2.1. Quantum K-theory of target stacks

We begin with recalling the definition of K-theoretic Gromov-Witten invariants of Deligne-Mumford stacks, as given in ref. [Reference Tonita and Tseng24]. Let $\mathcal{Z}$ be a smooth proper Deligne-Mumford stack with projective coarse moduli space $Z$ . The moduli stack of $n$ -pointed genus $g$ degree $d$ stable maps to $\mathcal{Z}$ is denoted by ${\mathcal{K}}_{g,n}(\mathcal{Z},d)$ . The detailed definition can be found in [Reference Abramovich and Vistoli4]. It is known that ${\mathcal{K}}_{g,n}(\mathcal{Z},d)$ is a proper Deligne-Mumford stack equipped with a perfect obstruction theory, see [Reference Abramovich and Vistoli4], [Reference Abramovich, Graber and Vistoli3]. Applying the recipe of [Reference Lee20] to this perfect obstruction theory yields a virtual structure sheaf ${\mathcal{O}}_{{\mathcal{K}}_{g,n}(\mathcal{Z},d)}^{vir}$ . There are evaluation maps $ev_i\,:\,{\mathcal{K}}_{g,n}(\mathcal{Z}, d)\to \bar{I}\mathcal{Z}$ , where $\bar{I}\mathcal{Z}$ is the rigidified inertia stack of $\mathcal{Z}$ . See [Reference Abramovich, Graber and Vistoli3] for more details on the construction of evaluation maps.

K-theoretic Gromov-Witten invariants of $\mathcal{Z}$ are Euler characteristics of the following form:

(2.1) \begin{equation} \chi \!\left ({\mathcal{K}}_{g,n}(\mathcal{Z},d),{\mathcal{O}}_{{\mathcal{K}}_{g,n}(\mathcal{Z},d)}^{vir}\otimes \bigotimes _{i=1}^n ev_i^* \alpha _i\right )\in \mathbb{Z}, \quad \alpha _1,\ldots,\alpha _n\in K^*(\bar{I}\mathcal{Z}). \end{equation}

2.2. Structure morphism

Sending a stable map $f\,:\, \left({\mathcal{C}}, \{\Sigma _i\}_{i=1}^n\right)/S\to \mathcal{Z}$ to the induced map $\bar{f}\,:\, \left(C, \{\bar{\Sigma }_i\}_{i=1}^n\right){/}S\to Z$ between coarse moduli spaces yields a morphism

(2.2) \begin{equation} {\mathcal{K}}_{g,n}(\mathcal{Z},d)\to{\mathcal{K}}_{g,n}(Z,d). \end{equation}

We examine (2.2) in the special case $\mathcal{Z}=\sqrt [r]{L/X}$ and $g=0$ .

As explained in [Reference Andreini, Jiang and Tseng5, Section 3.1], the rigidified inertia stack of $\sqrt [r]{L/X}$ is a disjoint union of components $\bar{I}\!\left(\sqrt [r]{L/X}\right)_g$ indexed by $g\in \mu _r$ . As in [Reference Andreini, Jiang and Tseng5, Definition 3.3], for $g_1,\ldots,g_n\in \mu _r$ , put

\begin{equation*} {\mathcal {K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec {g}}=\bigcap _{i=1}^{n}ev_i^{-1}\!\left(\bar {I}(\sqrt [r]{L/X})_{g_i}\right). \end{equation*}

In order for ${\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}}$ to be non-empty, the elements $g_1,\ldots,g_n$ are required to satisfy certain condition, see [Reference Andreini, Jiang and Tseng5, Section 3.1].

We consider the restriction of (2.2) to ${\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}}$ :

(2.3) \begin{equation} p\,:\,{\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}}\to{\mathcal{K}}_{0,n}(X,d). \end{equation}

The structure of the map $p$ has been analyzed in ref. [Reference Andreini, Jiang and Tseng5]. We reproduce [Reference Andreini, Jiang and Tseng5, Diagram (26)] as follows:

(2.4)

Here, $\mathfrak{M}_{0,n}$ is the stack of $n$ -pointed genus $0$ prestable curves (see e.g. [Reference Bae and Schmidt7] for a discussion), and $\mathfrak{M}_{0,n}^{tw}$ is the stack of $n$ -pointed genus $0$ prestable twisted curves (see [Reference Olsson21]). $\mathfrak{M}_{0,n,d}$ and $\mathfrak{M}_{0,n,d}^{tw}$ are variants of $\mathfrak{M}_{0,n}$ and $\mathfrak{M}_{0,n}^{tw}$ parametrizing prestable (twisted) curves weighted by $d\in H^2(X,\mathbb{Z})$ , see [Reference Andreini, Jiang and Tseng5, Section 3.2] for an introduction and [Reference Bae and Schmidt7] and [Reference Tseng26] for further details.

