2.1 Setup

1. (1) a finite-dimensional $\mathbb{Z}$ -graded complex vector space,

   $$V=\bigoplus_{i \in\mathbb{Z}_{\geq 0}}V_i,$$
   where the grading is induced by the action of a one-dimensional torus $\mathbb{C}^*_R \subseteq GL(V)$ (called the R-charge);

2. (2) an action by a linearly reductive group $G \subseteq GL(V)$ ;

3. (3) a choice of character $\theta \in \widehat G_\mathbb{Q} := \mathrm{Hom}(G,\mathbb{C}^*) \otimes_\mathbb{Z} \mathbb{Q}$ ;

4. (4) a G-invariant polynomial function $w \colon V \to \mathbb{C}$ , homogeneous of degree $d_w \gt 0$ with respect to the grading action $\mathbb{C}^*_R$ ,

satisfying the following requirements:

1. –G and $\mathbb{C}^*_R$ commute, $g\cdot \lambda = \lambda \cdot g$ for all $g \in G$ and $\lambda \in \mathbb{C}^*_R$ ;

2. –the intersection $G \cap \mathbb{C}^*_R$ is cyclic of order $d_w$ with generator J, the diagonal element in GL(V) that acts on V by multiplying $V_n$ by $e^{2 \pi i n/d_w}$ ;

3. –there are no strictly semistable points for the linearization of the G-action on V given by $\theta$ ,

   (2.1) \begin{equation}V^{\rm ss}(\theta)=V^s(\theta);\end{equation}

4. –the critical locus $Z(dw) \subset Y$ is a non-empty proper over $\operatorname{Spec} \mathbb{C}$ ;

5. –the graded pieces $V_i$ of V are empty for $i\lt 0$ and $i\gt d_w$ ,

   $$V = \bigoplus_{0 \leq i \leq d_w} V_i;$$

6. –and the group $\mathbb{C}^*_R$ preserves the $\theta$ -semistable locus $V^{\rm ss}(\theta)$ .

We denote by $\Gamma$ the group $G \cdot \mathbb{C}^*_R.$

Let Y denote the GIT stack quotient

$$Y := [V \mathbin{/\mkern-6mu/}_\theta G].$$

The function $w\colon V \to \mathbb{C}$ descends to a function on $Y\!$ , called the potential function. By abuse of notation, we will also denote the potential function by w.

Given a smooth orbifold Y and a function $w: Y \to \mathbb{C}$ , the pair (Y, w) is known as a Landau–Ginzburg (LG) model. A map of LG models

$$j': (Y, w) \to (Y', w')$$

is a morphism $j: Y \to Y'$ such that

$$w' \circ j = w.$$

We say that the GLSM $(V, G, \theta, w)$ represents the LG model (Y, w) if $[V \mathbin{/\mkern-6mu/}_\theta G] = Y$ and the potential $w: Y \to \mathbb{C}$ is induced by the G-equivariant function on V of the same name. We note that two different GLSMs can represent the same LG model. In this case, one expects a relationship between the associated GLSM invariants. By abuse of notation, we sometimes speak of the GLSM invariants of (Y, w) when the particular GLSM $(V, G, \theta, w)$ representing (Y, w) has been fixed.

We record for future use the following map of exact sequences

(2.2)

where the homomorphism $\xi$ is defined by $\xi(g \cdot \lambda) = \lambda^{d_w}$ for $g \in G$ , $\lambda \in \mathbb{C}^*_R$ .

2.2 The state space

In this section, we introduce the state spaces used for GLSM and quasimap invariants.

Definition 2.4. The GLSM state space of $(V, G, \theta, w)$ is defined as

$$\mathcal{H}(Y, w) := H^*_{\rm CR}(Y, w^{+\infty}) = H^*(IY, w^{+\infty}),$$

where $w^{+\infty} := (\mathrm{Re}(w))^{-1}( M, \infty)$ for M a sufficiently large real number.

The (Gromov–Witten theory) state space of Y is given by

$$H^*_{\rm CR}(Y) := H^*(IY).$$

There is a map

$$\phi^w\colon \mathcal{H}(Y, w) \to H^*_{\rm CR}(Y),$$

defined by pullback via the inclusion of pairs $(IY, \emptyset) \hookrightarrow (IY, w^{+\infty})$ . The kernel of this map is sometimes referred to as the broad sectors.

Consider a component $Y_g$ of $IY\!$ , with g a representative of a conjugacy class of G. Let $j\colon Z \hookrightarrow Y_g$ be a smooth and proper substack of $Y_g$ . Assume that $w|_Z = 0$ . In this case, there exists a map of LG models:

$$ j'\colon (Z, 0) \hookrightarrow (Y_g, w) \hookrightarrow (IY, w).$$

By capping with the fundamental class $\mu_{Y_g} \in H_*(Y_g)$ , we obtain an isomorphism as in [Reference FultonFul13, § 19.1]:

(2.3) \begin{equation}H^*(Y_g, Y_g\setminus Z) \xrightarrow{\cong} H_*(Z).\end{equation}

The pushforward $j_*\colon H^*(Z) \to H^*_{\rm CR}(Y)$ may be defined as the composition

$$H^*(Z) \xrightarrow{\cong} H_*(Z) \xrightarrow{\cong} H^*(Y_g, Y_g\setminus Z) \to H^*(Y_g) \subset H^*_{\rm CR}(Y),$$

where the first map is Poincaré duality, the second is the inverse to (2.3), and the third is the natural pullback of inclusion of pairs.

By assumption on Z, we have the inclusion $w^{+\infty} \subset IY \setminus Z$ . This allows the following definition.

Definition 2.5. Define the pushforward

$$j'_*\colon H^*(Z) \to \mathcal{H}(Y, w)$$

as the composition

$$H^*(Z) \xrightarrow{\cong} H_*(Z) \xrightarrow{\cong} H^*(Y_g, Y_g\setminus Z) \to H^*(Y_g, w^{+\infty}) \subset \mathcal{H}(Y, w).$$

Let

$$\phi^{\rm cs}\colon H^*_{\rm CR,cs}(Y) \to H^*_{\rm CR}(Y),$$

denote the natural map from compactly supported Chen–Ruan cohomology $H^*_{\rm CR,cs}(Y)$ to Chen–Ruan cohomology.

We observe the following.

Lemma 2.8. For Z a smooth, proper substack of $w^{-1}(0)$ in $IY\!$ , the pushforward $j_*\colon H^*(Z) \to H^*_{\rm CR}(Y)$ factors as

$$j_* = \phi^w \circ j'_*.$$

Proof. The result is immediate from the above definitions and the obvious commutativity of the following diagram.

If $j\colon Z \hookrightarrow IY$ is the inclusion of a smooth proper substack, then $j_*(\alpha)$ lies in $H^*_{\rm CR,ct}(Y)$ . By the above lemma, this implies that

$$ \phi^w (\mathcal{H}_{\rm ct}(Y, w)) \subseteq H^*_{\rm CR,ct}(Y).$$

2.3 Compact-type cup product and pairing

Here, we recall some facts about compact-type cohomology $H^*_{\rm CR,ct}(Y)$ from [Reference ShoemakerSho21]. Note that in [Reference ShoemakerSho21], the compact-type cohomology has been referred to as narrow cohomology. In this paper, we have adopted the language that appears in [Reference Fan, Jarvis and RuanFJR18].

Definition 2.9. Given a class $\gamma \in H^*_{\rm CR,ct}(Y)$ , we say that $\widetilde{\gamma} \in H^*_{\rm CR,cs}(Y)$ is a lift of $\gamma$ if

$$\phi^{\rm cs}(\widetilde{\gamma}) = \gamma.$$

In addition to the usual cup products,

\begin{align*}\cup \colon &H^*_{\rm CR}(Y) \times H^*_{\rm CR}(Y) \to H^*_{\rm CR}(Y), \\\cup \colon &H^*_{\rm CR,cs}(Y) \times H^*_{\rm CR}(Y) \to H^*_{\rm CR,cs}(Y),\end{align*}

there is a cup product $\cup_{\rm cs}\colon H^*_{\rm CR,ct}(Y) \times H^*_{\rm CR,ct}(Y) \to H^*_{\rm CR,cs}(Y)$ that multiplies a pair of elements from compact-type cohomology to an element of compactly supported cohomology. It is given by

$$\gamma_1 \cup_{\rm cs} \gamma_2 := \widetilde{\gamma}_1 \cup\gamma_2,$$

where $\gamma_1, \gamma_2 \in H^*_{\rm CR,ct}(Y)$ and $\widetilde{\gamma}_1$ is a lift of $\gamma_1$ . One can check that if $\kappa \in \ker(\phi^{\rm cs})$ and $\gamma \in H^*_{\rm CR,ct}(Y)$ , then $\kappa \cup \gamma = 0$ in $H^*_{\rm CR,ct}(Y)$ , from which it follows that $\cup_{\rm cs}$ is well defined (for details and a proof in de Rham cohomology when Y is smooth, see [Reference ShoemakerSho21]; a similar argument holds in singular cohomology).

Recall the Chen–Ruan pairing $\langle -,-\rangle^Y\!$ . For $\widetilde{\gamma}_1 \in H^*_{\rm CR,cs}(Y)$ and $ \gamma_2 \in H^*_{\rm CR}(Y)$ ,

$$\langle\widetilde{\gamma}_1, \gamma_2\rangle^Y :=\int_{IY} \widetilde{\gamma}_1 \cup I_*( \gamma_2),$$

where $I\colon IY \to IY$ is the natural involution. By [Reference Chen and RuanCR04], this pairing is non-degenerate. We also define a compact-type pairing $\langle -,-\rangle^{Y, {\rm cs}}$ on $H^*_{\rm CR,ct}(Y)$ by

$$\langle\gamma_1, \gamma_2\rangle^{Y, {\rm cs}} := \int_{IY} \gamma_1 \cup_{\rm cs} I_*(\gamma_2),$$

By [Reference ShoemakerSho21, Corollary 2.7], this pairing is non-degenerate.

It is immediate from the definitions that the product and pairing on $H^*_{\rm CR,ct}(Y)$ are compatible with the map $\phi^{\rm cs}$ . That is, for $\widetilde{\gamma}_1, \widetilde{\gamma}_2 \in H^*_{\rm CR,cs}(Y)$ ,

(2.4) \begin{equation}\widetilde{\gamma}_1 \cup \widetilde{\gamma}_2 = \phi^{\rm cs}(\widetilde{\gamma}_1) \cup \widetilde{\gamma}_2 = \phi^{\rm cs}(\widetilde{\gamma}_1) \cup_{\rm cs} \phi^{\rm cs}(\widetilde{\gamma}_2)\end{equation}

and

(2.5) \begin{equation}\langle \widetilde{\gamma}_1, \phi^{\rm cs}(\widetilde{\gamma}_2)\rangle^Y = \langle \phi^{\rm cs}(\widetilde{\gamma}_1) , \phi^{\rm cs}(\widetilde{\gamma}_2)\rangle^{Y, {\rm cs}},\end{equation}

where $\langle -,-\rangle^Y$ denote the usual Chen–Ruan pairing between $H^*_{\rm CR,cs}(Y)$ and $H^*_{\rm CR}(Y)$ .

3.1 Review of quasimap invariants

We denote the moduli space of genus-h, n-pointed, degree-d quasimaps to $Y = [V \mathbin{/\mkern-6mu/}_\theta G]$ by $Q_{h,n}(Y, d)$ . In this paper, we only work with $0^+$ -stability, so we omit it from the notation. We give a construction of $Q_{h,n}(Y, d)$ and the corresponding quasimap invariants below. This particular presentation of the moduli space and virtual class appears in [Reference Ciocan-Fontanine and KimCFK10] in the toric setting. In this generality, the moduli space and virtual class have originally been constructed in [Reference Ciocan-Fontanine, Kim and MaulikCFKM14].

Let $\mathfrak{M}_{h,n}(BG)$ denote the moduli space of genus-h, n-pointed, prestable orbi-curves $\mathcal{C}$ , together with a principal G-bundle $\mathcal{P}$ such that the induced morphism $[\mathcal{P}]\colon \mathcal{C} \to BG$ is representable.

Define the degree of a principal G-bundle $\mathcal{P} \to \mathcal{C}$ to be the element $d \in \hom_\mathbb{Z}(\widehat G, \mathbb{Q})$ such that, for each character $\zeta \in \widehat G$ ,

$$d(\zeta) = \deg(\mathcal{P} \times_G \mathbb{C}_\zeta),$$

where $\mathbb{C}_\zeta$ is the representation corresponding to $\zeta$ .

For $d \in \hom_\mathbb{Z}(\widehat G, \mathbb{Q})$ , define $\mathfrak{M}_{h,n}(BG, d)$ to be the open and closed substack of $\mathfrak{M}_{h,n}(BG)$ for which the principal bundle has degree d. Denote by $\mathfrak{M}_{h,n}(BG, d)^\theta$ the open substack consisting of pairs $(\mathcal{C}, \mathcal{P})$ such that $\mathcal{C}$ has no rational tails and, on each irreducible component $\mathcal{C} '$ of $\mathcal{C}$ for which $\omega_{\mathcal{C}, \mathrm{log}}|_{\mathcal{C} '}$ is trivial, the degree of

$$\mathcal{L}_\theta := \mathcal{P} \times_G\mathbb{C}_\theta$$

restricted to $\mathcal{C} '$ is positive. Let $\pi\colon \mathfrak{C} \to \mathfrak{M}_{h,n}(BG, d)^\theta$ denote the universal curve and let $\mathcal{P}_G \to \mathfrak{C}$ be the universal principal G-bundle. Denote by $\mathcal{V}_G$ the vector bundle $ \mathcal{P}_G \times_G V$ .

Definition 3.2. Define the space of sections of $\mathcal{V}_G$ to be

\[ \operatorname{tot}(\pi_*\mathcal{V}_G) := \operatorname{Spec} ( \mathrm{Sym} ( \mathbb{R}^1 \pi_*(\omega_\pi \otimes \mathcal{V}_G^\vee))),\]

lying over $\mathfrak{M}_{h,n}(BG, d)^\theta$ .

