2.1 Quasimap invariants

We define the quasimap invariants in this section following [Reference Abramovich, Graber and Vistoli1, Reference Cheong, Ciocan-Fontanine and Kim9, Reference Ciocan-Fontanine and Kim11, Reference Ciocan-Fontanine, Kim and Maulik14]. Consider an algebraic torus T action on W, which commutes with the given $G-$ action on W. Here, T can be the identity group. Assume further that the $T-$ fixed loci $\underline {X} _0^T$ of the affine quotient $\underline {X}_0=\text {Spec}(\mathbb {C}[W]^{G})$ is $0-$ dimensional. We also denote $K:=\mathbb {Q}(\{\lambda _i\})$ by the rational localized T-equivariant cohomology of $\operatorname {\mathrm {Spec}} \mathbb {C}$ , with $\{\lambda _1,\dots ,\lambda _{\text {rank}(T)}\}$ corresponding to a basis for the characters of T. Denote

$$\begin{align*}\Lambda _K := K [[\text{Eff}(W, G, \theta ) ]] \end{align*}$$

to be the corresponding Novikov ring. We write $q^\beta $ for the element corresponding to $\beta $ in $\Lambda _K$ so that $\Lambda _K$ is the q-adic completion.

Given any two elements $\alpha _{1},\alpha _{2}$ in the $T-$ equivariant Chen-Ruan cohomology of X,

$$ \begin{align*}H^*_{\text{CR}, T} (X, \mathbb{Q}):=H ^*_T(\bar{I}_{\mu}X , \mathbb{Q} ) ,\end{align*} $$

we can define the Poincaré pairing in the non-rigidified cyclotomic inertia stack $I_\mu X$ of X:

$$ \begin{align*}\langle \alpha_{1} , \alpha_{2} \rangle _{\text{orb}} := \int _{\sum _{r\in \mathbb{N}_{\geq 1} } r^{-1} [\bar{I}_{\mu _r}X]} \alpha_{1} \cdot \iota ^* \alpha_{2} .\end{align*} $$

Here, $\iota $ is the involution of $\bar {I}_{\mu }X $ obtained from the inversion automorphisms. Therefore, the diagonal class $[\Delta _{\bar {I}_{\mu _r}X}] $ obtained via pushforward of the fundamental class by $(\text {id}, \iota ): \bar {I}_{\mu _r}X \rightarrow \bar {I}_{\mu _r}X \times \bar {I}_{\mu _r}X$ can be written as

$$ \begin{align*}\sum _{r=1}^{\infty} r[ \Delta _{\bar{I}_{\mu _r}X}] = \sum _{\alpha} \phi_{\alpha} \otimes \phi^{\alpha} \text{ in } H^*(\bar{I}_{\mu}X \times \bar{I}_{\mu}X, \mathbb{Q}),\end{align*} $$

where $\{ \phi _{\alpha }\}$ is a basis of $H^*_{\text {CR}, T} (X, \mathbb {Q})$ with $\{\phi ^{\alpha }\}$ the dual basis with respect to the Poincaré pairing defined above.

Denote by $\bar {\psi }_i$ the first Chern class of the universal cotangent line whose fiber at $((C, q_1, ..., q_m), [x])$ is the cotangent space of the coarse moduli $\underline {C}$ of C at i-th marking $\underline {q} _i$ . For nonnegative integers $a_i$ and classes $\alpha _i \in H ^*_T (\bar {I}_{\mu }X , \mathbb {Q})$ , $\delta _{j}\in H_{T}^{*}(\mathfrak {X},\mathbb {Q})$ , we write

$$ \begin{align*} \langle \alpha _1\bar{\psi} ^{a_1}, ..., \alpha _m \bar{\psi}^{a_m};\delta_{1},\cdots,\delta_{p} \rangle^{\theta',\boldsymbol{\varepsilon}} _{0, m|p, \beta} & := \int _{[Q^{\theta',\boldsymbol{\varepsilon}}_{0,m|p}(\mathfrak{X}, \beta )]^{\text{vir} }} \prod _i ev _i ^* (\alpha _i) \bar{\psi} ^{a_i}_i \prod_{j}\hat{ev}_{j}^{*}(\delta_{j}). \end{align*} $$

When $\varepsilon $ is empty, $\theta '=\epsilon \theta $ for a integer character $\theta $ , we will also write this as

$$ \begin{align*}\langle \alpha _1\bar{\psi} ^{a_1}, ... , \alpha _m\bar{\psi}^{a_m}\rangle^{\epsilon}\ ;\end{align*} $$

when $\epsilon $ is sufficiently large, the above formula recovers the usual Gromov-Witten invariants, in which case, we simply write this as

$$ \begin{align*}\langle \alpha _1\bar{\psi} ^{a_1}, ... , \alpha _m\bar{\psi}^{a_m}\rangle.\end{align*} $$

We will also need the morphism

(2.2) $$ \begin{align} (\widetilde {ev _j})_* = \iota _*(\boldsymbol{r} _{j}(ev_{j})_*),\end{align} $$

where $\boldsymbol {r} _{j}$ is the order function of the band of the gerbe structure at the marking $q_{j}$ . Define a class in $H_*^T(\bar {I}_{\mu }X )\cong H^*_T(\bar {I}_{\mu }X )$ by

(2.3) $$ \begin{align} \langle \alpha _1,..., \alpha _m, - \rangle ^{\epsilon}_{0, m+1,\beta} &:=(\widetilde{ev_{m+1}})_* \left((\prod ev _i ^* \alpha _i) \cap [Q^{\epsilon}_{0, m+1}(X, \beta)]^{\text{vir} }\right)\qquad \end{align} $$

(2.4) $$ \begin{align} &\qquad=\sum _{\alpha}\phi^{\alpha} \langle\alpha_1,...,\alpha_m,\phi_{\alpha}\rangle^{\epsilon}_{0,m+1,\beta}. \end{align} $$

3.1 Two special cases of the mirror theorem

Using Corollary 3.4, we consider two interesting special cases of the I-function. The first case is when Y is a hypersurface with respect to a line bundle $L:=L_{\tau }$ for some character $\tau $ . The mirror formula (1.1) becomes

(3.8)

Here, $\text {exp}$ is short for $\text {exp}\big (\frac {1}{z}\sum _{i=1}^{l}t_{i}u_{i}(c_{1}(L_{\pi _j})+\beta (L_{\pi _j})z)\big )$ , and $\big [[Y^{ss}_{\beta }/(G/\langle {g^{-1}_{\beta }} \rangle )]\big ]$ is the fundamental class of $[Y^{ss}_{\beta }/(G/\langle {g^{-1}_{\beta }} \rangle )]$ in $H^{*}(\bar {I}_{g_{\beta }^{-1}}Y )$ . Note that we only use Corollary 3.4 in the first and third summand of (3.8), while in the second summand, we use the fact that the Gysin pullback $s^{!}_{E_{\beta },loc}$ is just the identity morphism, as the vector bundle $E_{\beta }$ is of rank zero when $\beta (L)\in \mathbb {Z}_{<0}$ .

Remark 3.7. The reader may wonder whether we can express the cohomology class $\big [[Y^{ss}_{\beta }/(G/\langle {g^{-1}_{\beta }} \rangle )]\big ]$ as the product of and $D_{\rho }$ like in other cases. Note that this will in particular imply that $\big [[Y^{ss}_{\beta }/(G/\langle {g^{-1}_{\beta }} \rangle )]\big ]$ is an ambient cohomology class (i.e., a cohomology class pulled back from the Chen-Ruan cohomology $H^{*}(\bar {I}_{\mu }X)$ of the ambient toric stack). However, $\big [[Y^{ss}_{\beta }/(G/\langle {g^{-1}_{\beta }} \rangle )]\big ]$ is not an ambient cohomology class in general. For example, let $X=\mathbb {P}^{3}$ and Y be a quadratic hypersurface of X. We will choose a GIT presentation of X and degree $\beta $ such that $\big [[Y^{ss}_{\beta }/(G/\langle {g^{-1}_{\beta }} \rangle )]\big ]$ can be the line $\{[0,*,*,0]\in \mathbb {P}^{3}\}$ . To achieve this, we choose a non-standard GIT presentation of $\mathbb {P}^{3}$ : Let $W=\mathbb {C}^{5}$ , $G=(\mathbb {C}^{*})^{2}$ so that G acts on W via the right action

$$ \begin{align*}(x_{1},x_{2},x_{3},x_{4},x_{5})\cdot (t_{1},t_{2})= (t_{1}x_{1},t_{1}t_{2}x_{2},t_{1}t_{2}x_{3},t_{1}x_{4},t_{2}x_{5}) ,\end{align*} $$

where $(x_{1},x_{2},x_{3},x_{4},x_{5})\in W$ and $(t_{1},t_{2})\in G$ . If we choose the stablity condition $\theta (t_{1},t_{2})=t_{1}t_{2}^{2}\in \chi (G)$ , we have $W^{ss}(\theta )=(\mathbb {C}^{4}\backslash \{0\})\times \mathbb {C}^{*}$ . Let Y be the quadratic hypersurface cut off by the polynomial $x_{1}x_{2}-x_{3}x_{4}$ , and we choose degree $\beta \in \text {Eff}(W,G,\theta )$ defined by $\beta (L_{t_{1}})=-1$ and $\beta (L_{t_{2}})=1$ . It is a very interesting question to use this to calculate the GW invariants with insertion of non-ambient cohomology classes, and we will explain how to do it elsewhere.

The second case is when all the line bundles $L_{\tau _{b}}$ are all semi-positive (i.e., $\beta (L_{\tau _{b}})\geq 0$ for all $\beta \in \text {Eff}(W,G,\theta )$ and b). Then the I-function specializes to

(3.9)

The above formulae match the formula for positive hypersurfaces in toric stacks for which the convexity holds [Reference Coates, Corti, Iritani and Tseng18, §5] and the formula for a ray divisor (given by a coordinate function corresponding to the ray) of a toric stack for which the convexity may fail [Reference Coates, Corti, Iritani and Tseng18, Reference Cheong, Ciocan-Fontanine and Kim9]. See §7 for a non-positive example where the convexity fails.

4.1 Construction of master space I

In this section, we will construct a master space which is a root stack modification of the twisted graph space considered in [Reference Clader, Janda and Ruan15]. Let $(AY,G,\theta )$ be the GIT data which gives rise to a complete intersection in the toric stack $X=[W^{ss}(\theta )/ G]$ as in previous sections. Since a positive rational scaling of the stability character $\theta $ will not change the GIT quotient. Without loss of generality, let’s assume that the line bundle $L_{\theta }$ on $Y=[AY^{ss}(\theta )/G]$ is the pullback of an ample line bundle on the coarse moduli space $\underline {Y}$ of Y. First, we will consider the following quotient stack

$$ \begin{align*}\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p}=[(AY \!\times\! \mathbb{C}^{p} \!\times\!\mathbb{C}^2)/(G \!\times\! (\mathbb{C}^{*})^{p}\!\times\!\mathbb{C}^{*})]\end{align*} $$

defined by the following (right) action

$$ \begin{align*}(\vec{x},\vec{y},z_{1},z_{2})\cdot (g,h,t)=(\vec{x}\cdot g,(h_{j}y_{j})_{j=1}^{p},\theta(g)^{-1}(\prod_{j=1}^{p}h_{j}^{-1})t^{r}z_{1},tz_{2}) ,\end{align*} $$

where $(g,h=(h_{j})_{j=1}^{p},t)\in G \!\times \! (\mathbb {C}^{*})^{p} \!\times \!\mathbb {C}^{*}$ , $(\vec {x},\vec {y}=(y_{j})_{j=1}^{p},z_{1},z_{2})\in AY \!\times \! \mathbb {C}^{p}\!\times \! \mathbb {C}^{2}$ . For simplicity, we will write $AY_{p}:=AY \!\times \! \mathbb {C}^{p}$ , and $G_{p}:=G \!\times \! (\mathbb {C}^{*})^{p}$ . Let $\theta _{p}$ be the character of $G_{p}$ defined by

$$ \begin{align*}\theta_{p}(g,h)=\theta(g)\prod_{j=1}^{p}h_{j}\; \text{for all}\; (g,h)\in G_{p}.\end{align*} $$

Fix a positive rational number $\epsilon \in \mathbb {Q}_{>0}\cap (0,1]$ . We consider the stability given by the rational character of $G_{p} \!\times \! \mathbb {C}^{*}$ defined by

$$ \begin{align*}\widetilde{\theta}(g,h,t)=\theta_{p}(g,h)^{\epsilon}t^{3r}\end{align*} $$

for $(g,h,t)\in G_{p}\!\times \! \mathbb {C}^{*}$ . Then the GIT stack quotient $[(AY_{p} \!\times \! \mathbb {C}^{2})^{ss}(\widetilde {\theta })/(G_{p} \!\times \! \mathbb {C}^{*})]$ is the root stack of the $\mathbb {P}^{1}-$ bundle $\mathbb {P}_{Y}(L_{-\theta }\oplus \mathbb {C})$ over Y by taking r-th root of the infinity divisor $D_{\infty }$ given by $z_{2}=0$ . We will denote the GIT stack quotient $[(AY_{p} \!\times \! \mathbb {C}^2)^{ss}(\widetilde {\theta })/ (G_{p} \!\times \! \mathbb {C}^{*})]$ to be $\mathbb {P} Y^{\frac {1}{r}}$ , which is equipped with the infinity section $\mathcal {D}_{\infty }$ given by $z_{2}=0$ and the zero section $\mathcal {D}_{0}$ given by $z_{1}=0$ . Note that this GIT quotient is independent of the integer p as the semistable(=stable) loci $(AY_{p} \!\times \! \mathbb {C}^2)^{ss}(\widetilde {\theta })=AY^{ss}(\theta ) \!\times \! (\mathbb {C}^{*})^{p} \!\times \! (\mathbb {C}^{2}\backslash \{0\})$ . We will take $p=0$ as our standard GIT quotient reference for $\mathbb PY^{\frac {1}{r}}$ , which will be canonically identified with other GIT quotients from $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ by choosing the embedding $AY\subset AY_{p}$ as $AY\cong AY_{p}\cap \{y_{i}=1| i=1,\ldots ,p \}$ .

