{rel}}}}} \big) \to\\& \to {\mathcal{E}}xt_\pi ^1\big( {{{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big) \to {\mathcal{E}}xt_\pi ^1\big( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big) \to {\mathcal{E}}xt_\pi ^1\big( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}}\!\left( { - C \times E} \right),{{\mathcal{Q}}^{{\rm{rel}}}}} \big) \to . \ldots \end{align*}

Since $Q$ is scheme-theoretically supported on the zero section, the map ${\mathcal{H}}o{m_\pi }\big( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big) \to {\mathcal{H}}o{m_\pi }( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}}\!\left( { - C \times E} \right),{{\mathcal{Q}}^{{\rm{rel}}}}})$ is zero; hence, the map

\begin{align*}{\mathcal{H}}o{m_\pi }\big( {{{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big) \to {\mathcal{H}}o{m_\pi }\big( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big)\end{align*}

is an isomorphism. This also implies that

\begin{align*}{\mathcal{H}}o{m_\pi }( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}}\!\left( { - C \times E} \right),{{\mathcal{Q}}^{{\rm{rel}}}}}) \to {\mathcal{E}}xt_\pi ^1\big( {{{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big)\end{align*}

is injective. Since we are considering quotients of the stable sheaf ${\rm{ex}}{{\rm{t}}^2}\left( {K,Q} \right) = 0$ , hence

\begin{align*}{\rm{ex}}{{\rm{t}}^0}\left( {K,Q} \right) - {\rm{ex}}{{\rm{t}}^1}\left( {K,Q} \right) = {\rm{ch}}\!\left( {{K^ \vee }} \right) \cdot {\rm{ch}}(Q) = 0,\end{align*}

we conclude that ${\mathcal{H}}o{m_\pi }( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}}\!\left( { - C \times E} \right),{{\mathcal{Q}}^{{\rm{rel}}}}}) \to {\mathcal{E}}xt_\pi ^1\big( {{{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big)$ is in fact an isomorphism. Using the $\mathbb{C}_t^{\rm{*}}$ -equivariant identification (14), we obtain that

\begin{align*}{\mathcal{E}}xt_\pi ^1\big( {{{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big) & \cong {\mathcal{H}}o{m_\pi }( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}}\!\left( { - C \times E} \right),{{\mathcal{Q}}^{{\rm{rel}}}}})\\&\cong {\mathcal{H}}o{m_\pi }\big( {{{\bar {\mathcal{K}}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big) \boxtimes {\omega _C}{\bf{t}} \cong {\mathcal{H}}o{m_\pi } \big( {{{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big) \boxtimes{\omega _C}{\bf{t}}.\end{align*}

The sheaf ${\mathcal{H}}o{m_\pi }\big( {{{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}} \big)$ is the $C$ -relative tangent bundle of ${\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ , and due to the decomposition (12), its $C$ -relative tangent bundle is exactly the pull-back of the tangent bundle of ${\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})$ .

4.4 Cosections

Cosections of the obstruction theory of ${\rm{Fl}}{({\mathsf{a}},{{\mathsf{w}}})^{{\rm{rel}}}}$ are constructed in exactly the same way as in [Reference Pandharipande and ThomasPT16, Section 5.4] (see also [Reference OberdieckObe21, Section 4] and [Reference NesterovNes21, Section 10.2]). However, since we work relative to $C$ , the cosections map is not to a trivial line bundle but to ${\omega _C}$ . For example, this can already be seen in Lemma 4.6. This is because the relative canonical sheaf of

\begin{align*}{K_{C \times E}} \to C\end{align*}

is the pull-back of $\omega _C^ \vee $ . By the argument from [Reference NesterovNes21, Section 10.2], which uses Serre’s duality, we therefore get a $C$ -relative cosection to $\omega _C^{ \oplus 2}$ instead of the trivial bundle ${{\mathcal{O}}^{ \oplus 2}}$ (here, we use the identification (9)):

\begin{align*}\sigma = \left( {{\sigma _1},{\sigma _2}} \right):{h^1}\left( {T_{{\rm{Fl}}{{({\mathsf{a}},{{\mathsf{w}}})}^{{\rm{rel}}}}}^{{\rm{vir}}}} \right) \to \omega _C^{ \oplus 2}{\bf{t}},\end{align*}

where

\begin{align*}{\bf{t}}: = {\rm{weight}}\ 1\ {\rm{representation\ of}}\ \mathbb{C}_t^{\rm{*}}\ {\rm{on}}\ \mathbb{C}.\end{align*}

As in [Reference NesterovNes21, Proposition 10.6], we have the following result:

By the description of the obstruction theory of ${W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ in terms of flags from [Reference OberdieckObe21, Section 4], we can compute its virtual dimension. Indeed, for any two sheaves ${F_1}$ and ${F_2}$ supported on the zero section of ${K_E}$ , we have

(15) \begin{align} \sum_i (-1)^i\mathrm{ext}^i(F_1, F_2) ={\mathrm{ch}}(F_1^{\vee}) \cdot {\mathrm{ch}}(F_2) =0,\end{align}

and the $C$ -relative virtual dimension of ${\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ is therefore 0. Hence the absolute virtual dimension of ${\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ is 1.

Both the virtual normal bundle and the cosections are ${{\rm{\Phi }}_{\mathsf{a}}}$ -equivariant by the construction; hence, they descend to the quotients

\begin{align*}[ {{\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} ].\end{align*}

This, in conjunction with Proposition 4.8, implies that the virtual fundamental cycles of quotients $\left[ {{\rm{F}}{{\rm{l}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]$ , when restricted to Quot schemes and to their complements, are of the following corollary:

Corollary 4.9. We have

\begin{align*}& {{{[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}]}^{{\rm{vir}}}}} {}{ = {\mathsf{B}} \boxtimes [ {{\rm{pt}}}] \in {H_0}\!\left( {W( {{\mathsf{a}},{{\mathsf{w}}}}) \times C,\mathbb{Q}} \right)[t],}\\& {{{[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})^{c}/{{\rm{\Phi }}_{\mathsf{a}}}]}^{{\rm{vir}}}}} {}{ = {t\mathsf{B}}^{\prime} \boxtimes [ {{\rm{pt}}}] \in {H_2}\!\left( {W( {{\mathsf{a}},{{\mathsf{w}}}}) \times C,\mathbb{Q}} \right)[t],}\end{align*}

