Verlinde/Grassmannian Correspondence and Rank 2 $\delta$-Wall-Crossing

Abstract

Motivated by Witten’s work, we propose a K-theoretic Verlinde/Grassmannian correspondence which relates the GL Verlinde numbers to the K-theoretic quasimap invariants of the Grassmannian. We recover these two types of invariants by imposing different stability conditions on the gauged linear sigma model associated with the Grassmannian. We construct two families of stability conditions connecting the two theories and prove two wall-crossing results. We confirm the Verlinde/Grassmannian correspondence in the rank two case.

Appendix A: K-Theoretic I-Function of the Grassmannian and Wall-Crossing Contributions

In this appendix, we compute the function $\mu ^{W_1}_{d'}(q)$ which shows up in the wall-crossing contributions in Theorem 6.13. This function is closely related to the K-theoretic small I-function of the Grassmannian. The level-0 K-theoretic small I-function of $\text {Gr}(n,N)$ was computed in [65] and the computation was generalized to the case for arbitrary level structure in [72]. Both computations are generalizations of [8] on the cohomological small I-function of $\text {Gr}(n,N)$.

Recall that $QG^{\epsilon =0+}_{g,k}(\mathrm {Gr}(n,N),d)$ denotes the $(\epsilon =0+)$-stable quasimap graph space and there is a $\mathbb {C}^{*}$-action on it defined by (52). The K-group valued rational function $\mu ^{W_1}_{d'}(q)$ in the proof of Theorem 6.13 is defined using this $\mathbb {C}^{*}$-action on $QG^{0+}_{0,1}:=QG^{\epsilon =0+}_{0,1}(\mathrm {Gr}(n,N),d)$. Recall that $F_{d}\subset QG^{0+}_{0,1}$ is the distinguished $\mathbb {C}^*$-fixed point locus parametrizing quasimaps

$$\begin{aligned} (\mathbb {P}^1,q_\bullet ,E,s), \end{aligned}$$

where the only marking $q_\bullet$ lies at $\infty \in \mathbb {P}^1$ and the only base point of s is at 0. We want to analyze $N^{\text {vir},\text {rel}}_{F_{d}/QG^{0+}_{0,1}}$, i.e., the moving part of the relative perfect obstruction theory (58).

In fact, we only need to study the quasimap graph space $QG^{0+}_{0,0}:=QG^{0+}_{0,0}(\text {Gr}(n,N),d)$ with no marking. We denote by $F'_d$ the fixed-point component of $QG^{0+}_{0,0}$ parametrizing the quasimaps of degree d

$$\begin{aligned} (\mathbb {P}^1,E,s) \end{aligned}$$

with E a vector bundle of rank n and degree d, and $s:\mathbb {P}^1\rightarrow E\otimes \mathcal {O}^{\oplus N}_{\mathbb {P}^1}$ a section such that $s(x)\ne 0$ for $x\ne 0\in \mathbb {P}^1$ and $0\in \mathbb {P}^1$ is a base point of length d. As explained in [17, §4.1], there is an evaluation morphism $\rho : F_{d}\rightarrow \text {Gr}(n,N)$ at the generic point of $\mathbb {P}^{1}$.

There is an obvious isomorphism between $F_{d}'$ and $F_{d}$ that identifies the universal N-pairs over their trivial universal curves. Hence, the determinant line bundles defined by (54) are the same for both spaces under the obvious isomorphism. We can also identify $\rho$ with $\mathrm {ev}_{\infty }$. By abuse of notation, we will use the notation $F_{d}$ to denote both distinguished $\mathbb {C}^{*}$-fixed point loci. Let $N^{\mathrm {vir}}_{F_{d}/QG^{0+}_{0,0}}$ be the virtual normal bundle of $F_{d}$ in $QG^{0+}_{0,0}$, i.e., the moving part of the restriction of the absolute perfect obstruction theory of $QG^{0+}_{0,0}$ induced by (58). In the computation of $N^{\mathrm {vir}}_{F_{d}/QG^{0+}_{0,0}}$, the contribution from the automorphisms moving the unmarked points at 0 and $\infty$ cancels with that of the deformation of the parametrization $\varphi$. Hence we have $N^{\mathrm {vir},\text {rel}}_{F_{d}/QG^{0+}_{0,1}}\cong N^{\mathrm {vir}}_{F_{d}/QG^{0+}_{0,0}}$. In sum, we can use $F_{d'}$ and $QG^{0+}_{0,0}(\text {Gr}(n,N),d')$ to compute $\mu ^{W_1}_{d'}(q)$.