Closed points of $\operatorname{tot}(\pi_*\mathcal{V}_G)$ consist of a marked curve $\mathcal{C}$ together with a section $u_G \in \Gamma(\mathcal{C}, \mathcal{V}_G|_\mathcal{C})$ , which defines a map $[u_G]\colon \mathcal{C} \to [V/G]$ to the stack quotient. The basepoints of $[u_G]$ consist of the preimage of the unstable locus $[u_G]^{-1}( [V^{\rm us}(\theta)/G])$ .

The moduli space $Q_{h,n}(Y, d)$ admits a natural relative obstruction theory over $\mathfrak{M}_{h,n}(BG, d)^\theta$ given by $\mathbb{R} \pi_*(\mathcal{V}_G)^\vee.$ This defines a virtual class

$$[Q_{h,n}(Y, d)]^{\rm vir}$$

by [Reference Behrend and FantechiBF97]. By choosing an appropriate $\pi$ -acyclic resolution of $\mathcal{V}_G$ , we can realize this virtual class more concretely. We make use of the following.

1. – $\mathcal{A}$ and $\mathcal{B}$ are $\pi$ -acyclic; and

2. –there exists a restriction map $\mathcal{A} \to \mathcal{V}_G|_{\Sigma}$ such that the composition $\mathcal{V}_G \to \mathcal{A} \to \mathcal{V}_G|_{\Sigma}$ is the usual restriction, and $\pi_*(\mathcal{A}) \to \pi_*(\mathcal{V}_G|_{\Sigma})$ is surjective.

Let $A = \pi_*(\mathcal{A})$ and $B = \pi_*(\mathcal{B})$ . Let $p_A\colon \operatorname{tot}(A) \to \mathfrak{M}_{h,n}(B\Gamma, d)^\theta$ denote the projection. There exists an open subset $U \subset \textrm{tot}(A)$ such that $Q_{h,n}(Y, d)$ can be realized as $\{\beta = 0\} \subset U$ , where $\beta = p_A^*(\delta) \circ \operatorname{taut}_A $ is the section of $ E = p_A^*(B)$ induced by $\delta$ (see, e.g., [Reference Ciocan-Fontanine and KimCFK10, proof of Theorem 3.2.1]).

The two-term complex

$$[E^\vee\xrightarrow{p_A^*(\delta)^\vee} p_A^*(A)^\vee = \Omega^1_{U/\mathfrak{M}_{h,n}(BG, d)^\theta}]$$

gives a resolution of $\mathbb{R} \pi_*(\mathcal{V}_G)^\vee$ over U, and thus restricts to the relative perfect obstruction theory described above. In this case, the virtual class may be realized as a localized top Chern class (for details, see [Reference Behrend and FantechiBF97, § 6]). Consider the fiber square

where $\beta$ and 0 are the maps induced by the corresponding sections of E.

Definition 3.6 Define the virtual class $[Q_{h,n}(Y, d)]^{\rm vir}$ to be the localized top Chern class

$$[Q_{h,n}(Y, d)]^{\rm vir}:= e_{\rm loc}(E, \beta) = \operatorname{cl}(0^!( [U])),$$

where $0^!: A_*(U) \to A_*(Q_{h,n}(Y, d))$ is the Gysin pullback and $\operatorname{cl}: A_*(Q_{h,n}(Y, d)) \to H_*(Q_{h,n}(Y, d))$ is the map, described in [Reference FultonFul13, § 19.1], which gives the corresponding class in Borel–Moore homology.

We take this as the definition. It coincides with the (image in homology of the) virtual class of [Reference Behrend and FantechiBF97] using the relative obstruction theory given above, and thus is independent of the resolution $[A \to B]$ chosen in Proposition 3.5.

We define quasimap invariants by integrating against the virtual class. We require the following lemma.

Proof. By [Reference Cheong, Ciocan-Fontanine and KimCCFK15, Theorem 2.7], $Q_{h,n}(Y, d)$ is a proper Deligne–Mumford stack over the affine quotient $\operatorname{Spec}( (\mathrm{Sym}^\bullet V^\vee)^G)$ . The map

$$Q_{h,n}(Y, d) \to \operatorname{Spec}( (\mathrm{Sym}^\bullet V^\vee)^G)$$

factors through $ev_i$ ; therefore by [Sta20, Lemma 01W6], $ev_i$ is proper.

For $1 \leq i\leq n$ , let $\boldsymbol{r}_i$ be the locally constant function on $Q_{h,n}(Y, d)$ with value given by the order of the automorphism group at the ith marking on each connected component (see [Reference TsengTse10, § 2.1]). Let $\boldsymbol{r}$ denote the product

$$\boldsymbol{r} = \prod_{i=1}^n \boldsymbol{r}_i.$$

The quasimap invariant $\langle \gamma_1 \psi^{k_1}, \ldots, \gamma_n \psi^{k_n}\rangle_{h,n,d}^Y$ is well defined if either:

1. • at least one class $\gamma_i$ lies in compactly supported cohomology; or

2. • at least two classes $\gamma_i, \gamma_j$ lie in compact-type cohomology.

Definition 3.8. Given $\gamma_1 \in H^*_{\rm CR,cs}(Y)$ and $\gamma_2, \ldots, \gamma_n \in H^*_{\rm CR}(Y)$ , define the quasimap invariant $\langle \gamma_1 \psi^{k_1}, \ldots, \gamma_n \psi^{k_n}\rangle_{h,n,d}^Y$ to be

$$ \int_{[Q_{h,n}(Y, d)]^{\rm vir}}\boldsymbol{r} \cdot{ev_1^c}^*(\gamma_1) \cup \psi_1^{k_1} \cup \prod_{2 \leq i \leq n} ev_i^*(\gamma_i) \cup \psi_i^{k_i}, $$

where ${ev_1^c}^*$ is the pullback in compactly supported cohomology. Given $\gamma_1, \gamma_2 \in H^*_{\rm CR,ct}(Y)$ and $\gamma_3, \ldots, \gamma_n \in H^*_{\rm CR}(Y)$ , define $\langle \gamma_1 \psi^{k_1}, \ldots, \gamma_n \psi^{k_n}\rangle_{h,n,d}^Y$ to be

$$ \int_{[Q_{h,n}(Y, d)]^{\rm vir}}\boldsymbol{r} \cdot{ev_1}^*(\gamma_1) \cup_{\rm cs} {ev_2}^*(\gamma_2) \cup \psi_1^{k_1} \cup \psi_2^{k_2} \cup \prod_{3 \leq i \leq n} ev_i^*(\gamma_i) \cup \psi_i^{k_i}. $$

3.2 A virtual class for GLSMs

Given a GLSM $(V, G, \theta, w)$ , the moduli stack of $\epsilon$ -stable genus-h, n-pointed, degree-d LG maps, $LG_{h,n}^{\epsilon}(Y, d)$ has been defined in [Reference Fan, Jarvis and RuanFJR18]. In this paper, we focus on the $\epsilon = 0^+$ stability condition. The moduli space $LG_{h,n}^{0^+}(Y, d)$ will be denoted by $QLG_{h,n}(Y, d)$ . We call this the moduli space of LG quasimaps.

Via the exact sequences given in (2.2), the group $\hom_\mathbb{Z}(\hat{\Gamma}, \mathbb{Q})$ is canonically isomorphic to

$$\hom_\mathbb{Z}(\widehat G, \mathbb{Q}) \times \hom_\mathbb{Z}(\hat{\mathbb{C}}^*_R, \mathbb{Q}).$$

As in Definition 3.1, a principal $\Gamma$ -bundle $\mathcal{P} \to \mathcal{C}$ naturally defines an element of $\hom_\mathbb{Z}(\hat{\Gamma}, \mathbb{Q})$ . Unless otherwise specified, the degree of a principal $\Gamma$ -bundle will refer the image of this element under the projection map $\hom_\mathbb{Z}(\hat{\Gamma}, \mathbb{Q}) \to \hom_\mathbb{Z}(\widehat G, \mathbb{Q})$ .

Definition 3.10. For $c \in \mathbb{Z}$ , $h, n \in \mathbb{Z}_{\geq 0}$ and $d \in \hom_\mathbb{Z}(\widehat G, \mathbb{Q})$ , let $\mathfrak{M}_{h,n}(B\Gamma, d)_{\omega_{\mathcal{C}, \mathrm{log}}^c}$ denote the moduli stack of genus-h, n-pointed, prestable orbi-curves $\mathcal{C}$ with a degree-d principal $\Gamma$ -bundle $\mathcal{P} \to \mathcal{C}$ , and an isomorphism

$$\eta\colon \xi_*(\mathcal{P}) \cong (\omega_{\mathcal{C}, \mathrm{log}}^{\otimes c})^\circ,$$

such that the induced morphism $[\mathcal{P}]\colon \mathcal{C} \to B\Gamma$ is representable.

Let $\nu \in \widehat \Gamma$ be a lift of $\theta$ ; that is, $\nu|_G = \theta$ . Denote by $\mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}^c}$ the open substack of $\mathfrak{M}_{h,n}(B\Gamma, d)_{\omega_{\mathcal{C}, \mathrm{log}}^c}$ consisting of pairs $(\mathcal{C}, \mathcal{P})$ such that $\mathcal{C}$ has no rational tails and, on each irreducible component $\mathcal{C} '$ of $\mathcal{C}$ for which $\omega_{\mathcal{C}, \mathrm{log}}|_{\mathcal{C} '}$ is trivial, the degree of

$$\mathcal{L}_\nu := \mathcal{P} \times_\Gamma \mathbb{C}_\nu$$

restricted to $\mathcal{C} '$ is positive. We note that for two choices $\nu$ or $\nu '$ of lift, the line bundles $\mathcal{L}_\nu$ and $\mathcal{L}_{\nu '}$ differ by a power of $\omega_{\mathcal{C}, \mathrm{log}}$ . Therefore, the definition of $\mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}^c}$ is independent of the lift of $\theta$ (see [Reference Fan, Jarvis and RuanFJR18, Corollary 4.2.15]).

Let $\pi\colon \mathfrak{C} \to \mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}^c}$ denote the universal curve and let $\mathcal{P}_\Gamma \to \mathfrak{C}$ denote the universal principal $\Gamma$ -bundle. Let $\mathcal{V}_\Gamma := \mathcal{P}_\Gamma \times_\Gamma V$ .

The construction of the moduli space $QLG_{h,n}(Y, d)$ and the virtual class is identical to § 3.1 after replacing $\mathfrak{M}_{h,n}(BG, d)^\theta$ with $\mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}}$ . As in Definition 3.2, define $\operatorname{tot}(\pi_*\mathcal{V}_\Gamma)$ , this time over $\mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}}$ . Closed points of $\operatorname{tot}(\pi_*\mathcal{V}_\Gamma)$ consist of a marked curve $\mathcal{C}$ together with a section $u_\Gamma \in \Gamma(\mathcal{C}, \mathcal{V}_\Gamma|_\mathcal{C})$ , which defines a map $[u_\Gamma]\colon \mathcal{C} \to [V/\Gamma]$ to the stack quotient. The stack $QLG_{h,n}(Y, d)$ is the open substack of $\operatorname{tot}(\pi_*\mathcal{V}_\Gamma)$ over $\mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}}$ satisfying the same condition that the basepoint locus $[u_\Gamma]^{-1}( [V^{\rm us}(\theta)/\Gamma])$ is a finite set disjoint from nodes and markings.

As described in [Reference Fan, Jarvis and RuanFJR18, § 4.4], there exist evaluation maps $ev_i\colon QLG_{h,n}(Y, d) \to IY$ which are defined as follows. Let $\Sigma_i \to QLG_{h,n}(Y, d)$ denote the ith-marked-point gerbe (recall that by the definition of $ \mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}}$ this is a trivial gerbe). The restriction $[u_\Gamma|_{\Sigma_i}]\colon \Sigma_i \to [V/ \Gamma]$ determines a map $QLG_{h,n}(Y, d) \to I[V/\Gamma]$ . In fact, this map factors through $[V/G]$ as follows. The isomorphism $\eta\colon \xi_*(\mathcal{P}) \cong (\omega_{\mathcal{C}, \mathrm{log}})^\circ$ induces a map

$$ \mathcal{P}|_{\Sigma_i} \to (\omega_{\mathcal{C}, \mathrm{log}})^\circ|_{\Sigma_i} \xrightarrow{=} \mathcal{O}_{QLG_{h,n}(Y, d)}^\circ|_{\Sigma_i},$$

where the second map is the residue at $\Sigma_i$ . The kernel ${\mathcal{P}}_i '$ gives a principal G-bundle on $\Sigma_i$ . There is an isomorphism ${\mathcal{P}}_i ' \times_G \Gamma \cong \mathcal{P}|_{\Sigma_i} $ , from which it follows that ${\mathcal{P}}_i ' \times_G V = \mathcal{P}|_{\Sigma_i} \times_\Gamma V$ . Therefore, the restriction $u|_{\Sigma_i} $ defines a section of ${\mathcal{P}}_i ' \times_G V$ , which in turn defines a map $\Sigma_i \to [V/G]$ . Due to the condition that basepoints are disjoint from markings, in fact $\Sigma_i $ maps to $Y= [V\mathbin{/\mkern-6mu/}_\theta G]$ , which allows us to define evaluation maps

$$ev_i\colon QLG_{h,n}(Y, d) \to IY.$$

As before, the relative obstruction theory over $\mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}}$ is given by $\mathbb{R} \pi_*(\mathcal{V}_\Gamma)^\vee.$ Proposition 3.5 holds over $\mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}}$ as well. Thus, we can define a smooth Deligne–Mumford stack $U \to \mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}}$ , a vector bundle $E \to U$ , and a natural section $\beta \in \Gamma(U, E)$ such that $QLG_{h,n}(Y, d) = \{\beta = 0\}$ . As in the previous section, we make the following definition.

Definition 3.11. Define the virtual class $[QLG_{h,n}(Y, d)]^{\rm vir}$ to be

$$[QLG_{h,n}(Y, d)]^{\rm vir} := e_{\rm loc}(E, \beta) = \operatorname{cl}(0^!( [U])),$$

where $0^!: A_*(U) \to A_*(QLG_{h,n}(Y, d))$ is the Gysin pullback via the following diagram.