4.5 Master space

Let ${\epsilon _0} \in {\mathbb{Q}_{ \gt 0}}$ be a wall of $\epsilon $ -stabilities for quasisections, and let ${\epsilon ^ + }$ and ${\epsilon ^ - }$ be the values close to the wall ${\epsilon _0}$ from the right-hand side and from the left-hand side, respectively. Consider the master space $M{Q^{{\epsilon _0}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ for the wall-crossing around the wall ${\epsilon _0};$ we refer to [Reference ZhouZho22, Section 4] for the construction of the master space. Let ${{{\mathsf{w}}}_0} = 1/{\epsilon _0}$ . By construction, there is a $\mathbb{C}_z^{\rm{*}}$ -action on $M{Q^{{\epsilon _0}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ . In what follows, we use the identification

\begin{align*}M_E^{{\rm{rel}}}( {\mathsf{a}}) = \left\{ {\left( {G,0} \right)} \right\} \times C = C.\end{align*}

Following the terminology of [Reference ZhouZho22], we define ${\tilde Q^{{\epsilon ^ + }}}( {{\mathsf{a}},{{\mathsf{w}}}})$ to be the pull-back of $Q_k^{{\epsilon ^ + }}( {{\mathsf{a}},{{\mathsf{w}}}})$ to the moduli space of entangled semistable degenerations ${\tilde{\mathfrak{M}}_{C,k,{{\mathsf{w}}}}}$ , which are constructed in [Reference ZhouZho22, Section 2.2] as a blow-up of the moduli space of weighted semistable degenerations ${\mathfrak{M}_{C,k,{{\mathsf{w}}}}}$ ,

\begin{align*}\tilde Q_k^{{\epsilon ^ + }}( {{\mathsf{a}},{{\mathsf{w}}}}): = Q_k^{{\epsilon ^ + }}( {{\mathsf{a}},{{\mathsf{w}}}}){ \times _{{\mathfrak{M}_{C,k,{{\mathsf{w}}}}}}}\ {\tilde{\mathfrak{M}}_{C,k,{{\mathsf{w}}}}}.\end{align*}

We also define $\tilde Q_k^{{\epsilon ^ + }}( {{\mathsf{a}},{{\mathsf{w}}}})^{\prime}$ to be a $k$ -root stack of $\tilde Q_k^{{\epsilon ^ + }}( {{\mathsf{a}},{{\mathsf{w}}}})$ associated to the calibration bundle $\mathbb{M}$ of $\tilde Q_k^{{\epsilon ^ + }}( {{\mathsf{a}},{{\mathsf{w}}}})$ , defined in [Reference ZhouZho22, Section 2.8]. By the analysis of [Reference ZhouZho22, Section 6], the $\mathbb{C}_z^{\rm{*}}$ -fixed locus of $M{Q^{{\epsilon _0}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ then has the following form,

(16) \begin{align} MQ^{\epsilon_0}({\mathsf{a}}, \mathsf{w} )^{{\mathbb{C}}_z^*} = \widetilde{Q}^{\epsilon^+}({\mathsf{a}}, \mathsf{w} ) \cup Q^{\epsilon^-}({\mathsf{a}}, \mathsf{w} )\cup \coprod_{k } \!\left(\widetilde{Q}_{k}^{\epsilon^+}({\mathsf{a}}, \mathsf{w} _1)^{\prime} \times_{C^k} W^{\mathrm{rel}}({\mathsf{a}},\mathsf{w} _0)^k \right),\end{align}

such that ${{\mathsf{w}}} = {{{\mathsf{w}}}_1} + k{{{\mathsf{w}}}_0}$ .

The group ${{\rm{\Phi }}_{\mathsf{a}}}$ acts on the master space $M{Q^{{\epsilon _0}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ . Since the action of ${{\rm{\Phi }}_{\mathsf{a}}}$ and $\mathbb{C}_z^{\rm{*}}$ on $M{Q^{{\epsilon _0}}}{({\mathsf{a}},{{\mathsf{w}}})^{}}$ commute, the operations of taking the quotient by ${{\rm{\Phi }}_{\mathsf{a}}}$ and taking the $\mathbb{C}_z^{\rm{*}}$ -fixed locus also commute. We therefore obtain

(17) \begin{align} \left[MQ^{\epsilon_0}({\mathsf{a}}, \mathsf{w} )^{{\mathbb{C}}_z^*}/\Phi_{\mathsf{a}}\right]=\widetilde{Q}^{\epsilon^+}({\mathsf{a}},\mathsf{w} )^\bullet \cup Q^{\epsilon^-}({\mathsf{a}}, \mathsf{w} )^\bullet\cup \coprod_{k} \!\left( \left[ \widetilde{Q}_k^{\epsilon^+}({\mathsf{a}}, \mathsf{w} _1)^{\prime} \times_{C^k} W^{\mathrm{rel}}({\mathsf{a}},\mathsf{w} _0)^k/\Phi_{\mathsf{a}}\right] \right),\end{align}

such that the action on the wall-crossing components (the components on the right-hand side in the expression above) is given by the diagonal action of ${{\rm{\Phi }}_{\mathsf{a}}}$ . By [Reference ZhouZho22, Section 6], the wall-crossing formula is obtained by taking residues of the localisation formula associated with (17). Let ${{\mathcal{N}}^{{\rm{vir}}}}$ be the virtual normal bundle of wall-crossing components. The wall-crossing invariants are therefore given by the following residues:

\begin{align*}{\rm{Re}}{{\rm{s}}_{z = 0}}{\rm{\;\;}}\left( {\frac{{{{\left[ {\tilde Q_k^{{\epsilon ^ + }}\left( {{\mathsf{a}},{{{\mathsf{w}}}_1}} \right)^{\prime}\,{ \times _{{C^k}}}\,{W^{{\rm{rel}}}}{{({\mathsf{a}},{{{\mathsf{w}}}_0})}^k}/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}^{{\rm{vir}}}}}}{{{e_{\mathbb{C}_{z,t}^{\rm{*}}}}\!\left( {{{\mathcal{N}}^{{\rm{vir}}}}} \right)}}} \right).\end{align*}

We now show that most of the wall-crossing invariants essentially vanish by the second-cosection argument, except that our cosections are twisted, as explained in Section 4.4, which forces us to work a bit harder to obtain the vanishing.