When the integer r is prime to the orders of isotropy groups of all points for X, which happens, in particular, as r is a sufficiently large prime, the rigidified inertia stack $\bar {I}_{\mu } \mathbb {P} Y^{\frac {1}{r}} $ of $\mathbb {P} Y^{\frac {1}{r}}$ can be decomposed as the disjoint union

$$ \begin{align*}\underbrace{ \mathbb{P} (\bar{I}_{\mu} Y)^{\frac{1}{r}}}_{1}\; \bigsqcup \; \underbrace{\sqcup_{j=1}^{r-1} \bar{I}_{\mu}Y}_{2}.\end{align*} $$

Let $(x,(g,t))$ represent a $\mathbb {C}-$ point of $\bar {I}_{\mu }\mathbb {P} Y^{\frac {1}{r}}$ where x is a $\mathbb {C}-$ point of $\mathbb {P} Y^{\frac {1}{r}}$ and $(g,t)\in G\times \mathbb {C}^{*}$ represents an automorphism of x in the isotropy group of x in $\mathbb {P} Y^{\frac {1}{r}}$ . If $(x,(g,t))$ appears in the first factor of the decomposition above, then the element $(g,t)$ is in the subgroup $G \!\times \! \{1\}\subset G \!\times \! \mathbb {C}^{*}$ , and the space $\mathbb {P} (\bar {I}_{\mu } Y)^{\frac {1}{r}}$ can be further decomposed as $\mathbb {P} (\bar {I}_{\mu } Y)^{\frac {1}{r}}=\sqcup _{g\in G} \mathbb {P} (\bar {I}_{g}Y)^{\frac {1}{r}}$ , where $\mathbb {P} (\bar {I}_{g}Y)^{\frac {1}{r}}$ is defined as the quotient stack

$$ \begin{align*}\mathbb{P} (\bar{I}_{g}Y)^{\frac{1}{r}}:=[(AY(\widetilde{\theta})^{g} \!\times\!(\mathbb{C}\backslash \{0\})^2)/((G/\langle{g}\rangle) \!\times\!\mathbb{C}^{*})]\end{align*} $$

via the action similar to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},0}$ as above; if $(x,(g,t))$ occurs in the second factor of the decomposition above, the automorphism $(g,t)$ lies in $G \!\times \! \{\mu _{r}^{j}: 1\leq j\leq r-1\}\subset G \!\times \! \boldsymbol {\mu }_{r}$ , and the point x belongs to the infinity section $\mathcal {D}_{\infty }$ defined by $z_{2}=0$ . Here, $\mu _{r}=\text {exp}(\frac {2\pi \sqrt {-1}}{r})\in \mathbb {C}^{*}$ and $\boldsymbol {\mu }_{r}$ is the cyclic group generated by $\mu _{r}$ .

For $(g,t)\in G \!\times \! \boldsymbol {\mu }_{r}$ , we will use the notation $\bar {I}_{(g,t)}\mathbb {P} Y^{\frac {1}{r}}$ to mean the rigidified inertia stack component of $\bar {I}_{\mu }\mathbb {P} Y^{\frac {1}{r}} $ corresponding to the isotropy element $(g,t)$ .

Consider the moduli stack of $\widetilde {\theta }-$ stable quasimaps to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ :

$$ \begin{align*}Q^{\widetilde{\theta}}_{0,m}(\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p},(d,1^p,\frac{\delta}{r})).\end{align*} $$

More concretely,

$$\begin{align*}Q^{\widetilde{\theta}}_{0,m}(\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p},(d,1^{p},\frac{\delta}{r})) = \{(C;q_1, \ldots, q_m; L_1,\cdots,L_{k+p}, N; \vec{x},\vec{y}, z_1, z_2)\},\end{align*}$$

where $(C; q_1, \ldots , q_m)$ is a m-pointed prestable balanced orbifold curve of genus $0$ with possible nontrivial isotropy only at special points, that is, marked gerbes or nodes, the line bundles $(L_j:1\leq j\leq k+p)$ and N are orbifold line bundles on C with

(4.1) $$ \begin{align} \mathrm{deg}([\vec{x}])=d\in Hom(Pic(\mathfrak{Y}),\mathbb{Q}), \; \; \; \mathrm{deg}(N)=\frac{\delta}{r} , \end{align} $$

(4.2) $$ \begin{align} \mathrm{deg}(L_{k+j})=1,\; \; 1 \leq j\leq p , \end{align} $$

and

$$\begin{align*}(\vec{x},\vec{y},\vec{z}) := (x_1, \ldots, x_n,y_{1},\ldots,y_{p}, z_1, z_2) \in \Gamma\,\left(\bigoplus_{i=1}^{n} L_{\rho_{i}}\oplus \bigoplus_{j=1}^{p}L_{k+j} \oplus (L_{-\theta_{p}}\otimes N^{\otimes r}) \oplus N\right).\end{align*}$$

Here, for $1\leq i\leq n$ , the line bundle $L_{\rho _{i}}$ is equal to

$$ \begin{align*}\otimes_{j=1}^{k}L_{j}^{\otimes m_{ij}} ,\end{align*} $$

where $(m_{ij})$ ( $1\leq i\leq n$ , $1\leq j\leq k+p$ ) is given by the unique relation $\rho _{i}=\sum _{j=1}^{k}m_{ij}\pi _{j}$ . The same construction applies to the line bundle $L_{-\theta _{p}}$ on C. Note that here $\delta $ is an integer when $Q^{\widetilde {\theta }}_{0,m}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(d,1^{p},\frac {\delta }{r}))$ is nonempty, as $N^{\otimes r}$ is the pullback of some line bundle on the coarse moduli curve $\underline {C}$ of C.

We require that the following conditions are satisfied for the above data:

• • Representability: For every $q \in C$ with isotropy group $G_q$ , the homomorphism $\mathbb {B}G_q \rightarrow \mathbb {B}(G_{p}\times \mathbb {C}^{*})$ induced by the restriction of line bundles $(L_{j}:1\leq j\leq k+p)$ and N to q is representable. Note that the image of the homomorphism lies in the subgroup $G\!\times \! \mathbb {C}^{*}\subset G_{p}\!\times \! \mathbb {C}^{*}$ .

• • Nondegeneracy: The sections $z_1$ and $z_2$ never simultaneously vanish. Furthermore, for each point q of C at which $z_2(q) \neq 0$ , the stability condition 2.3

  $$ \begin{align*}l_{\widetilde{\theta}}(q)\leq 1\end{align*} $$
  for $\widetilde {\theta }$ -stable map to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ becomes the stability condition
  (4.3) $$ \begin{align} l_{\epsilon\theta_{p}}(q)\leq 1, \end{align} $$
  for the prestable quasimap $[\vec {x},\vec {y}]:C\rightarrow \mathfrak {Y}\!\times \! [\mathbb {C}/\mathbb {C}^{*}]^{p}$ . For each point q of C at which $z_2(q) = 0$ , we have
  (4.4) $$ \begin{align} \text{ord}_q(\vec{x})=\text{ord}_{q}(\vec{y}) = 0. \end{align} $$
  We note that this can be phrased as the length condition (2.1) bounding the order of contact of $(\vec {x},\vec {y}, \vec {z})$ with the unstable loci of $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ as in [Reference Ciocan-Fontanine and Kim13, §2.1].

• • Stability: The $\mathbb {Q}-$ line bundle

  $$ \begin{align*}(\phi_{*}(L_{\theta}))^{\otimes \epsilon}\otimes \bigotimes_{j=1}^{p}\phi_{*}(L_{k+j})^{\otimes \epsilon}\otimes \phi_{*}(N^{\otimes 3r})\otimes \omega_{\underline{C}}^{log}\end{align*} $$
  on the coarse curve $\underline {C}$ is ample. Here, $\phi :C\rightarrow \underline {C}$ is the coarse moduli map. Note that here, the line bundles $L_{\theta }$ , $(L_{k+j})_{j=1}^{p}$ and $N^{\otimes 3r}$ are the pullback of line bundles on the coarse moduli of $\underline {C}$ .

• • Vanishing: The image of $[\vec {x}]:C\rightarrow \mathfrak {X}$ lies in $\mathfrak {Y}$ .

Let $\vec {m}=(v_{1},\cdots ,v_{m})\in (G \!\times \! \boldsymbol {\mu }_{r})^{m}$ . We will denote $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(d,1^p,\frac {\delta }{r}))$ to be

$$ \begin{align*}Q^{\widetilde{\theta}}_{0,m}(\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p},(d,1^p,\frac{\delta}{r}))\cap ev_{1}^{-1}(\bar{I}_{v_{1}}\mathbb{P} Y^{\frac{1}{r}})\cap \cdots \cap ev_{m}^{-1}(\bar{I}_{v_{m}}\mathbb{P} Y^{\frac{1}{r}} ) ,\end{align*} $$

where

$$ \begin{align*} ev_i: Q^{\widetilde{\theta}}_{0,\vec{m}}(\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p},(d,1^p,\frac{\delta}{r}))\rightarrow \bar{I}_{\mu}\mathbb{P} Y^{\frac{1}{r}} \end{align*} $$

are natural evaluation maps by evaluating the sections $(\vec {x},\vec {z})$ at ith marking $q_i$ . Evaluating the section $\vec {x}$ at the vanishing loci of the section $y_{j}$ of the degree one line bundle $L_{k+j}$ for $1\leq j\leq p$ , which corresponds to a smooth non-orbifold point on C (as it must be a base point), one has another tuple of evaluation maps

(4.5) $$ \begin{align} \hat{ev}_{j}: Q^{\widetilde{\theta}}_{0,\vec{m}}(\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p},(d,1^p,\frac{\delta}{r}))\rightarrow \mathfrak{Y} , \end{align} $$

for $1\leq j\leq p$ .

Because $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(d,1^{p},\frac {\delta }{r}))$ is the moduli space of stable quasimaps to a proper lci GIT quotient, it is a proper Deligne-Mumford stack equipped with a natural perfect obstruction theory relative to the Artin stack $\mathfrak {M}^{tw}_{0,m}$ of prestable twisted curves by [Reference Ciocan-Fontanine, Kim and Maulik14]. This relative perfect obstruction theory has the form

(4.6) $$ \begin{align} \mathbb{E}:=R^{\bullet}\pi_{*}(f^*\mathbb{T}_{\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p}}). \end{align} $$

Here, we denote the universal family over $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(d,1^{p},\frac {\delta }{r}))$ by

The obstruction theory (4.6) can be obtained as the cone of the morphism of complexes

(4.7) $$ \begin{align} R^{\bullet}\pi_{*}(\mathcal{O}_{\mathcal{C}}\otimes \mathfrak{g}_{r,p})\rightarrow R^{\bullet}\pi_{*}\big(\mathcal{V}\oplus(\oplus_{j=1}^{p}\mathcal{L}_{k+j})\oplus (\mathcal{L}_{-\theta_{p}}\otimes\mathcal{N}^{\otimes r})\oplus \mathcal{N}\big) , \end{align} $$

which is induced from applying $R^{\bullet }\pi _{*}$ to the distinguished triangle (see [Reference Ciocan-Fontanine, Kim and Maulik14, §5.1]) of the tangent complex $\mathbb T_{\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}}$ of $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$

$$ \begin{align*}AY_{r,p} \times_{G_{r,p}} \mathfrak{g}_{r,p}\rightarrow AY_{r,p}\times_{G_{r,p}}\mathbb T_{AY_{r,p}}\rightarrow \mathbb T_{\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p}}.\end{align*} $$