In general, the virtual class of Definition 3.11 is not sufficient for defining enumerative GLSM invariants because the moduli space $QLG_{h,n}(Y, d)$ is not usually proper; nor are the evaluation maps. Furthermore, this virtual class does not take into account the potential $w\colon Y \to \mathbb{C}$ .

3.3 GLSM invariants via cosection localization

In this section, we recall the definition of compact-type GLSM invariants proposed by Fan, Jarvis, and Ruan in [Reference Fan, Jarvis and RuanFJR18, Reference Fan, Jarvis and RuanFJR20].Footnote ¹ For compact-type insertions $\gamma_1, \ldots, \gamma_n \in \mathcal{H}_{\rm ct}(Y, w)$ satisfying Assumption 3.12 below, they define the GLSM invariant

$$\langle \gamma_1 \psi^{k_1}, \ldots, \gamma_n \psi^{k_n}\rangle_{h,n,d}^{(Y, w)}$$

in terms of a cosection-localized virtual class. This section serves as motivation for our eventual definition of compact-type GLSM invariants in the case of proper evaluation maps (Definition 3.17); however, it is not strictly necessary for the rest of the paper.

Let $\gamma_1, \ldots, \gamma_n \in \mathcal{H}_{\rm ct}(Y, w)$ be compact-type insertions. By definition of $\mathcal{H}_{\rm ct}(Y, w)$ , $\gamma_i$ may be expressed as ${j'_i}_*(\beta_i)$ for $j_i\colon Z_i \hookrightarrow IY$ an inclusion of a smooth proper substack into the locus $\{w = 0\}$ and $j'_i\colon (Z_i,0) \to (IY, w)$ the map of LG models. Without loss of generality, we may assume that $Z_i$ is contained in a single twisted sector $Y_{g_i} = [(V^{g_i})^{\rm ss} / Z_G(g_i)]$ . Following [Reference Fan, Jarvis and RuanFJR20, § 6.1], we assume the following for $1 \leq i \leq n$ .

Let

$$\underline {ev}: QLG_{h,n}(Y, d) \to IY \times\cdots \times IY $$

denote the product of the evaluation maps. Denote by $\underline Z$ the product $Z_1 \times \cdots \times Z_n \subset IY \times \cdots \times IY\!$ . Define the substacks

$$QLG_{\underline Z}(Y, d) : = \underline {ev}^{-1}(\underline Z)_{\vphantom{\big(}}$$

and

$$QLG_{\underline Z}(\mathrm{Crit}(w), d) = QLG_{\underline Z}(Y, d)\cap QLG_{h,n}(\mathrm{Crit}(w), d),_{\vphantom{\big(}}$$

where $QLG_{h,n}(\mathrm{Crit}(w), d)$ denotes the locus of LG quasimaps to the critical locus $[\{dw = 0\} /\Gamma] \subset [V/\Gamma]$ . As explained in [Reference Fan, Jarvis and RuanFJR20], there is a modified perfect obstruction theory on $QLG_{\underline Z}(Y, d)$ , which we recall below.

First, we restrict from $\mathfrak{M}_{h,n}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}}$ to $\mathfrak{M}_{h,(\underline g)}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}}$ , the open and closed substack defined by the condition that the generator of the isotropy at the ith marked point maps to $g_i \in G$ . Let $\pi_{\underline g}: \mathfrak{C}_{\underline g} \to \mathfrak{M}_{h,(\underline g)}(B\Gamma, d)^\theta_{\omega_{\mathcal{C}, \mathrm{log}}}$ denote the universal curve. On $\mathfrak{C}_{\underline g}$ , consider the restriction map $\mathcal{V}_\Gamma \to \mathcal{V}_\Gamma|_{\Sigma_i} = {\sigma_i}_* \sigma_i^* \mathcal{V}_\Gamma$ . The vector bundle $\sigma_i^* \mathcal{V}_\Gamma$ splits into eigen-bundles with respect to the action of $g_i$ . Thus, there is a canonical projection $\sigma_i^* \mathcal{V}_\Gamma \to (\sigma_i^* \mathcal{V}_\Gamma)^{g_i}$ . Define

$$\mathcal{V}_\Gamma|_{\Sigma_i}^{g_i}:={\sigma_i}_* (\sigma_i^* \mathcal{V}_\Gamma)^{g_i}.$$

Let $\mathcal{H}_i$ denote the vector bundle $\mathcal{P}_\Gamma \times_\Gamma H_i$ on $\mathfrak{C}_{\underline g}$ . Let $\mathcal{H}_i^\perp$ denote the quotient $(\sigma_i^* \mathcal{V}_\Gamma)^{g_i}/ \sigma_i^* H_i$ . By composing the map $\mathcal{V}_\Gamma \to \mathcal{V}_\Gamma|_{\Sigma_i}$ with

(3.1) \begin{equation}{\sigma_i}_* \sigma_i^* \mathcal{V}_\Gamma \to {\sigma_i}_* (\sigma_i^* \mathcal{V}_\Gamma)^{g_i} \to {\sigma_i}_* \mathcal{H}_i^\perp\end{equation}

for each i, we obtain the following surjective morphism of sheaves:

(3.2) \begin{equation} \mathcal{V}_\Gamma \to \bigoplus_{i=1}^n {\sigma_i}_* \mathcal{H}_i^\perp.\end{equation}

Denote the kernel by $\widetilde{\mathcal{V}}_\Gamma$ . A local computation shows that $\widetilde{\mathcal{V}}_\Gamma$ is a vector bundle on $\mathfrak{C}_{\underline g}$ (see the discussion following [Reference Fan, Jarvis and RuanFJR20, Definition 6.1.2]).

Observe that $QLG_{\underline Z}(Y, d) $ is the locus in $QLG_{h,(\underline g)}(Y, d)$ for which the section of $\mathcal{V}_\Gamma$ vanishes outside of $ \mathcal{H}_i$ after restricting to $\Sigma_i$ . In other words, $QLG_{\underline Z}(Y, d) $ is defined as the intersection of

$$QLG_{h,(\underline g)}(Y, d) := \underline { ev}^{-1}(Y_{g_1} \times \cdots \times Y_{g_n})$$

with

$$\textrm{tot} \pi_*( \widetilde{\mathcal{V}}_\Gamma) \subset\textrm{tot} \pi_* (\mathcal{V}_\Gamma).$$

On $QLG_{\underline Z}(Y, d) $ , there is a relative perfect obstruction theory given by $\mathbb{R} \pi_*(\widetilde{\mathcal{V}}_\Gamma)^\vee.$ By [Reference Fan, Jarvis and RuanFJR20, Lemma 6.1.5], on the corresponding absolute obstruction theory there exists a cosection

$$\textrm{Obs}_{QLG_{\underline Z}(Y, d)} \to \mathcal{O}_{QLG_{\underline Z}(Y, d)}.$$

The associated cosection-localized virtual class [Reference Kiem and LiKL20] is supported on the proper stack $QLG_{\underline Z}(\mathrm{Crit}(w), d)$ . We denote the cosection-localized virtual class by

$$[QLG_{\underline Z}(\mathrm{Crit}(w), d)]^{\rm vir} := [QLG_{\underline Z}(Y, d)]^{\rm vir}_{\rm loc}.$$

With this setup, one can make the following definition.

Definition 3.13. Let $\gamma_1, \ldots, \gamma_n$ be classes arising as the pushforward of $\beta_1, \ldots, \beta_n$ via maps $H^*(Z_i) \to \mathcal{H}_{\rm ct}(Y, w)$ . Under Assumption 3.12, define the GLSM invariant $\langle \gamma_1 \psi^{k_1}, \ldots, \gamma_n \psi^{k_n}\rangle_{h,n,d}^{Y,w}$ as

(3.3) \begin{equation}\int_{[QLG_{\underline Z}(\mathrm{Crit}(w), d)]^{\rm vir}} \boldsymbol{r} \cdot \prod_{i=1}^n (ev_i^*(\beta_i) \cup \psi_i^{k_i} ).\end{equation}

Consider the following diagram

(3.4)

where $U_{\underline Z}$ is defined as the fiber product. Because the map $A \to \pi_*( \mathcal{V}_\Gamma|_\Sigma)$ is surjective, $\underline {\widetilde{ev}}$ is smooth. We have the following relationship between $ [QLG_{\underline Z}(\mathrm{Crit}(w), d)]^{\rm vir}$ and the virtual class $[QLG_{h,n}(Y, d)]^{\rm vir}$ of Definition 3.11.

Proposition 3.14. Let $c\colon QLG_{\underline Z}(\mathrm{Crit}(w), d) \to QLG_{\underline Z}(Y, d)$ denote the inclusion. Then,

$$ c_*([QLG_{\underline Z}(\mathrm{Crit}(w), d)]^{\rm vir}) = e_{\rm loc}(\widetilde j^* E, \widetilde j^*\beta)$$

in $H_*(QLG_{\underline Z}(Y, d))$ .

Recall the discussion preceding Definition 3.11. Let $\mathcal{A} \to \mathcal{B}$ be a resolution of $\mathcal{V}_\Gamma|_{\mathfrak{C}_{\underline g}}$ satisfying the properties of Proposition 3.5. Compose the surjective map $\mathcal{A} \to \mathcal{V}_\Gamma|_{\Sigma_i}$ with (3.1) and let $\widetilde{\mathcal{A}}$ denote the kernel. We have the following commutative diagram of sheaves on $\mathfrak{C}_{\underline g}$ , where all columns and rows are short exact sequences.

(3.5)

It follows from the second bullet in Proposition 3.5 that the map

$${\pi_{\underline g}}_*(\mathcal{A}) \to {\pi_{\underline g}}_*\bigg(\! \bigoplus_{i=1}^n {\sigma_i}_* \mathcal{H}_i^\perp\!\bigg)$$

is surjective; therefore $\mathbb{R}^1 {\pi_{\underline g}}_*(\widetilde {\mathcal{A}}) = 0$ . Let $\widetilde A$ denote the vector bundle ${\pi_{\underline g}}_*(\widetilde {\mathcal{A}})$ .

Pushing forward (3.5), we observe that $U_{\underline Z}$ is the intersection of U with $\textrm{tot} ( \widetilde A) \subset \textrm{tot} (A)$ . Furthermore, the two-term complex

$$[\widetilde j^* E^\vee \xrightarrow{\beta^\vee} \widetilde j^* p_A^*(\widetilde A)^\vee]$$

gives a resolution of $\mathbb{R} \pi_*(\widetilde{\mathcal{V}}_\Gamma)^\vee$ on $U_{\underline Z}$ , and $QLG_{\underline Z}(Y, d)$ is the zero locus of the section $\widetilde j^* \beta \in \Gamma( U_{\underline Z}, \widetilde j^* E)$ . By the general discussion in [Reference Behrend and FantechiBF97, § 6], it follows that $[QLG_{\underline Z}(Y, d)]^{\rm vir}$ is the localized top Chern class $e_{\rm loc}(\widetilde j^* E, \widetilde j^* \beta)$ as desired.

3.4 The special case of proper evaluation maps

In this section we explain how, if the evaluation maps $ ev_i\colon QLG_{h,n}(Y, d) \to IY$ are proper, there is a simple expression for compact-type GLSM invariants that does not require use of a cosection.

Proposition 3.15. Let $\gamma_1, \ldots, \gamma_n$ be classes arising as the pushforward of $\beta_1, \ldots, \beta_n$ via maps $H^*(Z_i) \to \mathcal{H}_{\rm ct}(Y, w)$ as in § 3.3, with each $Z_i$ satisfying Assumption 3.12. Assume that $ev_i\colon QLG_{h,n}(Y, d) \to IY$ is proper for $1 \leq i \leq n$ . Then, (3.3) is equal to

(3.6) \begin{equation}\int_{[QLG_{h,n}(Y, d)]^{\rm vir}} \boldsymbol{r} \cdot {ev_1^c}^*({j_1}^c_* \beta_1) \cup \cdots \cup {ev_n^c}^*({j_n}^c_* \beta_n) \cup \prod_{i=1}^n ( \psi_i^{k_i} ),\end{equation}

where ${ev_i^c}^*$ and ${j_i^c}_*$ denote the pullback and pushforward in compactly supported cohomology, and $[QLG_{h,n}(Y, d)]^{\rm vir}$ is the (non-localized) virtual class of Definition 3.11.

Proof. By assumption,

$$QLG_{\underline Z}(Y, d) = U_{\underline Z} \cap QLG_{h,n}(Y, d) = \underline {ev}^{-1}(\underline Z)$$

is proper.

By Proposition 3.14, (3.3) is equal to

(3.7) \begin{align}&\int_{QLG_{\underline Z}(Y, d)} \boldsymbol{r} \cdot \prod_{i=1}^n ev_i^*(\beta_i) \cup \prod_{i=1}^n ( \psi_i^{k_i} )\cap e_{\rm loc}(\widetilde j^* E,\widetilde j^* \beta),\end{align}

which simplifies to

(3.8) \begin{align}&\int_{QLG_{\underline Z}(Y, d)} \boldsymbol{r} \cdot \prod_{i=1}^n ev_i^*(\beta_i) \cup \prod_{i=1}^n ( \psi_i^{k_i} )\cap \widetilde j^! (e_{\rm loc}( E, \beta)) \nonumber\\[5pt]&\quad=\int_{QLG_{\underline Z}(Y, d)} \boldsymbol{r} \cdot \prod_{i=1}^n ev_i^*(\beta_i) \cup \prod_{i=1}^n ( \psi_i^{k_i} )\cap j^! (e_{\rm loc}( E, \beta)) \nonumber\\[5pt]&\quad=\int_{\underline Z} \boldsymbol{r} \cdot\underline {ev}_*\bigg({\underline {ev}}^*(\underline {\beta}) \cup \prod_{i=1}^n (\psi_i^{k_i} )\cap j^! (e_{\rm loc}( E, \beta))\!\bigg)\nonumber\\[5pt]&\quad=\int_{\underline Z} \boldsymbol{r} \cdot \underline {\beta} \cap \underline {ev}_* j^! \bigg( \prod_{i=1}^n ( \psi_i^{k_i} ) \cap e_{\rm loc}( E, \beta)\!\bigg)\nonumber\\[5pt]&\quad=\int_{\underline Z} \boldsymbol{r} \cdot \underline {\beta} \cap j^*{\underline {ev}}_* \bigg( \prod_{i=1}^n ( \psi_i^{k_i} ) \cap e_{\rm loc}( E, \beta)\!\bigg).\end{align}

The equality of (3.7) with the first line is by definition of $e_{\rm loc}(E, \beta)$ and [Reference FultonFul13, Proposition 14.1 (d)(ii)]. The first equality is compatibility of Gysin pullbacks [Reference FultonFul13, Theorem 6.2 (c)]. The final equality is [Reference FultonFul13, Theorem 6.2 (a)]. Here, $\underline {\beta}$ denotes the exterior product of $\beta_1$ through $\beta_n$ .