Theorem 4.10. If ${\epsilon _0} = 1/{{\mathsf{w}}}$ , then

\begin{align*}{\rm{deg}}{\Big[{Q^{{\epsilon ^ + }}}{({\mathsf{a}},{{\mathsf{w}}})^ \bullet }\Big]^{{\rm{vir}}}} - {\rm{deg}}{[{Q^{{\epsilon ^ - }}}{({\mathsf{a}},{{\mathsf{w}}})^ \bullet }]^{{\rm{vir}}}} = {\rm{deg\ Re}}{{\rm{s}}_{z = 0}}\left( {\frac{{{{[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}]}^{{\rm{vir}}}}}}{{{e_{\mathbb{C}_{z,t}^{\rm{*}}}}\!\left( {{{\mathcal{N}}^{{\rm{vir}}}}} \right)}}} \right).\end{align*}

Otherwise,

\begin{align*}{\rm{deg}}{\Big[{Q^{{\epsilon ^ + }}}{({\mathsf{a}},{{\mathsf{w}}})^ \bullet }\Big]^{{\rm{vir}}}} = {\rm{deg}}{[{Q^{{\epsilon ^ - }}}{({\mathsf{a}},{{\mathsf{w}}})^ \bullet }]^{{\rm{vir}}}}.\end{align*}

4.6 Proof of Theorem 4.10

By [Reference ZhouZho22, Section 6], we have to analyse the wall-crossing components in the decomposition (21); see also [Reference NesterovNes21, Section 6], [Reference NesterovNes23, Section 10] and [Reference Lee and WenLW23].

Assuming that $k \geqslant 2$ or ${{{\mathsf{w}}}_1} \ne 0$ , then the space

\begin{align*}X: = \left[\tilde Q_k^{{\epsilon ^ + }}\!\left( {{\mathsf{a}},{{{\mathsf{w}}}_1}} \right)^{\prime}\,{ \times _{{C^k}}}\,{W^{{\rm{rel}}}}\!\left( {\mathsf{a}},{{{\mathsf{w}}}_0}\right){^k}/{{\rm{\Phi }}_{\mathsf{a}}} \right]\end{align*}

has an additional action of ${{\rm{\Phi }}_{\mathsf{a}}}$ coming from any of the components of the product. To distinguish it from the diagonal action of ${{\rm{\Phi }}_{\mathsf{a}}}$ , we denote it by ${{\rm{\Phi }}^{\prime}_{\mathsf{a}}}$ . The obstruction theory of the quotient $\left[ {X/{{\rm{\Phi }}^{\prime}_{\mathsf{a}}}} \right]$ is compatible with the obstruction theory of $X$ ; hence,

\begin{align*}{\pi ^{\rm{*}}} [ X/{{\rm{\Phi }}^{\prime}_{\mathsf{a}}}]^{\rm{vir}} = [X]^{\rm{vir}}\end{align*}

for the quotient map $\pi :X \to \left[ {X/{{\rm{\Phi }}^{\prime}_{\mathsf{a}}}} \right]$ . Moreover, the virtual normal bundles from [Reference ZhouZho22, Section 6] are ${{\rm{\Phi }}^{\prime}_{\mathsf{a}}}$ -equivariant; hence, they descend to the quotient $\left[ {X/{{\rm{\Phi }}^{\prime}_{\mathsf{a}}}} \right]$ . Overall, we obtain that the wall-crossing class is a pull-back of some class ${\rm{A}}$ from the quotient $\left[ {X/{{\rm{\Phi }}^{\prime}_{\mathsf{a}}}} \right]$ ,

\begin{align*}\frac{{\left[ {\tilde Q_k^{{\epsilon ^ + }}\!\left( {{\mathsf{a}},{{{\mathsf{w}}}_1}} \right)^{\prime}\,{ \times _{{C^k}}}\,{W^{{\rm{rel}}}}{{({\mathsf{a}},{{{\mathsf{w}}}_0})}^k}/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}}{{{e_{\mathbb{C}_{z,t}^{\rm{*}}}}\!\left( {{{\mathcal{N}}^{{\rm{vir}}}}} \right)}} = {\pi ^{\rm{*}}}{\mathsf{A}};\end{align*}

that its degree is therefore 0 and that it does not contribute to the wall-crossing formula,

\begin{align*}{\rm{deg\ Re}}{{\rm{s}}_{z = 0}}\left( {\frac{{\left[ {\tilde Q_k^{{\epsilon ^ + }}\left( {{\mathsf{a}},{{{\mathsf{w}}}_1}} \right)^{\prime}\,{ \times _{{C^k}}}\,{W^{{\rm{rel}}}}{{({\mathsf{a}},{{{\mathsf{w}}}_0})}^k}/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}}{{{e_{\mathbb{C}_{z,t}^{\rm{*}}}}\!\left( {{{\mathcal{N}}^{{\rm{vir}}}}} \right)}}} \right) = 0.\end{align*}

It remains for us to determine the contribution of terms

\begin{align*}\Big[ \tilde Q_1^ + \left( {{\mathsf{a}},0} \right)^{\prime}{ \times _C}\,{W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}\Big] = \Big[Q_1^ + ( {{\mathsf{a}},0})\,{ \times _C}\,{W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}} \Big].\end{align*}

Since $Q_1^ + \left( {{\mathsf{a}},0} \right) = C$ , we obtain that

\begin{align*} \Big[ Q_1^ + ( {{\mathsf{a}},0} )\,{ \times _C}\,{W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}\Big] = [{W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}} ].\end{align*}

We will now show that the complement of ${\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}) \subset {W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ does not contribute. Ideally, one would say that this statement follows from the double cosection argument. However, in this case, the cosections are twisted due to the relative setup; hence, one has to do a little bit of additional work. By [Reference ZhouZho22, Lemma 6.5.6] and the dimension constraint, degrees of the following residues are equal,

(18) \begin{align}{\rm{deg\ Re}}{{\rm{s}}_{z = 0}}\left( {\frac{{{{[{W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}]}^{{\rm{vir}}}}}}{{{e_{\mathbb{C}_{z,t}^{\rm{*}}}}\!\left( {{N^{{\rm{vir}}}}} \right)}}} \right) = {\rm{deg\ Re}}{{\rm{s}}_{z = 0}}\left( {\frac{{{{[{W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}]}^{{\rm{vir}}}}}}{{{e_{\mathbb{C}_{z,t}^{\rm{*}}}}\!\left( {{{\mathcal{N}}^{{\rm{vir}}}}} \right)}}} \right),\end{align}

where ${N^{{\rm{vir}}}}$ is the normal bundle of ${W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ inside $GQ( {{\mathsf{a}},{{\mathsf{w}}}})$ , whose expression is given in Theorem 4.4.