Here, we use the GIT representation $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}=[AY_{r,p}/G_{r,p}]$ as constructed before, whereFootnote ⁹ $AY_{r,p}:=AY \times \mathbb {C}^{p} \times \mathbb {C}^{2}$ and $G_{r,p}:=G_{p}\!\times \! \mathbb {C}^{*}$ ( $\mathfrak {g}_{r.p}$ is the Lie algebra of $G_{r,p}$ ). Here, $\mathcal {L}_{j}$ ( $1\leq j\leq k+p$ ) and $\mathcal {N}$ are the universal line bundles over the universal curve $\mathcal {C}$ , and

$$ \begin{align*} \mathcal{V} \subset \oplus_{i=1}^{n}\mathcal{L}_{\rho_{i}} \end{align*} $$

is the subsheaf of sections taking values in the affine cone $AY$ of Y. Somewhat more explicitly, the sub-obstruction-theory $\mathbb {E}_{sub}:=R^{\bullet }\pi _{*}(\mathcal {V})$ comes from the deformations and obstructions of the sections $\vec {x}$ . Then $\mathbb {E}_{sub}$ fits into the following distinguished triangle:

(4.8)

Here, $ds=\oplus _{b=1}^{c} ds_{b}$ , where $ds_{b}:R^{\bullet }\pi _{*}(\oplus _{i=1}^{n}\mathcal {L}_{\rho _{i}})\rightarrow R^{\bullet }\pi _{*}\mathcal {L}_{\tau _{b}}$ is induced from the vector bundle map

$$ \begin{align*}\oplus_{i=1}^{n}\mathcal{L}_{\rho_{i}}\rightarrow \mathcal{L}_{\tau_{b}},\end{align*} $$

which sends $\vec {x}=(x_{i})_{i=1}^{n}$ to $s_{b}(\vec {x})$ . We note that we can interpret $R^{\bullet }\pi _{*}(\mathcal {O}_{\mathcal {C}}\otimes \mathfrak {g}_{r,p})$ as the deformation theory of line bundles $(L_{j})_{j=1}^{k+p}$ and N, and interpret the summand $R^{\bullet }\pi _{*}\big ( (\oplus _{j=1}^{p}\mathcal {L}_{k+j})\oplus (\mathcal {L}_{-\theta _{p}} \otimes \mathcal {N}^{\otimes r})\oplus \mathcal {N}\big )$ of $\mathbb {E}$ as the deformation theory of sections $\vec {y}$ and $z_{1},z_{2}$ .

4.2 $\mathbb {C}^{*}$ -action and fixed loci

Consider the (left) $\mathbb {C}^{*}$ -action on $AY_{p} \!\times \! \mathbb {C}^{2}$ defined by

$$ \begin{align*}\lambda(\vec{x},\vec{y},z_{1},z_{2})=(\vec{x},\vec{y},\lambda z_{1},z_{2}).\end{align*} $$

This action descends to be an action on $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ . We will denote $\lambda $ to be the equivariant class corresponding to the $\mathbb {C}^{*}$ -action of weight 1. Let’s first state a criteria for a morphism to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ to be $\mathbb {C}^{*}$ -equivariant (see also [Reference Chang, Li, Li and Liu8, §2.2]), which will be important in the analysis of localization computations.

Remark 4.2. (Equivariant morphism to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ ) Fix a stack S over $Spec(\mathbb {C})$ with a left $\mathbb {C}^{*}$ -action. Then a $\mathbb {C}^{*}$ -equivariant morphism from S to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ is equivalent to the following data: there exists $k+p+1 \mathbb {C}^{*}$ -equivariant line bundles on S

$$ \begin{align*}L_{1},\cdots, L_{k+p},N\end{align*} $$

together with $\mathbb {C}^{*}$ -invariant sections

$$ \begin{align*} (\vec{x},\vec{y}, \vec{z}) &:= (x_1, \ldots, x_n,y_{n+1},\ldots,y_{n+p}, z_1, z_2)\\ &\in \Gamma\left(\oplus_{i=1}^{n} L_{\rho_{i}}\oplus(\oplus_{j=1}^{p}L_{k+j}) \oplus (L_{-\theta_{p}}\otimes N^{\otimes r}\otimes \mathbb{C}_{\lambda}) \oplus N\right)^{\mathbb{C}^{*}}. \end{align*} $$

Here, $L_{\rho _{i}} (1\leq i\leq n)$ and $L_{-\theta _{p}}$ are constructed from $(L_{j})_{1\leq j\leq k+p}$ as explained before, and $\mathbb {C}_{\lambda }$ is the trivial line bundle over S with $\mathbb {C}^{*}-$ linearization of weight 1. These sections should also satisfy the vanishing condition imposed by the affine cone $AY$ of Y as above.

Fix a degree $\beta \in \text {Eff}(W,G,\theta )$ and a tuple of nonnegative integers $(\delta _{1},\cdots ,\delta _{m})\in \mathbb {N}^{m}$ . Consider the tuple of multiplicities $\vec {m}=(v_{1},\cdots ,v_{m})\in (G\!\times \! \boldsymbol {\mu }_{r})^{m}$ , where $v_{i}=(g_{i},\mu _{r}^{\delta _{i}})$ , we will denote $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta ,1^{p},\frac {\delta }{r}))$ to be

$$ \begin{align*}\bigsqcup_{\substack{d\in \text{Eff}(AY,G,\theta)\\ (i_{\mathfrak{Y}})_{*}(d)=\beta }}Q^{\widetilde{\theta}}_{0,\vec{m}}(\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p},(d,1^{p},\frac{\delta}{r})) ,\end{align*} $$

where $i_{\mathfrak {Y}}: \mathfrak {Y}\rightarrow \mathfrak {X}$ is the inclusion morphism. Thus, $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta ,1^{p},\frac {\delta }{r}))$ inherits a $\mathbb {C}^{*}$ -action from the $\mathbb {C}^{*}-$ action on $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ .

We can index the components of $\mathbb {C}^{*}-$ fixed loci of $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta ,1^{p},\frac {\delta }{r}))$ by decorated graphs. A decorated graph $\Gamma $ consists of vertices, edges and m legs, and we decorate it as follows:

• • Each vertex v is associated with an index $j(v) \in \{0, \infty \}$ , a degree $\beta (v) \in \text {Eff}(W,G,\theta )$ and a subset $J_{v}\subset \{1,\cdots ,p\}$ .

• • Each edge $e=\{h,h'\}$ consists of a pair of half edges, and it is equipped with a degree $\beta (e)\in \text {Eff}(W,G,\theta )$ , a subset $J_{e}\subset \{1,\cdots ,p\}$ and $\delta (e) \in \mathbb {Z}_{> 0}$ . Each half edge h (or $h'$ ) is incident to a unique vertex.

• • Each half-edge h and each leg l has an element (called multiplicity) $m(h)$ or $m(l)$ in $G \!\times \! \boldsymbol {\mu }_{r}$ .

• • The legs are labeled with the numbers $\{1, \ldots , m\}$ , and each leg is incident to a unique vertex.

By the ‘valence’ of a vertex v, denoted $\text {val}(v)$ , we mean the total number of incident half-edges and legs.

For any $\mathbb {C}^{*}-$ fixed stable quasimap $f:(C,q_{1},\cdots ,q_{m})\rightarrow \mathbb {P} \mathfrak {Y}^{\frac {1}{r},p} $ over $\mathbb {C}$ in $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta ,1^{p},\frac {\delta }{r}))$ , since the base points on C are isolated and away from special points (i.e., nodes or markings), the image of the generic point of each irreducible component of C under f must lie in

1. 1. the $\mathbb {C}^{*}-$ fixed components of $\mathbb {P} Y^{\frac {1}{r}}$ , which is $\mathcal {D}_{0}$ or $\mathcal {D}_{\infty }$ ;

2. 2. or the generic point of an orbi- $\mathbb {P}^{1}$ fiber of $ \mathbb {P} Y^{\frac {1}{r}}$ over Y.

In the second case, the irreducible component is a genus-zero curve with possible base points on it; we note that if the base point q exists, we have that $z_{2}(q)\neq 0$ by the nondegeneracy condition (4.4) and q is the unique base point on this component. Based on the above observations, we can associate a decorated graph $\Gamma $ to f as follows, where the vertex is either stable or unstable.

• • Each edge e corresponds to a genus-zero irreducible component $C_e$ of C such that it maps constantly to the base Y with possible basepoints on $C_{e}$ , and the generic point of $C_{e}$ maps to the generic point of a fiber of $\mathbb {P} Y^{\frac {1}{r}}$ over Y. Then the decorated degrees $\delta (e), \beta (e)$ and $J_{e}$ are determined by the conditions $\deg (N|_{C_{e}}) = \frac {\delta (e)}{r}$ , $deg(L_{j}|_{C_{e}})=\beta (e)(L_{\pi _{j}})$ ( $1\leq j\leq k$ ), and $deg(L_{k+j}|_{C_{e}})=1$ if and only if $j\in J_{e}$ and $0$ otherwise. We denote $1^{J_{e}}$ to be the degree coming from the line bundles $(L_{k+j}:1\leq j\leq p)$ . There are two distinguished points $q_{0}$ and $q_{\infty }$ on $C_{e}$ such that $q_{\infty }$ is the only point on $C_{e}$ at which $z_{2}$ vanishes, and $q_{0}$ is the only point on $C_{e}$ determined by the following conditions:

  • – if $C_{e}$ has base points on it, then $q_{0}$ is the only base point on $C_{e}$ ;

  • – if $C_{e}$ does not have base points on it, then $q_{0}$ is the only point on $C_{e}$ at which $z_{1}$ vanishes.

  We will also call $q_{0},q_{\infty }$ the $\text {ramification points}$ ,Footnote ¹⁰ and all of degree $(\beta (e),1^{J_{e}})$ is concentrated at the ramification point $q_{0}$ . That is,

  $$\begin{align*}\text{when} \; x_{i}|_{C_{e}}\neq 0,\; \text{we have}\; \text{ord}_{q_{0}}(x_{i})=\beta(e)(L_{\rho_{i}}),\; \text{and}\; \text{ord}_{q_{0}}(y_{i})=1\; \text{if}\; j\in J_{e}.\end{align*}$$
  For each ramification point of $C_{e}$ , we associate a half edge.

• • Each stable vertex v for which $j(v) = 0$ corresponds to a maximal sub-curve $C_v$ of C over which $z_1 \equiv 0$ , and each vertex v for which $j(v) = \infty $ corresponds to a maximal sub-curve $C_{v}$ of C over which $z_2 \equiv 0$ . The label $\beta (v)$ denotes the degree coming from the restriction map $[\vec {x}]|_{C_{v}}$ . Note that here we count the degree $\beta (v)$ in $\text {Eff}(W,G,\theta )$ , but not in $\text {Eff}(AY,G,\theta )$ . The subset $J_{v}$ is equal to the set $\{ j | deg(L_{k+j}|_{C_{v}})=1,\; 1\leq j\leq p\}$ . We denote $1^{J_{v}}$ to be the ordered tuple $(deg(L_{k+j}|_{C_{v}}))_{j=1}^{p}$ . The legs incident to v indicate the marked points on $C_{v}$ .

• • Each unstable vertex v corresponds to a point on $C\backslash (\cup _{v \text { stable}} C_{v})$ which appears as a ramification point on some edge curve $C_{e}$ . In this case, the corresponding point q may be a node at which $C_e$ meets another edge curveFootnote ¹¹ $C_{e'}$ , a marked point of $C_e$ , an unmarked point, or a base point on $C_e$ . Denote $j(v)=0$ if v corresponds to the ramification point $q_{0}$ and $j(v)=\infty $ if v corresponds to the ramification point $q_{\infty }$ . Note that the base point only appears as a vertex v labeled by $0$ due to the nondegeneracy condition for quasimaps. We always set the decorated degree $\beta (v)$ to be zero and $J_{v}=\emptyset $ if v is unstable.

• • The index $m(l)$ on a leg l indicates the rigidified inertia stack component $\bar {I}_{m(l)}\mathbb {P} Y^{\frac {1}{r}}$ of $\mathbb {P} Y^{\frac {1}{r}}$ on which the marked point corresponding to the leg l is evaluated. This is determined by the multiplicity of $L_{1},\cdots , L_{k}, N$ at the corresponding marked point.

• • Let h be a half-edge of an edge e with $q \in C_{e}$ the corresponding ramification point. If q is not a base point, then $m(h)$ indicates the rigidified inertia component $\bar {I}_{m(h)}\mathbb {P} Y^{\frac {1}{r}}$ of $\mathbb {P} Y^{\frac {1}{r}}$ on which the ramification point q associated with h is evaluated. If q is a base point, we take $m(h)=(1,1)\in G \!\times \! \boldsymbol {\mu }_{r}$ .