Let $j_*^c\colon H^*(\underline Z) \to H^*_{\rm cs}(\prod_{i=1}^n I Y)$ denote the pushforward to compactly supported cohomology. Then, (3.8) is equal to

(3.9) \begin{align}&\int_{\prod_{i=1}^n I Y} j_*^c(\underline {\beta}) \cap {\underline {ev}}_* \bigg(\!\boldsymbol{r} \cdot \prod_{i=1}^n ( \psi_i^{k_i} )\cap e_{\rm loc}( E, \beta)\!\bigg)\nonumber\\&\quad=\int_{QLG_{h,n}(Y, d)} \boldsymbol{r} \cdot {\underline {ev}^c}^*j_*^c(\underline {\beta}) \cup \prod_{i=1}^n ( \psi_i^{k_i} )\cap e_{\rm loc}( E, \beta)\nonumber\\&\quad=\int_{QLG_{h,n}(Y, d)} \boldsymbol{r} \cdot{\underline {ev}^c}^*\bigg(\!\bigotimes_{i=1}^n {j_i}_*^c({\beta_i}) \!\bigg) \cup \prod_{i=1}^n ( \psi_i^{k_i} )\cap e_{\rm loc}( E, \beta)\nonumber\\&\quad=\int_{QLG_{h,n}(Y, d)} \boldsymbol{r} \cdot {ev_1^c}^*({j_1}^c_* \beta_1) \cup \cdots \cup {ev_n^c}^*({j_n}^c_* \beta_n) \cup \prod_{i=1}^n ( \psi_i^{k_i} )\cap e_{\rm loc}( E, \beta).\end{align}

The first and second lines are each applications of the projection formula (with respect to the maps j and $\underline {ev}$ , respectively). The second equality is simply by functoriality of the Kunneth formula, which holds for compactly supported cohomology as well.

The last line of (3.9) is exactly (3.6), finishing the proof.

Proof. As shown in [Reference ShoemakerSho21], for a proper map f, $\phi^{\rm cs} \circ {f^c}^* = f^* \circ \phi^{\rm cs}$ , and $\phi^{\rm cs} \circ f^c_* = f_* \circ \phi^{\rm cs}$ . Then, under our assumption that $n \geq 2$ , by (2.4) we have

\begin{align*}&{ev_1^c}^*({j_1}^c_* \beta_1) \cup {ev_2^c}^*({j_2}^c_* \beta_2) \\&\quad=\phi^{\rm cs}({ev_1^c}^*({j_1}^c_* \beta_1)) \cup_{\rm cs} \phi^{\rm cs}({ev_2^c}^*({j_2}^c_* \beta_2)) \\&\quad= ev_1^*({j_1}_*\phi^{\rm cs}( \beta_1)) \cup_{\rm cs} ev_2^*({j_2}_*\phi^{\rm cs}( \beta_2))\\&\quad= ev_1^*({j_1}_*( \beta_1)) \cup_{\rm cs} ev_2^*({j_2}_*( \beta_2))\\&\quad= ev_1^*(\phi^w(\gamma_1)) \cup_{\rm cs} ev_2^*(\phi^w( \gamma_2)).\\[-30pt]\end{align*}

Note that in the presence of proper evaluation maps, (3.6) is defined regardless of Assumption 3.12. Motivated then by Proposition 3.15, we use (3.6) and Lemma 3.16 to define compact-type GLSM invariants in the case in which the evaluation maps are proper.

Definition 3.17 Let $\gamma_1, \ldots, \gamma_n$ be classes in $\mathcal{H}_{\rm ct}(Y, w)$ . Assume that $n\geq 2$ and the evaluation maps $ev_i$ are proper for $1 \leq i \leq 2$ . Define the GLSM invariant $\langle \gamma_1 \psi^{k_1}, \ldots, \gamma_n \psi^{k_n}\rangle_{h,n,d}^{Y,w}$ as

(3.10) \begin{equation}\int_{ [QLG_{h,n}(Y, d)]^{\rm vir}} \boldsymbol{r} \cdot ev_1^*(\phi^w(\gamma_1)) \cup_{\rm cs} ev_2^*(\phi^w( \gamma_2)) \cup \cdots \cup ev_n^*(\phi^w( \gamma_n)) \cup \prod_{i=1}^n ( \psi_i^{k_i} ).\end{equation}

Elements of the kernel of the map

$$\phi^w\colon \mathcal{H}(Y, w) \to H^*_{\rm CR}(Y)$$

are often referred to as broad sectors. There are conjectures (see, e.g., [Reference Clader, Janda and RuanCJR21, Conjecture 2.13]) about when GLSM invariants with broad insertions are zero. Such results are known as broad vanishing. The following ‘corollary’ is immediate from our definition.

In practice, Definition 3.17 is quite restrictive. Evaluation maps from moduli spaces of LG maps are rarely proper. In the next section, we show that in the special case in which $h=0$ and $n=2$ , the evaluation maps are always proper and the definition applies. We will see in later sections that this case is of special relevance to mirror symmetry.

5.1 Quasimap invariants

In [Reference Ciocan-Fontanine and KimCFK16], Ciocan-Fontanine and Kim define quasimap invariants with light points, and use these to define a new class of generating functions of quasimap invariants. These functions are shown to be related to more standard generating functions of Gromov–Witten invariants via wall crossing, but they have the benefit of being readily computable.

In the language of quasimaps, light points are obtained by replacing the moduli space $ Q_{h,n}(Y, d)$ with $ Q_{h,n}(Y \times (\mathbb{P}^0)^k, (d,1^k))$ , where $(d, 1^k)$ is shorthand for $(d, 1, \ldots, 1) \in \hom_\mathbb{Z}( \widehat{(G \times (\mathbb{C}^*)^k)} , \mathbb{Q})$ . The basepoint of a degree-one quasimap to $\mathbb{P}^0:= [\mathbb{C}\mathbin{/\mkern-6mu/}_{\operatorname{id}} \mathbb{C}^*]$ marks a single point on the source curve. We call these points light because they may collide with one another in the moduli space of stable quasimaps.

Definition 5.1. Define the moduli space of genus-h, n-pointed, degree-d quasimaps with k light points to be

$$Q_{h,n|k}(Y, d):= Q_{h,n}(Y \times (\mathbb{P}^0)^k, (d,1^k)).$$

Define

$$[Q_{h,n|k}(Y, d)]^{\rm vir}:= [Q_{h,n}(Y \times (\mathbb{P}^0)^k, (d,1^k))]^{\rm vir}.$$

Label the light marked points as $q_1, \ldots, q_k$ . Note that light marked points may collide with each other, but not with heavy points or nodes. On the quasimap moduli space $Q_{h,n|k}(Y, d)$ , there exist evaluation maps $\hat{ev}_j$ at the light marked points that map to the stack quotient [Reference Ciocan-Fontanine and KimCFK16, § 2.3]. They may be defined as the composition

$$\hat{ev}_j^Q \colon Q_{h,n|k}(Y, d) \xrightarrow{\hat{s}_j} \mathcal{C} \xrightarrow{[u_G]} [V/G],$$

where $\hat{s}_j\colon Q_{h,n|k}(Y, d) \to \mathcal{C} $ is the section of the universal curve given by the jth light point. We note that the light markings defined here have no orbifold structure.

Quasimap invariants with light-point insertions are defined in [Reference Ciocan-Fontanine and KimCFK16]. The following is a slight modification of the definition in [Reference Ciocan-Fontanine and KimCFK16] to the case in which Y is not assumed to be proper.

Definition 5.2. Given $\gamma_1 \in H^*_{\rm CR,cs}(Y)$ , $\gamma_2, \ldots, \gamma_n \in H^*_{\rm CR}(Y)$ , and $\alpha_1, \ldots, \alpha_k \in H^*([V/G])$ , define $\langle \gamma_1 \psi^{k_1}, \ldots, \gamma_n \psi^{k_n}| \alpha_1, \ldots, \alpha_k\rangle_{h,n|k,d}^{Y}$ to be

$$ \int_{[Q_{h,n}(Y, d)]^{\rm vir}}\boldsymbol{r} \cdot{ev_1^c}^*(\gamma_1) \cup \psi_1^{k_1} \cup \prod_{2 \leq i \leq n} ( ev_i^*(\gamma_i) \cup \psi_i^{k_i}) \cup \prod_{j=1}^k \hat{ev}^{Q *}(\alpha_j), $$

where ${ev_1^c}^*$ is the pullback in compactly supported cohomology.

Given $\gamma_1, \gamma_2 \in H^*_{\rm CR,ct}(Y)$ , $\gamma_3, \ldots, \gamma_n \in H^*_{\rm CR}(Y)$ , and $\alpha_1, \ldots, \alpha_k \in H^*([V/G])$ , define

$$\langle \gamma_1 \psi^{k_1}, \ldots, \gamma_n \psi^{k_n}| \alpha_1, \ldots, \alpha_k\rangle_{h,n|k,d}^{Y}$$

to be

$$ \int_{[Q_{h,n|k}(Y, d)]^{\rm vir}}\boldsymbol{r} \cdot{ev_1}^*(\gamma_1) \cup_{\rm cs} {ev_2}^*(\gamma_2) \cup \psi_1^{k_1} \cup \psi_2^{k_2} \cup \prod_{3 \leq i \leq n} (ev_i^*(\gamma_i) \cup \psi_i^{k_i} )\cup \prod_{j=1}^k \hat{ev}^{Q *}_j(\alpha_j). $$

5.2. GLSM invariants

In this section, we use the same type of modification of the target $[V/G]$ to define LG quasimaps with light points and their corresponding GLSM invariants in the special case of $h=0, n=2$ .

Definition 5.3. Define the space of genus-h, n-pointed, degree-d LG quasimaps with k light points to be

$$QLG_{h,n|k}(Y, d):= QLG_{h,n}(Y \times (\mathbb{P}^0)^k, (d,1^k)).$$

On the moduli space of LG quasimaps, there exist light evaluation maps to $[V_0 /(G/\langle J\rangle)]$ , where $V_0$ is the $\mathbb{C}^*_R$ -fixed locus of V. The evaluation map at the jth light point is defined as

(5.1) \begin{equation}\hat{ev}^{QLG}_j\colon QLG_{h,n|k}(Y, d)\xrightarrow{\hat s_j} \mathcal{C} \xrightarrow{[u_\Gamma]} [V/\Gamma] \xrightarrow{\operatorname{proj}} [V_0/\Gamma] \to [V_0 /(G/\langle J\rangle)],\end{equation}

where the last map is induced by the homomorphism $\Gamma \to G/\langle J \rangle$ (note that $\langle J \rangle$ acts trivially on $V_0$ ).

Remark 5.4. (Motivating the space of LG quasimaps with light points.) Let $Y^0_J$ denote the $\mathbb{C}^*_R$ -invariant locus of the Jth twisted sector $Y_J$ . Assume that $Y^0_J = [V_0 \mathbin{/\mkern-6mu/}_\theta G]$ and consider the substack of $QLG_{h,n+k}(Y, d)$ of LG quasimaps, where the last k marked points each map via the evaluation map to $Y^0_J$ :

$$QLG_{h,n+k}(Y, d)(Y^0_J)_{j=1}^k := \bigcap_{j=1}^k ev_j^{-1}(Y^0_J).$$

The space $QLG_{h,n+k}(Y, d)(Y^0_J)_{j=1}^k$ contains an open subset $QLG_{h,n+k}(Y, d)^\circ(Y^0_J)_{j=1}^k$ defined by the condition that $\omega_{\mathcal{C}}(p_1 + \cdots + p_n) \otimes L_\theta^{\otimes \epsilon}$ is ample for all $\epsilon \gt 0 $ .

On the other hand, let $QLG_{h,n|k}(Y, d)^\circ$ denote the open substack of $QLG_{h,n|k}(Y, d)$ consisting of quasimaps for which the light points are disjoint from each other and from basepoints, and $\omega_{\mathcal{C}}(p_1 + \cdots + p_n) \otimes L_\theta^{\epsilon}$ is ample for all $\epsilon \gt 0 $ .

Following the arguments of [Reference Favero and KimFK20, § 5.4], by applying a Hecke modification and forgetting the orbifold structure at the last k marked points, one can define a map

$$QLG_{h,n+k}(Y, d)^\circ(Y^0_J)_{j=1}^k \to QLG_{h,n|k}(Y, d)^\circ,$$

which, due to the gerbe structure at the last k marked points, is a $\mu_{d_w}^k$ -gerbe (in particular, the map on coarse spaces is an isomorphism).

Furthermore, one can check using the proof of [Reference Favero and KimFK20, Lemma 5.5 (1)] that the evaluation maps are compatible, i.e. for $1 \leq j\leq k$ the following diagram commutes.

Thus, under mild assumptions, the space $QLG_{h,n|k}(Y, d)$ may be viewed, up to a $\mu_{d_w}^k$ -gerbe structure, as an alternative compactification of $QLG_{h,n+k}(Y, d)^\circ(Y^0_j)$ where the last k points are allowed to coincide. This is the motivation behind Definition 5.3. We expect these invariants to be related to GLSM invariants with all heavy marked points via a wall-crossing formula as in [Reference PinharryPin20, Reference ZhouZho20].