We argue that ${\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}{({\mathsf{a}},{{\mathsf{w}}})^c} \subset {W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ does not contribute because the quantity (18) is a multiple of ${t^2}$ . As taking quotient by ${{\rm{\Phi }}_{\mathsf{a}}}$ can be exchanged with taking an insertion, it is enough to show that (18) is a multiple of ${t^2}$ before taking quotient. By Corollary 4.6, we know that ${[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}{({\mathsf{a}},{{\mathsf{w}}})^c}]^{{\rm{vir}}}}$ is a multiple of $t$ , and we therefore have to show that the residue of ${e_{\mathbb{C}_{z,t}^{\rm{*}}}}{({N^{{\rm{vir}}}})^{ - 1}}$ is a multiple of $t$ , too. By Theorem 4.4, the class ${e_{\mathbb{C}_{z,t}^{\rm{*}}}}{({N^{{\rm{vir}}}})^{ - 1}}$ leads to the following expression:

(19) \begin{align} e_{{\mathbb{C}}_{z,t}^*}(N^{\mathrm{vir}})^{-1}&= \prod_{i\geqslant 1, k\geqslant 1} \frac{e_{{\mathbb{C}}_{z,t}^*} ( R {\mathcal H} om_{\pi}({\mathcal F}_{i+k}/{\mathcal F}_{i+k-1},{\mathcal F}_i) \mathbf z^{k})}{e_{{\mathbb{C}}_{z,t}^*} ( R {\mathcal H} om_{\pi}({\mathcal F}_{i+k+1}/ {\mathcal F}_{i+k},{\mathcal F}_i)^{\vee} \mathbf z^{-k})} .\end{align}

According to (15), the rank of $R{\mathcal{H}}o{m_\pi }\!\left( {{{\mathcal{F}}_{i + k}}/{{\mathcal{F}}_{i + k - 1}},{{\mathcal{F}}_i}} \right)$ is 0; hence,

\begin{align*}e_{{\mathbb{C}}_{z,t}^*} ( R {\mathcal H} om_{\pi}({\mathcal F}_{i+k} /{\mathcal F}_{i+k-1},{\mathcal F}_i) \otimes \mathbf z^{k}) &=\sum_{j\geqslant 0}(kz)^{-j}\mathrm{c}_j(R {\mathcal H} om_{\pi}({\mathcal F}_{i+k} /{\mathcal F}_{i+k-1},{\mathcal F}_i)) \\ &=1-(kz)^{-1}\mathrm{c}_1(R {\mathcal H} om_{\pi}({\mathcal F}_{i+k+1} /{\mathcal F}_{i+k},{\mathcal F}_i))+ O(z^{-2}), \end{align*}

the same applies to the denominator of (19). We therefore obtain that

(20) \begin{align} e_{{\mathbb{C}}_{z,t}^*}(N^{\mathrm{vir}})^{-1} & = 1-\sum_{i,k} (kz)^{-1}\mathrm{c}_1(R {\mathcal H} om_{\pi}({\mathcal F}_{i+k} /{\mathcal F}_{i+k-1},F_i))\nonumber\\& - \sum_{i,k} (kz)^{-1}\mathrm{c}_1(R {\mathcal H} om_{\pi}({\mathcal F}_{i+k+1} /{\mathcal F}_{i+k},{\mathcal F}_i))^{\vee} + O(z^{-2}). \end{align}

By Lemma 4.5, we obtain that

\begin{align*}{{\rm{c}}_1}(R{\mathcal{H}}o{m_\pi }\!\left( {{{\mathcal{F}}_j}/{{\mathcal{F}}_{j - 1}},{{\mathcal{F}}_i}} \right) = {\mathsf{A}}t + {\mathsf{A}} \cdot {{\rm{c}}_1}( {{\omega _C}}) \in {H^2}\left( {{\rm{Quo}}{{\rm{t}}^c},\mathbb{Q}} \right),\end{align*}

for some class ${\mathsf{A}}$ of cohomological degree 0. Using Corollary 4.6 and (20), we conclude that

\begin{align*}{\rm{Re}}{{\rm{s}}_{z = 0}}\left( {\frac{{{{[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}{{({\mathsf{a}},{{\mathsf{w}}})}^c}]}^{{\rm{vir}}}}}}{{{e_{\mathbb{C}_{z,t}^{\rm{*}}}}\!\left( {{N^{{\rm{vir}}}}} \right)}}} \right) = {\mathsf{A}^{\prime}}{t^2} \in {H_2}\!\left( {{\rm{Quo}}{{\rm{t}}^c},\mathbb{Q}} \right)[t],\end{align*}

for some class ${\mathsf{A}^{\prime}}$ of homological degree 2. Taking the degree of the class above, we obtain 0. This finishes the proof of Theorem 4.10.

4.7 Contributions from Quot schemes

We now have to determine the contributions of ${\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}) \subset {W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ . Firstly, by [Reference OberdieckObe21, Section 4], the component ${\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}) \subset {W^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})$ is composed of the following Quot schemes

(21) \begin{equation} \mathrm{Quot}^{\mathrm{rel}}({\mathsf{a}},\mathsf{w} ) = \coprod_{{\mathsf{m}} | \mathsf{w} }\mathrm{Quot}^{\mathrm{rel}}({\mathsf{a}}, u_{\mathsf{m}}),\end{equation}

and the classes ${u_{\rm{m}}}$ are defined as follows:

\begin{align*}u_{\mathsf{m}}:=h_{\mathsf{m}}{\mathsf{v}}-\frac{\check{\mathsf{w} }}{{\mathsf{m}}},\end{align*}

where $\check{\mathsf{w}} $ is given by (7) and ${h_{\mathsf{m}}}$ is the unique integer such that

\begin{align*}h_{\mathsf{m}}{\mathsf{r}} -\frac{\check{\mathsf{w} }_1}{{\mathsf{m}}}\in [0, {\mathsf{r}}-1].\end{align*}

We therefore obtain the following proposition:

Proposition 4.11. We have

\begin{align*}[\mathrm{Quot}^{\mathrm{rel}}({\mathsf{a}},\mathsf{w} )/\Phi_{\mathsf{a}}]^{\mathrm{vir}}= \sum_{{\mathsf{m}} | \mathsf{w} } [\mathrm{Quot}^{\mathrm{rel}}({\mathsf{a}},u_{\mathsf{m}})/\Phi_{\mathsf{a}}]^{\mathrm{vir}}.\end{align*}