In particular, we note that the decorations at each stable vertex v yield a tuple

$$ \begin{align*}\vec{m}(v) \in (G\times \boldsymbol{\mu}_{r})^{\text{val}(v)}\end{align*} $$

recording the multiplicities of $L_1,\cdots , L_{k},N$ at every special point of $C_v$ .Footnote ¹² We have the following remarks:

Remark 4.3. The crucial observation, now, is the following. For a stable vertex v such that $j(v) = 0$ , we have $z_1|_{C_v} \equiv 0$ , so the stability condition (4.3) implies that $l_{\epsilon \theta _{p}}(q)\leq 1$ for each $q \in C_v$ . That is, the restriction of $(C; q_1, \ldots , q_m; L_1,\cdots ,L_{k+p};\vec {x},\vec {y})$ to $C_v$ gives rise to a $\epsilon \theta _{p}$ -stable quasimap to the quotient stack $\mathfrak {Y}_{p}:=[AY/G] \times [\mathbb {C}/\mathbb {C}^{*}]^{p} $ (cf. Definition 2.3) in

$$ \begin{align*} Q^{\epsilon\theta_{p}}_{0,\vec{m}(v)}(\mathfrak{Y}_{p},(\beta(v),1^{J_{v}})):=\bigsqcup_{\substack{d\in \text{Eff}(AY,G,\theta)\\ (i_{\mathfrak{Y}})_{*}(d)=\beta(v)}}Q^{\epsilon\theta_{p}}_{0,\vec{m}(v)}(\mathfrak{Y}_{p},(d,1^{J_{v}})). \end{align*} $$

In this case, let $j\in J_{v}$ . The evaluation map considered in (4.5) coincides with $\hat {ev}_{j}$ for $Q^{(\epsilon \theta ,\epsilon ^{|J_{v}|})}_{0,\vec {m}(v)|\; |J_{v}|}(\mathfrak {Y},\beta (v))$ in Remark 2.8.Footnote ¹³ However, for a stable vertex v such that $j(v) = \infty $ , we have $z_2|_{C_v} \equiv 0$ , so the stability condition (4.4) implies that $\text {ord}_q(\vec {x}) =\text {ord}_{q}(\vec {y})=0$ for each $q \in C_v$ . Thus, the restriction of $(C; q_1, \ldots , q_m; L_1,\cdots ,L_{k};\vec {x})$ to $C_v$ gives rise to a usual twisted stable map in

$$ \begin{align*} \mathcal{K}_{0,\vec{m}(v)}(\sqrt[r]{L_{\theta}/Y},\beta(v)):=\bigsqcup_{\substack{d\in \text{Eff}(AY,G,\theta)\\ (i_{\mathfrak{Y}})_{*}(d)=\beta(v)}}\mathcal{K}_{0,\vec{m}(v)}(\sqrt[r]{L_{\theta}/Y},d). \end{align*} $$

Here, $\sqrt [r]{L_{\theta }/Y}$ is the root gerbe of Y by taking r-th root of $L_{\theta }$ .

Remark 4.4. For each edge e, the restriction of $(\vec {x},\vec {y})$ to $C_e$ defines a constant map to Y (possibly with an additional basepoint at the ramification point $q_{0}$ ). So if there is no basepoint on $C_{e}$ , the restriction of $(\vec {x},\vec {y},\vec {z})$ to $C_{e}$ defines a representable map

$$ \begin{align*}C_{e}\rightarrow \mathbb{B} G_{y}\times \mathbb{P}_{r,1},\end{align*} $$

where $y\in Y$ comes from $\vec {x}$ , $G_{y}$ is the isotropy group of $y\in Y$ . Then we have $m(q_{0})=(g^{-1},1)$ and $m(q_{\infty })=(g,\mu _{r}^{\delta (e)})$ for some $g\in G_{y}$ . Note that when r is a sufficiently large prime comparing to $\delta (e)$ , assuming that the order of g is equal to a, we have $C_{e}\cong \mathbb {P}^{1}_{ar,a}$ , and the ramification point $q_{\infty }$ must be a special point. Here, $\mathbb {P}^{1}_{ar,a}$ is the unique Deligne-Mumford stack with coarse moduli $\mathbb {P}^{1}$ with isotropy group $\boldsymbol {\mu }_{a}$ at $0\in \mathbb {P}^{1}$ , isotropy group $\boldsymbol {\mu }_{ar}$ at $\infty \in \mathbb {P}^{1}$ , and generic trivial stabilizer.

If $q_{0}$ is a basepoint of degree $(\beta ,1^{J_{e}})$ (we write $\beta =\beta (e)$ for short), the ramification point $q_{0}$ cannot be an orbifold point; thus, $m(q_{0})=(1,1)\in G\!\times \! \boldsymbol {\mu }_{r}$ . When r is a sufficiently large prime, we have $m(q_{\infty })=(g,\mu _{r}^{\delta (e)})$ for some $g\in G_{y}$ . Let a be the order of g. By the representable condition, we have $C_{e}\cong \mathbb {P}_{ar,1}$ . Note that the restriction of $(\vec {x},\vec {y})$ to $C_{e}$ defines an element in the space $F_{(\beta ,1^{J_{e}})}(Y)$ of the stacky loop space $Q_{\mathbb {P}_{ar,1}}(Y, (\beta ,1^{J_{e}}))$ (see §3; note that here we use the GIT model $[AY_{p}/G_{p}]$ of Y.). Then the restriction of $(\vec {x},\vec {y},\vec {z})$ to $C_{e}$ defines a quasimap f which can be explicitly described as follows. Write $\mathbb {P}_{ar,1}$ as the quotient stack $[U/C^{*}]$ , where $U:=\mathbb {C}^{2}\setminus \{0\}$ and $\mathbb {C}^{*}$ acts on U with weights $[ar,1]$ . We define a map F from U to $AY_{p} \!\times \! U$ to be

$$ \begin{align*} \begin{aligned} (x,y)& \in U \mapsto\\ &\big(\big(x_{1}x^{\beta(L_{\rho_{1}})},\cdots,x_{n}x^{\beta(L_{\rho_{n}})}\big),(x)_{j\in J_{e}},x^{\delta(e)-\beta(L_{\theta})-|J_{e}|},y^{a\delta(e)}\big)\in AY_{p} \!\times\! U. \end{aligned} \end{align*} $$

Here, $(x)_{j\in J_{e}}$ is an element belonging to $\mathbb {C}^{p}$ so that the $j-$ th component is $1$ if $j\notin J_{e}$ and the other component is x. Notice that F is equivariant with respect to the group homomorphism

$$ \begin{align*} t\in \mathbb{C}^{*}_{t} \mapsto\big(t^{ar\beta(L_{\pi_{1}})},\cdots, t^{ar\beta(L_{\pi_{k}})}\big),(t^{ar})_{j\in J_{e}},t^{a\delta(e)}\big)\in G_{p} \!\times\! \mathbb{C}^{*}. \end{align*} $$

Then F descends to be the desired morphism f from $\mathbb {P}_{ar,1}$ to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ . For F to exist, we must have $g=g_{\beta }$ , and $(x_{1},\cdots ,x_{n})$ must belong to the space $Y^{ss}_{\beta }$ defined in §3, thus defining a unique point in the $ F_{(\beta ,1^{J_{e}})}(Y)\cong [Y^{ss}_{\beta }/G]$ . Conversely, when given a point in $ F_{(\beta ,1^{J_{e}})}(Y) $ , we can always construct a unique map in the above way up to $2-$ isomorphisms.

4.3 Localization analysis

Fix $\beta \in \text {Eff}(W,G,\theta )$ and $\delta \in \mathbb {Z}_{\geq 0}$ . We will consider the space $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta ,1^{p},\frac {\delta }{r}))$ . The reason why we assume that the third degree is $\frac {\delta }{r}$ is that $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta ,1^{p},\frac {\delta }{r}))$ corresponds to $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y},(\beta ,\delta )) $ , here $\mathbb {P} \mathfrak {Y}$ is equal to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ for $r=1$ and $p=0$ . In the remaining section, we will always assume that r is a sufficiently large prime.

By virtual localization formula of Graber–Pandharipande [Reference Graber and Pandharipande27], we can write

$$ \begin{align*}[Q^{\widetilde{\theta}}_{0,\vec{m}}(\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p},(\beta,1^{p},\frac{\delta}{r}))]^{ \text{vir}} ,\end{align*} $$

in terms of contributions from each decorated graph $\Gamma $ :

(4.9) $$ \begin{align} [Q^{\widetilde{\theta}}_{0,\vec{m}}(\mathbb{P} \mathfrak{Y}^{\frac{1}{r},p},(\beta,1^{p},\frac{\delta}{r}))]^{\text{vir}} = \sum_{\Gamma} \frac{1}{\mathbb A_{\Gamma}}\iota_{\Gamma*} \left(\frac{[F_{\Gamma}]^{\text{vir}}}{e^{\mathbb{C}^{*}}(N_{\Gamma}^{\text{vir}})}\right). \end{align} $$

Here, for each graph $\Gamma $ , we will associate a space $F_{\Gamma }$ which parameterizesFootnote ¹⁴ $\mathbb {C}^{*}-$ fixed quasimaps of associated graph $\Gamma $ such that the induced morphism $\iota _{\Gamma }:F_{\Gamma } \rightarrow Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta ,1^{p},\frac {\delta }{r})) $ is a finite étale map from $F_{\Gamma }$ into the corresponding open and closed $\mathbb {C}^{*}$ -fixed substack $i_{\Gamma }(F_{\Gamma })$ ; $[F_{\Gamma }]^{\text {vir}}$ is obtained via the $\mathbb {C}^*$ -fixed part of the restriction to the fixed loci of the obstruction theory on $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta ,1^{p},\frac {\delta }{r}))$ and $e^{\mathbb {C}^{*}}(N_{\Gamma }^{\text {vir} })$ is the equivariant Euler class of the $\mathbb {C}^*$ -moving part of this restriction. Besides, $\mathbb A_{\Gamma }$ is the automorphism factor for the graph $\Gamma $ , which represents the degree of $F_{\Gamma }$ into the corresponding open and closed $\mathbb {C}^{*}$ -fixed substack $i_{\Gamma }(F_{\Gamma })$ in $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta ,1^{p},\frac {\delta }{r}))$ . In our case, $\mathbb A_{\Gamma }$ will be the product of the size of the automorphism group $Aut(\Gamma )$ of the graph $\Gamma $ and degrees from each edge moduli $\mathcal {M}_{e}$ over the corresponding fixed loci.

We will do an explicit computation for the contributions of each graph $\Gamma $ as follows. As for the contribution of a graph $\Gamma $ to (4.9), one can first apply the normalization exact sequence

$$ \begin{align*}0\rightarrow \mathcal{O}_{C}\rightarrow \bigoplus_{stable\; vertices}\mathcal{O}_{C_{v}}\oplus \bigoplus_{edges} \mathcal{O}_{C_{e}} \rightarrow \bigoplus_{nodes}\mathcal{O}_{q_{node}} \rightarrow 0\end{align*} $$

to the relative obstruction theory (4.6) and (4.7), which decomposes the contribution from $\Gamma $ to (4.9) into contributions from vertex, edge and node factors. This includes all but the automorphisms and deformations within $\mathcal {M}^{tw}_{0,\vec {m}}$ . The latter are distributed in the vertex, edge and node factors as deformations of the vertex components, deformations of the edge components and deformations of smoothing the nodes, respectively. As a result, for each decorated graph $\Gamma $ , we will associate each stable vertex v (resp. edge e) a moduli space $\mathcal {M}_{v}$ (resp. $\mathcal {M}_{e}$ ) over which there is a family of $\mathbb {C}^{*}-$ fixed stable quasimap to $ \mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ with the decorated degree. Let $F_{\Gamma }$ be the fiber product

(4.10) $$ \begin{align} \prod_{\substack{v\; stable \\j(v)=0}}\mathcal{M}_{v}\times_{\bar{I}_{\mu}\mathcal{D}_{0}}\prod_{e\in E}\mathcal{M}_{e}\times_{\bar{I}_{\mu}\mathcal{D}_{\infty}} \prod_{\substack{v\; stable\\j(v)=\infty}} \mathcal{M}_{v} , \end{align} $$

where the fiber product is taken by gluing the two branches at each nodes; see §4.4 for more details. And we can associate a virtual cycle $[\mathcal {M}_{v}]^{\text {vir} }$ (resp. $[\mathcal {M}_{e}]^{\text {vir}}$ ) to each stable vertex moduli $\mathcal {M}_{v}$ (resp. $\mathcal {M}_{e}$ ). Then we can write $[F_{\Gamma }]^{\text {vir}}$ to be the fiber product:

$$ \begin{align*} \begin{aligned} \prod_{\substack{v\; stable\\j(v)=0}}[\mathcal{M}_{v}]^{\text{vir}} \times_{\bar{I}_{\mu}Y}\prod_{e\in E}[\mathcal{M}_{e}]^{\text{vir}}\times_{\bar{I}_{\mu}\sqrt[r]{L_{\theta}/Y}} \prod_{\substack{v\; stable\\ j(v)=\infty}}[\mathcal{M}_{v}]^{\text{vir}}\; \end{aligned} \end{align*} $$

and we can write $e^{\mathbb {C}^{*}}(N_{\Gamma }^{\text {vir}})$ as the product:

$$ \begin{align*}e^{\mathbb{C}^{*}}(N_{\Gamma}^{\text{vir}}):=\prod_{\text{stable vertics}} e^{\mathbb{C}^{*}}(N_{v}^{\text{vir}}) \cdot\big( \prod_{edges} e^{\mathbb{C}^{*}}(N_{e}^{\text{vir}})\big ) \cdot \prod_{nodes} e^{\mathbb{C}^{*}}(N_{node}^{\text{vir}}) ,\end{align*} $$

where we describe $ e^{\mathbb {C}^{*}}(N_{v}^{\text {vir}})$ , $ e^{\mathbb {C}^{*}}(N_{e}^{\text {vir}})$ and $ e^{\mathbb {C}^{*}}(N_{node}^{\text {vir}})$ in the subsections §4.3.1, §4.3.2 (see also §4.3.3) and §4.3.4 respectively.

