Double $J$-function of stable quasimap invariants for complete intersection in Grassmannian

TL;DR

This paper derives explicit formulas for the $J$-function and double $J$-function of stable quasimaps to complete intersections in Grassmannians using localization techniques.

Contribution

It provides new explicit formulas for $J$-functions and double $J$-functions in the context of stable quasimaps to Grassmannians, advancing computational methods in enumerative geometry.

Findings

Explicit formulas for $J$-function of stable quasimaps.

Explicit formulas for double $J$-function of stable quasimaps.

Application of localization methods to Grassmannian quasimaps.

Abstract

Using localization methods on the moduli space of stable quasimaps to Grassmannian, we give explicit formulas of $J$-function and double $J$-function of stable quasimaps for complete intersection in Grassmannian.

Equations

$$H^{\ast }(\mathbf{Gr})\cong \mathbb{Q}[e_1,e_2]/⟨h_{n-1},h_n⟩,$$

$$[\Delta ]=\sum _{k,j}{\gamma }_j^k(x)\otimes {\gamma }_j^{2(n-2)-k}(x)\in H^{\ast }(\mathbf{Gr})\otimes H^{\ast }(\mathbf{Gr}),$$

$${\mathrm{ev}}_i:{\overline{Q}}_{0,2}(\mathbf{Gr},d)\to \mathbf{Gr}$$

$$\pi :\mathcal{C}\to {\overline{Q}}_{0,2}(\mathbf{Gr},d)$$

$$\mathcal{S}\subset {\mathbb{C}}^n\otimes {\mathcal{O}}_{\mathcal{C}}$$

$$|\mathbf{a}|=\sum _{k=1}^{\ell }{a}_k,⟨\mathbf{a}⟩=\prod _k{a}_k,\mathbf{a}!=\prod _k{a}_k!,{\mathbf{a}}^{\mathbf{a}}=\prod _{k=1}^{\ell }{a}_k^{a_k}.$$

$${\cal V}_{n;\mathbf{a}}^{(d)}:=\bigoplus_{k}R^{0}\pi_{*}\big{(}{\mathscr{L}}^{\otimes a_{k}}\big{)}\longrightarrow\overline{Q}_{0,2}(\mathbf{Gr},d),$$

$$\dot{\cal V}_{n;\mathbf{a}}^{(d)}:=\bigoplus_{k}R^{0}\pi_{*}\big{(}{\mathscr{L}}^{\otimes a_{k}}(-\sigma_{1})\big{)}\longrightarrow\overline{Q}_{0,2}(\mathbf{Gr},d)$$

$$\ddot{\cal V}_{n;\mathbf{a}}^{(d)}:=\bigoplus_{k}R^{0}\pi_{*}\big{(}{\mathscr{L}}^{\otimes a_{k}}(-\sigma_{2})\big{)}\longrightarrow\overline{Q}_{0,2}(\mathbf{Gr},d).$$

$$\dot{Z}_{n;\mathbf{a}}(x,\hbar,q):=1+\sum_{d=1}^{\infty}q^{d}\mathrm{ev}_{1*}\!\!\left[\frac{\mathbf{e}(\dot{\cal V}_{n;\mathbf{a}}^{(d)})}{\hbar\!-\!\psi_{1}}\right]\in H^{*}(\mathbf{Gr})[\hbar^{-1}]\big{[}\big{[}q\big{]}\big{]},$$

$$\ddot{Z}_{n;\mathbf{a}}(x,\hbar,q):=1+\sum_{d=1}^{\infty}q^{d}\mathrm{ev}_{1*}\!\!\left[\frac{\mathbf{e}(\ddot{\cal V}_{n;\mathbf{a}}^{(d)})}{\hbar\!-\!\psi_{1}}\right]\in H^{*}(\mathbf{Gr})[\hbar^{-1}]\big{[}\big{[}q\big{]}\big{]}.$$

$$\dot{Z}_{n;\mathbf{a}}(\hbar_{1},\hbar_{2},q):=\frac{[\Delta]}{\hbar_{1}+\hbar_{2}}+\sum_{d=1}^{\infty}q^{d}\big{\{}\mathrm{ev}_{1}\!\times\!\mathrm{ev}_{2}\}_{*}\!\!\left[\frac{\mathbf{e}(\dot{\cal V}_{n;\mathbf{a}}^{(d)})}{(\hbar_{1}\!-\!\psi_{1})(\hbar_{2}\!-\!\psi_{2})}\right]$$

$$\in H^{*}(\mathbf{Gr})[\hbar_{1}^{-1},\hbar_{2}^{-1}]\big{[}\big{[}q\big{]}\big{]}.$$

$$\begin{split}\dot{Z}_{\gamma^{k}_{j}}(\hbar,q)&:=\gamma^{k}_{j}(x)+\sum_{d=1}^{\infty}\!q^{d}\mathrm{ev}_{1*}\!\!\left[\frac{\mathbf{e}(\dot{\cal V}_{n;\mathbf{a}}^{(d)})\mathrm{ev}_{2}^{*}\gamma^{k}_{j}(x)}{\hbar\!-\!\psi_{1}}\right]\in H^{*}(\mathbf{Gr})[\hbar^{-1}]\big{[}\big{[}q\big{]}\big{]},\\ \ddot{Z}_{\gamma^{k}_{j}}(\hbar,q)&:=\gamma^{k}_{j}(x)+\sum_{d=1}^{\infty}\!q^{d}\mathrm{ev}_{1*}\!\!\left[\frac{\mathbf{e}(\ddot{\cal V}_{n;\mathbf{a}}^{(d)})\mathrm{ev}_{2}^{*}\gamma^{k}_{j}(x)}{\hbar\!-\!\psi_{1}}\right]\in H^{*}(\mathbf{Gr})[\hbar^{-1}]\big{[}\big{[}q\big{]}\big{]}.\end{split}$$

$${\dot{Y}}_{n;\mathbf{a}}(x,\hslash ,q)=\sum _{d\ge 0}{q}^d{\dot{Y}}_d(\hslash ),$$

$${\ddot{Y}}_{n;\mathbf{a}}(x,\hslash ,q)=\sum _{d\ge 0}{q}^d{\ddot{Y}}_d(\hslash ),$$