Let us analyse ${N^{{\rm{vir}}}}$ over each component $\left[ {{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}\!\left( {{\mathsf{a}},{u_{\rm{m}}}} \right)\!/{{\rm{\Phi }}_{\mathsf{a}}}} \right]$ . In what follows, we use the notation from Section 5. By [Reference OberdieckObe21, Section 4.4], the equivariant Euler class of the virtual normal bundle ${N^{{\rm{vir}}}}$ can be expressed as follows:

\begin{align*}{e_{\mathbb{C}_{z,t}^{\rm{*}}}}{({N^{{\rm{vir}}}})^{ - 1}} & = {e_{\mathbb{C}_{z,t}^{*}}}\!\big( {R{\mathcal{H}}o{m_\pi }\big( {{{\mathcal{Q}}^{{\rm{rel}}}},{{\mathcal{K}}^{{\rm{rel}}}}} \big){{\bf{z}}^{\mathsf{m}}}} \big)\\& = {e_{\mathbb{C}_{z,t}^{\rm{*}}}}\!\big(R{\mathcal{H}}o{m_\pi }\big( {{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}\big){^ \vee }{\bf{t}}{{\bf{z}}^{\mathsf{m}}} \big)\\ {}&=\sum_{k \in {\mathbb{Z}}}({\mathsf{m}} z)^{-k}\mathrm{c}_{k}(R{\mathcal H} om_\pi({\mathcal K}^{\mathrm{rel}},{\mathcal Q}^{\mathrm{rel}})^{\vee}\mathbf t),\end{align*}

where ${{\mathcal{K}}^{{\rm{rel}}}}$ and ${{\mathcal{Q}}^{{\rm{rel}}}}$ are as in Section 4.2. We are interested in the residue of ${e_{\mathbb{C}_{z,t}^{\rm{*}}}}{({N^{{\rm{vir}}}})^{ - 1}}$ ,

\begin{align*}{\rm{Re}}{{\rm{s}}_{z = 0}}({e_{\mathbb{C}_{z,t}^{\rm{*}}}}\!\left( {{N^{{\rm{vir}}}}{)^{ - 1}}} \right) & = {{\mathsf{m}}^{ - 1}}{{\rm{c}}_1}\big(R{\mathcal{H}}o{m_\pi }( {{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}){^ \vee }{\bf{t}} \big)\\&= {{\mathsf{m}}^{ - 1}}{\rm{rk}}\big({\mathcal{H}}o{m_\pi }( {{\mathcal{K}}^{{\rm{rel}}}},{{\mathcal{Q}}^{{\rm{rel}}}}){^ \vee } \big)( {{{\rm{c}}_1}( {{\omega _C}}) + t} )\\&= {{\mathsf{m}}^{ - 1}}\,{\rm{dim}}\!\left( {{\rm{Quot}}( {{\mathsf{a}},{u_{\mathsf{m}}}})} \right)\left( {{{\rm{c}}_1}( {{\omega _C}}) + t} \right).\end{align*}

Using Corollary 4.9, the total residue then takes the following form:

\begin{align*}{\rm{Re}}{{\rm{s}}_{z = 0}}\left( {\frac{{{{[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{u_{\mathsf{m}}}})/{{\rm{\Phi }}_{\mathsf{a}}}]}^{{\rm{vir}}}}}}{{{e_{\mathbb{C}_{z,t}^{\rm{*}}}}\!\left( {{N^{{\rm{vir}}}}} \right)}}} \right) = {{\mathsf{m}}^{ - 1}}\,{\rm{dim}}\!\left( {{\rm{Quot}}( {{\mathsf{a}},{u_{\mathsf{m}}}})} \right){[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{u_{\mathsf{m}}}})/{{\rm{\Phi }}_{\mathsf{a}}}]^{{\rm{vir}}}}t.\end{align*}

Assume that $( {{\mathsf{r}},{\mathsf{a}}}) = ( {2,1});$ using the analysis from Corollaries 5.3 and 5.4, we get

\begin{align*}{\rm{deg\ Re}}{{\rm{s}}_{z = 0}}\left( {\frac{{{{[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{u_{\mathsf{m}}}})/{{\rm{\Phi }}_{\mathsf{a}}}]}^{{\rm{vir}}}}}}{{{e_{{\mathbb{C}^{\rm{*}}}}}\!\left( {{N^{{\rm{vir}}}}} \right)}}} \right) = ( {2g - 2}){{\mathsf{m}}^{ - 1}}t.\end{align*}

Now, applying Theorem 4.10 repeatedly and using the fact that ${Q^ + }( {{\mathsf{a}},{{\mathsf{w}}}})$ is empty for ${{\mathsf{w}}} \ne 0$ , we obtain the following result (for how ${\mathsf{d}}$ and ${{\mathsf{w}}}$ are related, see (7)):

Theorem 4.12. If $( {{\mathsf{r}},{\mathsf{a}}}) = ( {2,1})$ , then

\begin{align*} \mathsf{QM} (E)_{1, \mathsf{w} }^{{\mathsf{d}}, \bullet}=\begin{cases} (2g-2) \sum_{{\mathsf{m}}| \mathsf{w} }{\mathsf{m}}^{-1}t, \quad &\text{ if } \mathsf{w} ={\mathsf{d}} \ \mathrm{mod}\ 2 \\ 0, &\text{ otherwise.} \end{cases}\end{align*}

Using Corollary 3.7, we obtain the desired quasimap invariants.

Theorem 4.13. If $\left( {\mathsf{r},\mathsf{a}} \right) = ( {2,1})$ , then

\begin{align*} \mathsf{QM} (C)_{{\mathsf{d}},\mathsf{w} }^{1, \bullet} = \begin{cases} (2g-2)2^{2g} \sum_{{\mathsf{m}}| \mathsf{w} }{\mathsf{m}}^{-1}t, \quad &\text{ if } \mathsf{w} ={\mathsf{d}} \ \mathrm{mod}\ 2 \\ 0, &\text{ otherwise.} \end{cases} \end{align*}

This gives us Theorems 1.1 and 1.2 (after passing to reduced invariants; that is, after dividing by $t$ ).