Let $j\colon [V_0/(G/\langle J\rangle)] \to [ V^J/(G/\langle J\rangle)]$ denote the obvious inclusion of stacks. The $(G/\langle J\rangle)$ -equivariant analogue of (2.3) and Definition 2.5 defines the pushforwards

$$j_*\colon H^*([V_0/(G/\langle J\rangle)]) \to H^*([ V^J/(G/\langle J\rangle)])$$

and

$$j'_*\colon H^*( [V_0/(G/\langle J\rangle)]) \to H^*([V^J/(G/\langle J\rangle)], w^{+\infty}).$$

Viewing j as the zero section of a vector bundle, we note that $j_*\circ j^*(\gamma)= e(V_{d_w}) \cup \gamma$ , where here $V_{d_w}$ denotes the vector bundle $[(V^J \times V_{d_w})/(G/\langle J\rangle)] \to [V^J/(G/\langle J\rangle)]$ . As before, if

$$\phi^w\colon H^*([ V^J/(G/\langle J\rangle)], w^{+\infty}) \to H^*([ V^J/(G/\langle J\rangle)])$$

is the $(G/\langle J\rangle)$ -equivariant pullback of the map of pairs $(V_0, \emptyset) \hookrightarrow (V^J, w^{+\infty})$ , then

(5.2) \begin{equation} j_* = \phi^w \circ j'_*.\end{equation}

Definition 5.5. Given classes $\gamma_1, \gamma_2 \in \mathcal{H}_{\rm ct}(Y, w)$ and

$$\tilde \sigma_1, \ldots, \tilde \sigma_k \in H^*([V/(G/\langle J\rangle)], w^{+\infty})$$

arising as the pushforward of $\sigma_1, \ldots, \sigma_k \in H^*([V_0/(G/\langle J\rangle)])$ via $j'_*$ , define $\langle\gamma_1 \psi^{k_1}, \gamma_2\psi^{k_2}| \tilde \sigma_1, \ldots, \tilde \sigma_k\rangle_{0,2|k,d}^{(Y, w)}$ to be

\[ \int_{ [QLG_{0,2|k}(Y, d)]^{\rm vir}} \boldsymbol{r} \cdot ev_1^*(\phi^{\rm cs}(\gamma_1)) \cup_{\rm cs} ev_2^*(\phi^{\rm cs}(\gamma_2)) \cup \psi^{k_1} \cup \psi^{k_2} \cup \prod_{j=1}^k \hat{ev}^{QLG *}_j(\sigma_j). \]

In the toric setting under a natural assumption, we verify that the above invariants are well defined. This is the main setting in which we perform computations (see Corollary 6.19).

Proof. We claim that $j'_*$ is injective. By (5.2), $j'_*$ is injective if $j_*$ is injective. Note that

$$j^*\colon H^*([V^J/(G/\langle J\rangle)]) \to H^*([V_0/(G/\langle J\rangle)])$$

is an isomorphism; thus the claim holds whenever $j_* \circ j^* = e(V_{d_w}) \cup ( - )$ is injective.

Because G is a torus, $V_{d_w}$ splits as a sum of one-dimensional representations. We claim that none of these are trivial. Suppose on the contrary that $V_{d_w}$ contains a trivial rank-one G-representation L’ as a summand. Let p be the degree- $d_w$ homogeneous coordinate on L’. Because w is degree $d_w$ , we may write it as

$$w = p \cdot w_p + w',$$

where $w_p$ is a an element of $\mathrm{Sym}^\bullet(V_0^\vee)$ and w’ does not depend on p. Because p is G-invariant and each monomial of W is G-invariant, we conclude that $w_p$ is G-invariant. The assumption $(\mathrm{Sym}^\bullet(V^\vee))^\Gamma = \mathbb{C}$ implies that $\mathrm{Sym}^\bullet(V_0^\vee)^G = \mathbb{C}$ . Consequently $w_p$ is constant. If $w_p \neq 0$ , then the critical locus Z(dw) is empty, as it lies in $\partial w/\partial p = w_p = 0$ . Thus $w_p$ must be zero, so $w = w'$ . Because L’ is a trivial representation, the critical locus Z(dw) is therefore isomorphic to $Z(dw'|_{p=0}) \times \mathbb{C}$ , where $Z(dw'|_{p=0})$ is the critical locus in $\{p = 0\} \subset V$ . This contradicts the assumption that Z(dw) is proper over $\operatorname{Spec} \mathbb{C}$ .

Because $V_{d_w}$ has no trivial summands, its Euler class is a non-zero polynomial in $H^*([ V^J/(G/\langle J\rangle)]) \cong \mathbb{C}[t_1, \ldots, t_k]$ . This proves that $j_* \circ j^* = e(V_{d_w}) \cup ( - )$ is injective.

Let $i\colon [V_0/G] \to [V/G]$ denote the inclusion and let $r\colon [V_0/G] \to [V_0/(G/\langle J\rangle)]$ denote the $\langle J \rangle$ -gerbe. The pullback $r^*$ is an isomorphism by the Lyndon–Hochschild–Serre spectral sequence for the extension

$$1 \to \langle J\rangle \to G \to G/\langle J\rangle \to 1$$

and the fact that $H^*(B\langle J\rangle, \mathbb{C}) = \mathbb{C}$ . Define the map

\begin{align*}\tau\colon H^*([V/G]) \to & H^*([ V^J/(G/\langle J\rangle)], w^{+\infty}) \\ \alpha \mapsto &j'_*((r^*)^{-1}i^*\alpha).\end{align*}

We have the following comparison between quasimap invariants and GLSM invariants with light points.

Theorem 5.8. Given classes $\gamma_1, \gamma_2 \in \mathcal{H}_{\rm ct}(Y, w)$ and $\alpha_1, \ldots, \alpha_k \in H^*([V/G])$ ,

\[\langle\gamma_1 \psi^{k_1}, \gamma_2\psi^{k_2}| \tau(\alpha_1), \ldots, \tau(\alpha_k)\rangle_{0,2|k,d}^{(Y, w)} = \langle\phi^w(\gamma_1)\psi_1^{k_1},\phi^w(\gamma_2)\psi_2^{k_2}| \alpha_1, \ldots, \alpha_k\rangle_{0,2|k,d}^{Y}.\]

Proof. First recall that, by definition,

\begin{align*}QLG_{0,2|k}(Y, d) &= QLG_{0,2}(Y \times (\mathbb{P}^0)^k, (d,1^k)),\\Q_{0,2|k}(Y, d) & = Q_{0,2}(Y \times (\mathbb{P}^0)^k, (d,1^k)).\end{align*}

By Proposition 4.4 applied to the spaces on the right, $QLG_{0,2|k}(Y, d)$ and $Q_{0,2|k}(Y, d)$ are canonically isomorphic and their virtual classes are identified.

Next, we compare the light evaluation maps. For $1 \leq j\leq k$ , we have the following commuting diagram.

We see that

$$ \hat{ev}_j^{Q *} \alpha = \hat{ev}_j^{QLG *}(r^*)^{-1}i^*\alpha.$$

The result then follows from Definition 5.5.

6.1 The S-operator

Definition 6.1. Let $\mathbb{C}[[q]]$ denote the Novikov ring:

$$\mathbb{C}[[q]] := \mathbb{C}[[ \mathrm{Eff}(V, G, \theta)]],$$

where we denote the element $d \in \mathbb{C}[[ \mathrm{Eff}(V, G, \theta)]]$ by $q^d$ .

Let ${\boldsymbol{t}} = \sum_{1 \leq j \leq m} t^j T_j$ , where $T_j\in H^*([V/G])$ and $t^j$ are formal parameters. For $\gamma_1 \in H^*_{\rm CR,cs}(Y)$ , $ \gamma_2 \in H^*_{\rm CR}(Y)$ and $k_1, k_2 \geq 0$ , define the generating function

\[\langle\!\langle \gamma_1 \psi^{k_1}, \gamma_2\psi^{k_2}\rangle\!\rangle_{0,2}^{Y}({\boldsymbol{t}}) := \sum_{d \in\mathrm{Eff}(V, G, \theta)} \sum_{k \geq 0} \frac{q^d}{k!} \langle \gamma_1\psi^{k_1}, \gamma_2\psi^{k_2} | {\boldsymbol{t}}, \ldots, {\boldsymbol{t}}\rangle_{0, 2|k, d}^Y,\]

where

\[\langle \gamma_1\psi^{k_1}, \gamma_2\psi^{k_2} | {\boldsymbol{t}}, \ldots, {\boldsymbol{t}}\rangle_{0, 2|k, d}^Y := \int_{[Q_{0, 2|k}(Y, d)]^{\rm vir}} \boldsymbol{r} \cdot{ev_1^c}^*(\gamma_1) \cup ev_2^*(\gamma_2)\cup \psi_1^{k_1} \cup \psi_2^{k_2}\prod_{j=1}^{k} \hat{ev}_j^{Q *}({\boldsymbol{t}}).\]

Alternatively, for $\gamma_1, \gamma_2 \in H^*_{\rm CR,ct}(Y)$ , $\langle\!\langle \gamma_1\psi^{k_1}, \gamma_2\psi^{k_2}\rangle\!\rangle_{0,2}^{Y}({\boldsymbol{t}})$ is defined similarly, but replacing ${ev_1^c}^*(\gamma_1) \cup ev_2^*(\gamma_2)$ with $ev_1^*(\gamma_1) \cup_{\rm cs} ev_2^*(\gamma_2)$ .

Definition 6.2. Let $\{ \tilde \gamma_i\}_{i \in \tilde I}$ be a basis for $H^*_{\rm CR}(Y)$ and let $\{ \tilde \gamma^i\}_{i \in \tilde I}$ be the dual basis in $H^*_{\rm CR,cs}(Y)$ . The (light-point, quasimap) S-operator on $H^*_{\rm CR}(Y)[[q]][[t^i]][z, z^{-1}]$ is defined by

\[S^{Y}(q, {\boldsymbol{t}}, z)(\gamma) := \sum_{i \in \tilde I} \tilde\gamma_i \left\langle\!\!\left\langle \frac{\tilde \gamma^i}{z-\psi}, \gamma\right\rangle\!\!\right\rangle_{\!0, 2}^{\!\!Y}({\boldsymbol{t}}).\]

As shown in [Reference Ciocan-Fontanine and KimCFK16], the inverse of S is given by $L^{Y}(q, {\boldsymbol{t}}, z) = {S^Y}^*(q, {\boldsymbol{t}}, -z)$ ,

\[L^{Y}(q, {\boldsymbol{t}}, z)(\gamma) := \sum_{i \in \tilde I} \tilde\gamma_i \left\langle\!\!\left\langle\tilde \gamma^i,\frac{\gamma}{-z- \psi}\right\rangle\!\!\right\rangle_{\!0, 2}^{\!\!Y}({\boldsymbol{t}}).\]

Recall from § 2.3 that there is a perfect pairing on $H^*_{\rm CR,ct}(Y)$ . Let $\{ \gamma_i\}_{i \in I}$ be a basis for $H^*_{\rm CR,ct}(Y)$ and let $\{ \gamma^i\}_{i \in I}$ be the dual basis.

Lemma 6.3. The operator $S^{Y}(q, {\boldsymbol{t}}, z)$ preserves the compact-type subspace, i.e. if $\gamma \in H^*_{\rm CR,ct}(Y)[[q]][[t^i]][z, z^{-1}]$ , then $S^{Y}(q, {\boldsymbol{t}}, z)(\gamma)\in H^*_{\rm CR,ct}(Y)[[q]][[t^i]][z, z^{-1}]$ . The same holds for $L^{Y}(q, {\boldsymbol{t}}, z)$ . For $\gamma$ as above, we have

$$S^{Y}(q, {\boldsymbol{t}}, z)(\gamma) = \sum_{i \in I} \gamma_i\left\langle\!\!\left\langle \frac{\gamma^i}{z- \psi},\gamma\right\rangle\!\!\right\rangle_{\!0, 2}^{\!\!Y}({\boldsymbol{t}}).$$

Proof. The generating function $S^{Y}(q, {\boldsymbol{t}}, z)(\gamma)$ may be alternatively defined as

\[\gamma + \sum_{d \in \operatorname{Eff}} \sum_{k \geq 0} \frac{q^d}{k!} \widetilde{ev_1}_* \bigg(\frac{ev_2^*(\gamma)}{z - \psi} \prod_{j=1}^{k} \hat{ev}_j^{Q *}({\boldsymbol{t}}) \cap [Q_{0, 2|k}(Y, d)]^{\rm vir}\! \bigg), \]

where $\widetilde{ev_1}$ is the composition of ${ev_1}$ with the involution $I\colon IY \to IY\!$ . Both $ev_1$ and $ev_2$ are proper by Lemma 3.7. By [Reference ShoemakerSho21, Proposition 2.5], $\widetilde{ev_1}_*$ and ${ev_2}^*$ preserve a cohomology of compact type. This proves the first statement. The same argument applies to $L^{Y}(q, {\boldsymbol{t}}, z)$ .

The final statement follows from the first together with the observation that, for $\tilde \gamma^i \in H^*_{\rm CR,cs}(Y)$ and $\gamma \in H^*_{\rm CR,ct}(Y)$ ,

\begin{align*}{ev_1^c}^*(\tilde \gamma^i) \cup ev_2^*(\gamma) = {ev_1}^*(\phi^{\rm cs}(\tilde \gamma^i)) \cup_{\rm cs} ev_2^*(\gamma).\\[-30pt]\end{align*}

For the remainder of the paper, we will assume the following.

If Assumption 6.4 holds, then $ \phi^w (\mathcal{H}_{\rm ct}(Y, w)) = H^*_{\rm CR,ct}(Y).$ We expect it to hold quite generally.

Let $\Sigma$ denote the toric fan for $Y\!$ , which may be obtained from the GIT description by, for instance, [Reference Coates, Iritani and JiangCIJ18]. Following the description by Borisov and Horja of compactly supported cohomology [Reference Borisov and Paul HorjaBPH15, Proposition 2.4], the compact-type cohomology $H^*_{\rm ct}(Y)$ is generated by the images of the maps $H^*(Y_I) \to H^*(Y)$ for I ranging over all subsets of $ \{1, \ldots, r\}$ such that $\sigma_I := \operatorname{cone}\{v_i\}_{i \in I}$ is a cone of $\Sigma$ and the interior $\sigma_I^\circ$ is contained in $|\Sigma|^\circ$ . One can check, for these choices of I, that $Y_I$ is proper over $\operatorname{Spec} \mathbb{C}$ . Thus, to verify Assumption 6.4 in this case, it suffices to show that for each such choice of I, the restriction of w to $Y_I$ is identically zero.