4.3.1 Vertex contributions

First of all, the vertex moduli $\mathcal {M}_{v}$ for the stable vertex v over $\infty $ corresponds to the moduli stack $\mathcal {K}_{0,\vec {m}(v)}(\sqrt [r]{L_{\theta }/Y},\beta (v))$ , which parameterizes twisted stable maps to the root gerbe $\sqrt [r]{L_{\theta }/Y}$ over Y.

Let $\pi :\mathcal {C}_{\infty }\rightarrow \mathcal {K}_{0,\vec {m}(v)}(\sqrt [r]{L_{\theta }/Y},\beta (v))$ be the universal curve over $\mathcal {K}_{0,\vec {m}(v)}(\sqrt [r]{L_{\theta }/Y},\beta (v))$ . In this case, on $\mathcal {C}_{\infty }$ , we have $\mathcal {L}_{-\theta }\otimes \mathcal {N}^{\otimes r}\otimes \mathbb {C}_{\lambda }\cong \mathcal {O}_{\mathcal {C}_{\infty }}$ as $z_{1}|_{\mathcal {C}_{\infty }}\equiv 1$ ; hence, we have $\mathcal {N}\cong \mathcal {L}_{\theta }^{\frac {1}{r}}\otimes \mathbb {C}_{-\frac {\lambda }{r}}$ . Here, $\mathcal {L}_{\theta }^{\frac {1}{r}}$ is the line bundle over $\mathcal {C}_{\infty }$ that is the pullback of the universal root bundle over $\sqrt [r]{L_{\theta }/Y}$ along the universal map $f:\mathcal {C}_{\infty }\rightarrow \sqrt [r]{L_{\theta }/Y}$ . The movable part of the perfect obstruction theory comes from the deformation of $z_{2}$ ; thus, the inverse of Euler class $ e^{\mathbb {C}^{*}}(N_{v}^{\text {vir}})$ of the virtual normal bundle is equal to

$$ \begin{align*} e^{\mathbb{C}^{*}}((-R^{\bullet}\pi_{*}\mathcal{L}_{\theta}^{\frac{1}{r}})\otimes \mathbb{C}_{-\frac{\lambda}{r}}). \end{align*} $$

When r is a sufficiently large prime and the multiplicity $m(l)$ corresponding to each leg l incident to v is equal to $(g_{l}, \mu _{r}^{f_{l}})$ for some prefixed number $f_{l}\in \mathbb {Z}_{\geq 0}$ (note this implies $f_{l}\ll r$ ) and $g_{l}\in G$ , following [Reference Janda, Pandharipande, Pixton and Zvonkine31] to the orbifold case, the above Euler class has a representation

(4.11) $$ \begin{align} \sum_{d\geq 0}c_{d}(-R^{\bullet}\pi_{*}\mathcal{L}_{\theta}^{\frac{1}{r}})(\frac{-\lambda}{r})^{|E(v)|-1-d} . \end{align} $$

Here, the virtual bundle $-R^{\bullet }\pi _{*}\mathcal {L}_{\theta }^{\frac {1}{r}}$ has virtual rank $|E(v)|-1$ , where $|E(v)|$ is the number of edges incident to the vertex v. The fixed part of the perfect obstruction theory yields the virtual cycle

$$ \begin{align*}[\mathcal{K}_{0,\vec{m}(v)}(\sqrt[r]{L_{\theta}/Y},\beta(v))]^{\text{vir}}.\end{align*} $$

For the stable vertex v over $0$ , the vertex moduli $\mathcal {M}_{v}$ corresponds to the moduli space $Q^{\epsilon \theta _{p}}_{0,\vec {m}(v)}(\mathfrak {Y}_{p},(\beta (v),1^{J_{v}}))$ . Let $\pi :\mathcal {C}_{0}\rightarrow Q^{\epsilon \theta _p}_{0,\vec {m}(v)}(\mathfrak {Y}_p,(\beta (v),1^{J_{v}}))$ be the universal curve over $Q^{\epsilon \theta _{p}}_{0,\vec {m}(v)}(\mathfrak {Y}_{p},(\beta (v),1^{J_{v}}))$ . In this case, the fixed part of the obstruction theory of the vertex moduli over $0$ yields the virtual cycle

$$ \begin{align*}[Q^{\epsilon\theta_{p}}_{0,\vec{m}(v)}(\mathfrak{Y}_{p},(\beta(v),1^{J_{v}}))]^{\text{vir}}.\end{align*} $$

Note that $\mathcal {N}|_{\mathcal {C}_{0}}=\mathcal {O}_{\mathcal {C}_{0}}$ as $z_{2}|_{\mathcal {C}_{0}}\equiv 1$ ; therefore, the virtual normal bundle comes from the movable part of the infinitesimal deformations of the section $z_{1}$ , which is a section of the line bundle $\mathcal {L}_{-\theta _{p}}$ over $\mathcal {C}_{0}$ , whose Euler class $ e^{\mathbb {C}^{*}}(N_{v}^{\text {vir}})$ is equal to

(4.12) $$ \begin{align} e^{\mathbb{C}^{*}}((R^{\bullet}\pi_{*}\mathcal{L}_{-\theta_{p}})\otimes \mathbb{C}_{\lambda}). \end{align} $$

4.3.2 Edge contributions: basepoint case

When there is a base point on the edge curve, it has degree $(\beta (e),1^{J_{e}},\frac {\delta (e)}{r})$ with $\beta (e)\neq 0$ and $\delta (e)\geq \beta (e)(L_{\theta })+|J_{e}|$ by Remark 4.5. We will write $\beta (e)$ as $\beta $ only in this subsection for simplicity unless stated otherwise. Then the multiplicity at $q_{\infty }\in C_{e}$ is equal to $(g,\mu _{r}^{\delta (e)})\in G \!\times \! \boldsymbol {\mu }_{r}$ , where $g=g_{\beta }$ is defined in §3. Let a (or $a_{e}$ ) be the minimal positive integer associated to $\beta $ as in §3, which is also the order of $g_{\beta }$ . When r is a sufficiently large prime, due to Remark 4.4, $C_{e}$ must be isomorphic to $\mathbb {P}^{1}_{ar,1}$ where the ramification point $q_{0}$ for which $z_{1}=0$ is an ordinary point, and the ramification point $q_{\infty }$ for which $z_{2}=0$ must be a special point, which is isomorphic to $\mathbb {B}\boldsymbol {\mu }_{ar}$ .

Recall that

$$ \begin{align*}F_{\beta}(Y)\cong [Y^{ss}_{\beta}/G]\cong[(Z^{ss}_{\beta}\cap AY)/G]\end{align*} $$

in §3. We now define the edge moduli $\mathcal {M}_{e}$ to be

$$ \begin{align*}\sqrt[a\delta(e)]{L_{-\theta}/[Y^{ss}_{\beta}/G]} ,\end{align*} $$

which is the root gerbe over the stack $[Y^{ss}_{\beta }/G]$ by taking $a\delta (e)$ th root of the line bundle $L_{-\theta }$ on $[Y^{ss}_{\beta }/G]$ .

The root gerbe $\sqrt [a\delta (e)]{L_{-\theta }/[Y^{ss}_{\beta }/G]}$ admits a representation as a quotient stack:

$$ \begin{align*} [(Y^{ss}_{\beta} \!\times\! \mathbb{C}^{*}) / (G \!\times\! \mathbb{C}^{*}_{w})], \end{align*} $$

where the (right) action is defined by

$$ \begin{align*}(\vec{x},v)\cdot (g,w)=(\vec{x}\cdot g,\theta(g)vw^{a\delta(e)}) ,\end{align*} $$

for all $(g,w)\in G \!\times \! \mathbb {C}^{*}_{w}$ and $(\vec {x},v)\in A(Y)^{g} \!\times \! \mathbb {C}^{*}$ . Here, $ \vec {x}\cdot g$ is given by the action as in the definition of $[AY/G]$ . For every character $\rho $ of G, we can define a new character of $G \!\times \! \mathbb {C}^{*}_{w}$ by composing the projection map $\mathrm {pr}_{G}:G \!\times \! \mathbb {C}^{*}_{w} \rightarrow G$ . By an abuse of notation, we will continue to use the notation $\rho $ to name the new character of $G \!\times \! \mathbb {C}^{*}_{w} $ . Then the new character $\rho $ will determine a line bundle $L_{\rho }:=[(Y^{ss}_{\beta }\!\times \! \mathbb {C}^{*} \times \mathbb {C}_{\rho }) /(G \!\times \!\mathbb {C}^{*}_{w})]$ on $\sqrt [a\delta (e)]{L_{-\theta }/[Y^{ss}_{\beta }/G]}$ .

By virtue of its universal property of the root gerbe $\sqrt [a\delta (e)]{L_{-\theta }/[Y^{ss}_{\beta }/G]}$ , there is a line bundle $\mathcal {R}$ called root bundle that is the $a\delta (e)$ th root of line bundle $L_{-\theta }$ over the root gerbe. This root line bundle $\mathcal {R}$ can also constructed by the Borel construction; that is, $\mathcal {R}$ is associated to the character

$$ \begin{align*}\mathrm{pr}_{\mathbb{C}^{*}_{w}}:G \!\times\! \mathbb{C}^{*}_{w} \rightarrow \mathbb{C}^{*}_{w} \quad (g,w)\in G \!\times\! \mathbb{C}^{*}_{w} \mapsto w\in \mathbb{C}^{*}_{w}.\end{align*} $$

We have the relation

$$ \begin{align*}L_{-\theta}=\mathcal{R}^{\otimes a\delta(e)}.\end{align*} $$

Then the coordinate function $(\vec {x},v)\in Y^{ss}_{\beta } \!\times \! \mathbb {C}^{*} $ descents to be a tautological section of vector bundle $\bigoplus _{i=1}^{n}L_{\rho _{i}}\oplus (L_{\theta }\otimes \mathcal {R}^{\otimes a\delta (e)})$ on $\sqrt [a\delta (e)]{L_{-\theta }/[Y^{ss}_{\beta }/G]}$ .

We will construct a universal family of $\mathbb {C}^{*}-$ fixed quasimaps to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ of degree $(\beta ,1^{J_{e}},\frac {\delta (e)}{r})$ over the edge moduli $\mathcal {M}_{e}$ , which takes the form

The universal curve $\mathcal {C}_{e}$ over the edge moduli $\mathcal {M}_{e}$ is constructed as a quotient stack:

$$ \begin{align*}\mathcal{C}_{e}=[(Y^{ss}_{\beta} \!\times\! \mathbb{C}^{*} \!\times\! U)/ (G \!\times\! \mathbb{C}^{*}_{w} \!\times\! \mathbb{C}^{*}_{t})] ,\end{align*} $$

where the right action is defined by

$$ \begin{align*}(\vec{x},v,x,y)\cdot (g,w,t)=(\vec{x}\cdot g,\theta(g)vw^{a\delta(e)},w^{a}t^{ar}x,ty) ,\end{align*} $$

for all $(g,w,t)\in G \!\times \! \mathbb {C}^{*}_{w} \!\times \! \mathbb {C}^{*}_{t}$ and $(\vec {x},v,(x,y))=((x_{1},\cdots , x_{n}),v,(x,y))\in Y^{ss}_{\beta } \!\times \! \mathbb {C}^{*} \!\times \! U$ .