$$\displaystyle\dot{Y}_{d}(\hbar)=(-1)^{d}\sum_{d_{1}+d_{2}=d}\frac{(x_{1}-x_{2}+(d_{1}-d_{2})\hbar)\prod_{k=1}^{\ell}\prod_{l=1}^{a_{k}d}(a_{k}(x_{1}+x_{2})+l\hbar)}{(x_{1}-x_{2})\prod_{i=1}^{2}\prod_{l=1}^{d_{i}}\bigg{(}(x_{i}+l\hbar)^{n}-x_{i}^{n}\bigg{)}},$$

$$\displaystyle\ddot{Y}_{d}(\hbar)=(-1)^{d}\sum_{d_{1}+d_{2}=d}\frac{(x_{1}-x_{2}+(d_{1}-d_{2})\hbar)\prod_{k=1}^{\ell}\prod_{l=0}^{a_{k}d-1}(a_{k}(x_{1}+x_{2})+l\hbar)}{(x_{1}-x_{2})\prod_{i=1}^{2}\prod_{l=1}^{d_{i}}\bigg{(}(x_{i}+l\hbar)^{n}-x_{i}^{n}\bigg{)}}.$$

$${\dot{I}}_{n;\mathbf{a}}(q):=\{\begin{array}{ll}1& \text{if }|\mathbf{a}|<n\\ {\dot{Y}}_{n;\mathbf{a}}(0,0,1,q)& \text{if }|\mathbf{a}|=n\end{array},$$

$$\dot{Z}_{n;\mathbf{a}}(x,\hbar,q)=\frac{\dot{Y}_{n;\mathbf{a}}(x,\hbar,q)}{\dot{I}_{n;\mathbf{a}}(q)}\in H^{*}(\mathbf{Gr})[\hbar^{-1}]\big{[}\big{[}q\big{]}\big{]},$$

$$\ddot{Z}_{n;\mathbf{a}}(x,\hbar,q)=\frac{\ddot{Y}_{n;\mathbf{a}}(x,\hbar,q)}{\ddot{I}_{n;\mathbf{a}}(q)}\in H^{*}(\mathbf{Gr})[\hbar^{-1}]\big{[}\big{[}q\big{]}\big{]}.$$

$${\dot{Z}}_{{\gamma }_j^k}(\hslash ,q)={\dot{Y}}_{{\gamma }_j^k}(\hslash ,q)\in H^{\ast }(\mathbf{Gr})[{\hslash }^{-1}][[q]],$$

$${\ddot{Z}}_{{\gamma }_j^k}(\hslash ,q)={\ddot{Y}}_{{\gamma }_j^k}(\hslash ,q)\in H^{\ast }(\mathbf{Gr})[{\hslash }^{-1}][[q]],$$

$${\dot{Y}}_{{\gamma }_j^k}(\hslash ,q):=D^{k,j}{\dot{Y}}_{n;\mathbf{a}}(x,\hslash ,q)+\sum _{t=1}^k\sum _{s=0}^{k-t}{\dot{C}}_{k,j;s,i}^{(t)}(q){\hslash }^{k-t-s}{D}^{s,i}{\dot{Y}}_{n;\mathbf{a}}(x,\hslash ,q)\in \mathbb{Q}(x,\hslash )[[q]],$$

$${\ddot{Y}}_{{\gamma }_j^k}(\hslash ,q):=D^{k,j}{\ddot{Y}}_{n;\mathbf{a}}(x,\hslash ,q)+\sum _{t=1}^k\sum _{s=0}^{k-t}{\ddot{C}}_{k,j;s,i}^{(t)}(q){\hslash }^{k-t-s}{D}^{s,i}{\ddot{Y}}_{n;\mathbf{a}}(x,\hslash ,q)\in \mathbb{Q}(x,\hslash )[[q]].$$

$${\dot{Z}}_{n;\mathbf{a}}({\hslash }_1,{\hslash }_2,q)=\frac{1}{{\hslash }_1+{\hslash }_2}\sum _{k,j}{\dot{Y}}_{{\gamma }_j^k}({\hslash }_1,q){\ddot{Y}}_{{\gamma }_j^{2(n-2)-k}}({\hslash }_2,q).$$

$${\pi }_i:({\mathbb{P}}^{\infty }{)}^n⟶{\mathbb{P}}^{\infty }and\gamma ⟶{\mathbb{P}}^{\infty }$$

$$H_{\mathbb{T}}^{\ast }(\mathbf{Gr})\cong \mathbb{Q}[{\alpha }_1,\cdots ,{\alpha }_n][e_1(\mathbf{x}|\alpha ),e_2(\mathbf{x}|\alpha )]/<h_{n-1}(\mathbf{x}|\alpha ),h_n(\mathbf{x}|\alpha )>,$$

$${\varphi }_{ij}=\prod _{k\ne i,j}({\mathbf{x}}_1-{\alpha }_k)\prod _{l\ne i,j}({\mathbf{x}}_2-{\alpha }_l).$$

$${\int }_{\mathbf{Gr}}\eta =\frac{1}{2}\sum _{ij\in {\mathbf{n}}^2}{\int }_{p_{ij}}\frac{\eta {|}_{p_{ij}}}{\mathbf{e}(N_{p_{ij}/\mathbf{Gr}})},\text{for all }\eta \in H_{\mathbb{T}}^{\ast }(\mathbf{Gr}).$$

$$\eta {|}_{p_{ij}}={\int }_{\mathbf{Gr}}\eta {\varphi }_{ij}.$$

Full text

1. introduction

The moduli space of stable quotients was first constructed and studied by Marian, Oprea and Pandharipande [7]. Cooper and Zinger [5] calculated the $J$-function of stable quotients for projective complete intersections, and proved that it is related to the genus zero Gromov-Witten invariants by mirror map. Later Zinger [10] calculated the double and the triple $J$-functions of stable quotients for projective complete intersections. Ciocan-Fontanine, Kim and Maulik [4] generalized the definition of stable quotients to arbitrary GIT quotients by the notion of stable quasimaps. Ciocan-Fontanine and Kim proved that the genus zero stable quasimap invariants (including twisted cases) are related to stable map by mirror map in [2], and for the higher genus cases of projective complete intersections in [3]. In this paper, using the abelian non-abelian relation between Grassamannian and product of projective spaces as in [1], we analyze the $J$-function and the double $J$-function of stable quasimaps for complete intersection in Grassmannian cases, which should play a key role in the computation of the genus one stable quasimap invariants for complete intersection in Grassmannian.