Let $\pi :\mathcal {C}\rightarrow QG^{0+}_{0,0}(\text {Gr}(n,N),d)$ be the universal curve and let $\mathcal {O}^{\oplus N}_{\mathcal {C}}\rightarrow \mathcal {E}$ be the universal N pair. Recall that $\rho : F_{d}\rightarrow \text {Gr}(n,N)$ is the evaluation map at the generic point of $\mathbb {P}^{1}$. The K-theoretic small I-function of $\text {Gr}(n,N)$ of level l is defined by

$$\begin{aligned} \mathcal {I}^{l}(q,Q)=1+\sum _{d>0}Q^d\rho _{*}\bigg (\mathcal {O}_{F_d}^{\text {vir}}\otimes \bigg (\frac{\mathcal {D}^{l}}{\lambda _{-1}^{\mathbb {C}^*}(N^{\mathrm {vir},\vee }_{F_{d}/QG^{0+}_{0,0}})}\bigg )\bigg )\otimes \text {det}^{-l}(S), \end{aligned}$$

where

• Q is a formal variable called the Novikov variable,

• $\mathcal {D}^{l}$ is the restriction of the level-l determinant line bundle $(\text {det}\,R\pi _*(\mathcal {E}))^{-l}$ to $F_{d}$,

• $N^{\mathrm {vir},\vee }_{F_{d}/QG^{0+}_{0,0}}$ denotes the dual of the virtual normal bundle $N^{\mathrm {vir}}_{F_{d}/QG^{0+}_{0,0}}$, and

• S is the tautological subbundle over $\text {Gr}(n,N)$.

Since $\mathbb {C}^{*}$ acts trivially on $F_{d}$, we have $K_{0}^{\mathbb {C}^{*}}(F_{d})=K_{0}(F_{d})[q,q^{-1}]$. By definition, $\mathcal {I}^{l}(q,Q)$ is a formal power series in Q and the coefficient of $Q^{i}, i>0$ is a $K^{0}(\mathrm {Gr}(n,N))$-valued rational function in q.

Now let us give explicit descriptions of $QG^{0+}_{0,0}(\text {Gr}(n,N),d)$ and $F_{d}$. Let $\text {Quot}_{\mathbb {P}^1,d}(\mathbb {C}^N,N-n)$ be the Grothendieck’s Quot scheme parametrizing quotients $\mathcal {O}^{\oplus N}_{\mathbb {P}^1}\rightarrow Q\rightarrow 0$, where Q is a coherent sheaf on $\mathbb {P}^{1}$ of rank $N-n$ and degree d. Let X be a scheme. Suppose $\mathcal {O}^{\oplus N}_{\mathbb {P}^1\times X}\rightarrow {\tilde{Q}}\rightarrow 0$ is a flat quotient over $\mathbb {P}^1\times X$. The kernel $\mathcal {K}$ of the quotient map is locally free due to flatness. Let $\mathcal {E}$ be the dual of $\mathcal {K}$. By dualizing the injection $0\rightarrow \mathcal {K}\rightarrow \mathcal {O}^{\oplus N}_{\mathbb {P}^1\times X}$, we obtain a morphism $\mathcal {O}^{\oplus N}_{\mathbb {P}^1\times X}\rightarrow \mathcal {E}$ satisfying that, for any closed point $x\in X$, the restriction of the morphism to $\mathbb {P}^1\times \{x\}$ is surjective at all but a finite number of points. It is easy to check that $\mathcal {O}^{\oplus N}_{\mathbb {P}^1\times X}\rightarrow \mathcal {E}$ is a flat family of quasimaps with one parametrized rational component. Therefore, we obtain a morphism from $\text {Quot}_{\mathbb {P}^1,d}(\mathbb {C}^N,N-n)$ to $QG^{0+}_{0,0}(\text {Gr}(n,N),d)$. In fact, this morphism is an isomorphism. Indeed, for any quasimap $(C',E,s,\varphi )$ in $QG^{0+}_{0,0}(\text {Gr}(n,N),d)$, the underlying curve must be $\mathbb {P}^1$ due to the stability condition. In sum, we have an isomorphism

$$\begin{aligned} QG^{0+}_{0,0}(\text {Gr}(n,N),d)\cong \text {Quot}_{\mathbb {P}^1,d}(\mathbb {C}^N,N-n). \end{aligned}$$

Note that the above Quot scheme is a smooth, projective variety. Hence its virtual structure sheaf is the same as the structure sheaf.