In (2.4), the stack $\mathfrak{Y}_{0,n,d}^{\vec{g}}$ is constructed in [Reference Andreini, Jiang and Tseng5, Definition 3.12] by applying the root construction to a certain divisor of $\mathfrak{M}_{0,n,d}$ . It follows that the composition $\mathfrak{Y}_{0,n,d}^{\vec{g}}\to \mathfrak{M}_{0,n,d}^{tw}\to \mathfrak{M}_{0,n,d}$ is proper and birational. The stacks $P$ and $P_n^{\vec{g}}$ are defined by cartesian squares. The map $s$ is defined by [Reference Andreini, Jiang and Tseng5, Lemma 3.18].

2.3. Pushforward

We now examine obstruction theories. Since the map $s^{\prime\prime}$ is étale, the standard obstruction theory on ${\mathcal{K}}_{0,n}(X,d)$ relative to $\mathfrak{M}_{0,n}$ can be viewed as a obstruction theory $E_{{\mathcal{K}}_{0,n}(X,d)}^\bullet \to L_q^\bullet$ on ${\mathcal{K}}_{0,n}(X,d)$ relative to the morphism $q$ . The stack $P$ can be equipped with an obstruction theory relative to the morphism $q^{\prime}$ by pulling back $E_{{\mathcal{K}}_{0,n}(X,d)}^\bullet$ . The stack $P_n^{\vec{g}}$ can be equipped with an obstruction theory relative to $\mathfrak{Y}_{0,n,d}^{\vec{g}}$ by pulling back the obstruction theory on $P$ .

Since both maps $s^{\prime}$ and $\mathfrak{Y}_{0,n,d}^{\vec{g}}\to \mathfrak{M}_{0,n,d}^{tw}$ are étale [Reference Andreini, Jiang and Tseng5, Lemma 3.15], the standard obstruction theory on ${\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}}$ relative to $\mathfrak{M}_{0,n}^{tw}$ can be viewed as an obstruction theory $E_{{\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}}}^\bullet \to L_s^\bullet$ on ${\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}}$ relative to the morphism $s$ .

By [Reference Andreini, Jiang and Tseng5, Lemma 4.1], $E_{{\mathcal{K}}_{0,n}(X,d)}^\bullet$ pulls back to $E_{{\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}}}^\bullet$ . We then have the following results on virtual structure sheaves.

1. 1. Since $\mathfrak{Y}_{0,n,d}^{\vec{g}}\to \mathfrak{M}_{0,n,d}$ is proper and birational, by [Reference Chou, Herr and Lee11, Theorem 1.12], we have

   (2.5) \begin{equation} (r\circ r^{\prime})_*\!\left[{\mathcal{O}}_{P_n^{\vec{g}}}^{vir}\right]=\left[{\mathcal{O}}_{{\mathcal{K}}_{0,n}(X,d)}^{vir}\right]. \end{equation}

2. 2. By [Reference Andreini, Jiang and Tseng5, Theorem 3.19], the map $t\,:\,{\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}}\to P_n^{\vec{g}}$ is a $\mu _r$ -gerbe. Hence, by [Reference Chou, Herr and Lee11, Proposition 1.9], we have

   (2.6) \begin{equation} t_*\!\left[{\mathcal{O}}_{{\mathcal{K}}_{0,n}\left(\sqrt [r]{L/X},d\right)^{\vec{g}}}^{vir}\right]=\left[{\mathcal{O}}_{P_n^{\vec{g}}}^{vir}\right]. \end{equation}

2.4. Invariants

The evaluation maps on ${\mathcal{K}}_{0,n}(X,d)$ and ${\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}}$ fit into the following commutative diagram:

Consider the descendant line bundles $L_1, \ldots, L_n\to{\mathcal{K}}_{0,n}(X,d)$ associated to the marked points. The following is the main result of this note:

Proposition 2.2. For $\alpha _1,\ldots,\alpha _n\in K^*(X)$ and $k_1,\ldots,k_n\in \mathbb{Z}$ , we have

\begin{align*} & \chi \!\left ({\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}},{\mathcal{O}}_{{\mathcal{K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec{g}}}^{vir}\otimes \bigotimes _{i=1}^n \!\left((p^*L_i)^{\otimes k_i}\otimes ev_i^*\left(\left(\bar{I}\rho \right)^*\alpha _i\right)\right) \right )\\ & \qquad =\chi \!\left ({\mathcal{K}}_{0,n}(X,d),{\mathcal{O}}_{{\mathcal{K}}_{0,n}(X,d)}^{vir}\otimes \bigotimes _{i=1}^n \left(L_i^{\otimes k_i}\otimes ev_i^*(\alpha _i)\right) \right ). \end{align*}

Proof. Since $\bar{I}\rho \circ ev_i=ev_i\circ p$ , projection formula gives

\begin{equation*} p_*\left ({\mathcal {O}}_{{\mathcal {K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec {g}}}^{vir}\otimes \bigotimes _{i=1}^n \left(\left(p^*L_i\right)^{\otimes k_i}\otimes ev_i^*\left(\left(\bar {I}\rho \right)^*\alpha _i\right)\right)\right )=p_*\left({\mathcal {O}}_{{\mathcal {K}}_{0,n}\!\left(\sqrt [r]{L/X},d\right)^{\vec {g}}}^{vir}\right)\!\otimes\! \bigotimes _{i=1}^n \left(L_i^{\otimes k_i}\otimes ev_i^*(\alpha _i)\right). \end{equation*}

Since $p=r\circ r^{\prime}\circ t$ , the result follows from (2.5) and (2.6).

2.5. Banded abelian gerbes

Suppose $G$ is a finite abelian group. Suppose $\mathcal{G}\to X$ is a gerbe banded by $G$ . Then the isomorphism class of $\mathcal{G}\to X$ is classified by the cohomology group $H^2(X, G)$ , where $G$ is viewed as a constant sheaf on $X$ . We say that $\mathcal{G}\to X$ is essentially trivial if the image of its class is trivial for maps $H^2(X,G)\to H^2(X,\mathbb{C}^*)$ induced by group homomorphisms $G\to \mathbb{C}^*$ . Examples of essentially trivial gerbes include toric gerbes [Reference Tseng25].

Let $\mathcal{G}\to X$ be an essentially trivial gerbe over $X$ . Then by [Reference Andreini, Jiang and Tseng5, Lemma A.2], $\mathcal{G}$ is of the form

(2.7) \begin{equation} \mathcal{G}\simeq \sqrt [r_1]{L_1/X}\times _X\sqrt [r_1]{L_2/X}\times _X\ldots \times _X\sqrt [r_k]{L_k/X} \end{equation}

where $L_1,\ldots,L_k$ are line bundles over $X$ and $r_1,\ldots,r_k$ are natural numbers.

Consider the morphism (2.2) in this case:

(2.8) \begin{equation} {\mathcal{K}}_{0,n}(\mathcal{G},d)\to{\mathcal{K}}_{0,n}(X,d). \end{equation}

By the alaysis of [Reference Andreini, Jiang and Tseng5, Appendix A], (2.8) also fits into diagram like (2.4), with a factorization

(2.9) \begin{equation}{\mathcal{K}}_{0,n}(\mathcal{G},d)^{\vec{g}}\to P_n^{\vec{g}}\to{\mathcal{K}}_{0,n}(X,d). \end{equation}

Here, $\vec{g}$ is defined in [Reference Andreini, Jiang and Tseng5, Definition A.5].