Let $\mathbf{Gr}:=\mathbf{Gr}(2,n)$ be the space of all $2$-dimensional subspaces of $\mathbb{C}^{n}$. Recall that the cohomology of $\mathbf{Gr}$ has the presentation:

[TABLE]

where $e_{1},\,e_{2}$ are the elementary symmetric polynomials of two variables $x_{1},x_{2}$ (the Chern roots of the dual universal bundle), and $h_{i}$ is the $i$-th complete symmetric polynomial (sums of all monomials of degree $i$) of $x_{1},x_{2}$. Denote by $x=(x_{1},x_{2})$. Let $\{\gamma^{k}_{j}\}$ be a basis of $H^{*}(\mathbf{Gr})$ such that the Poincaré dual of the diagonal class $[\Delta]$ can be written as

[TABLE]

where $\gamma^{k}_{j}$ is a symmetry polynomial of degree $k$ in $x_{1},x_{2}$.

Denote by $\overline{Q}_{0,2}(\mathbf{Gr},d)$ the moduli of genus zero and degree $d$ stable quasimaps to $\mathbf{Gr}$ with two marked points. Let

[TABLE]

be the evaluation map at the $i$-th marked point, see [2]. For each $i=1,2$, let $\psi_{i}$ be the first Chern class of the universal cotangent line bundle for the $i$-th marked point. Let

[TABLE]

be the universal family, and $\sigma_{1},\sigma_{2}$ be the sections given by the marked points. Let

[TABLE]

be the universal bundle. Let ${\mathscr{L}}:=\det{\cal S}^{\vee}$.

For $\ell\!\in\!\mathbb{Z}^{\geq 0}$ and $\ell$-tuple $\mathbf{a}\!=\!(a_{1},\ldots,a_{\ell})\!\in\!(\mathbb{Z}^{>0})^{\ell}$ of positive integers, denote

[TABLE]

Let

[TABLE]

[TABLE]

and

[TABLE]

In particular, ${\cal V}_{n;\mathbf{a}}^{(d)}$, $\dot{\cal V}_{n;\mathbf{a}}^{(d)}$ and $\ddot{\cal V}_{n;\mathbf{a}}^{(d)}$ are locally free sheaves. The $J$-functions associated to stable quotients are defined as

[TABLE]

[TABLE]

The double $J$-function is defined as

[TABLE]

[TABLE]

Let

[TABLE]

Let

[TABLE]

where

[TABLE]

Denote by

[TABLE]

$\ddot{I}_{n;\mathbf{a}}(q):=1$. Let $D^{k,j}$, $\dot{C}^{(r)}_{k,i;s,j},\ddot{C}^{(r)}_{k,i;s,j}\!\in\!\mathbb{Q}[[q]]$ be defined as in Section 4.

Theorem 1.1**.**

If $\ell\!\in\!\mathbb{Z}^{\geq 0}$, $n\!\in\!\mathbb{Z}^{+}$, and $\mathbf{a}\!\in\!(\mathbb{Z}^{>0})^{\ell}$ are such that $|\mathbf{a}|\!\leq\!n$, then

[TABLE]

and

[TABLE]

Theorem 1.2**.**

If $\ell\!\in\!\mathbb{Z}^{\geq 0}$, $n\!\in\!\mathbb{Z}^{+}$, and $\mathbf{a}\!\in\!(\mathbb{Z}^{>0})^{\ell}$ are such that $|\mathbf{a}|\!\leq\!n$, then

[TABLE]

and

[TABLE]

where

[TABLE]

[TABLE]

Theorem 1.3**.**

If $\ell\!\in\!\mathbb{Z}^{\geq 0}$, $n\!\in\!\mathbb{Z}^{+}$, and $\mathbf{a}\!\in\!(\mathbb{Z}^{>0})^{\ell}$ are such that $|\mathbf{a}|\!\leq\!n$, then

[TABLE]

Acknowledgment: The author thanks Wanmin Liu for helpful discussions. This work was supported by Start-up Fund of Hunan University.

2. Equivariant cohomology

Let $\mathbf{Gr}:=\mathbf{Gr}(2,n)$ be the space of all $2$-dimensional subspaces of $\mathbb{C}^{n}$. It has a torus action induced by the standard torus $\mathbb{T}=(\mathbb{C}^{*})^{n}$ action on $\mathbb{C}^{n}$. The $\mathbb{T}$ fixed points of $\mathbf{Gr}$ are the 2-planes spanned by $\langle e_{i},e_{j}\rangle$, where $\{e_{i}\}_{i=1}^{n}$ is the standard basis of $\mathbb{C}^{n}$. We label $\langle e_{i},e_{j}\rangle$ by $p_{ij}$. Let $\mathbf{n}^{2}$ be the set of all the pairs $ij$ with $1\leq i\neq j\leq n$. Thus $\{p_{ij}\}$ with $ij\in\mathbf{n}^{2}$ is the collection of all the fixed points.

Denote by $H^{*}(B\mathbb{T},\mathbb{Q}):=\mathbb{Q}[\alpha_{1},\cdots,\alpha_{n}]$, where $\alpha_{i}\!=\!\pi_{i}^{*}c_{1}(\gamma^{*})$. Here

[TABLE]

are the projection onto the $i$-th component and the tautological line bundle respectively. Denote by $\alpha=(\alpha_{1},\cdots,\alpha_{n})$. Let $\mathbb{Q}_{\alpha}$ be the fractional field of $\mathbb{Q}[\alpha_{1},\cdots,\alpha_{n}]$.

Let $\mathbf{x}_{1},\mathbf{x}_{2}$ be the equivariant Chern root of the dual universal bundle, who’s non-equivariant counterpart are $x_{1},x_{2}$. Let $\mathbf{x}:=(\mathbf{x}_{1},\mathbf{x}_{2})$, then

[TABLE]

where $e_{i}(\mathbf{x}|\alpha)$ and $h_{i}(\mathbf{x}|\alpha)$ are factorial elementary and complete symmetric functions which symmetric in $\mathbf{x}$, see [8, Proposition 5.2].