Here $Y_I$ may be constructed as a GIT quotient of $\operatorname{Spec} ( \mathbb{C}[x_i]_{\{i \notin I\}})$ by G. Because $Y_I$ is proper over $\operatorname{Spec} \mathbb{C}$ , any algebraic function on $Y_I$ is constant. As a result, any G-invariant function in $\mathbb{C}[x_i]_{\{i \notin I\}}$ is constant on a non-empty open subset of $\operatorname{Spec} ( \mathbb{C}[x_i]_{\{i \notin I\}})$ , and therefore on all of $\operatorname{Spec} ( \mathbb{C}[x_i]_{\{i \notin I\}})$ . It follows that $\mathbb{C}[x_i]_{\{i \notin I\}}^G = \mathbb{C}$ . Because w is homogeneous of degree $d_w \gt 0$ with respect to the R-charge, it contains no constant term. Therefore, w is zero on $\operatorname{Spec} \mathbb{C}$ . Since $w|_{Y_I}: Y_I \to \mathbb{C}$ factors through $\operatorname{Spec} (\mathbb{C} = \mathbb{C}[x_i]_{\{i \notin I\}}^G)$ , we conclude that $Y_I \subset w^{-1}(0)$ .

In the general case in which Y is a stack, we must consider the Chen–Ruan cohomology $H^*_{\rm CR,ct}(Y) = H^*_{\rm ct}(IY)$ . Observe that:

1. (1) each component $Y_g \subset IY$ is itself a toric stack arising as a GIT quotient; and

2. (2) the cohomology of $Y_g$ (with coefficients in $\mathbb{C}$ ) is equal to the cohomology of its coarse space, the simplicial toric variety $|Y_g|$ .

Working component by component, the result then follows from the previous case.

By Assumption 6.4, the map $\phi^w\colon \mathcal{H}_{\rm ct}(Y, w) \to H^*_{\rm CR,ct}(Y)$ is surjective. Choose a splitting

$$\sigma^w\colon H^*_{\rm CR,ct}(Y) \to \mathcal{H}_{\rm ct}(Y, w).$$

Denote the image of $\sigma^w$ by $ \mathcal{R}_c(Y, w)$ . Although there is no canonical choice of $ \mathcal{R}_c(Y, w)$ , by the broad vanishing result of Corollary 3.19, the $h=0$ , $n=2$ GLSM invariants with heavy-point insertions in $\mathcal{H}_{\rm ct}(Y, w)$ are fully determined by the GLSM invariants with insertions from $ \mathcal{R}_c(Y, w)$ .

Definition 6.6. Let $\{ \gamma_i\}_{i \in I}$ be a basis for $H^*_{\rm CR,ct}(Y)$ and let $\{ \gamma^i\}_{i \in I}$ be the dual basis. Define the (light-point, LG quasimap) S-operator on $\mathcal{H}_{\rm ct}(Y, w)[[q]][[t^i]][z, z^{-1}]$ by

\[S^{(Y, w)}(q, {\boldsymbol{t}}, z)(\gamma) := \sum_{i \in I}\sigma^w(\gamma_i) \bigg\langle\!\!\bigg\langle \frac{\sigma^w(\gamma^i)}{z- \psi},\gamma\bigg\rangle\!\!\bigg\rangle_{\!0, 2}^{\!\!(Y, w)}({\boldsymbol{t}}),\]

where ${\boldsymbol{t}} = \sum_{1 \leq j \leq m} t^j T_j$ with $T_j\in H^*([ V^J/(G/\langle J\rangle)], w^{+\infty})$ , and the variables $t^j$ are formal parameters.

Proposition 6.7. Whenever $P(q, {\boldsymbol{t}}, z) \in H^*_{\rm CR,ct}(Y)[[q]][[t^i]][z, z^{-1}]$ ,

\[S^{(Y, w)}(q, \tau({\boldsymbol{t}}), z)(\sigma^w(P(q, {\boldsymbol{t}}, z))) = \sigma^w(S^{Y}(q, {\boldsymbol{t}}, z)(P(q, {\boldsymbol{t}}, z))).\]

6.2 I-functions

Definition 6.8. A $(0^+, 0^+)$ I-function for Y is any function $I(q, {\boldsymbol{t}}, z)$ of the form

$$S^{Y}(q, f({\boldsymbol{t}}), z)P(q, {\boldsymbol{t}}, z)$$

where $f \in H^*([V/G])[[t^i]]$ is a formal cohomology-valued function satisfying $f(0) = 0$ , and $P(q, {\boldsymbol{t}}, z)$ lies in $H^*_{\rm CR}(Y)[[q]][[t^i]][z]$ (in particular, it contains only positive powers of z).

The relationship between the I-functions for toric stacks defined in [Reference Cheong, Ciocan-Fontanine and KimCCFK15, Reference Coates, Corti, Iritani and TsengCCI⁺15] and other $(0^+, 0^+)$ I-functions is clarified in Proposition 6.14. The connection to GLSM I-functions (as defined below) is given in Theorem 6.15.

Definition 6.8 can be generalized to the setting of GLSMs. As usual we restrict our attention to compact-type insertions.

Definition 6.11. A $(0^+, 0^+)$ GLSM I-function for $(V, G, \theta, w)$ is any function of the form

$$S^{(Y, w)}(q, f({\boldsymbol{t}}), z)P(q, {\boldsymbol{t}}, z),$$

where $f \in H^*([ V^J/(G/\langle J\rangle)], w^{+\infty})[[t^i]]$ is a formal cohomology-valued function satisfying $f(0) = 0$ , and $P(q, {\boldsymbol{t}}, z) $ lies in $ \mathcal{H}_{\rm ct}(Y, w)[[q]][[t^i]][z].$

Next, we recall the definition of the Big I-function, $\mathbb{I}^Y(q, {\boldsymbol{t}}, z)$ , of [Reference Ciocan-Fontanine and KimCFK16, Reference Cheong, Ciocan-Fontanine and KimCCFK15]. The $0^+$ -quasimap graph space $QG_{0,n|k}(Y, d)$ is the space of genus-zero quasimaps (with k light points) to $[V\mathbin{/\mkern-6mu/}_\theta G] \times [\mathbb{C}^2 \mathbin{/\mkern-6mu/}_{\operatorname{id}} \mathbb{C}^*]$ of degree (d, 1) that are $0^+$ -stable on the first factor and $\infty$ -stable (Kontsevich-stable) on the second factor:

$$QG_{0,n|k}(Y, d) := Q_{0,n|k}^{0^+, \infty}(Y \times \mathbb{P}^1, (d,1)).$$

The graph space may be viewed as the moduli space of quasimaps to Y such that each source curve has a distinguished component whose coarse underlying curve is equipped with a parametrization, induced by the degree-one map to $\mathbb{P}^1$ .

Consider the $\mathbb{C}^*$ -action on $\mathbb{P}^1$ that scales the first homogeneous coordinate. This induces a natural $\mathbb{C}^*$ -action on the graph space. The evaluation maps $ev_i:QG_{0,n|k}(Y, d) \to IY$ for $1 \leq i \leq n$ and $\hat{ev}_j^{Q}\colon QG_{0,n|k}(Y, d) \to [V/G]$ for $1 \leq j \leq k$ are equivariant, where the action of $\mathbb{C}^*$ on IY and $[V/G]$ is trivial. Let $z \in H^*_{\mathbb{C}^*}(pt)$ denote the equivariant parameter

$$z := e_{\mathbb{C}^*}(T_0\mathbb{P}^1),$$

the first Chern class of $T\mathbb{P}^1$ at $0 \in \mathbb{P}^1$ . Denote by $p_0$ and $p_\infty$ the classes in $H^*_{\mathbb{C}^*}(\mathbb{P}^1)$ determined by the conditions

$$p_0|_0 = z, p_0|_\infty = 0, p_\infty|_0 = 0, p_\infty|_\infty =-z.$$

Fix $ n=1$ . Define $F^{k,d}_{1,0}$ to be the $\mathbb{C}^*$ -fixed locus such that the source curve is irreducible, the heavy marked point $p_1$ lies over $\infty$ , there is a basepoint of degree $d(L_\theta)$ over 0, and all light points are concentrated over 0. Let $\iota_d\colon F^{k,d}_{1,0} \to QG_{0,n|k}(Y, d)$ denote the inclusion.

Definition 6.13. Define

$$ \operatorname{Res}_{F^{k,d}_{1,0}}({\boldsymbol{t}}^k) := \frac{ev_1^*(p_\infty) \cup \prod_{j=1}^k \hat{ev}_j^{Q *}({\boldsymbol{t}}) \cap [F^{k,d}_{1,0}]^{\rm vir}}{e_{\mathbb{C}^*}( N^{\rm vir}_{F^{k,d}_{1,0}})},$$

where $N^{\rm vir}_{F^{k,d}_{1,0}}$ is the virtual normal bundle in the sense of [Reference Graber and PandharipandeGP99]. Define

(6.1) \begin{equation}\mathbb{I}^Y(q, {\boldsymbol{t}}, z) := \sum_{d \in\mathrm{Eff}(V, G, \theta)} \sum_{k \geq 0} \frac{q^d}{k!} (\widetilde{ev_1} \circ {\iota_d})_*\operatorname{Res}_{F^{k,d}_{1,0}}({\boldsymbol{t}}^k).\end{equation}

We view $\mathbb{I}^Y(q, {\boldsymbol{t}}, z)$ as a power series in the q and $t^i$ variables, whose coefficients lie in the localized equivariant cohomology ring $H^*_{CR, \mathbb{C}^*, \operatorname{loc}}(Y) = H^*_{\rm CR}(Y)[z, z^{-1}].$

By [Reference Cheong, Ciocan-Fontanine and KimCCFK15, Remark 4.3], $\mathbb{I}^Y(q, {\boldsymbol{t}}, z)$ is equal to the I-function defined in [Reference Cheong, Ciocan-Fontanine and KimCCFK15, Definition 4.1]. In the toric case, if ${\boldsymbol{t}} \in H^2([V/G])$ , this coincides with the ‘S-extended stacky I-function’ of [Reference Coates, Corti, Iritani and TsengCCI⁺15]. By [Reference Cheong, Ciocan-Fontanine and KimCCFK15, (4.2)], it is a $(0^+, 0^+)$ I-function in the sense of Definition 6.8.

Given $\rho \in H^*([V/G])$ , define

$$z\partial _\rho \mathbb{I}^Y(q, {\boldsymbol{t}}, z) := z\frac{\partial }{\partial s} \mathbb{I}^Y(q, {\boldsymbol{t}} + s \rho, z)|_{s = 0}.$$

Let $\{ \tilde \gamma_i\}_{i \in \tilde I}$ be a basis for $H^*_{\rm CR}(Y)$ and let $\{ \tilde \gamma^i\}_{i \in \tilde I}$ be the dual basis in $H^*_{\rm CR,cs}(Y)$ . Define

$$P_{\rho}(q, {\boldsymbol{t}}, z) := \sum_{i \in \tilde I} \tilde \gamma_i \langle\!\langle\tilde \gamma^i \otimes p_\infty| \rho \otimes p_0\rangle\!\rangle_{0, 1|1}^{QG(Y)^{0^+, 0^+}}({\boldsymbol{t}}),$$

where $\langle\!\langle -\rangle\!\rangle^{QG(Y)^{0^+, 0^+}}({\boldsymbol{t}})$ is the double-bracket generating function of $(0^+, 0^+)-stable$ graph space invariants as in [Reference Kiem and LiKL20, § 2.2]. Because $P_{\rho}(q, {\boldsymbol{t}}, z)$ is defined before localization, it lies in $H^*_{\rm CR}(Y)[[q]][[t^i]][z]$ (i.e. it contains only positive powers of the equivariant parameter z).

After localization on the graph space, we obtain

$$P_\rho(q, {\boldsymbol{t}}, z) = {S^{Y}}^*(q, {\boldsymbol{t}}, -z)(z \partial _\rho \mathbb{I}^Y(q, {\boldsymbol{t}}, z)).$$

Applying the operator $S^{Y}(q, {\boldsymbol{t}}, z)$ to both sides, we conclude that

\begin{align*}z \partial _\rho \mathbb{I}^Y(q, {\boldsymbol{t}}, z) = S^{Y}(q, {\boldsymbol{t}}, z) P_\rho(q, {\boldsymbol{t}}, z).\\[-33pt]\end{align*}

We arrive at one of the main theorems of the paper, which describes how $(0^+, 0^+)$ GLSM I-functions arise as derivatives of the big I-function of $Y\!$ .

Proof. By Proposition 6.14, we can write $z \partial _\rho \mathbb{I}^Y(q, {\boldsymbol{t}}, z)$ as $S^{Y}(q, {\boldsymbol{t}}, z) P_\rho(q, {\boldsymbol{t}}, z)$ for $ P_\rho(q, {\boldsymbol{t}}, z) \in H^*_{\rm CR}(Y)[[q]][[t^i]][z]$ . By Lemma 6.12, $ P_\rho(q, {\boldsymbol{t}}, z)$ lies in a compact-type cohomology: $ P_\rho(q, {\boldsymbol{t}}, z) \in H^*_{\rm CR,ct}(Y)[[q]][[t^i]][z]$ . Applying Proposition 6.7, we conclude that

\begin{align*}\sigma^w(z \partial _\rho \mathbb{I}^Y(q, {\boldsymbol{t}}, z)) &= \sigma^w(S^{Y}(q, {\boldsymbol{t}}, z) P_\rho(q, {\boldsymbol{t}}, z)) \\&= S^{(Y, w)}(q, \tau({\boldsymbol{t}}), z)(\sigma^w( P_\rho(q, {\boldsymbol{t}}, z))),\end{align*}

which is of the form given in Definition 6.11.

6.3 The toric case

In this section, we focus on the case in which G is a torus, and give specific conditions for when $z \partial _\rho \mathbb{I}^Y(q, {\boldsymbol{t}}, z)$ lies in a cohomology of compact type. We then use Theorem 6.15 and the explicit formula for $\mathbb{I}^Y(q, {\boldsymbol{t}}, z)$ from [Reference Cheong, Ciocan-Fontanine and KimCCFK15] to obtain an explicit I-function for any toric GLSM.