The distinguished $\mathbb {C}^*$-fixed-point loci $F_d$ is explicitly identified in [8]. Consider a collection of integers $\{d_i\}_{1\le i\le n}$ which satisfies

$$\begin{aligned} \sum d_i=d\quad \text {and}\quad 0\le d_1\le d_2\le \dots \le d_n. \end{aligned}$$

(61)

Suppose $0\le d_1=\dots =d_{n_1}<d_{n_1+1}=\dots =d_{n_2}<\dots <d_{n_{j}+1}=\dots =d_{n}$. Then the jumping index of $\{d_i\}$ is defined as the collection of integers $\{n_i\}_{1\le i\le j}$. According to [8, Lemma 1.2], the irreducible components of $F_d$ are indexed by collections of integers satisfying (61). More precisely, the irreducible components of $F_d$ are the images of flag varieties:

$$\begin{aligned} \iota _{\{d_i\}}:\text {Fl}(n_1,\dots ,n_j;S)\hookrightarrow \text {Quot}_{\mathbb {P}^1,d}(\mathbb {C}^N,N-n). \end{aligned}$$

Here $\text {Fl}(n_1,\dots ,n_j;S)$ denotes the relative flag variety of type $\{n_i\}$ and we refer the reader to [8, §1] for the precise definition of the embedding $\iota _{\{d_i\}}$. The (restriction of) evaluation map $\rho : F_{d}\rightarrow \text {Gr}(n,N)$ can be identified with the flag bundle map $\text {Fl}(n_1,\dots ,n_j;S)\rightarrow \text {Gr}(n,N)$.

Consider the universal sequence of sheaves on $\text {Quot}:=\text {Quot}_{\mathbb {P}^1,d}(\mathbb {C}^N,N-n)$:

$$\begin{aligned} 0\rightarrow \mathcal {K}\rightarrow \mathbb {C}^N\otimes \mathcal {O}_{\text {Quot}\times \mathbb {P}^1}\rightarrow \mathcal {Q}\rightarrow 0. \end{aligned}$$

Let $\pi :\text {Quot}\times \mathbb {P}^1\rightarrow \text {Quot}$ be the projection. For simplicity, we denote $\text {Fl}:=\text {Fl}(n_1,\dots ,n_j;S)$ and set $n_{j+1}:=n$. Let $\rho :\text {Fl}\rightarrow \text {Gr}(n,N)$ be the flag bundle map and let

$$\begin{aligned} 0\subset S_{n_1}\subset S_{n_2}\subset \dots \subset S_{n_j}\subset S_{n_j+1} =\rho ^*S \end{aligned}$$

be the universal flag on $\text {Fl}$. Let $z=\mathrm {Fl}\times 0\subset \mathrm {Fl}\times \mathbb {P}^1$. According to [8], the restriction of $\mathcal {K}$ from $\text {Quot}\times \mathbb {P}^1$ to $\text {Fl}\times \mathbb {P}^1$ has an increasing filtration $0=\mathcal {K}_0\subset \dots \subset \mathcal {K}_j\subset \mathcal {K}_{j+1}=\mathcal {K}$ with

$$\begin{aligned} \mathcal {K}_a/\mathcal {K}_{a-1}\cong \pi ^*(S_{n_a}/S_{n_{a-1}})(-d_{n_a}z). \end{aligned}$$

Therefore, in the K-group $K^0(\text {Fl}\times \mathbb {P}^1)$, we have

$$\begin{aligned}{}[\mathcal {K}^\vee ]=\sum _{a=1}^{j+1}[(S_{n_a}/S_{n_{a-1}})^\vee (d_{n_a})]. \end{aligned}$$

(62)

Using the splitting principle, we write $\sum _{s=1}^{n_a-n_{a-1}}\mathcal {L}_{n_{a-1}+s}=(S_{n_a}/S_{n_{a-1}})^\vee .$

To obtain the explicit formula of the small I-function of level l, we need to compute three factors: $\det ^{-l}(S)$, $\lambda _{-1}^{\mathbb {C}^*}(N^\vee _{\text {Fl}/\text {Quot}})$, and $\mathcal {D}^{l}$. First, we have

$$\begin{aligned} \mathrm {det}^{-l}(S)=\prod _{a=1}^{j+1}\prod _{s=1}^{n_a-n_{a-1}}(\mathcal {L}_{n_{a-1}+s})^l. \end{aligned}$$