Let

[TABLE]

Then $\phi_{ij}$ is the equivariant Poincaré dual of the points $p_{ij}\in H_{\mathbb{T}}^{*}(\mathbf{Gr})$. The Artiyah-Bott localization theorem states that

[TABLE]

Therefore

[TABLE]

So for $\eta\in H^{*}_{\mathbb{T}}(\mathbf{Gr})$

[TABLE]

Let ${\cal E}\!\longrightarrow\!\mathbf{Gr}$ be the universal bundle. Thus, the action of $\mathbb{T}$ on $\mathbf{Gr}$ naturally lifts to an action on ${\cal E}$ and the Euler class

[TABLE]

The $\mathbb{T}$-action on $\mathbf{Gr}$ also has a natural lift to the tangent bundle $T_{\mathbf{Gr}}\!\longrightarrow\!\mathbf{Gr}$ so that there is a short exact sequence

[TABLE]

Therefore

[TABLE]

3. Algebraic calculation

In this section we describe a number of properties of hypergeometric series which determine them uniquely.

If $R$ is a ring, denote by

[TABLE]

the $R$-algebra of Laurent series in $\hbar^{-1}$ (with finite principal part). For $f\!\in\!R[[q]]$ and $d\!\in\!\mathbb{Z}^{\geq 0}$, let $\llbracket{f}\rrbracket_{q;d}\!\in\!R$ be the coefficient of $q^{d}$ in $f$. If

[TABLE]

for some ${\cal F}_{d}^{(r)}\!\in\!R$, we define

[TABLE]

i.e. we drop $\hbar^{-p}$ and higher powers of $\hbar^{-1}$, instead of higher powers of $\hbar$. If $R$ is a field, let

[TABLE]

be the embedding given by taking the Laurent series of rational functions at $\hbar^{-1}\!=\!0$.

If $f\!=\!f(z)$ is a rational function in $z$ and possibly with some other variables, then for any $z_{0}\!\in\!\mathbb{P}^{1}\!\supset\!\mathbb{C}$, denote the residue of the 1-form $f\text{d}z$ at $z_{0}$ by

[TABLE]

where the integral is taken over a positively oriented loop around $z\!=\!z_{0}$ with no other singular points of $f\mathrm{d}z$. For any collection of points $z_{1},\ldots,z_{k}\!\in\!\mathbb{P}^{1}$, let

[TABLE]

By the Residue Theorem on $S^{2}$,

[TABLE]

for every rational function $f\!=\!f(z)$ on $S^{2}\!\supset\!\mathbb{C}$. If $f$ is regular at $z\!=\!0$, let \left\llbracket f\right\rrbracket_{z;p} be the coefficient of $z^{p}$ in the power series expansion of $f$ around $z\!=\!0$.

Definition 3.1**.**

Let $C:=(C_{ij}^{ik}(d))_{d\in\mathbb{Z}^{+}}\cup(C_{ij}^{kj}(d))_{d\in\mathbb{Z}^{+}}$ be any collection of elements of $\mathbb{Q}_{\alpha}$, with $ij,ik,kj\in\mathbf{n}^{2}$. A symmetric power series ${\cal F}\!\in\!\mathbb{Q}_{\alpha}[\mathbf{x}_{1},\mathbf{x}_{2}]\llceil{\hbar}\rrceil[[q]]$ is $C$-recursive if the following holds: if $d^{*}\!\in\!\mathbb{Z}^{\geq 0}$ is such that

[TABLE]

and \big{\llbracket}{{\cal F}(\alpha_{i},\alpha_{j},\hbar,q)}\big{\rrbracket}_{q;d} is regular at $\hbar\!=\!(\alpha_{k}\!-\!\alpha_{i})/d$ and $\hbar\!=\!(\alpha_{k}\!-\!\alpha_{j})/d$ for all $d\!<\!d^{*}$, $k\!\neq\!i$ and $k\!\neq\!j$, then

[TABLE]

Thus, if ${\cal F}\!\in\!\mathbb{Q}_{\alpha}[\mathbf{x}_{1},\mathbf{x}_{2}]\llceil{\hbar}\rrceil[[q]]$ is $C$-recursive, for any collection $C$, then

[TABLE]

as can be seen by induction on $d$. Moreover,

[TABLE]

for some ${\cal F}_{ij}^{r}(d)\!\in\!\mathbb{Q}_{\alpha}$. The nominal issue with defining $C$-recursivity by (3), as is normally done, is that a priori the evaluation of ${\cal F}(\alpha_{i},\alpha_{j},\hbar,q)$ at $\hbar\!=\!(\alpha_{k}\!-\!\alpha_{i})/d$ and $\hbar\!=\!(\alpha_{k}\!-\!\alpha_{j})/d$ need not be well-defined, since ${\cal F}(\alpha_{i},\alpha_{j},\hbar,q)$ is a power series in $q$ with coefficients in the Laurent series in $\hbar^{-1}$; a priori they may not converge anywhere. However, taking the coefficient of each power of $q$ in (3) shows by induction on the degree $d$ that this evaluation does make sense. This is the substance of Definition 3.1.

Definition 3.2**.**

Let $\eta\!\in\!\mathbb{Q}[\mathbf{x}]$ be a homogeneous polynomial such that taking value at $\mathbf{x}\!=\!(\alpha_{i},\alpha_{j})$, $\eta(\alpha_{i},\alpha_{j})\!\in\!\mathbb{Q}_{\alpha}$ is well-defined and nonzero for every $1\leq i,j\leq n$. For any ${\cal F},{\cal F}^{\prime}\in\mathbb{Q}_{\alpha}[\mathbf{x}_{1},\mathbf{x}_{2}]\llceil{\hbar}\rrceil[[q]]$, and symmetric on $\mathbf{x}_{1},\mathbf{x}_{2}$, define

[TABLE]

which is in $\mathbb{Q}_{\alpha}\llceil{\hbar}\rrceil[[z,q]]$. The pair $({\cal F},{\cal F}^{\prime})$ satisfies $\eta$-mutual polynomiality condition ($\eta$-MPC) if $\Phi^{\eta}_{{\cal F},{\cal F}^{\prime}}\in\mathbb{Q}_{\alpha}[\hbar][[z,q]]$. If ${\cal F}={\cal F}^{\prime}$, we call that ${\cal F}$ satisfies $\eta$-self polynomiality condition ($\eta$-SPC).