Let $(V, G, \theta, w)$ be a GLSM such that $G = (\mathbb{C}^*)^k$ . Let

$$\rho_1, \ldots, \rho_r \in \widehat G $$

denote the characters of G that define the action on $V = \mathbb{C}^r$ . By abuse of notation, we also use $\rho_1, \ldots, \rho_r $ to denote the corresponding cohomology classes in $H^2([V/G])$ . We identify the coordinates $x_1, \ldots, x_r$ on V with the homogeneous coordinates on $[V/G]$ and $[V\mathbin{/\mkern-6mu/}_\theta G]$ .

In [Reference Ciocan-Fontanine and KimCFK16, Reference Cheong, Ciocan-Fontanine and KimCCFK15], $\mathbb{I}^Y$ is computed explicitly for toric varieties. Choose characters $\eta_1, \ldots, \eta_l \in \widehat G$ that generate $H^*([V/G])$ as an algebra. Then, in the expression ${\boldsymbol{t}} = \sum_{1 \leq j \leq m} t^j T_j$ , each $T_j\in H^*([V/G])$ may be expressed as

$$T_j = p_j(\eta_1, \ldots, \eta_l) =: p_j(\eta_s),$$

for some polynomial $p_j(x_s):=p_j(x_1, \ldots, x_l)$ in l variables. For $g \in G$ , let $\mathbb{1}_g$ denote the fundamental class of the twisted sector $Y_g$ , and for $d \in\mathrm{Eff}(V, G, \theta)$ , let

$$g_d = ( e^{2 \pi i \langle d, \rho_1\rangle}, \ldots, e^{2 \pi i \langle d, \rho_r\rangle}) \in G.$$

With this notation, the big I-function is given by

(6.2) \begin{align}\mathbb{I}^Y({\boldsymbol{t}}, q, z) =& \sum_{d \in\mathrm{Eff}(V, G,\theta)}q^d \exp \bigg( \frac{1}{z} \sum_{j=1}^l t^j p_j(\eta_s +z\langle d, \eta_s\rangle) \!\bigg) \nonumber\\& \times \frac{\prod_{i| \langle d, \rho_i\rangle\lt 0} \prod_{ \langle d, \rho_i\rangle \leq \nu\lt 0} (\rho_i + ( \langle d, \rho_i\rangle - \nu)z)}{\prod_{i| \langle d, \rho_i\rangle \gt 0} \prod_{ 0 \leq \nu \lt \langle d, \rho_i\rangle} (\rho_i + ( \langle d, \rho_i\rangle - \nu)z)} \mathbb{1}_{g_d^{-1}}, \end{align}

where the products on the second line run over all integers $\nu$ in the specified range [Reference Cheong, Ciocan-Fontanine and KimCCFK15, Corollary 5.6 (2)].

Lemma 6.16. Fix $d \in\mathrm{Eff}(V, G, \theta)$ and $k \geq 0$ . The composition

$$F^{k,d}_{1,0} \xrightarrow{\iota_d} QG_{0, 1|k} (Y, d) \to \operatorname{Spec} ( \mathbb{C}[x_i]_{\{1\leq i\leq r\}}^G)$$

factors through $\operatorname{Spec} ( \mathbb{C}[x_i]_{\{i|\langle d, \rho_i\rangle = 0\}}^G)$ .

Proof. By [Reference Cheong, Ciocan-Fontanine and KimCCFK15, Theorem 2.7], $QG_{0, 1|k} (Y, d)$ is proper over $\operatorname{Spec} ( \mathbb{C}[x_i]_{\{1\leq i\leq r\}}^G)$ . Because G is abelian, the vector bundle $\mathcal{P} \times_G V \to \mathcal{C}$ splits as

$$\mathcal{P} \times_G V = \prod_{i=1}^r \mathcal{L}_i.$$

Here, $\mathcal{L}_i$ is the line bundle obtained as the pullback of $[(V \times \mathbb{C}_{\rho_i})/G] \to [V/G]$ via the map $\mathcal{C} \to [V/G]$ .

Restrict the universal curve to $F^{k,d}_{1,0}$ and denote it by

$$\pi_F: \mathcal{C}_F \to F^{k,d}_{1,0}.$$

By definition of $F^{k,d}_{1,0}$ and $0^+$ -stability, each fiber of $\pi_F$ is irreducible. If $\langle d, \rho_i\rangle \lt 0$ , then $\mathcal{L}_i \to \mathcal{C}_F$ has negative degree on each fiber. Therefore, the universal section of $\mathcal{L}_i \to \mathcal{C}_F$ is zero, i.e. $\mathcal{C}_F$ maps to the locus $\{x_i = 0\} \subset [V/G]$ . Consequently, $F^{k,d}_{1,0}$ maps properly to the closed subset $\operatorname{Spec} ( \mathbb{C}[x_i]_{\{i|\langle d, \rho_i\rangle \geq 0\}}^G)$ of $\operatorname{Spec} ( \mathbb{C}[x_i]_{\{1\leq i\leq r\}}^G)$ .

Next, we describe how the degree $d\in \hom_\mathbb{Z} ( \widehat G, \mathbb{Q})$ defines a subgroup of G. By possibly replacing d with a multiple n d for some $n \in \mathbb{N}$ , we can assume that d lies in $\hom_\mathbb{Z} ( \widehat G, \mathbb{Z})$ . Consider the isomorphism

\begin{align*}\hom(\mathbb{C}^*, G) &\to \hom_\mathbb{Z} ( \widehat G, \mathbb{Z}), \\f &\mapsto (\chi \mapsto \chi \circ f \in \hom(\mathbb{C}^*, \mathbb{C}^*) \cong \mathbb{Z}),\end{align*}

which is canonical up to a choice of isomorphism $\hom(\mathbb{C}^*, \mathbb{C}^*) \cong \mathbb{Z}$ . Under this identification, $d(\mathbb{C}^*)$ is a subgroup of G. Consequently,

$$\mathbb{C}[x_i]_{\{i|\langle d, \rho_i\rangle \geq 0\}}^G \subset \mathbb{C}[x_i]_{\{i|\langle d, \rho_i\rangle \geq 0\}}^{d(\mathbb{C}^*)} = \mathbb{C}[x_i]_{\{i|\langle d, \rho_i\rangle = 0\}}.$$

The last equality follows from the fact that the $d(\mathbb{C}^*)$ -invariant polynomials in $\mathbb{C}[x_i]_{\{i|\langle d, \rho_i\rangle \geq 0\}}$ are exactly the polynomials that are homogeneous of degree zero with respect to the grading: $\deg_d(x_i) := \langle d, \rho_i\rangle$ . Thus,

$$\mathbb{C}[x_i]_{\{i|\langle d, \rho_i\rangle \geq 0\}}^G = \mathbb{C}[x_i]_{\{i|\langle d, \rho_i\rangle = 0\}}^G.$$

Combining this with the previous paragraph concludes the proof.

Next, we give conditions for when a derivative $(\prod_{i \in {\hat I}}z\partial _{\rho_i} ) \mathbb{I}^Y$ is supported in a cohomology of compact type.

Proof. Fix $d \in \hom_\mathbb{Z} ( \widehat G, \mathbb{Q})$ , $k \geq 0$ and $g \in G$ . Let $QG_{0, (g)|k} (Y, d)$ denote the open and closed subset of $QG_{0, 1|k} (Y, d)$ that maps to the twisted sector $Y_g$ under $ev_1$ . Let $F^{k,d}_{g,0}$ denote the intersection of the fixed locus $F^{k,d}_{1,0}$ with $QG_{0, (g)|k} (Y, d)$ . We prove the statement for each the summand

$$\mathbb{I}^Y_{d,k,g} := \widetilde{ev_1}_*\operatorname{Res}_{F^{k,d}_{g,0}}({\boldsymbol{t}}^k)$$

of $\mathbb{I}^Y(q, {\boldsymbol{t}}, z)$ .

Let $I(g) \subset \{1, \ldots, r\}$ denote the set of indices such that g acts trivially on $\mathbb{C}_{\rho_i}$ , i.e. $\rho_i(g) = 1$ . The twisted sector $Y_g$ is equal to the vanishing locus $\{x_i = 0\}_{i \notin I(g)} \subset Y\!$ . The map $QG_{0, (g)|k} (Y, d) \to \operatorname{Spec} ( \mathbb{C}[x_i]_{\{1\leq i\leq r\}}^G)$ factors as

$$QG_{0, (g)|k} (Y, d) \to Y_g \to \operatorname{Spec} ( \mathbb{C}[x_i]^G_{i \in I(g)}) \to \operatorname{Spec} ( \mathbb{C}[x_i]_{\{1\leq i\leq r\}}^G).$$

Let ${\hat I}^0_d(g) \subset {\hat I}$ denote the set of indices $i \in {\hat I} \cap I(g)$ such that $\langle d, \rho_i\rangle = 0$ . and let ${\hat I}^1_d(g) \subset {\hat I}$ denote the set of indices $i \in {\hat I}\cap I(g)$ such that $\langle d, \rho_i\rangle \neq 0$ . Using the explicit formula for $\mathbb{I}^Y$ given in [Reference Cheong, Ciocan-Fontanine and KimCCFK15], we observe that for $i \in {\hat I}^0_d(g)$ ,

$$\partial _{\rho_i} \mathbb{I}^Y_{d,k,g} = \rho_i \mathbb{I}^Y_{d,k,g}.$$

Let $Z_{{\hat I}^0_d(g)} := \{x_i = 0\}_{i \in {\hat I}^0_d(g)} \subset IY$ and let $s_0\colon Z_{{\hat I}^0_d(g)} \to IY$ denote the inclusion. Consider the following commutative diagram

where $F_{{\hat I}^0_d(g)}$ is the fiber product of $\widetilde{ev_1} \circ \iota_d$ and $s_0$ . Then,

(6.3) \begin{align} \bigg(\prod_{i \in {\hat I}^0_d(g)}\partial _{\rho_i}\!\bigg)\mathbb{I}^Y_{d,k,g} &= \bigg(\prod_{i \in {\hat I}^0_d(g)} \rho_i\!\bigg)\mathbb{I}^Y_{d,k,g} \\ \nonumber&= {s_0}_*{s_0}^* \mathbb{I}^Y_{d,k,g}\\ \nonumber&= {s_0}_*{s_0}^* (\widetilde{ev_1} \circ {\iota_d})_*\operatorname{Res}_{F^{k,d}_{g,0}}({\boldsymbol{t}}^k) \\ \nonumber&= {s_0}_* (\widetilde{ev_1} \circ {\iota_d})'_* {s_0}^! \operatorname{Res}_{F^{k,d}_{g,0}}({\boldsymbol{t}}^k) .\end{align}

The map $c_{{\hat I}^0_d(g)}$ is seen to be proper because $\widetilde{ev_1} \circ {\iota_d}$ , $s_0$ , and $c_g$ are [Sta20, Lemma 01W6]. In addition, since $\operatorname{Spec} ( \mathbb{C}[x_i]^G_{i \notin {\hat I}})$ is assumed to be a point, so is $\operatorname{Spec} ( \mathbb{C}[x_i]^G_{i \in I(g) \setminus {\hat I}})$ . We conclude that $F_{{\hat I}^0_d(g)}$ is proper.

Finally, (6.3) gives us

\begin{align*}\bigg(\prod_{i \in {\hat I}}z\partial _{\rho_i}\! \bigg) \mathbb{I}^Y_{d,k,g} & = z^{|{\hat I}|} \prod_{i \in {\hat I} \setminus {\hat I}^0_d(g)}(\partial _{\rho_i}) \prod_{i \in {\hat I}^0_d(g)}(\partial _{\rho_i}) \mathbb{I}^Y_{d,k,g} \\& = z^{|{\hat I}|} \prod_{i \in {\hat I} \setminus {\hat I}^0_d(g)}(\partial _{\rho_i}) {s_0}_* (\widetilde{ev_1} \circ {\iota_d})'_* {s_0}^! \operatorname{Res}_{F^{k,d}_{g,0}}({\boldsymbol{t}}^k) \\&= {s_0}_*(\widetilde{ev_1} \circ {\iota_d})'_* {s_0}^! (z^{|{\hat I}|} \prod_{i \in {\hat I} \setminus {\hat I}^0_d(g)}(\partial _{\rho_i}) \operatorname{Res}_{F^{k,d}_{g,0}}({\boldsymbol{t}}^k) ).\end{align*}

This exhibits $(\prod_{i \in {\hat I}}z\partial _{\rho_i} ) \mathbb{I}^Y_{d,k,g} $ as the pushforward of a class on the proper stack $F_{{\hat I}^0_d(g)}$ . We conclude that $(\prod_{i \in {\hat I}}z\partial _{\rho_i} ) \mathbb{I}^Y_{d,k,g} $ lies in $ H^*_{\rm ct}(Y_g)[[q]][[t^i]][z, z^{-1}] $ .

Let $c_i$ denote the degree of $x_i$ with respect to the $\mathbb{C}^*_R$ -action. A natural choice of ${\hat I}$ is given by ${\hat I} = \{i | c_i \neq 0\}$ . In this case, $\mathbb{C}[x_i]_{i \notin {\hat I}} = (\mathbb{C}[x_i]_{1\leq i\leq r})^{\mathbb{C}^*_R}$ , and $\mathbb{C}[x_i]_{i \notin {\hat I}}^G = (\mathbb{C}[x_i]^{\mathbb{C}^*_R}_{1\leq i\leq r})^G = (\mathbb{C}[x_i]_{1\leq i\leq r})^\Gamma$ .