Second, according to [8, §1], the tangent bundle $T_{\mathrm {Quot}}$ of the Quot scheme is given by $\pi _{*}(\mathcal {K}^{\vee }\otimes \mathcal {Q})$, and it fits into the following long exact sequence:

$$\begin{aligned} 0\rightarrow \pi _{*}(\mathcal {K}^{\vee }\otimes \mathcal {K})\rightarrow \pi _{*}\mathcal {K}^{\vee }\otimes \mathbb {C}^{N}\rightarrow T_{\mathrm {Quot}}\rightarrow R^{1}\pi _{*}(\mathcal {K}^{\vee }\otimes \mathcal {K})\rightarrow 0. \end{aligned}$$

By restricting the above exact sequence to $\mathrm {Fl}$ and then taking the $\mathbb {C}^{*}$-moving parts, we obtain the following identity in $K^{0}_{\mathbb {C}^{*}}(\mathrm {Fl})$:

$$\begin{aligned}{}[N_{\text {Fl}/\text {Quot}}]=[(\pi _{*}\mathcal {K}^{\vee }\otimes \mathbb {C}^{N})|_{\mathrm{Fl}}^{\mathrm{mv}}]+[(R^{1}\pi _{*}(\mathcal {K}^{\vee }\otimes \mathcal {K}))|_{\mathrm{Fl}}^{\mathrm{mv}}]-[(\pi _{*}(\mathcal {K}^{\vee }\otimes \mathcal {K}))|_{\mathrm{Fl}}^{\mathrm{mv}}]. \end{aligned}$$

Here, the superscript “mv” denotes taking the moving part. By the definition of the $\mathbb {C}^{*}$-action (52), it is straightforward to check that the $\mathbb {C}^{*}$-weights of $H^{0}(\mathbb {P}^{1},\mathcal {O}_{\mathbb {P}^{1}}(d))$ are $1,q^{-1},q^{-2},\dots ,q^{-d}$ for $d\ge 0$ and the $\mathbb {C}^{*}$-weights of $H^{1}(\mathbb {P}^{1},\mathcal {O}_{\mathbb {P}^{1}}(-d))$ are $q,q^{2},\dots ,q^{d-1}$ for $d>0$. Using this fact and the decomposition (62), we can easily check:

$$\begin{aligned}{}[(\pi _{*}\mathcal {K}^{\vee }\otimes \mathbb {C}^{N})|_{\mathrm{Fl}}^{\mathrm{mv}}]&=N\sum _{a=1}^{j+1}\sum _{s=1}^{r_a}\sum _{b=1}^{d_{n_a}}(\mathcal {L}_{n_{a-1}+s}q^{-b}), \\ [(\pi _{*}(\mathcal {K}^{\vee }\otimes \mathcal {K}))|_{\mathrm{Fl}}^{\mathrm{mv}}]&=\sum _{1\le a<b\le j+1}\sum _{\substack{1\le s\le r_{a} \\ 1\le t\le r_{b}}} \sum _{i=1}^{d_{ba}}\mathcal {L}_{n_{b-1}+t}\mathcal {L}^{\vee }_{n_{a-1}+s}q^{-i}, \\ [(R^{1}\pi _{*}(\mathcal {K}^{\vee }\otimes \mathcal {K}))|_{\mathrm{Fl}}^{\mathrm{mv}}]&=\sum _{1\le a<b\le j+1}\sum _{\substack{1\le s\le r_{a}\\1\le t\le r_{b}}} \sum _{i=1}^{d_{ba}-1}\mathcal {L}_{n_{b-1}+t}^{\vee }\mathcal {L}_{n_{a-1}+s}q^{i}, \end{aligned}$$