Proposition 3.3** ([10, Proposition 6.3]).**

Let ${\cal F},{\cal F}^{\prime}\in\mathbb{Q}_{\alpha}[\mathbf{x}_{1},\mathbf{x}_{2}]\llceil{\hbar}\rrceil[[q]]$. If ${\cal F}$ and ${\cal F}^{\prime}$ are $C$-recursive, for some collection $C=(C_{ij}^{ik}(d))_{d\in\mathbb{Z}^{+};ij,ik\in\mathbf{n}^{2}}$ of elements of $\mathbb{Q}_{\alpha}$, and $({\cal F},{\cal F}^{\prime})$ satisfies $\eta$-MPC, with

[TABLE]

for all $ij\!\in\!\mathbf{n}^{2}$, then ${\cal F}^{\prime}\cong 0~{}(\mathrm{mod}\ \hbar^{-1})$ if and only if ${\cal F}^{\prime}=0$.

Proposition 3.4**.**

Let

[TABLE]

and

[TABLE]

where $h_{d},g_{d}$,$\tilde{h}_{d},\tilde{g}_{d}$ are polynomials in $\mathbb{Q}_{\alpha}[\mathbf{x}_{1},\mathbf{x}_{2},\hbar]$ symmetric on $\mathbf{x}_{1},\mathbf{x}_{2}$, and $g_{d}$, $\tilde{g}_{d}$ are the product of \bigg{(}\prod_{i=1}^{n}(\mathbf{x}_{1}-\alpha_{i}+l\hbar)-\prod_{i=1}^{n}(\mathbf{x}_{1}-\alpha_{i})\bigg{)} and \bigg{(}\prod_{i=1}^{n}(\mathbf{x}_{2}-\alpha_{i}+l\hbar)-\prod_{i=1}^{n}(\mathbf{x}_{2}-\alpha_{i})\bigg{)} with $l\in\mathbb{Z}^{>0}$, $1\leq i\leq n$. Let $\eta$ be defined as in Definition 3.2, then

[TABLE]

Proof.

Denote by

[TABLE]

Since $g_{d}$ and $\tilde{g}_{d}$ are not zero when $(\mathbf{x}_{1},\mathbf{x}_{2})=(\alpha_{i},\alpha_{j})$ for all $i,j$,

[TABLE]

Thus, by the Residue Theorem on $S^{2}$,

[TABLE]

Since $g_{d}$ and $\tilde{g}_{d}$ are not zero when $(\mathbf{x}_{1},\mathbf{x}_{2})=(0,0)$,

[TABLE]

Therefore

[TABLE]

On the other hand

[TABLE]

where $B_{1}(\mathbf{x}_{1},\hbar)$ is a product of \bigg{(}\prod_{i=1}^{n}(\mathbf{x}_{1}-\alpha_{i}+l\hbar)-\prod_{i=1}^{n}(\mathbf{x}_{1}-\alpha_{i})\bigg{)}, which is not zero when $\mathbf{x}_{1}=0$. Thus $\mathrm{Res}_{\mathbf{x}_{1}=0}\mathrm{Res}_{\mathbf{x}_{2}=\infty}(\widetilde{\Phi})=0$, and the $(d_{1},d_{2},p)$-summand of $\mathrm{Res}_{\mathbf{x}_{1}=\infty}\mathrm{Res}_{\mathbf{x}_{2}=\infty}(\widetilde{\Phi})$ is $q^{d_{1}+d_{2}}$ times an element of \mathbb{Q}_{\alpha}[\hbar]\left\llbracket z\right\rrbracket. Therefore

[TABLE]

∎

3.1. Recursive

Definition 3.5**.**

Let $\mathfrak{C}:=(\mathfrak{C}_{ij}^{ik}(d))_{d\in\mathbb{Z}^{+}}\cup(\mathfrak{C}_{ij}^{kj}(d))_{d\in\mathbb{Z}^{+}}$ be any collection of elements of $\mathbb{Q}_{\alpha}$, with $ij,ik,kj\in\bar{\mathbf{n}}^{2}$, where $\bar{\mathbf{n}}^{2}$ is the collection of the pairs $ij$ with $1\leq i,j\leq n$. A power series ${\cal F}\!\in\!\mathbb{Q}_{\alpha}[\mathbf{x}_{1},\mathbf{x}_{2}]\llceil{\hbar}\rrceil[[\mathbf{q}]]$ is $\mathfrak{C}$-recursive if the following holds:

[TABLE]

with $i\neq j$.

For ${\cal F}\in\mathbb{Q}_{\alpha}[\mathbf{x}_{1},\mathbf{x}_{2}]\llceil{\hbar}\rrceil[[\mathbf{q}]]$ which is symmetry on $\mathbf{x}$, Let

[TABLE]

Proposition 3.6**.**

If ${\cal F}$ is a symmetry function on $\mathbf{x}$ and $\mathfrak{C}$-recursive as in Definition 3.5, then $\overline{{\cal F}}$ is $C$-recursive as in Definition 3.1 with

[TABLE]

Proof.

${\cal F}$ is $\mathfrak{C}$-recursive, thus

[TABLE]

[TABLE]

Therefore

[TABLE]

So

[TABLE]

[TABLE]

By the Residue Theorem on $S^{2}$,

[TABLE]

So $\overline{{\cal F}}$ is $C$-recursive with relations as (3.7). ∎

Denote by $\mathbb{Q}_{\tilde{\alpha}}=\mathbb{Q}(\alpha_{1;1},\cdots,\alpha_{1;n},\alpha_{2;1},\cdots,\alpha_{2;n})$. Let $\dot{{\cal A}},\ddot{{\cal A}}\in\mathbb{Q}_{\tilde{\alpha}}[\mathbf{x}_{1},\mathbf{x}_{2}]\llceil{\hbar}\rrceil[[\mathbf{q}]]$ as

[TABLE]

with $a_{k,i}\in\mathbb{Z}^{\geq 0}$. Let

[TABLE]

Note that $\dot{{\cal A}},\ddot{{\cal A}}$ can be considered as the associated $J$-functions of $\mathbb{P}^{n-1}\times\mathbb{P}^{n-1}$. By [10, Section 6, Page 477], we have the following result.