The universal map $ev$ from $\mathcal {C}_{e}$ to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ can be presented as follows: define the morphism

$$ \begin{align*}\widetilde{ev}: Y^{ss}_{\beta} \!\times\! \mathbb{C}^{*} \!\times\! U\rightarrow AY_{p} \!\times\! U\ \end{align*} $$

by

(4.13) $$ \begin{align} \begin{aligned} (\vec{x},v,(x,y))&\in Y^{ss}_{\beta} \!\times\! \mathbb{C}^{*} \!\times\! U \mapsto\\ &\big(\big(x_{1}x^{\beta(L_{\rho_{1}})},\cdots,x_{n}x^{\beta(L_{\rho_{n}})}\big),(x)_{j\in J_{e}},v^{-1}x^{\delta(e)-\beta(L_{\theta})-|J_{e}|},y^{a\delta(e)}\big)\in AY_{p} \!\times\! U. \end{aligned} \end{align} $$

Here, $(x)_{j\in J_{e}}$ is an element belonging to $\mathbb {C}^{p}$ so that the $j-$ th component is $1$ if $j\notin J_{e}$ and all the other components are x. Note that when $\beta (L_{\rho _{i}})\notin \mathbb {Z}_{\geq 0}$ for some i, we must have $x_{i}=0$ as $\vec {x}\in Y^{ss}_{\beta }$ , so $\tilde {ev}$ is well defined. Then $\tilde {ev}$ is equivariant with respect to the group homomorphism from $G \!\times \! \mathbb {C}^{*}_{w} \!\times \! \mathbb {C}^{*}_{t} $ to $G_{p} \!\times \! \mathbb {C}^{*}$ defined by

(4.14) $$ \begin{align} \begin{aligned} (g,w,t)&\in G \!\times\! \mathbb{C}^{*}_{w} \!\times\! \mathbb{C}^{*}_{t} \mapsto\\ &\big(g\cdot \big(t^{ar\beta(L_{\pi_{1}})}w^{a\beta(L_{\pi_{1}})},\cdots, t^{ar\beta(L_{\pi_{k}})}w^{a\beta(L_{\pi_{k}})}\big),(w^{a}t^{ar})_{j\in J_{e}},t^{a\delta(e)}\big)\in G_{p} \!\times\! \mathbb{C}^{*}. \end{aligned} \end{align} $$

Here, $(w^{a}t^{ar})_{j\in J_{e}}$ is the element belonging to $(\mathbb {C}^{*})^{p}$ so that the $j-$ th component is $1$ if $j\notin J_{e}$ and all the other components are $w^{a}t^{ar}$ . This gives the universal morphism f from $\mathcal {C}_{e}$ to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ by descent.

There is a tautological line bundle $\mathcal {O}_{C_{e}}(1)$ on $C_{e}$ associated to the character $\mathrm {pr}_{\mathbb {C}^{*}_{t}}$ of $G \!\times \! \mathbb {C}^{*}_{w} \!\times \! \mathbb {C}^{*}_{t}$ by the Borel construction. Here, $\mathrm {pr}_{\mathbb {C}^{*}_{t}}$ is the projection map from $G \!\times \! \mathbb {C}^{*}_{w} \!\times \! \mathbb {C}^{*}_{t}$ to $\mathbb {C}^{*}_{t}$ .

We will define a (quasiFootnote ¹⁵ -left) $\mathbb {C}^{*}-$ action on $\mathcal {C}_{e}$ such that the map $ev$ constructed above is $\mathbb {C}^{*}-$ equivariant. Define a (left) $\mathbb {C}^{*}-$ action on $\mathcal {C}_{e}$ which is induced from the $\mathbb {C}^{*}-$ action on $Y^{ss}_{\beta } \!\times \! \mathbb {C}^{*} \!\times \! U$ :

$$ \begin{align*}m:\mathbb{C}^{*}\times Y^{ss}_{\beta} \!\times\! \mathbb{C}^{*} \!\times\! U\rightarrow Y^{ss}_{\beta} \!\times\! \mathbb{C}^{*} \!\times\! U ,\end{align*} $$

$$ \begin{align*}t\cdot (x,v,(x,y))= (x,v,(x,t^{\frac{-1}{ar\delta(e)}}y)).\end{align*} $$

Note that the morphism $\pi $ is also $\mathbb {C}^{*}$ -equivariant, where $\mathcal {M}_{e}$ is equipped with the trivial $\mathbb {C}^{*}$ -action. By the universal property of the projectivized bundle $\mathcal {C}_e$ over $\mathcal {M}_{e}$ , the line bundle $\mathcal {O}_{\mathcal {C}_e}(1)$ is equipped with a tautological section

$$ \begin{align*} (x, y) \in H^0\big(\big(\mathcal{O}_{\mathcal{C}_e}(ar) \otimes \mathcal \pi^{*}\mathcal{R}^{\otimes a}\big) \oplus (\mathcal{O}_{\mathcal{C}_e}(1)\otimes \mathbb{C}_{\frac{-1}{ar\delta(e)}})\big) , \end{align*} $$

which is also a $\mathbb {C}^{*}-$ invariant section.

Now we can check that $ev$ is a $\mathbb {C}^{*}-$ equivariant morphism from $\mathcal {C}_{e}$ to $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ with respect to the $\mathbb {C}^{*}-$ actions for $\mathcal {C}_{e}$ and $\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}$ . According to Remark 4.2, $ev$ is equivalent to the following data:

1. 1. $k+p+1\ \mathbb {C}^{*}$ -equivariant line bundles on $\mathcal {C}_{e}$ :

   $$ \begin{align*}\mathcal{L}_{j} := \pi^*L_{\pi_{j}} \otimes \mathcal{O}_{\mathcal{C}_e}(ar\beta(L_{\pi_{j}}))\otimes \pi^{*}\mathcal{R}^{\otimes a\beta(L_{\pi_{j}})}, 1\leq j\leq k ,\end{align*} $$
   $$ \begin{align*}\mathcal{L}_{k+j}:=\pi^{*}\mathcal{R}^{\otimes a}\otimes \mathcal{O}_{\mathcal{C}_{e}}(ar), j\in J_{e},\;\text{and}\; \mathcal{L}_{k+j}:=\mathbb{C},\; j\notin J_{e}\end{align*} $$
   and
   $$ \begin{align*}\mathcal{N}:=\mathcal{O}_{\mathcal{C}_{e}}(a\delta(e))\otimes \mathbb{C}_{\frac{-\lambda}{r}} ,\end{align*} $$
   where the line bundles $L_{\pi _{j}}$ , $\mathcal {R}$ are the standard $\mathbb {C}^{*}$ -equivariant line bundle on $\mathcal {M}_{e}$ by the Borel construction;

2. 2. a universal section

   (4.15) $$ \begin{align} \begin{aligned} \big(\vec{x},\vec{y},(\zeta_1, \zeta_2)\big) :=& \big((x_{1}x^{\beta(L_{\rho_{1}})},\cdots,x_{n}x^{\beta(L_{\rho_{n}})}),(x)_{J_{e}},(v^{-1}x^{\delta(e) - \beta(L_{\theta})-|J_{e}|}, y^{a\delta(e)})\big)\\ &\in H^0\big(\mathcal{C}_{e}, (\oplus_{i=1}^{n} \mathcal{L}_{\rho_{i}})\oplus(\oplus_{j=1}^{p}\mathcal{L}_{k+j})\oplus (\mathcal{L}_{-\theta_{p}}\otimes \mathcal{N}^{\otimes r}\otimes \mathbb{C}_{\lambda})\oplus \mathcal{N}\big)^{\mathbb{C}^{*}} , \end{aligned} \end{align} $$
   where the line bundles $\mathcal {L}_{-\theta _{p}}$ and $\mathcal {L}_{\rho _{i}}$ are induced from line bundles $\mathcal {L}_{j}$ as before.

Assume that $j\in J_{e}$ , analogous to the definition of $\hat {ev}_{j}$ in (4.5). We can define a morphism

$$ \begin{align*}\hat{ev}_{j,e}:\mathcal{M}_{e}\rightarrow \mathfrak{Y}\end{align*} $$

by evaluating $[\vec {x}]$ at the ramification point $q_{0}$ . More explicitly, use the setting in §4.3.2. Denote $\underline {ev}:=\mathrm {pr}_{r,p}\circ ev:\mathcal {C}_{e}\rightarrow \mathfrak {Y}$ , where $\mathrm {pr}_{r,p}:\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p}\rightarrow \mathfrak {Y}$ is the natural projection map. Let $D_{0}$ be the zero section of $\mathcal {C}_{e}$ over $\mathcal {M}_{e}$ given by $x=0$ . Then $\hat {ev}_{j,e}=\underline {ev}|_{D_{0}}$ .

However, the restriction morphism $\hat {ev}_{j}|_{F_{\Gamma }}$ to $F_{\Gamma }$ (see (4.10)) coming from the whole space $Q^{\widetilde {\theta }}_{0,\vec {m}}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta ,1^{p},\frac {\delta }{r}))$ factors through the projection $\mathfrak {p}: F_{\Gamma }\rightarrow \mathcal {M}_{e}$ from $F_{\Gamma }$ to the factor $\mathcal {M}_{e}$ . Then we have that

$$ \begin{align*}\hat{ev}_{j,e}\circ \mathfrak{p}= \hat{ev}_{j}|_{F_{\Gamma}}.\end{align*} $$

Let u be a polynomial on $k\;(=\text {rk}(G))$ variables. Write $u(c_{1}(L_{\pi _{i}}))$ for

$$ \begin{align*}u\big(c_{1}(L_{\pi_{1}}),\cdots,c_{1}(L_{\pi_{k}})\big)\end{align*} $$

for simplicity. Thus, when we want to apply virtual localization to compute $\hat {ev}^{*}_{j}(u(c_{1}(L_{\pi _{i}})))$ , we only need to compute $(\hat {ev}_{j.e})^{*}( u(c_{1}(L_{\pi _{i}})))$ . More explicitly, we have the following:

Proposition 4.6. Using the above notation, we have that

$$ \begin{align*}(\hat{ev}_{j,e})^{*}( u(c_{1}(L_{\pi_{i}})))=u\bigg(c_{1}(L_{\pi_{i}})+ \frac{\beta(L_{\pi_{i}})(\lambda-D_{\theta})}{\delta(e)}\bigg).\end{align*} $$

Here, $u \bigg (c_{1}(L_{\pi _{i}})+ \frac {\beta (L_{\pi _{i}})(\lambda -D_{\theta })}{\delta (e)}\bigg ) $ is short for

$$ \begin{align*}u \bigg(c_{1}(L_{\pi_{1}})+ \frac{\beta(L_{\pi_{1}})(\lambda-D_{\theta})}{\delta(e)},\cdots, c_{1}(L_{\pi_{k}})+ \frac{\beta(L_{\pi_{k}})(\lambda-D_{\theta})}{\delta(e)} \bigg)\end{align*} $$

for simplicity and $D_{\theta }=c_{1}(L_{\theta })$ .

Proof. Note that we have

$$ \begin{align*}\underline{ev}^{*}(L_{\tau})=\pi^{*}(L_{\tau}\otimes \mathcal{R}^{a\beta(L_{\tau})})\otimes \mathcal{O}_{\mathcal{C}_{e}}(ar\beta(L_{\tau}))\end{align*} $$

for any character $\tau $ of G, $\mathcal {O}_{\mathcal {C}_{e}}(1)|_{D_{0}}=\mathbb {C}_{\frac {\lambda }{ar\delta (e)}}$ and $\mathcal {R}^{a\delta (e)}=L_{-\theta }$ . Then we have $c_{1}(\hat {ev}^{*}_{j,e}(L_{\tau }))=c_{1}(L_{\tau })+\frac {\beta (L_{\tau })(\lambda -D_{\theta })}{\delta (e)}$ . Then the claim follows.

From the description of $\mathcal {M}_{e}$ with the associated family map $ev$ , we see that $\mathcal {M}_{e}$ allows a finite étale map of degreeFootnote ¹⁶ $\frac {1}{a}$ into the corresponding fixed loci in $Q^{\widetilde {\theta }}_{0,1}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta (e),1^{J_{e}},\frac {\delta (e)}{r}))$ where the marking corresponds to the ramification point $q_{\infty }$ . Then the fixed part of the restriction of the prefect obstruction theory of $Q^{\widetilde {\theta }}_{0,1}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta (e),1^{J_{e}},\frac {\delta (e)}{r}))$ to $\mathcal {M}_{e}$ yields the virtual cycle $[\mathcal {M}_{e}]^{vir}$ of $\mathcal {M}_{e}$ , while the movable part yields the virtual normal bundle $N^{\text {vir}}_{e}$ whose $\mathbb {C}^{*}-$ equivariant class is $e^{\mathbb {C}^{*}}(N^{\text {vir}}_{e})$ .

Let $\omega :\mathcal {M}_{e}\rightarrow \bar {I}_{g_{\beta }}Y$ be the composition

$$ \begin{align*}\mathcal{M}_{e} \rightarrow \bar{I}_{(g_{\beta},\mu_{r}^{\delta(e)})}\mathbb PY^{\frac{1}{r}} \rightarrow \bar{I}_{g_{\beta}}Y ,\end{align*} $$

where the first arrow is the evaluation map at the ramification point $q_{\infty }$ and the second arrow induced form the projection map from $\mathbb PY^{\frac {1}{r}}$ to Y; in particular, the second map is an isomorphism when r is a sufficiently large prime. We also note that $\omega $ can be obtained as the composition of the following three maps:

$$ \begin{align*} \mathcal{M}_{e}\rightarrow [Y^{ss}_{\beta}/G]\rightarrow I_{g_{\beta}}Y \rightarrow \bar{I}_{g_{\beta}}Y. \end{align*} $$

where the first map (denoted by $i_{\mathcal {M}_{e}}$ on Lemma 4.9) is obtained by forgetting root structure of $\mathcal {M}_{e}$ , the second map is taking inclusion $ [Y^{ss}_{\beta }/G] \rightarrow [AY^{ss}(\theta )^{g_{\beta }}/G]\cong I_{g_{\beta }}Y$ and the third map is the rigidified map.