where $r_a:=n_a-n_{a-1}$ and $d_{ba}:=d_{n_b}-d_{n_a}$. Note that⁵

$$\begin{aligned} \begin{aligned} \frac{\lambda ^{\mathbb {C}^{*}}_{-1}([(\pi _{*}(\mathcal {K}^{\vee }\otimes \mathcal {K}))|_{\mathrm{Fl}}^{\mathrm{mv}}])}{\lambda ^{\mathbb {C}^{*}}_{-1}([(R^{1}\pi _{*}(\mathcal {K}^{\vee }\otimes \mathcal {K}))|_{\mathrm{Fl}}^{\mathrm{mv}}])}&=\frac{\prod _{1\le a<b\le j+1}\prod _{\substack{ 1\le s\le r_{a} \\ 1\le t\le r_{b}}}\prod _{i=1}^{d_{ba}}(1-\mathcal {L}_{n_{b-1}+t}^\vee \mathcal {L}_{n_{a-1}+s}q^{d_{i}})}{\prod _{1\le a<b\le j+1}\prod _{\substack{1\le s\le r_{a} \\ 1\le t\le r_{b} }}\prod _{i=1}^{d_{ba}-1}(1-\mathcal {L}_{n_{b-1}+t}\mathcal {L}_{n_{a-1}+s}^{\vee }q^{d_{-i}})} \\&={\prod _{1\le a<b\le j+1}\prod _{\begin{subarray}{c} 1\le s\le r_{a} \\ 1\le t\le r_b \end{subarray}}\prod _{c=1}^{d_{ba}-1}\mathcal {L}^\vee _{n_{b-1}+s}\mathcal {L}_{n_{a-1}+t}q^{c}} \\&\quad \cdot \prod _{1\le a<b\le j+1}\prod _{\substack{1\le s\le r_{a} \\ 1\le t\le r_{b}}}(-1)^{r_ar_b(d_{ba}-1)}(1-\mathcal {L}_{n_{b-1}+t}^\vee \mathcal {L}_{n_{a-1}+s}q^{d_{ba}}). \end{aligned} \end{aligned}$$

(63)

Hence, we obtain

$$\begin{aligned} \frac{1}{\lambda ^{\mathbb {C}^{*}}_{-1}(N^\vee _{\text {Fl}/\text {Quot}})}&={\prod _{1\le a<b\le j+1}\prod _{\substack{ 1\le s\le r_{a} \\ 1\le t\le r_b}}\prod _{c=1}^{d_{ba}-1}\mathcal {L}^\vee _{n_{b-1}+s}\mathcal {L}_{n_{a-1}+t}q^{c}} \\&\quad \cdot \frac{\prod _{1\le a<b\le j+1}\prod _{\substack{ 1\le s\le r_{a} \\ 1\le t\le r_{b}}}(-1)^{r_ar_b(d_{ba}-1)}(1-\mathcal {L}_{n_{b-1}+t}^\vee \mathcal {L}_{n_{a-1}+s}q^{d_{ba}})}{\prod _{a=1}^{j+1}\prod _{s=1}^{r_a}\prod _{b=1}^{d_{n_a}}(1-\mathcal {L}^\vee _{n_{a-1}+s}q^b)^N}. \end{aligned}$$

Finally, recall that $\mathcal {E}=\mathcal {K}^\vee$. Then it follows from the definition $\mathcal {D}^{l}=\det ^{-l}R\pi _*(\mathcal {E})$ that

$$\begin{aligned} \mathcal {D}^{l}=\prod _{a=1}^{j+1}\prod _{s=1}^{r_a}\prod _{b=0}^{d_{n_a}}(\mathcal {L}^\vee _{n_{a-1}+s}q^b)^l. \end{aligned}$$

The last ingredient that we need to recall is the pushforward lemma [65, Lemma 6]:

$$\begin{aligned} \rho _{*}([\mathcal {F}])=\sum _{w}w\left( \frac{[\mathcal {F}]}{\lambda ^{\mathbb {C}^{*}}_{-1}(T_{\rho }^{\vee })} \right) , \end{aligned}$$

(64)

where $\mathcal {F}$ is a Laurent polynomial in $\mathcal {L}_{i}$’s, $T_{\rho }^{\vee }$ is the relative conormal bundle, and the summation is over all $w\in S_n/(S_{r_1}\times \dots \times S_{r_{j+1}})$, which acts on the indices of $\mathcal {L}_{i}$’s. We have

$$\begin{aligned}{}[T_{\rho }^{\vee }]=\sum _{1\le a<b\le j+1}\sum _{\substack{ 1\le s\le r_a \\ 1\le t\le r_b}}\mathcal {L}_{n_{b-1}+t}^\vee \mathcal {L}_{n_{a-1}+s}. \end{aligned}$$