Lemma 3.7**.**

If $\sum_{r,i}a_{r,i}\!\leq\!n$, the power series $\dot{{\cal A}},\ddot{{\cal A}}$ are $\dot{{\mathscr{C}}}_{i_{1}i_{2}}^{j}(s,d)$ and $\ddot{{\mathscr{C}}}_{i_{1}i_{2}}^{j}(s,d)$-recursive respectively.

Therefore

[TABLE]

similarly for $\ddot{\cal A}(\alpha_{1,i_{1}},\alpha_{2,i_{2}},\hbar,\mathbf{q})$.

For all $i,j,r$, denote

[TABLE]

Under the conditions

[TABLE]

$\dot{{\mathscr{C}}}_{i_{1}i_{2}}^{j}(s;d)$ and $\ddot{{\mathscr{C}}}_{i_{1}i_{2}}^{j}(s;d)$ can be written as

[TABLE]

[TABLE]

just changing $\alpha_{s,i_{1}}$ to $\alpha_{i}$, $\alpha_{s,i_{2}}$ to $\alpha_{j}$, $\alpha_{s,k}$ to $\alpha_{k}$.

Therefore $\dot{{\cal K}}$ and $\ddot{{\cal K}}$ satisfy the $\mathfrak{C}$-recursive as defined in Definition 3.5.

Denote

[TABLE]

and

[TABLE]

Thus $\dot{Y},\ddot{Y}$ are the non-equivariant form of $\dot{{\cal Y}},\ddot{{\cal Y}}$

[TABLE]

Let

[TABLE]

[TABLE]

By Proposition 3.6 and Proposition 3.4, we have

Corollary 3.8**.**

The power series $\dot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)$, $\ddot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)$, are $C$-recursive as in (3.1) with $\dot{C}_{ij}^{ik}(d)$ and $\ddot{C}_{ij}^{ik}(d)$ given as above. Moreover, $\dot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)$ satisfies $\eta$-SPC with $\eta=\langle\mathbf{a}\rangle(\mathbf{x}_{1}+\mathbf{x}_{2})^{\ell}$, and $(\dot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),\ddot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q))$ satisfies $1$-MPC.

3.2. Differential operators on hypergeometric functions

Denote $\mathbf{q}=(q_{1},q_{2})$, and $\mathbf{x}=(\mathbf{x}_{1},\mathbf{x}_{2})$. Fix an element ${\cal Y}(\mathbf{x},\hbar,\mathbf{q})\!\in\!\mathbb{Q}_{\tilde{\alpha}}(\mathbf{x},\hbar)[[\mathbf{q}]]$. For $\mathbf{r}\!\in\!(\mathbb{Z}^{\geq 0})^{2}$, $\mathbf{x}^{\mathbf{r}}:=\mathbf{x}_{1}^{r_{1}}\mathbf{x}_{2}^{r_{2}}$. Denote

[TABLE]

for some homogeneous polynomials $f_{{\mathbf{d}}}(\mathbf{x},\hbar),g_{{\mathbf{d}}}(\mathbf{x},\hbar)\!\in\!\mathbb{Q}[\tilde{\alpha},\mathbf{x},\hbar]$ satisfying

[TABLE]

By [9, Section 4], there exists differential operators $\mathfrak{D}^{\mathbf{p}}$, where $\mathbf{p}\!\in\!(\mathbb{Z}^{\geq 0})^{2}$, such that $\mathfrak{D}^{\mathbf{p}}{\cal Y}(\mathbf{x},\hbar,\mathbf{q})$ in $\mathbb{Q}_{\tilde{\alpha}}(\mathbf{x},\hbar)[[\mathbf{q}]]$ satisfies the following properties:

1. There exist ${\cal C}^{(\mathbf{r})}_{\mathbf{p},s}\!:=\!{\cal C}^{(\mathbf{r})}_{\mathbf{p},s}({\cal Y})\!\in\!\mathbb{Q}[\tilde{\alpha}][[\mathbf{q}]]$ with $\mathbf{p},\mathbf{r}\!\in\!(\mathbb{Z}^{\geq 0})^{2}$, $s\!\in\!\mathbb{Z}^{\geq 0}$, such that \left\llbracket{\cal C}^{(\mathbf{r})}_{\mathbf{p},s}\right\rrbracket_{\mathbf{q};{\mathbf{d}}} is a homogeneous polynomial in $\alpha$ of degree $\!-\!|\mathbf{r}|\!+\!s$,

[TABLE]

[TABLE]

and

[TABLE]

Let $\gamma^{k}_{j}(\mathfrak{D})$ be a polynomial of operator $\mathfrak{D}$ such that

[TABLE]

Denote by

[TABLE]

[TABLE]

Define

[TABLE]

where $\mathbb{Q}[[q]][\mathbf{x}_{1},\mathbf{x}_{2}]^{S_{2}}_{k}$ is the set of all the symmetric polynomials of $\mathbf{x}=(\mathbf{x}_{1},\mathbf{x}_{2})$ with degree $k$. Then

[TABLE]

In particular $\overline{J}_{k}(\dot{{\cal Y}})$ as an element of $\mbox{End}(\mathbb{Q}[[q]][\mathbf{x}_{1},\mathbf{x}_{2}]^{S_{2}}_{k})$ is invertible, with $\bar{c}_{\gamma^{k}_{i},\gamma^{k}_{j}}$ given by

[TABLE]

We define

[TABLE]

Then

[TABLE]

and

[TABLE]

Define $\dot{{\cal C}}^{(r)}_{k,i;s,j}\!:=\!\dot{{\cal C}}^{(r)}_{k,i;s,j}(\dot{{\cal Y}}_{n;\mathbf{a}})\!\in\!\mathbb{Q}[\alpha][[q]]$ for $k,s$ and $r\!\in\!\mathbb{Z}^{\geq 0}$ with $s\!\leq\!k\!-\!r$ and $r\!\leq\!k$ by

[TABLE]

These equations indeed uniquely determine $\dot{{\cal C}}^{(r)}_{k,i;s,j}$, since

[TABLE]

Induction by (3.18) and (3.2), we obtain

[TABLE]

Similarly we can define $\ddot{{\cal C}}^{(r)}_{k,i;s,j}$ for $\ddot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)$.