4.8 Higher rank

By (7), if we assume that all divisors ${\mathsf{m}}$ of ${{\mathsf{w}}}$ are congruent to $0$ or ${\mathsf{a}}$ modulo ${\mathsf{r}}$ , then

\begin{align*}{u_{\mathsf{m}}} = \left( {0,{\mathsf{k}}} \right){\rm{\;or\;\;}}\left( {{\mathsf{r}} - 1,{\mathsf{k}}} \right);\end{align*}

therefore, using the arguments of the previous section, the analysis of Section 5.2 is enough to conclude the following:

Proposition 4.16. If ${\mathsf{r}}$ is prime, ${\mathsf{a}} \ne 0$ and all divisors ${\mathsf{m}}$ of ${{\mathsf{w}}}$ satisfy

\begin{align*}{\mathsf{m}} = 0\ \rm{or}\ \mathsf{a}\ \mathrm{mod}\ \mathsf{r},\end{align*}

then

\begin{align*}\mathsf{QM} (C)_{{\mathsf{d}},\mathsf{w} }^{{\mathsf{a}}, \bullet} = \begin{cases} (2g-2){\mathsf{r}}^{2g} \sum_{{\mathsf{m}}| \mathsf{w} }{\mathsf{m}}^{-1}t, \quad &\text{ if } \mathsf{w} = {\mathsf{d}} \cdot {\mathsf{a}} \ \mathrm{mod}\ {\mathsf{r}} \\ 0, &\text{ otherwise.} \end{cases} \end{align*}

This agrees with [Reference NesterovNes23, Conjecture E]. Our methods involving Quot schemes lead to an obvious conjectural extension of Proposition 4.16:

Conjecture 4.17. If $\mathrm{gcd}(\mathsf{r}, \mathsf{a}) = 1$ , then

\begin{align*} \mathsf{QM} (C)_{{\mathsf{a}}, \mathsf{w} }^{{\mathsf{d}}, \bullet}=\begin{cases} (2g-2) {\mathsf{r}}^{2g}\sum_{{\mathsf{m}}| \mathsf{w} }{\mathsf{m}}^{-1}t, \quad &\text{ if } \mathsf{w} ={\mathsf{d}} \cdot {\mathsf{a}} \ \mathrm{mod}\ {\mathsf{r}} \\ 0, &\text{ otherwise.} \end{cases} \end{align*}

5.1 Group actions on Quot schemes

The group ${{\rm{\Phi }}_{\mathsf{a}}}$ acts naturally on Quot schemes ${\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})$ . The stabilisers of the action are finite, as long as ${{\mathsf{w}}} \ne 0$ . The quotient stack $\left[ {{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]$ is therefore a Deligne–Mumford stack. Taking the quotient respects the identification (12),

\begin{align*}\big[ {\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}\big] = [{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}} ] \times C.\end{align*}

The obstruction theory of $\left[ {{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]$ is a descent of the obstruction theory of $\left[ {{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})} \right]$ . By Lemma 4.6, the $C$ -relative obstruction bundle of the quotient is therefore given by the descent of ${T_{{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})}} \boxtimes {\omega _C}{\bf{t}}$ . More precisely, let

\begin{align*}q:{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}}) \to \big[ {{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \big]\end{align*}

be the quotient map; then we have the ${{\rm{\Phi }}_{\mathsf{a}}}$ -equivariant identification

\begin{align*}{q^{\rm{*}}}{\rm{O}}{{\rm{b}}_{\left[ {{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}} \cong {\rm{O}}{{\rm{b}}_{{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})}},\end{align*}

such that ${\rm{O}}{{\rm{b}}_{{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})}}$ is ${{\rm{\Phi }}_{\mathsf{a}}}$ -equivariantly isomorphic to ${T_{{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})}}\boxtimes {\omega _C}{\bf{t}}$ . We will denote the descent of ${T_{{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})}}$ to $\left[ {{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]$ by the same symbol. Hence, by Lemma 4.6, we obtain the following corollary:

Corollary 5.1. There is a natural identification on $\left[ {\mathrm{Quot}^{\mathrm{rel}}({\mathsf{a},\mathsf{w}})/{\Phi _{\mathsf{a}}}} \right]$ ,

\begin{align*}{\rm{O}}{{\rm{b}}_{\left[ {{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}} \cong {T_{{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})}} \boxtimes {\omega _C}{\bf{t}}.\end{align*}

We are now ready to determine the virtual degree.

Proposition 5.2. For any ${\mathsf{r}}$ and ${\mathsf{a}}$ , we have

\begin{align*}{\rm{deg}}\big[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}\big]^{{\rm{vir}}} = ( {2g - 2})e\!\left( {{T_{\left[ {{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}}} \right).\end{align*}

Proof. Considering the map

\begin{align*}[ {\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}] \to [{\rm{pt}}/{{\rm{\Phi }}_{\mathsf{a}}}],\end{align*}

the associated sequence of tangent complexes takes the following form:

\begin{align*}0 \to {T_{{{\rm{\Phi }}_{\mathsf{a}}}}} \to {T_{{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})}} \to {T_{\left[ {{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}} \to 0.\end{align*}

Since ${T_{{{\rm{\Phi }}_{\mathsf{a}}}}}$ is trivial, we obtain that

\begin{align*}e\left( T_{[{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}}) ]}\right) & = 0,\\{{\rm{c}}_{{\mathsf{rk}} - 1}}\left( T_{[{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}}) ]}\right) & = e\!\left( {{T_{\left[ {{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}}} \right).\end{align*}

Using Corollary 5.1, we therefore obtain

\begin{align*}{[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}]^{{\rm{vir}}}} &= {e_{\mathbb{C}_t^{\rm{*}}}}\!\left( {{\rm{O}}{{\rm{b}}_{\left[ {{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}}} \right)\\&= e\!\left( {{T_{{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})}}} \right) + {{\rm{c}}_{{\rm{rk}} - 1}}\!\left( {T_{[{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}}) ]}}\right) \cdot ( {{{\rm{c}}_1}( {{\omega _C}} ) - t} ) + \ldots\\&= e\!\left( {{T_{\left[ {{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}}} \right) \cdot ( {{{\rm{c}}_1}( {{\omega _C}} ) - t} ) + \ldots .\end{align*}

Taking the degree, we arrive at the statement of the proposition,

\begin{align*}{\rm{deg}}{[{\rm{Quo}}{{\rm{t}}^{{\rm{rel}}}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}]^{{\rm{vir}}}} = ( {2g - 2})e\!\left( {{T_{\left[ {{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}}} \right).\end{align*}

We now have to compute $e\!\left( {{T_{\left[ {{\rm{Quot}}( {{\mathsf{a}},{{\mathsf{w}}}})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}}} \right)$ . The analysis might be split, depending on the Chern character of quotients in the decomposition (21).