Combining all the above results, we obtain

$$\begin{aligned}\mathcal {I}^{l}(q,Q)&=\sum _{d\ge 0}Q^d\sum _{\substack{ \{d_i\} \\ \sum d_i=d}}\rho _*\bigg (\frac{\mathcal {D}^{l}}{\lambda ^{\mathbb {C}}_{-1}(N^\vee _{\text {Fl}/\text {Quot}})}\bigg )\otimes \mathrm {det}^{-l}(S) \\ &=\sum _{d\ge 0}Q^d\sum _{\substack{\{d_i\} \\ \sum d_i=d }}\rho _*\Bigg (\prod _{1\le a<b\le j+1}\prod _{\substack{ 1\le s\le r_a \\ 1\le t\le r_b}}\prod _{c=1}^{d_{ba}-1}\mathcal {L}^\vee _{n_{b-1}+t}\mathcal {L}_{n_{a-1}+s}q^{c}\nonumber \\&\quad\cdot \prod _{1\le a<b\le j+1}\prod _{\substack{ 1\le s\le r_a \\ 1\le t\le r_b}}(-1)^{r_ar_b(d_{ba}-1)}(1-\mathcal {L}_{n_{b-1}+t}^\vee \mathcal {L}_{n_{a-1}+s}q^{d_{ba}})\nonumber \\&\quad \cdot \frac{\prod _{a=1}^{j+1}\prod _{s=1}^{r_a}\prod _{b=0}^{d_{n_a}}(\mathcal {L}^\vee _{n_{a-1}+s}q^b)^l }{\prod _{a=1}^{j+1}\prod _{s=1}^{r_a}\prod _{b=1}^{d_{n_a}}(1-\mathcal {L}^\vee _{n_{a-1}+s}q^b)^N}\Bigg )\otimes \mathrm {det}^{-l}(S) \\ &=\sum _{d\ge 0}Q^d\sum _{\substack{\{d_i\} \\ \sum d_i=d}}\sum _{w}\Bigg (\prod _{1\le a<b\le j+1}\prod _{\substack{1\le s\le r_a \\ 1\le t\le r_b }}\prod _{c=1}^{d_{ba}-1}\mathcal {L}^\vee _{n_{b-1}+t}\mathcal {L}_{n_{a-1}+s}q^{c}\nonumber \\&\quad \cdot \prod _{1\le a<b\le j+1}\prod _{\substack{ 1\le s\le r_a \\ 1\le t\le r_b }}(-1)^{r_ar_b(d_{ba}-1)} \frac{1-\mathcal {L}_{n_{b-1}+t}^\vee \mathcal {L}_{n_{a-1}+s}q^{d_{ba}}}{1-\mathcal {L}_{n_{b-1}+t}^\vee \mathcal {L}_{n_{a-1}+s}} \\&\quad \cdot \frac{\prod _{a=1}^{j+1}\prod _{s=1}^{r_a}\prod _{b=1}^{d_{n_a}}(\mathcal {L}^\vee _{n_{a-1}+s}q^b)^l }{\prod _{a=1}^{j+1}\prod _{s=1}^{r_a}\prod _{b=1}^{d_{n_a}}(1-\mathcal {L}^\vee _{n_{a-1}+s}q^b)^N} \Bigg ). \end{aligned}$$

In the last step, we use the identity

$$\begin{aligned} \frac{1}{1-\mathcal {L}^{\vee }_{i}q^{b}}=\sum _{m=0}^{\dim \mathrm {Fl}}\frac{q^{bm}}{(1-q^{b})^{m+1}} (\mathcal {L}^{\vee }_{i}-1)^{m} \end{aligned}$$

to remove the K-theory classes from the denominators and then apply the pushforward formula (64).

Note that we write the I-function in a way that all the exponents of q are positive. This will be convenient for the proof of Lemma A.1 below. If we allow negative exponents, the I-function can be written in a more symmetric way, see formulas (4.44) and (4.45) in [69].

Let $W_1=\mathbb {S}_{\lambda ^*}(S)$ with $\lambda \in \text {P}_l$. Recall in the proof of Theorem 6.13, we need to study

$$\begin{aligned} \mu ^{W_1}_{d'}(q)=(\text {ev}_\infty )_*\bigg (\frac{\mathcal {D}^l\otimes \widetilde{\text {ev}}_0^*(\mathbb {S}_{\lambda ^*}(S))}{\lambda _{-1}^{\mathbb {C}^*}(N^\vee )}\bigg )\otimes \text {det}^{-l}(S), \end{aligned}$$