Proposition 3.9**.**

The power series ${\cal D}^{k,j}\dot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)$ and ${\cal D}^{k,j}\ddot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)$ are $C$-recursive as in (3.1) with $\dot{C}_{ij}^{ik}$ and $\ddot{C}_{ij}^{ik}$ as (3.10) and (3.11). Moreover, $(\dot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),{\cal D}^{k,j}\dot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q))$ satisfies $\eta$-MPC with $\eta=\langle\mathbf{a}\rangle(\mathbf{x}_{1}+\mathbf{x}_{2})^{\ell}$, and $(\dot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),{\cal D}^{k,j}\ddot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q))$ satisfies $1$-MPC.

Proof.

By [9, Lemma 5.8 (a)], $\gamma_{j}^{k}(\mathfrak{D})\dot{{\cal K}}$ and $\gamma_{j}^{k}(\mathfrak{D})\ddot{{\cal K}}$ are $\mathfrak{C}$-recursive. Then by Proposition 3.6 and [9, Lemma 5.8(b)], ${\cal D}^{k,j}\dot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)$ and ${\cal D}^{k,j}\ddot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)$ are $C$-recursive as in Definition 3.1.

The second assertion follows from Proposition 3.4. ∎

4. localization formula

The standard $\mathbb{T}$-action on $\mathbb{C}^{n}$ (as well as any other representation) induces corresponding $\mathbb{T}$-actions on $\overline{Q}_{0,2}(\mathbf{Gr},d)$, ${\cal S}$, ${\cal V}_{n;\mathbf{a}}^{(d)}$, $\dot{\cal V}_{n;\mathbf{a}}^{(d)}$, and $\ddot{\cal V}_{n;\mathbf{a}}^{(d)}$. Thus, ${\cal V}_{n;\mathbf{a}}^{(d)}$, $\dot{\cal V}_{n;\mathbf{a}}^{(d)}$, and $\ddot{\cal V}_{n;\mathbf{a}}^{(d)}$ have well-defined equivariant Euler classes

[TABLE]

The universal cotangent line bundle for the $i$-th marked point also has a well-defined equivariant Euler class, which will still be denoted by $\psi_{i}$. Let

[TABLE]

[TABLE]

where $\mathrm{ev}_{1}:\overline{Q}_{0,2}(\mathbf{Gr},d)\longrightarrow\mathbf{Gr}$ is as before.

Let $[\overline{\Delta}]$ be the equivariant class of the diagonal

[TABLE]

where $g_{k,j;s,i}\in\mathbb{Q}[\alpha]$ are homogeneous polynomials.

Let

[TABLE]

where $\mathrm{ev}_{1},\mathrm{ev}_{2}\!:\overline{Q}_{0,2}(\mathbf{Gr},d)\!\longrightarrow\!\mathbf{Gr}$.

Let

[TABLE]

Let

[TABLE]

where $\omega\in H_{\mathbb{T}}^{*}(\mathbf{Gr})$.

Marian, Oprea and Pandharipande [7, Section 7.3] described the fixed loci of the $\mathbb{T}$-action on $\overline{Q}_{g,m}(\mathbf{Gr},d)$, which are indexed by connected decorated graphs. However, in the case $g=0,\,m\!=\!2$, the relevant graphs consist of a single strand (possibly consisting of a single vertex) with the two marked points attached at the opposite ends of the strand. Such a graph can be described by an ordered set $(\mathrm{Ver},<)$ of vertices, where $<$ is a strict order on the finite set $\mathrm{Ver}$. Given such a strand, denote by $v_{\min}$ and $v_{\max}$ its minimal and maximal elements and by Edg its set of edges, i.e. of pairs of consecutive elements. A decorated strand is a tuple

[TABLE]

where $(\mathrm{Ver},<)$ is a strand as above and

[TABLE]

are maps such that

[TABLE]

In Figure 1, the vertices of a decorated strand $\Gamma$ are indicated by dots in the increasing order, with respect to $<$, from left to right. Let $Q_{\Gamma}$ be the fixed locus of $\overline{Q}_{0,2}(\mathbf{Gr},d)$ corresponding to a decorated strand $\Gamma$, then $Q_{\Gamma}$ isomorphic to

[TABLE]

up to a finite quotient, where $\overline{M}_{0,2|\mathfrak{d}(v)}$ is the Hassett moduli space of weighted pointed stable curves. Let $\pi_{1}:{\cal C}_{\overline{M}_{0,2|\mathfrak{d}(v)}}\to\overline{M}_{0,2|\mathfrak{d}(v)}$ be the restriction of universal family $\pi:{\cal C}\to\overline{Q}_{0,2}(\mathbf{Gr},d)$. The universal bundle is split when restricting on ${\cal C}_{\overline{M}_{0,2|\mathfrak{d}(v)}}$. We denote by ${\cal S}^{\vee}|_{{\cal C}_{\overline{M}_{0,2|\mathfrak{d}(v)}}}\cong{\mathscr{L}}_{1}\oplus{\mathscr{L}}_{2}$.

Lemma 4.1**.**

$\dot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),\dot{{\cal Z}}^{b_{2},\omega}(\hbar,q)$* are $\dot{C}$-recursive. Moreover, $\dot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)$ satisfies $\eta$-SPC, and $(\dot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),\dot{{\cal Z}}^{b_{2},\omega}(\hbar,q))$ satisfy $\eta$-MPC with $\eta=\langle\mathbf{a}\rangle(\mathbf{x}_{1}+\mathbf{x}_{2})^{\ell}$. $\ddot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),\ddot{{\cal Z}}^{b_{2},\omega}(\hbar,q)$ are $\ddot{C}$-recursive, and $(\dot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),\ddot{{\cal Z}})$, $(\dot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),\ddot{{\cal Z}}^{b_{2},\omega}(\hbar,q))$ satisfy $1$-MPC.*

Proof.

The recursive of $\dot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),\ddot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),\dot{{\cal Z}}^{b_{2},\omega}(\hbar,q)$ and $\ddot{{\cal Z}}^{b_{2},\omega}(\hbar,q)$ can be proved parallel to [10, Lemma 6.5], or be obtained by just applying [3, Lemma 7.5.2] to the twisted case of $W\sslash G$, and $\mathbf{Gr}$ be considered as GIT qoutient $\mathbb{C}^{2n}\sslash GL_{2}(\mathbb{C})$.

The proofs of polynomiality are parallel to [10, Lemma 6.6]. We sketch it as follows.