The map $P_n^{\vec{g}}\to{\mathcal{K}}_{0,n}(X,d)$ is by construction virtually birational, hence we can apply [Reference Chou, Herr and Lee11, Theorem 1.12] to it. By [Reference Andreini, Jiang and Tseng5, Theorem A.6], the map ${\mathcal{K}}_{0,n}(\mathcal{G},d)^{\vec{g}}\to P_n^{\vec{g}}$ is also a gerbe, so we can apply [Reference Chou, Herr and Lee11, Proposition 1.9] to it. Therefore, we may repeat the arguments in Section 2.4 to extend Proposition 2.2 to essentially trivial banded abelian gerbes $\mathcal{G}\to X$ .

2.6. Root stacks

Let $D\subset X$ be a smooth irreducible divisor. For an integer $r\gt 0$ , one can construct the stack $X_{D,r}$ of $r$ -th roots of $X$ along $D$ , see [Reference Cadman9] and [Reference Abramovich, Graber and Vistoli3, Appendix B]. The natural map

(2.10) \begin{equation} X_{D,r}\to X \end{equation}

is an isomorphism over $X\setminus D$ and is a $\mu _r$ -gerbe over $D$ . Denote by $D_r\subset X_{D,r}$ the inverse image of $D$ under (2.10).

It is shown in ref. [Reference Abramovich, Cadman and Wise1] that genus $0$ relative Gromov-Witten invariants of the pair $(X,D)$ are the same as Gromov-Witten invariants of $X_{D,r}$ for $r$ sufficiently large. Here we explain how their method can be adapted to K-theoretic Gromov-Witten theory.

By [Reference Abramovich and Fantechi2], there is an isomorphism between moduli spacesFootnote 1 of stable relative maps,

\begin{equation*} \Psi \,:\, \overline {M}_{0,n}(X_{D,r}, D_r)\to \overline {M}_{0,n}(X,D), \end{equation*}

see also [Reference Abramovich, Cadman and Wise1, Theorem 2.1]. This implies an identification of virtual structure sheaves,

(2.11) \begin{equation} \Psi _*\big[\mathcal{O}_{\overline{M}_{0,n}(X_{D,r}, D_r)}^{vir}\big]=\big[\mathcal{O}_{\overline{M}_{0,n}(X,D)}^{vir}\big]. \end{equation}

There is a natural map that forgets the relative structure

\begin{equation*} \Phi \,:\,\overline {M}_{0,n}(X_{D,r}, D_r)\to \overline {M}_{0,n}(X_{D,r}). \end{equation*}

Assume that $r$ is sufficiently large. The proof of [Reference Abramovich, Cadman and Wise1, Theorem 2.2] implies that $\Phi$ is virtually birational. Hence, by [Reference Chou, Herr and Lee11, Theorem 1.12], we have

(2.12) \begin{equation} \Phi _*\big[\mathcal{O}_{\overline{M}_{0,n}(X_{D,r}, D_r)}^{vir}\big]=\big[\mathcal{O}_{\overline{M}_{0,n}(X_{D,r})}^{vir}\big]. \end{equation}

Evaluation maps of these moduli spaces are compatible with $\Psi$ and $\Phi$ , see [Reference Abramovich, Cadman and Wise1, Section 2.2]. It follows from (2.11) and (2.12) that for $\alpha _1,\ldots,\alpha _k\in K^*(X)$ , $\gamma _1,\ldots,\gamma _l\in K^*(D)$ , and $r$ sufficiently large, we have

(2.13) \begin{align} &\chi \!\left (\overline{M}_{0,n}(X_{D,r}),\mathcal{O}_{\overline{M}_{0,n}(X_{D,r})}^{vir}\otimes \bigotimes _{i=1}^k L_i^{k_i}\otimes ev_i^*(\alpha _i)\otimes \bigotimes _{j=1}^l L_j^{m_j}\otimes ev_j^*(\beta _j) \right )\nonumber\\& \quad = \chi \!\left (\overline{M}_{0,n}(X,D),\mathcal{O}_{\overline{M}_{0,n}(X,D)}^{vir}\otimes \bigotimes _{i=1}^k L_i^{k_i}\otimes ev_i^*(\alpha _i)\otimes \bigotimes _{j=1}^l L_j^{m_j}\otimes ev_j^*(\beta _j) \right ). \end{align}

We view (2.13) as a correspondence between genus $0$ K-theoretic Gromov-Witten invariants of $(X,D)$ and $X_{D,r}$ .