where $N=N^{\mathrm {vir},\text {rel}}_{F_{d'}/QG^{0+}_{0,1}}$ and $\widetilde{\text {ev}}_0:F_{d'}\rightarrow [M_{n\times N}/\text {GL}_n(\mathbb {C})]$ is the evaluation morphism at $0\in \mathbb {P}^1$. By the discussion at the beginning of this appendix, we can compute $\mu ^{W_1}_{d'}(q)$ via $QG^{0+}_{0,0}$ and $F_{d'}\cong \mathrm {Fl}$. Recall that the evaluation map $\mathrm {ev}_{\infty }$ is identified with the flag bundle map $\rho : F_{d'}\rightarrow \text {Gr}(n,N)$. Hence, by definition, the difference between $\mu ^{W_1}_{d'}(q)$ and the degree-$d'$ term of the I-function $\mathcal {I}^{l}(q,Q)$ is the insertion $\widetilde{\text {ev}}_0^*(\mathbb {S}_{\lambda ^*}(S))$.

Note that $\widetilde{\text {ev}}_0^*(\mathbb {S}_{\lambda ^*}(S))=\mathbb {S}_{\lambda ^*}(\mathcal {K}_0)$, where $\mathcal {K}_0$ denotes the restriction of $\mathcal {K}$ to $\text {Fl}\times \{0\}$. By the analysis of the I-function, we obtain the explicit formula of $\mu ^{W_1}_{d'}(q)$ as follows:

$$\begin{aligned} \begin{aligned}\mu ^{W_1}_{d'}(q) &=\sum _{\substack{ \{d_i\} \\ \sum d_i=d' }}\sum _{w}\Bigg ( \prod _{1\le a<b\le j+1}\prod _{\substack{ 1\le s\le r_a \\ 1\le t\le r_b}}\prod _{c=1}^{d_{ba}-1}\mathcal {L}^\vee _{n_{b-1}+t}\mathcal {L}_{n_{a-1}+s}q^{c} \\&\quad \cdot \prod _{1\le a<b\le j+1}\prod _{\substack{1\le s\le r_a \\ 1\le t\le r_b }}(-1)^{r_ar_b(d_{ba}-1)} \frac{1-\mathcal {L}_{n_{b-1}+t}^\vee \mathcal {L}_{n_{a-1}+s}q^{d_{ba}}}{1-\mathcal {L}_{n_{b-1}+t}^\vee \mathcal {L}_{n_{a-1}+s}} \\&\quad \cdot \frac{\prod _{a=1}^{j+1}\prod _{s=1}^{r_a}\prod _{b=1}^{d_{n_a}}(\mathcal {L}^\vee _{n_{a-1}+s}q^b)^l }{\prod _{a=1}^{j+1}\prod _{s=1}^{r_a}\prod _{b=1}^{d_{n_a}}(1-\mathcal {L}^\vee _{m_{a-1}+s}q^b)^N} \cdot \mathbb {S}_{\lambda ^*}(\mathcal {K}_0)\Bigg ). \end{aligned} \end{aligned}$$

(65)

Note that the $\mathbb {C}^*$-action (52) on the fiber of $\mathcal {O}_{\mathbb {P}^1}(-d)$ at 0 is given by the dth tensor power of the standard representation. Hence, $\mathcal {K}_0$ can be explicitly written as follows:

$$\begin{aligned} \mathcal {K}_0=\sum _{a=1}^{j+1}\sum _{s=1}^{r_{a}}\mathcal {L}^\vee _{n_{a-1}+s}q^{d_{n_a}}. \end{aligned}$$

(66)

Lemma A.1

If $N-n\ge 2l$, then $\mu ^{W_1}_{d'}(q)$ is regular at $q=0$ and vanishes at $q=\infty$.

Proof

It is clear that $\mu ^{W_1}_{d'}(q)$ has no pole at $q=0$. For any $\lambda \in \text {P}_l$, $\mathbb {S}_{\lambda ^*}(\mathcal {K}_0)$ is a polynomial in q whose degree is bounded above by $ld'$. For a fixed choice of $\{d_i\}$ and w, the degree of the numerator of (65) in q is bounded by

$$\begin{aligned} \begin{aligned}&\sum _{1\le a<b\le j+1}r_ar_bd_{ba} +\sum _{1\le a<b\le j+1}r_ar_b(d_{ba}-1)d_{ba}/2 +\sum _{a=1}^{j+1}r_a(d_{n_a}+1)d_{n_a}l/2+ld' \\&\quad =\sum _{1\le a<b\le j+1}r_ar_b(d_{ba}+1)d_{ba}/2 +\sum _{a=1}^{j+1}r_a(d_{n_a}+1)d_{n_a}l/2+ld' \end{aligned} \end{aligned}$$