5.2 Relevant Quot schemes

For this section we assume that ${\mathsf{a}} \ne 0$ . Consider firstly the class

\begin{align*}u = ( {{\mathsf{r}} - 1,{\mathsf{k}}}).\end{align*}

Let

\begin{align*}{\mathsf{dim}}: = {\mathrm{dim}}\!\left( {{\rm{Quot}}( {{\mathsf{a}},u})} \right) = {\mathsf{r}} \cdot ( {{\mathsf{k}} - {\mathsf{a}}}) + {\mathsf{a}};\end{align*}

then ${\rm{Quot}}( {{\mathsf{a}},u})$ is a ${\mathbb{P}^{{\mathsf{dim}} - 1}}$ -bundle over ${\rm{Jac}}(E)$ given by the natural projection

\begin{align*}{\rm{Quot}}( {{\mathsf{a}},u}) & \to {\rm{Pic}}(E)\\ {}\,[K \hookrightarrow G \twoheadrightarrow Q] &\mapsto K.\end{align*}

A fiber

\begin{align*}{\mathbb{P}^{{\mathsf{dim}} - 1}} \hookrightarrow {\rm{Quot}}( {{\mathsf{a}},u})\end{align*}

is a slice of the ${{\rm{\Phi }}_{\mathsf{a}}}$ -action on ${\rm{Quot}}( {{\mathsf{a}},u})$ . In other words, let ${{\rm{\Gamma }}_{\mathsf{k}}} \subset {{\rm{\Phi }}_{\mathsf{a}}}$ be a finite subgroup that fixes ${\mathbb{P}^{{\mathsf{dim}} - 1}}$ , then we have the following diagram:

The diagram above gives us that

(22) \begin{align}e\!\left( {{T_{\left[ {{\rm{Quot}}( {{\mathsf{a}},u})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}}} \right) = \frac{{\chi \!\left( {{\mathbb{P}^{{\mathsf{dim}} - 1}}} \right)}}{{\left| {{{\rm{\Gamma }}_{\mathsf{k}}}} \right|}} = \frac{{{\mathsf{dim}}}}{{\left| {{{\rm{\Gamma }}_{\mathsf{k}}}} \right|}}.\end{align}

It therefore remains to determine ${{\rm{\Gamma }}_{\mathsf{k}}}$ . The subgroup ${{\rm{\Gamma }}_{\mathsf{k}}} \subset {{\rm{\Phi }}_{\mathsf{a}}}$ that fixes the slice is exactly the subgroup that fixes a line bundle ${\mathcal{O}}( {{\mathsf{a}} - {\mathsf{k}}})$ of degree ${\mathsf{a}} - {\mathsf{k}}$ . A translation of ${\mathcal{O}}( {{\mathsf{a}} - {\mathsf{k}}})$ by ${\tau _p}$ that is associated to a point $p \in E$ can be described as follows:

\begin{align*}\tau _p^{\rm{*}}{\mathcal{O}}( {{\mathsf{a}} - {\mathsf{k}}}) = {\mathcal{O}}( {{\mathsf{a}} - {\mathsf{k}}}) \otimes L_p^{{\mathsf{a}} - {\mathsf{k}}},\end{align*}

where ${L_p}$ is line bundle corresponding to $p$ under the natural identification

\begin{align*}E & \mathop\to\limits^{\cong} {\rm{Jac}}(E)\\p & \mapsto {\mathcal{O}}\!\left( {{0_E}} \right) \otimes {\mathcal{O}}{(p)^{ - 1}} = :{L_p}.\end{align*}

By the definition of ${{\rm{\Phi }}_{\mathsf{a}}}$ from (2), determining the stabiliser of ${\mathcal{O}}( {{\mathsf{a}} - {\mathsf{k}}})$ in ${{\rm{\Phi }}_{\mathsf{a}}}$ therefore amounts to finding pairs

\begin{align*}\left( {p,L_p^{\frac{{\mathsf{a}}}{{\mathsf{r}}}}} \right) \in E \times {\rm{Jac}}(E)\end{align*}

such that

(23) \begin{align}L_p^{{\mathsf{a}} - {\mathsf{k}}} \otimes L_p^{ - \frac{{\mathsf{a}}}{{\mathsf{r}}}} = {{\mathcal{O}}_E}.\end{align}

Raising the expression to the power of ${\mathsf{r}}$ , we conclude that $L_p^{{\mathsf{dim}}} = {{\mathcal{O}}_E}$ . As a group, $E$ is isomorphic to $\mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}$ ; hence, with respect to the identification $E \cong \mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}$ , we obtain that

\begin{align*}{L_p} = \left( {\frac{{{n_1}}}{{{\mathsf{dim}}}},\frac{{{n_2}}}{{{\mathsf{dim}}}}} \right) \in \mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z},\end{align*}

and that $\mathsf{r}$ -roots of ${a_i} = \frac{{{n_i}}}{{{\mathsf{dim}}}}$ are given by the following elements:

\begin{align*}a_i^{\frac{1}{{\mathsf{r}}}} = \frac{{{n_i} + h{\mathsf{dim}}}}{{{\mathsf{rdim}}}},{\rm{\;\;\;\;}}h \in \left\{ {0, \ldots, {\mathsf{r}} - 1} \right\}.\end{align*}

However, only one satisfies the equation

\begin{align*}\frac{{( {{\mathsf{a}} - {\mathsf{k}}}){n_i}}}{{{\mathsf{dim}}}} - \frac{{{\mathsf{a}}\!\left( {{n_i} + h{\mathsf{dim}}} \right)}}{{{\mathsf{rdim}}}} = 0 \in \frac{\mathbb{R}}{\mathbb{Z}},\end{align*}

more specifically, $h$ is uniquely defined by the following equation:

\begin{align*}\frac{{\left( {{n_i} + h{\mathsf{a}}} \right){\mathsf{dim}}}}{{{\mathsf{rdim}}}} = 0 \in \mathbb{R}/\mathbb{Z}.\end{align*}

We therefore conclude that for all $p \in E$ , there exists a unique root $L_p^{\frac{{\mathsf{a}}}{{\mathsf{r}}}}$ that satisfies the equation (23) and that it must be ${\mathsf{dim}}$ -torsion; hence $p$ is also ${\mathsf{dim}}$ -torsion. We therefore obtain that

\begin{align*}{{\rm{\Gamma }}_{\mathsf{k}}} = E\!\left[ {{\mathsf{dim}}} \right] \cong \mathbb{Z}_{{\mathsf{dim}}}^{ \oplus 2},\end{align*}

or, in particular, that

\begin{align*}\left| {{{\rm{\Gamma }}_{\mathsf{k}}}} \right| = {\mathsf{dim}}^2.\end{align*}