Corollary 6.19. If $(\mathbb{C}[x_i]_{1\leq i\leq r})^\Gamma = \mathbb{C}$ , then

(6.4) \begin{align}\mathbb{I}^{(Y, w)}({\boldsymbol{t}}, q, z) :=& \sum_{d \in\mathrm{Eff}(V, G,\theta)}q^d \exp \bigg( \frac{1}{z} \sum_{j=1}^l t^j p_j(\eta_s +z\langle d, \eta_s\rangle) \!\bigg) \nonumber\\&\times \sigma^w\bigg( \frac{\prod_{i| c_i \neq 0, \langle d, \rho_i\rangle\leq 0} \prod_{ \langle d, \rho_i\rangle \leq \nu\leq 0} (\rho_i + ( \langle d, \rho_i\rangle - \nu)z)}{\prod_{i| c_i \neq 0, \langle d, \rho_i\rangle \gt 0} \prod_{ 0 \lt \nu \lt \langle d, \rho_i\rangle} (\rho_i + ( \langle d, \rho_i\rangle - \nu)z)} \nonumber \\& \qquad\quad \times\frac{\prod_{ i| c_i = 0, \langle d, \rho_i\rangle\lt 0} \prod_{ \langle d, \rho_i\rangle \leq \nu\lt 0} (\rho_i + ( \langle d, \rho_i\rangle - \nu)z)}{\prod_{ i| c_i = 0, \langle d, \rho_i\rangle \gt 0} \prod_{ 0 \leq \nu\lt \langle d, \rho_i\rangle} (\rho_i + ( \langle d, \rho_i\rangle -\nu)z)} \mathbb{1}_{g_d^{-1}} \!\bigg) \end{align}

is a GLSM I-function for (Y, w).

7.1 FJRW theory

The first example of a GLSM is the case in which G is finite and $w\colon [\mathbb{C}^n/ G] \to \mathbb{C}$ is a quasihomogeneous polynomial with an isolated singularity at the origin. These are sometimes called affine-phase GLSMs. The corresponding invariants are known as FJRW invariants and have been defined by Fan, Jarvis, and Ruan [Reference Fan, Jarvis and RuanFJR13] prior to defining enumerative invariants for more general GLSMs. FJRW mirror theorems have been proven in the case in which w is a Fermat polynomial [Reference Chiodo and RuanCR10, Reference Chiodo, Iritani and RuanCIR14, Reference Lee, Priddis and ShoemakerLPS16] and in the case in which w is a chain polynomial and G is maximal [Reference GuéréGuá16].

Here, we compute a $(0^+, 0^+)$ GLSM I-function for a GLSM representing the LG model (Y, w) in the case in which $Y = [\mathbb{C}^n / G]$ and G is a finite subgroup of $(\mathbb{C}^*)^n$ . Assume that $w = w(x_1, \ldots, x_n)$ is homogeneous of degree $d_w$ with respect to a $\mathbb{C}^*_R$ -action, where $\mathbb{C}^*_R$ acts with weight $c_i$ on $x_i$ . Under the assumption that Z(dw) is proper over $\operatorname{Spec} \mathbb{C}$ , w has an isolated singularity at the origin of $\mathbb{C}^n$ .

The first step is to present Y as a GIT quotient by a torus:

$$Y = [\mathbb{C}^{n+s} \mathbin{/\mkern-6mu/}_\theta (\mathbb{C}^*)^{s}].$$

The simplest case is when $G = \langle J \rangle$ . Let $\widetilde G = \mathbb{C}^*$ act on $\widetilde V = \operatorname{Spec} \mathbb{C}[x_1, \ldots, x_n, p]$ with weights $(c_1, \ldots, c_n, -d_w)$ . Then, $\widetilde w = p \cdot w$ is $\widetilde G$ -invariant. We extend the $\mathbb{C}^*_R$ -action from $\mathbb{C}^n$ to $\widetilde V$ by letting $\mathbb{C}^*_R$ act trivially on the p coordinate. Choose $\theta = (-1)$ ; then the GLSM $(\widetilde V, \widetilde G, \theta, \widetilde w)$ represents the LG model (Y, w).

Let $\eta_{p}\colon \widetilde G \to \mathbb{C}^*$ denote the homomorphism of weight $-d_w$ , viewed as an element of $H^*([\widetilde V/\widetilde G])$ . Define ${\boldsymbol{t}} = t \cdot \eta_{p}$ in the notation of (6.2). Then, the corresponding big I-function of (6.2) is given by

\begin{align*}\mathbb{I}^Y({\boldsymbol{t}}, q, z) = \sum_{k \geq 0} \frac{q^{k/d_w} e^{tk} \prod_{i=1}^n \prod_{0\lt \nu \leq c_i k/d_w} z(-c_i k/d_w + \nu) }{z^k k!} \mathbb{1}_{e^{2 \pi i k/d_w}}.\end{align*}

When Y is a finite quotient of affine space, the map $\phi^w\colon \mathcal{H}_{\rm ct}(Y, w) \to H^*_{\rm CR,ct}(Y)$ is an isomorphism, so $\sigma^w = (\phi^w)^{-1}$ . The class $\mathbb{1}_{e^{2 \pi i k/d_w}}$ is of compact type if and only if

$$\{i | c_i k/d_w \in \mathbb{Z}\} = \emptyset.$$

For such k, let $\varphi_{k-1} = \sigma^w(\mathbb{1}_{e^{2 \pi i k/d_w}})$ (note the shift in index).

We observe that $\mathbb{C}[x_1, \ldots, x_n]^G = \mathbb{C}$ . Therefore, by Theorem 6.18, a $(0^+, 0^+)$ GLSM I-function is obtained as

$$\mathbb{I}^{(Y, w)}({\boldsymbol{t}}, q, z) := \sigma^w (z\partial _{\eta_p}\mathbb{I}^Y)= \sigma^w\left(\!z \frac{\partial }{\partial t}\mathbb{I}^Y\!\right).$$

Here, one observes that the coefficient of $q^{k/d_w}$ in $\mathbb{I}^Y({\boldsymbol{t}}, q, z)$ is zero if $k\gt 0$ and $\{i | c_i k/d_w \in \mathbb{Z}\} \neq \emptyset$ . It follows that $z ({\partial }/{\partial t})\mathbb{I}^Y$ is supported in $ H^*_{\rm CR,ct}(Y)$ , thus verifying Proposition 6.17 directly in this case. Explicitly, we have

(7.1) \begin{equation}\mathbb{I}^{(Y, w)}({\boldsymbol{t}}, q, z) = \sum_{\substack{ k \geq 1 \\ \{i | c_i k/d_w \in \mathbb{Z}\} = \emptyset}} \frac{q^{k/d_w} e^{tk} \prod_{i=1}^n \prod_{0\lt \nu \leq c_i k/d_w} z(-c_i k/d_w + \nu) }{z^{k-1} (k-1)!} \varphi_{k-1}.\end{equation}

This function agrees with previously computed FJRW I-functions; for instance, with the non-equivariant limit of [Reference Chiodo, Iritani and RuanCIR14, Equation (40)] up to the substitution $u = -q^{1/d_w}e^t$ and a factor of $-z$ . In the case in which w is a Fermat polynomial, the mirror theorem of Chiodo, Iritani, and Ruan [Reference Chiodo, Iritani and RuanCIR14, Theorem 3.10] implies that (7.1) is equal to the FJRW J-function after a change of variables and a scaling. From our perspective, this may be interpreted as an $\epsilon$ -wall-crossing result from $(0^+, 0^+)$ -stability to $(\infty, \infty)$ -stability for the case in which w is Fermat. We emphasize, however, that (7.1) is a generating function of $(0^+, 0^+)$ -stable GLSM invariants for any choice of quasihomogeneous w with isolated singularity at the origin.

For a more general finite group G, write G as a product of cyclic groups, $G = \prod_{j=1}^s \mu_{r_j}$ . Choose generators $\omega_j \in \mu_{r_j}$ and suppose that

$$\omega_j \cdot (x_1, \ldots, x_n) = (e^{2 \pi i c_{1j}/r_j} x_1, \ldots , e^{2 \pi i c_{nj}/r_j} x_n).$$

For notational simplicity, let us assume that $\omega_1 = J$ , so $r_1 = d_w$ .Footnote ²

Define $\widetilde V = \operatorname{Spec} \mathbb{C}[x_1, \ldots,x_n, p_1, \ldots, p_s]$ by $\widetilde G = (\mathbb{C}^*)^s$ , where the jth factor of $\mathbb{C}^*$ acts on $\widetilde V$ with weights

$$(c_{1j}, \ldots, c_{nj}, 0 \ldots, -r_j, \ldots, 0),$$

where the term $-r_j$ occurs in the $(n+j)$ th coordinate. Then, $[\mathbb{C}^n /G]$ may be expressed as a GIT stack quotient $[\widetilde V \mathbin{/\mkern-6mu/}_\theta \widetilde G]$ , where

$$\theta (\lambda_1, \ldots, \lambda_s) = \bigg( \prod_{j=1}^s\lambda_j \!\bigg)^{\!{\kern-1pt}{-}1}.$$

If we define $\widetilde w$ to be $p_1\cdots p_s w(x_1, \ldots, x_n)$ , then $(\widetilde V, \widetilde G, \theta, \widetilde w)$ is a GLSM representing the LG model (Y, w).

Define $\eta_{p_1}\colon \widetilde G \to \mathbb{C}^*$ by $\eta_{p_1} (\lambda_1, \ldots, \lambda_s) = \lambda_1^{-r_1}$ and let ${\boldsymbol{t}} = t \cdot \eta_{p_1} \in H^*([\widetilde V/\widetilde G])$ . In this case, the big I-function of [Reference Ciocan-Fontanine and KimCFK16, Reference Cheong, Ciocan-Fontanine and KimCCFK15] (as well as the ‘S-extended stacky I-function’ of [Reference Coates, Corti, Iritani and TsengCCI⁺15]) is

(7.2) \begin{equation}\mathbb{I}^Y({\boldsymbol{t}}, q, z) = \sum_{d_1, \ldots, d_s \geq 0} \frac{ e^{td_1} \prod_{j=1}^s q_j^{d_j/r_j}}{ \prod_{j=1}^sd_j! z^{d_j}} \bigg( \prod_{i=1}^n \prod_{0\lt \nu \leq \sum_{j=1}^s c_{ij} d_j/r_j } z\bigg( \!{-} \sum_{j=1}^s c_{ij} d_j/r_j + \nu\bigg)\!\bigg) \mathbb{1}_{\!J^{d_1} \prod_{j=2}^s \omega_j^{d_j}}.\end{equation}

As before, we note that for $(d_1, \ldots, d_s) \neq (0, \ldots, 0)$ , the coefficient of $\prod_{j=1}^s q_j^{d_j}$ is zero unless $\{i | \sum_{j=1}^s c_{ij} d_j/r_j \in \mathbb{Z} \} = \emptyset$ , i.e. unless $ \mathbb{1}_{\!J^{d_1} \prod_{j=2}^s \omega_j^{d_j}} \in H^*_{\rm CR,ct}(Y)$ . Therefore, $z ({\partial }/{\partial t}) \mathbb{I}^Y(q, {\boldsymbol{t}}, z) $ lies in a compact-type cohomology.

Then, by Theorem 6.15, a $(0^+, 0^+)$ GLSM I-function is given by $\sigma^w( z ({\partial }/{\partial t}) \mathbb{I}^Y(q, {\boldsymbol{t}}, z) )$ :

(7.3) \begin{align}\mathbb{I}^{(Y, w)}({\boldsymbol{t}}, q, z) = \sum_{\substack{ d_1\geq 1 \\ d_2, \ldots, d_s \geq 0 \\ \{i | \sum_{j=1}^s c_{ij} d_j/r_j \in \mathbb{Z} \} = \emptyset}} &\bigg( \frac{ e^{td_1} \prod_{j=1}^s q_j^{d_j/r_j}}{ (d_1-1)!z^{d_1 - 1} \prod_{j=2}^s d_j! z^{d_j}} \nonumber\\ &\times \prod_{i=1}^n \prod_{0\lt \nu \leq \sum_{j=1}^s c_{ij} d_j/r_j } z\bigg( \!{-} \sum_{j=1}^s c_{ij} d_j/r_j + \nu\!\bigg)\!\bigg) \sigma^w\big( \mathbb{1}_{\!J^{d_1} \prod_{j=2}^s \omega_j^{d_j}}\!\big). \end{align}

Again, this formula agrees with FJRW I-functions previously computed in special cases (again up to a simple change of variables and scaling). For w a Fermat polynomial but G no longer necessarily cyclic, (7.3) appears in [Reference Acosta and ShoemakerAS18] (based on the work in [Reference Lee, Priddis and ShoemakerLPS16]). The restriction of (7.3) to $q_2 = \cdots = q_s = 0$ recovers the FJRW I-function for chain polynomials with $G = G_{\rm max}$ maximal as computed by Guéré in [Reference GuéréGuá16, Equation (98)].Footnote ³

7.2 Complete intersections

Let X be a smooth orbifold with a projective coarse moduli space arising as a toric GIT stack quotient $X = [V_1 \mathbin{/\mkern-6mu/}_\theta G]$ . An I-function for X, $\mathbb{I}^X({\boldsymbol{t}}, q, z)$ , has been given in (6.2). Choose characters $\tau_j \in \widehat G$ for $1 \leq i \leq n$ , and G-equivariant functions $f_j(\underline x)\colon V_1 \to \mathbb{C}_{\tau_j}$ . Assume that the intersection of $Z(f_1, \ldots, f_n) \subset V$ with the $\theta$ -semistable locus $V_1^{\rm ss}$ is smooth of codimension n. Let Z denote the corresponding complete intersection $[Z (f_1, \ldots, f_n) \cap V_1^{\rm ss} / G]$ in X. Let $L_{\tau_j} \to X$ denote the line bundle associated to the character $\tau_j$ . The quantum Lefschetz hyperplane theorem [Reference Kim, Kresch and PantevKKP03, Reference Ciocan-Fontanine, Kim and MaulikCFKM14] allows one to obtain an I-function for Z from an I-function for the ambient space X whenever the line bundles $L_{\tau_j}$ are each convex. If X is a smooth variety, $L_{\tau_j}$ is convex whenever it is semi-positive; however, this is no longer the case when X is an orbifold. This failure of convexity has presented a significant obstacle in proving a quantum Lefschetz statement for orbifold complete intersections [Reference Coates, Gholampour, Iritani, Jiang, Johnson and ManolacheCGI⁺12]; however, such a statement has recently been proven by Wang [Reference WangWan25] when X is toric.