We will show that the localization contribution from the edge moduli with basepoints yields the following:

Lemma 4.7. With the above notations, we have

$$ \begin{align*}\omega_{*}\big( \frac{[\mathcal{M}_{e}]^{\text{vir}}}{ e^{\mathbb{C}^{*}}(N^{\text{vir}}_{e})}\big)= \frac{1}{a^{2}\delta(e)}\iota_{*}\frac{\big(\frac{z}{z^{|J_{e}|}}\mathbb{I}_{\beta}(z)\big)|_{z=\frac{\lambda-D_{\theta}}{\delta(e)}}}{\prod_{m=1}^{\delta(e)-\beta(L_{\theta})-|J_e|}\frac{m}{\delta(e)}(-D_{\theta}+\lambda)} ,\end{align*} $$

where $\mathbb {I}_{\beta }(z)$ is the coefficient of $q^{\beta }$ of $\mathbb {I}(q,0,z)$ defined in the introduction 1.1.2 and $\iota _{*}:H^{*}(\bar {I}_{g^{-1}_{\beta }}Y)\rightarrow H^{*}(\bar {I}_{g_{\beta }}Y)$ is the isomorphism induced by inverting the bang structure of $\bar {I}_{\mu }Y$ .

Moreover, let $u_{1}, \cdots , u_{l}$ be l polynomials depending on k first chern class $c_{1}(L_{\pi _{1}}),\cdots , c_{1}(L_{\pi _{k}})$ and $t=\sum _{i=1}^{l}t_{i}u_{i}$ . Denote $Cont_{\mathcal {M}_{e}}\big (\prod _{j=1}^{p} \hat {ev}_{j}^{*}t\big ):= \big ( \prod _{j\in J_{e}} \hat {ev}_{j,e}^{*}t\big ) \cap \frac {[\mathcal {M}_{e}]^{\text {vir}}}{ e^{\mathbb {C}^{*}}(N^{\text {vir}}_{e}) } $ to be the edge contribution of the cohomology class $ \prod _{j=1}^{p} \hat {ev}_{j}^{*}t$ in the localization computation. By the discussion in Proposition 4.6, we have that

$$ \begin{align*}\omega_{*}\big(Cont_{\mathcal{M}_{e}} (\prod_{j=1}^{p} \hat{ev}_{j}^{*}t)\big)= \frac{1}{a^{2}\delta(e)}\iota_{*}\frac{\big(\frac{z\mathbf{t}^{|J_{e}|}}{z^{|J_{e}|}}\mathbb{I}_{\beta}(z)\big)|_{z=\frac{\lambda-D_{\theta}}{\delta(e)}}}{\prod_{m=1}^{\delta(e)-\beta(L_{\theta})-|J_e|}\frac{m}{\delta(e)}(-D_{\theta}+\lambda)} ,\end{align*} $$

where $\mathbf {t}=\sum _{i=1}^{l}t_{i}u_{i}\big (c_{1}(L_{\pi _1})+\beta (L_{\pi _1})z, \cdots , c_{1}(L_{\pi _{k}})+\beta (L_{\pi _k})z\big )$ .

The above lemma is based on the computation of the virtual cycle $[\mathcal {M}_{e}]^{\text {vir}}$ (see Lemma 4.9) and the $\mathbb {C}^{*}-$ equivariant Euler class $ e^{\mathbb {C}^{*}}(N_{e}^{\text {vir}})$ (see (4.16)), for which we now explain.

Based on the perfect obstruction theory (4.6) for quasimaps in $Q^{\widetilde {\theta }}_{0,1}(\mathbb {P} \mathfrak {Y}^{\frac {1}{r},p},(\beta (e),1^{J_{e}},\frac {\delta (e)}{r}))$ , the restriction of the prefect obstruction theory to $\mathcal {M}_{e}$ decomposes into three parts: (1) the deformation theory of source curve $\mathcal {C}_{e}$ ; (2) the deformation theory of the lines bundles $(\mathcal {L}_{j})_{1\leq j\leq k+p}$ and $\mathcal {N}$ ; (3) the deformation theory for the section

$$ \begin{align*}(\vec{x},\vec{y},(\zeta_{1},\zeta_{2}))\in \Gamma\left(\oplus_{i=1}^{n} \mathcal{L}_{\rho_{i}}\oplus (\oplus_{j=1}^{p}\mathcal{L}_{k+j}) \oplus (\mathcal{L}_{-\theta_{p}}\otimes \mathcal{N}^{\otimes r}\otimes \mathbb{C}_{\lambda}) \oplus \mathcal{N}\right).\end{align*} $$

The virtual normal bundle comes from the movable part of the three parts, and the fixed part will contribute to the virtual cycle of $\mathcal {M}_{e}$ . First, every fiber curve $C_{e}$ in $\mathcal {C}_{e}$ is isomorphic to $\mathbb {P}_{ar,1} $ , which is rational. Then the infinitesimal deformations/obstructions of $C_{e}$ and the line bundles $L_{j}:=\mathcal {L}_{j}|_{C_{e}},\; N:=\mathcal {N}|_{C_{e}}$ are zero. Hence, their contribution to the perfect obstruction theory solely comes from infinitesimal automorphisms. The infinitesimal automorphisms of $C_{e}$ come from the space of vector field on $C_{e}$ that vanishes on special points. Thus, the $\mathbb {C}^{*}-$ fixed part of the infinitesimal automorphisms of $C_{e}$ comes from the $1-$ dimensional subspace of vector fields on $C_{e}$ which vanish on the two ramification points, which, together with the infinitesimal automorphisms of line bundle N, will be canceled with the fixed part of infinitesimal deformation of sections $(z_{1},z_{2}):=(\zeta _{1},\zeta _{2})|_{C_{e}}$ . The movable part of infinitesimal automorphisms of $C_{e}$ is nonzero only if at least one of ramification points on $C_{e}$ is not a special point. By Remark 4.4, the ramification $q_{\infty }$ must be a special point since it has nontrivial stacky structure when r is sufficiently large, and the ramification point $q_{0}$ is not a special point. Then the movable part of infinitesimal automorphisms of $C_{e}$ contributes

$$ \begin{align*}\frac{\delta(e)}{\lambda-D_{\theta}}\end{align*} $$

to the virtual normal bundle.

Now let’s turn to the localization contribution from sections. As for the deformations of $z_{2}$ , we continue to use the tautological section $(x,y)$ in (4.3.2). Sections of N are spanned by monomials $(x^{m}y^{n})|_{C_{e}}$ with $arm+n=a\delta (e)$ and $m,n\in \mathbb {Z}_{\geq 0}$ . Note that $x^{m}y^{n}$ may not be a global section of $\mathcal {N}$ but always a global section of the line bundle $R^{\otimes am}\otimes \mathcal {N}\otimes \mathbb {C}_{\frac {m}{\delta (e)}\lambda }$ . Then $R^{\bullet }\pi _{*}\mathcal {N}$ will decompose as a direct sum of line bundles. Each corresponds to the monomial $x^{m}y^{n}$ , whose first chern class is

$$ \begin{align*}c_{1}(\mathcal{R}^{\otimes -am} \bigotimes \mathbb{C}_{\frac{-m}{\delta(e)}\lambda})=\frac{m}{\delta(e)}(D_{\theta}-\lambda).\end{align*} $$

So the total contribution is equal to

$$ \begin{align*}\prod_{m=0}^{\lfloor\frac{\delta(e)}{r}\rfloor}\bigg(\frac{m}{\delta(e)}(D_{\theta}-\lambda)\bigg).\end{align*} $$

The term corresponding to $m=0$ in the above product is the $\mathbb {C}^{*}-$ invariant part of $R^{\bullet }\pi _{*}\mathcal {N}$ . It will contribute to the virtual cycle of $\mathcal {M}_{e}$ . The rest contributes to the virtual normal bundle as

$$ \begin{align*}\prod_{m=1}^{\lfloor\frac{\delta(e)}{r}\rfloor}\bigg(\frac{m}{\delta(e)}(D_{\theta}-\lambda)\bigg).\end{align*} $$

Note that when r is sufficiently large, the above product becomes $1$ .

For the deformation of $z_{1}$ , arguing in the same way as $z_{2}$ , the Euler class of $R^{\bullet }\pi _{*}(\mathcal {L}_{-\theta _{p}}\otimes \mathcal {N}^{\otimes r}\otimes \mathbb {C}_{\lambda })$ is equal to

$$ \begin{align*}\prod_{m=0}^{\delta(e)-\beta(L_{\theta})-|J_e|}\bigg(\frac{m}{\delta(e)}(-D_{\theta}+\lambda)\bigg).\end{align*} $$

The factor for $m=0$ appearing in the above product is the $\mathbb {C}^{*}-$ fixed part of $R^{\bullet }\pi _{*}(\mathcal {L}_{-\theta }\otimes \mathcal {N}^{\otimes r}\otimes \mathbb {C}_{\lambda })$ . It will contribute to the virtual cycle of $\mathcal {M}_{e}$ . The rest contributes to the virtual normal bundle as

$$ \begin{align*}\prod_{m=1}^{\delta(e)-\beta(L_{\theta})-|J_e|}\bigg(\frac{m}{\delta(e)}(-D_{\theta}+\lambda)\bigg).\end{align*} $$

Finally, let’s turn to the localization contribution from the sections $\vec {x}$ and $\vec {y}$ . Before that, using the same argument above, one can prove the following lemma:

Lemma 4.8. When $n\in \mathbb {Z}_{\geq 0}$ , we have

$$ \begin{align*}e^{\mathbb{C}^{*}}\big(R^{\bullet}\pi_{*}(\mathcal{O}_{\mathcal{C}_{e}}(n))\big)=\prod_{m=0}^{\lfloor{\frac{n}{ar}}\rfloor}\bigg(\frac{m}{\delta(e)}(D_{\theta}-\lambda)+\frac{n}{ar\delta(e)}\lambda\bigg).\end{align*} $$

When $n\in \mathbb {Z}_{<0}$ , we have

$$ \begin{align*}e^{\mathbb{C}^{*}}\big(R^{\bullet}\pi_{*}(\mathcal{O}_{\mathcal{C}_{e}}(n))\big)=\prod_{\frac{n}{ar}< m<0}\frac{1}{\frac{m}{\delta(e)}(D_{\theta}-\lambda)+\frac{n}{ar\delta(e)}\lambda}.\end{align*} $$

Using the above lemma, we have the following description of $e^{\mathbb {C}^{*}}(R^{\bullet }\pi _{*}\mathcal {L}_{\rho _{i}})$ for $1\leq i\leq n$ . Then for each $\rho _{i}$ , we have the following:

1. 1. If $\beta (L_{\rho _{i}})\in \mathbb {Q}_{\geq 0}$ , one has

   $$ \begin{align*} \begin{aligned} e^{\mathbb{C}^{*}}\big(R^{\bullet}\pi_{*}(\mathcal{L}_{\rho_{i}})\big)&=e^{\mathbb{C}^{*}}\big(R^{\bullet}\pi_{*}(\pi^{*}(L_{\rho_{i}})\otimes \mathcal{O}_{\mathcal{C}_{e}}(ar\beta(L_{\rho_{i}}))\otimes \pi^{*}(\mathcal{R}^{\otimes a\beta(L_{\rho_{i}})}))\big)\\ &=e^{\mathbb{C}^{*}}\big(L_{\rho_{i}}\otimes \mathcal{R}^{\otimes a\beta(L_{\rho_{i}})} \otimes R^{0}\pi_{*}(\mathcal{O}_{\mathcal{C}_{e}}(ar\beta(L_{\rho_{i}})))\big)\\ &=\prod_{m=0}^{\lfloor {\beta(L_{\rho_{i}})}\rfloor}\bigg( D_{\rho_{i}}+\frac{\beta(L_{\rho_{i}})(-D_{\theta})}{\delta(e)}+\frac{m}{\delta(e)}(D_{\theta}-\lambda)+\frac{\beta(L_{\rho_{i}})}{\delta(e)}\lambda\bigg)\\ &=\prod_{m=0}^{\lfloor {\beta(L_{\rho_{i}})} \rfloor}\bigg( D_{\rho_{i}}+\frac{\beta(L_{\rho_{i}})-m}{\delta(e)}(\lambda-D_{\theta})\bigg). \end{aligned} \end{align*} $$

   Hence, we have

   $$ \begin{align*} e^{\mathbb{C}^{*}}((R^{\bullet}\pi_{*}\mathcal{L}_{\rho_{i}})^{\mathrm{mov}}) &=\prod_{0\leq m< \beta(L_{\rho_{i}})}\big( D_{\rho_{i}}+\frac{\beta(L_{\rho_i})-m}{\delta(e)}(\lambda-D_{\theta})\big). \end{align*} $$

   Note that the invariant part of $R^{\bullet }\pi _{*}\mathcal {L}_{\rho _{i}}$ is nonzero only when $\beta (L_{\rho _{i}})\in \mathbb {Z}_{\geq 0}$ .