Proposition 5.2 and (22) give us the following:

Corollary 5.3. If $u = ( {\mathsf{r} - 1,\mathsf{k}})$ , then

\begin{align*}{\rm{deg}}([{\rm{Quot}}( {{\mathsf{a}},u{)^{{\rm{rel}}}}/{{\rm{\Phi }}_{\mathsf{a}}}{]^{{\rm{vir}}}}} ) = ( {2g - 2}){\rm{dim}}{({\rm{Quot}}( {{\mathsf{a}},u}))^{ - 1}}.\end{align*}

Consider now the class

\begin{align*}u = ( {0,{\mathsf{k}}}).\end{align*}

In this case,

\begin{align*}{\rm{dim}}\!\left( {{\rm{Quot}}( {{\mathsf{a}},u})} \right) = {\mathsf{r}} \cdot {\mathsf{k}},\end{align*}

and ${\rm{Quot}}( {{\mathsf{a}},u})$ admits the natural projection

\begin{align*}{\rm{Quot}}( {{\mathsf{a}},u}) & \to {\rm{Pic}}(E)\\\,[G \twoheadrightarrow Q ]& \mapsto {\rm{det}}(Q),\end{align*}

which provides a slice of the action of ${{\rm{\Phi }}_{\mathsf{a}}}$ on ${\rm{Quot}}( {{\mathsf{a}},u})$ . More specifically, let

\begin{align*}{\rm{Quot}}{({\mathsf{a}},u)_0} \hookrightarrow {\rm{Quot}}( {{\mathsf{a}},u})\end{align*}

be the fibre of the projection, and let ${{\rm{\Gamma }}_{\mathsf{k}}} \subset {{\rm{\Phi }}_{\mathsf{a}}}$ be the stabiliser of ${\rm{Quot}}{({\rm{a}},u)_0}$ . We obtain that

(24) \begin{align}e\!\left( {{T_{\left[ {{\rm{Quot}}( {{\mathsf{a}},u})/{{\rm{\Phi }}_{\mathsf{a}}}} \right]}}} \right) = \frac{{\chi ({\rm{Quot}}\!\left( {{\mathsf{a}},u{)_0}} \right)}}{{\left| {{{\rm{\Gamma }}_{\mathsf{k}}}} \right|}}.\end{align}

The subgroup ${{\rm{\Gamma }}_{\mathsf{k}}} \in {{\rm{\Phi }}_{\mathsf{a}}}$ that fixes the slice is exactly the subgroup that fixes the determinant of a $0$ -dimension sheaf of degree ${\mathsf{k}}$ ; this means that it consists of pairs

\begin{align*}\left( {p,L_p^{\frac{{\mathsf{a}}}{{\mathsf{r}}}}} \right) \in E \times {\rm{Jac}}(E),\end{align*}

such that

\begin{align*}L_p^{\mathsf{k}} = {{\mathcal{O}}_E},\end{align*}

hence,

\begin{align*}\left| {{{\rm{\Gamma }}_{\mathsf{k}}}} \right| = {{\mathsf{r}}^2} \cdot {{\mathsf{k}}^2}.\end{align*}

Let us now determine the Euler characteristics of ${\rm{Quot}}{({\mathsf{a}},u)_0}$ . Firstly, any vector bundle on a curve can be deformed to a direct sum of line bundles. By the deformation invariance of the (virtual) Euler characteristics, we can assume that $G = \oplus _{i = 1}^{i = {\mathsf{r}}}{L_i}$ . In this case, ${\rm{Quot}}{({\mathsf{a}},u)_0}$ admits a torus-action of $T= \prod^{i={\mathsf{r}} }_{i=1} {\mathbb{C}}^*$ acting by scaling line bundles. The associated fixed locus has the following description:

\begin{align*} \mathrm{Quot}({\mathsf{a}}, u )_0^T= \coprod_{u_1\substack{+\dots +}u_{\mathsf{r}}=u} \left( \prod^{i={\mathsf{r}}}_{i=1} \mathrm{Quot}(L_i,u_i)\right)_0, \end{align*}

we refer to [Reference Monavari and RicolfiMR22, Section 3] for more details in case of usual Quot schemes, which extend in a straightforward manner to our slices. Let us now analyse Euler characteristics of the components in the decomposition above. Firstly, if at least two class ${u_k}$ and ${u_j}$ are nonzero, then $\left( \prod_i \mathrm{Quot}(L_i,u_i)\right)_0$ admits an extra fixed-point-free action of $E$ ; therefore,

\begin{align*} \chi\!\left( \left(\prod^{i={\mathsf{r}}}_{i=1}\mathrm{Quot}(L_i,u_i)\right)_0\right)=0, \quad \text{ if }u_k\neq 0 \text{ and } u_j \neq 0 . \end{align*}

If only one class ${u_j}$ is nonzero, then

\begin{align*}\left( \prod^{i={\mathsf{r}}}_{i=1} \mathrm{Quot}(L_i,u_i)\right)_0= \mathrm{Quot}(L_{j},u_{j})_0= {\mathbb{P}}^{{\mathsf{k}}-1}.\end{align*}

We therefore obtain

\begin{align*} \chi\!\left( \left( \prod^{i={\mathsf{r}}}_{i=1}\mathrm{Quot}(L_i,u_i)\right)_0\right)={\mathsf{k}}, \quad \text{ if only one } u_{j}\neq 0. \end{align*}

Overall,

(25) \begin{align}\chi ({\rm{Quot}}\!\left( {{\mathsf{a}},u{)_0}} \right) = e({\rm{Quot}}\!\left( {{\mathsf{a}},u)_0^T} \right) = {\mathsf{r}} \cdot {\mathsf{k}}.\end{align}

Combining Proposition 5.2, (24) and (25), we obtain the following:

Corollary 5.4. If $u = \left( {0,\mathsf{k}} \right)$ , then

\begin{align*}{\rm{deg}}([{\rm{Quot}}( {{\mathsf{a}},u{)^{{\rm{rel}}}}/{{\rm{\Phi }}_{\mathsf{a}}}{]^{{\rm{vir}}}}}) = ( {2g - 2}){\rm{dim}}{({\rm{Quot}}( {{\mathsf{a}},u}))^{ - 1}}.\end{align*}