Denote by the standard action $\mathbb{C}^{*}$ on $\mathbb{C}$ by $\hbar$. Let $V=\mathbb{C}\oplus\mathbb{C}$ be the representation of $\mathbb{C}^{*}$ with weight [math], and $-\hbar$. For $d\in\mathbb{Z}^{\geq 0}$, the action $\tilde{\mathbb{T}}:=(\mathbb{C}^{*})^{\frac{n(n-1)}{2}}\times\mathbb{C}^{*}$ on $\mathbb{C}^{(\frac{n(n-1)}{2})}\otimes\text{Sym}^{d}V^{*}$ induced an action on

[TABLE]

Let

[TABLE]

be the equivariant hyperplane class. Let

[TABLE]

Using Plücker embedding, we have the following morphism $\theta$

[TABLE]

Then applying localization formula we can prove

[TABLE]

with $\eta=\langle\mathbf{a}\rangle(x_{1}+x_{2})^{\ell}$, and

[TABLE]

which are in $\mathbb{Q}_{\alpha}[\hbar][[z,q]]$. ∎

Denote by

[TABLE]

$\ddot{{\cal I}}_{n;\mathbf{a}}(q):=1$.

Theorem 4.2**.**

If $\ell\!\in\!\mathbb{Z}^{\geq 0}$, $n\!\in\!\mathbb{Z}^{+}$, and $\mathbf{a}\!\in\!(\mathbb{Z}^{>0})^{\ell}$ are such that $|\mathbf{a}|\!\leq\!n$, then

[TABLE]

and

[TABLE]

Proof.

The proof is parallel to [5, Theorem 3].

When $|\mathbf{a}|\leq n-2$,

[TABLE]

For $d\in\mathbb{Z}^{+}$,

[TABLE]

Thus

[TABLE]

Therefore

[TABLE]

When $n-1\leq|\mathbf{a}|\leq n$. By Lemma 4.1, $\dot{{\cal Z}}(\hbar,q)$ is $C$-recursive, therefore

[TABLE]

for some $\dot{{\cal Z}}_{ij}^{r}(d)\!\in\!\mathbb{Q}_{\alpha}$.

Parallel to the proof of [5, Proposition 6.1], by applying localization formula to $\dot{{\cal Z}}(\hbar,q)$, for all $r<0$, we have

[TABLE]

Parallel to the proof of [5, Proposition 8.3], we have

[TABLE]

This means that $\dot{{\cal Z}}_{ij}^{r}(d)$ and $\frac{\dot{{\cal Y}}_{ij}^{r}(d)}{\dot{{\cal I}}}$ satisfy the same recursive relations. For their coefficients of $q^{0}$ are the same,

[TABLE]

[TABLE]

Then by [6, Lemma 30.3.2],

[TABLE]

Because $(\dot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),\ddot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q))$ satisfy 1-MPC, $(\dot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)=\frac{\dot{{\cal Y}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q)}{\dot{{\cal I}}_{n;\mathbf{a}}(q)},\ddot{{\cal Y}})$ satisfy 1-MPC. For $(\dot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q),\ddot{{\cal Z}}_{n;\mathbf{a}}(\mathbf{x},\hbar,q))$ satisfy 1-MPC, and

[TABLE]

By Proposition 3.3, we have

[TABLE]

∎

Theorem 4.3**.**

If $\ell\!\in\!\mathbb{Z}^{\geq 0}$, $n\!\in\!\mathbb{Z}^{+}$, and $\mathbf{a}\!\in\!(\mathbb{Z}^{>0})^{\ell}$ are such that $|\mathbf{a}|\!\leq\!n$, then

[TABLE]

where

[TABLE]

and $\dot{{\cal C}}^{(r)}_{k,i;s,j},\ddot{{\cal C}}^{(r)}_{k,i;s,j}$ are defined as in (3.19).

Proof.

By the definition of $\dot{{\cal C}}^{(r)}_{k,i;s,j}$,

[TABLE]

By Proposition 3.3 and Lemma 4.1,

[TABLE]

Similar argument holds for $\ddot{{\cal Z}}_{\gamma^{k}_{j}}(\hbar,q)$ and $\ddot{{\cal Y}}_{\gamma^{k}_{j}}(\hbar,q)$. ∎

Theorem 4.4**.**

If $\ell\!\in\!\mathbb{Z}^{\geq 0}$, $n\!\in\!\mathbb{Z}^{+}$, and $\mathbf{a}\!\in\!(\mathbb{Z}^{>0})^{\ell}$ are such that $|\mathbf{a}|\!\leq\!n$, then

[TABLE]

Proof.

By (2.3) and (2.2),

[TABLE]

for all $ij,kl\in\mathbf{n}^{2}$. By Lemma 4.1, the coefficient of $\hbar_{2}^{m}$ in $(\hbar_{1}+\hbar_{2})\dot{{\cal Z}}_{n;\mathbf{a}}(\hbar_{1},\hbar_{2},q)$ is $\dot{C}$-recursive and satisfies $\eta=\langle\mathbf{a}\rangle(\mathbf{x}_{1}+\mathbf{x}_{2})^{\ell}$-MPC with respect to $\dot{{\cal Z}}_{n;\mathbf{a}}(\hbar,q)$ for all $m\leq 0$. By Proposition 3.3, it follows that in order to prove (4.14) it suffices to show that

[TABLE]

The right-hand side of (4.16) mod $\hbar^{-1}_{1}$ is

[TABLE]

The left-hand side of (4.16) mod $\hbar^{-1}_{1}$ is

[TABLE]

By Lemma 4.1, (4.18) is $\ddot{C}$-recursive and satisfies $\eta=1$-MPC with respect to $\dot{{\cal Z}}_{n;\mathbf{a}}(\hbar_{2},q)$. Since (4.17) also satisfies these properties. The power series (4.18) and (4.17) agree if only if they agree mod $\hbar_{2}^{-1}$. The latter is the case since both are the equivariant Poincaré dual to the diagonal restricted to $p_{ij}\times p_{kl}$. ∎

Denote by

[TABLE]

[TABLE]

By restricting on $\alpha=0$, Theorem 4.2, Theorem 4.3 and Theorem 4.4 give us the non-equivariant Theorem 1.1, Theorem 1.2 and Theorem 1.3.