2. 2. If $\beta (L_{\rho _{i}})\in \mathbb {Q}_{< 0}$ , one has

   $$ \begin{align*} \begin{aligned} e^{\mathbb{C}^{*}}\big(R^{\bullet}\pi_{*}\mathcal{L}_{\rho_{i}}\big)&=e^{\mathbb{C}^{*}}\big(R^{\bullet}\pi_{*}\big(\pi^{*}L_{\rho_{i}}\otimes \mathcal{O}_{\mathcal{C}_{e}}(ar\beta(L_{\rho_{i}}))\otimes \pi^{*}\mathcal{R}^{\otimes a\beta(L_{\rho_{i}})}\big)\big)\\ &=\frac{1}{e^{\mathbb{C}^{*}}\big(L_{\rho_{i}}\otimes \mathcal{R}^{\otimes a\beta(L_{\rho_{i}})} \otimes R^{1}\pi_{*}(\mathcal{O}_{\mathcal{C}_{e}}(ar\beta(L_{\rho_{i}})))\big)}\\ &=\prod_{\beta(L_{\rho_{i}})<m<0}\frac{1}{ D_{\rho_{i}}+\frac{\beta(L_{\rho_{i}})(-D_{\theta})}{\delta(e)}+\frac{m}{\delta(e)}(D_{\theta}-\lambda)+\frac{\beta(L_{\rho_{i}})}{\delta(e)}\lambda}\\ &=\prod_{\beta(L_{\rho_{i}})<m<0}\frac{1}{ D_{\rho_{i}}+\frac{\beta(L_{\rho_{i}})-m}{\delta(e)}(\lambda-D_{\theta})} , \end{aligned} \end{align*} $$

   which implies that

   $$ \begin{align*} e^{\mathbb{C}^{*}}\big((R^{\bullet}\pi_{*}\mathcal{L}_{\rho_{i}})^{\mathrm{mov}}\big)&=e^{\mathbb{C}^{*}}\big(R^{\bullet}\pi_{*}\mathcal{L}_{\rho_{i}}\big)\\ &=\prod_{\beta(L_{\rho_{i}})<m<0}\frac{1}{ D_{\rho_{i}}+\frac{\beta(L_{\rho_{i}})-m}{\delta(e)}(\lambda-D_{\theta})}. \end{align*} $$

The movable part of the deformations of $\vec {y}$ contributes

$$ \begin{align*}e^{\mathbb{C}^{*}}\left( \oplus_{j=1}^{p}R^{\bullet}\pi_{*}(\mathcal{L}_{k+j})^{\mathrm{mov}}\right)=(\frac{\lambda-D_{\theta}}{\delta(e)})^{|J_{e}|}\end{align*} $$

to the virtual normal bundle, and the fixed part of the deformations of $\vec {y}$ will be canceled with the automorphisms of line bundles $(L_{k+j}:1\leq j\leq p)$ .

Recall that the complete intersection Y is cut off by the section $s:=\oplus _{b=1}^{c}s_{b}$ of the direct sum of the line bundles $E=\oplus _{b=1}^{c}L_{\tau _{b}}$ on X associated to the characters $\tau _{b}$ . There is also an obstruction corresponding to the infinitesimal deformations of $\vec {x}$ being moved away from $[AY^{ss}(\theta )/ G]\subset [W^{ss}(\theta )/ G]$ , which contributes to the virtual normal bundle as the movable part of

$$ \begin{align*} e^{\mathbb{C}^{*}}\big(-(\oplus_{b}R^{\bullet}\pi_{*}\mathcal{L}_{\tau_{b}})\big)&=\frac{e^{\mathbb{C}^{*}}\big(R^{1}\pi_{*}\oplus_{b:\beta(L_{\tau_{b}})<0}\mathcal{L}_{\tau_{b}})\big)}{e^{\mathbb{C}^{*}}\big(R^{0}\pi_{*}\oplus_{b:\beta(L_{\tau_{b}})\geq 0}\mathcal{L}_{\tau_{b}}\big)}\\ &=\frac{\prod_{b:\beta(L_{\tau_{b}})< 0}\prod_{\beta(L_{\tau_b})<m<0}\big( c_1(L_{\tau_b})+\frac{\beta(L_{\tau_b})-m}{\delta(e)}(\lambda-D_{\theta})\big)}{\prod_{b:\beta(L_{\tau_{b}})\geq 0}\prod_{0\leq m\leq \beta(L_{\tau_b})}\big( c_1(L_{\tau_b})+\frac{\beta(L_{\tau_b})-m}{\delta(e)}(\lambda-D_{\theta})\big)}. \end{align*} $$

Here, m are all integers.

Now we have the expression of virtual normal bundle from the movable part of curves, line bundles and sections as follows:

(4.16) $$ \begin{align} \begin{aligned} &e^{\mathbb{C}^{*}}(N_{e}^{\text{vir}})=\frac{\prod_{\rho:\beta(L_{\rho})> 0}\prod_{0\leq i< \beta(L_{\rho})}(D_{\rho}+(\beta(L_{\rho})-i)\frac{\lambda-D_{\theta}}{\delta(e)})}{\prod_{\rho:\beta(L_{\rho})< 0}\prod_{\lfloor {\beta(L_{\rho})+1 } \rfloor\leq i<0}(D_{\rho}+(\beta(L_{\rho})-i)\frac{\lambda-D_{\theta}}{\delta(e)})}\cdot (\frac{\lambda-D_{\theta}}{\delta(e)})^{|J_{e}|} \frac{ \delta(e)}{\lambda-D_{\theta}}\\ &\cdot \frac{\prod_{b:\beta(L_{\tau_{b}})< 0}\prod_{\beta(L_{\tau_b})<m<0}\big( c_1(L_{\tau_b})+\frac{\beta(L_{\tau_b})-m}{\delta(e)}(\lambda-D_{\theta})\big)}{\prod_{b:\beta(L_{\tau_{b}})\geq 0}\prod_{0\leq m< \beta(L_{\tau_{b}})}\big( c_1(L_{\tau_b})+\frac{\beta(L_{\tau_b})-m}{\delta(e)}(\lambda-D_{\theta})\big)}\cdot \prod_{m=1}^{\delta(e)-\beta(L_{\theta})-|J_e|}\bigg(\frac{m}{\delta(e)}(-D_{\theta}+\lambda)\bigg). \end{aligned} \end{align} $$

However, we can see that the fixed part of the perfect obstruction theory only comes from the summand corresponding to the terms b with $\beta (L_{\tau _{b}})\in \mathbb {Z}_{\geq 0}$ , for which there is one-dimensional $\mathbb {C}^{*}-$ fixed piece to each $-R^{\bullet }\pi _{*}\mathcal {L}_{\tau _b}$ , which contributes to the virtual cycle of $\mathcal {M}_{e}$ .

Now let’s move to the virtual cycle of $\mathcal {M}_{e}$ coming from the $\mathbb {C}^{*}-$ fixed part of the restriction of perfect obstruction theory. Let $E_{\beta }:=\oplus _{b:\beta (L_{\tau _{b}})\in \mathbb {Z}_{\geq 0}}L_{\tau _{b}}$ be the vector bundle over $[Z^{ss}_{\beta }/G]$ and ${s_{\beta }:=\oplus _{b:\beta (L_{\tau _{b}})\in \mathbb {Z}_{\geq 0}} s_{b}}$ be the section inside $E_{\beta }$ . Using Lemma 3.2. We can define the Gysin morphism

$$ \begin{align*}s^{!}_{E_{\beta},loc}:A_{*}([Z^{ss}_{\beta}/G])\rightarrow A_{*}([Y^{ss}_{\beta}/G])\end{align*} $$

as the localized top Chern class [Reference Fulton23, §14.1]. This Gysin morphism commutes with the one defined in 3.3 by the flat pullback $A^{*}([Y^{ss}_{\beta }/ (G/\langle g^{-1}_{\beta }\rangle )])\rightarrow A^{*}([Y^{ss}_{\beta }/G])$ on the target and the flat pullback $A^{*}([Z^{ss}_{\beta }/ (G/\langle g^{-1}_{\beta }\rangle )])\rightarrow A^{*}([Z^{ss}_{\beta }/G])$ on the source.

Lemma 4.9. We have the following:

$$ \begin{align*}[\mathcal{M}_{e}]^{\text{vir}}=i_{\mathcal{M}_{e}}^{*}\big(s_{E_{\beta},loc}^{!}([Z^{ss}_{\beta}/G])\big).\end{align*} $$

Here, $i^{*}_{\mathcal {M}_{e}}: A^{*}( [Y^{ss}_{\beta }/G]) \rightarrow A^{*}(\mathcal {M}_{e})$ is the isomorphism induced by the natural $\acute {e}tale$ morphism $i_{\mathcal {M}_{e}}:\mathcal {M}_{e}\rightarrow [Y^{ss}_{\beta }/G]$ by forgetting root structure.

Proof. By the previous discussion, the perfect obstruction theory of $\mathcal {M}_{e}$ solely comes from automorphisms of line bundles $(\mathcal {L}_{j})_{j=1}^{k}$ , the fixed part of deformations/obstructions of the section $\vec {x}$ . Note that the fixed part of the deformations of $\vec {y}$ cancels with the automorphisms of line bundles $(\mathcal {L}_{k+j}:1\leq j\leq p)$ , so we do not count. Using the distinguished triangle (4.8) and (4.7) in §4.2, the $\mathbb {C}^{*}-$ fixed part of the obstruction complex $\mathbb {E}^{\mathrm {fix}}$ over $\mathcal {M}_{e}$ is quasi-isomorphic to the complex

with the first term sitting in degree 0 and the second term sitting in degree 1, which also fits into the following distinguished triangle (from cone construction)

Here, $ds_{\beta }$ is the differential induced the section $s_{\beta }$ (cf. (4.8)), and $\mathbb T_{[Z^{ss}_{\beta }/G]}|_{\mathcal {M}_{e}}$ is the pullback of the tangent bundle $\mathbb T_{[Z^{ss}_{\beta }/G]}$ along the composition of morphisms

$$ \begin{align*}\mathcal{M}_{e}\rightarrow \sqrt[a\delta(e)]{L_{-\theta}/[Z^{ss}_{\beta}/G]}\rightarrow [Z^{ss}_{\beta}/G] ,\end{align*} $$

where the first arrow is the inclusion and the second arrow is the natural $\acute {e}$ tale morphism by forgetting root.

When we replace Y by X, and repeat the same localization analysis as above, we see the fixed part of the restriction of the obstruction theory to the edge moduli $\mathcal {M}_{e}(X):=\sqrt [a\delta (e)]{L_{-\theta }/[Z^{ss}_{\beta }/G]}$ of X is equal to the tangent complex of $\mathcal {M}_{e}(X)$ , which is a locally free sheaf sitting in degree zero as $\mathcal {M}_{e}(X)$ is a smooth Deligne-Mumford stack. Then we can view $\mathcal {M}_{e}$ as the zero loci of the section $s_{\beta }$ of the vector bundle $E_{\beta }$ over $\mathcal {M}_{e}(X)$ by Lemma 3.2. One has the following Cartesian diagram:

where the bottom arrow is the zero section. Then we have a morphism of two distinguished triangles in $D^{b}_{coh}(\mathcal {M}_{e})$ where all terms in the first low are perfect complexes with amplitude in $[-1,0]$

Here, the first and the second vertical maps are the dual perfect obstruction theory for $\mathcal {M}_{e}(X)$ and $\mathcal {M}_{e}$ (both restricted to $\mathcal {M}_{e}$ ), respectively, while the third vertical map is an obstruction theory for $\mathbb {C}^{*}-$ fixed quasimaps in $\mathcal {M}_{e}$ with the section $\vec {x}$ moving away from Y into X. A standard deformation theory argument (cf. [Reference Chang and Li7, Proposition 2.5]) shows the third vertical map $i^{*}$ is induced from the pullback of the conormal sheafs for the horizontal arrows in the above Cartesian square along the left arrow i. Then virtual cycle $[\mathcal {M}_{e}]^{vir}$ with respect to the dual perfect obstruction theory $(\mathbb {E}^{\mathrm {fix}})^{\vee }\rightarrow t_{\geq -1}\mathbb L_{\mathcal {M}_{e}}$ can be obtained by Manolache’s virtual pullback [Reference Manolache35, Construction 3.6], which is also identical to Gysin pullback $0^{!}=s_{E_{\beta },loc}^{!}$ (by the very of definition of localized top Chern class). Now the Lemma is immediate by flat pullback along $i_{\mathcal {M}_{e}}$ .