(67)

and the degree of the denominator in q is

$$\begin{aligned} \sum _{a=1}^{j+1}r_a(d_{n_a}+1)d_{n_a}N/2. \end{aligned}$$

(68)

Using $\sum _{a<b}r_{a}\le n-1$ and $0<d_{ba}\le d_{n_{b}}$, we obtain the following inequalities for the first term of the RHS of (67):

$$\begin{aligned}\sum _{1\le a<b\le j+1}r_ar_b(d_{ba}+1)d_{ba}/2& \le \sum _{1\le a<b\le j+1}r_{a}r_{b}(d_{n_{b}}+1)d_{n_{b}}/2 \\& \le \frac{n-1}{2}\sum _{1\le b\le j+1}r_{b}(d_{n_{b}}+1)d_{n_{b}}. \end{aligned}$$

It follows that the difference (68)−(67) is greater than or equal to

$$\begin{aligned} \sum _{a=1}^{j+1}r_a(d_{n_a}+1)d_{n_a}(N-l-(n-1))/2-ld'. \end{aligned}$$

(69)

Since

$$\begin{aligned} \sum _{a=1}^{j+1}r_a(d_{n_a}+1)d_{n_a}\ge 2 \sum _{a=1}^{j+1}d_{n_a}=2d', \end{aligned}$$

Formula (69) is bounded below by

$$\begin{aligned}&d'(N-l-(n-1))-ld' \\&\quad =d'(N-2l-(n-1)) \\&\quad >0. \end{aligned}$$

Hence, $\mu ^{W_1}_{d'}(q)$ vanishes at $q=\infty$ under the assumption that $N-n\ge 2l$. $\square$

Remark A.2

Recall there is a distinguished marked point $x_{0}$ on the curve C. As explained in Remark 6.6, the factor $\big (\text {det}\,\mathcal {E}_{x_0}\big )^{e^*}$ in the definition of the $(\epsilon =0+)$-stable GLSM invariant can be viewed as the pullback of the K-theory class $\text {det}(E)^{e^*}$ via the stacky evaluation map $\widetilde{\text {ev}}_0:\overline{\mathcal {M}}^{\epsilon =0+}_{C,k}(\text {Gr}(n,N),d)\rightarrow [M_{n\times N}/\text {GL}_n(\mathbb {C})]$. In principle, one can study the wall-crossing converting the distinguished marked point $x_{0}$ to a heavy point so that we obtain quantum K-invariants in the more classical sense. In the following discussion, we will use the same notation as in Sect. 6.2. If we carry out the computations as in the proof of Theorem 6.13, we will find that the wall-crossing contributions are determined by residues of classes of the form

$$\begin{aligned} \frac{\widetilde{\text {ev}}_0^*(\mu ^{\det }_{d'}(q))}{1-q^{-1}L}\otimes \mathcal {G}, \end{aligned}$$

(70)

where L is a cotangent line bundle, $\mathcal {G}$ is some K-theory class constant in q, and

$$\begin{aligned} \mu ^{\det }_{d'}(q)=(\text {ev}_\infty )_*\bigg (\frac{\mathcal {D}^l\otimes \widetilde{\text {ev}}_0^*( \text {det}(E)^{e^*})}{\lambda _{-1}^{\mathbb {C}^*}(N^\vee )}\bigg ). \end{aligned}$$

Note that $\widetilde{\text {ev}}_0^*( \text {det}(E)^{e^*})= \text {det}(\mathcal {K}_{0})^{-e^*}$, where $\mathcal {K}_{0}$ is given by (66). Hence, we have

$$\begin{aligned} \widetilde{\text {ev}}_0^*( \text {det}(E)^{e^*})=\bigg (\prod _{a=1}^{j+1}\prod _{s=1}^{r_{a}}\mathcal {L}^{e^*}_{n_{a-1}+s}\bigg )q^{-e^*\sum _{a}r_{a}d_{n_a}}. \end{aligned}$$

Unlike $\mu ^{W_1}_{d'}(q)$ in Lemma A.1, the above expression is not regular at $q=0$, and therefore, the residue of (70) is not necessarily equal to zero. It will be interesting to explicitly compute these contributions.

The above analysis also explains why we choose insertions of the form ${\mathbb {S}}_{\lambda ^*}(S)$ instead of ${\mathbb {S}}_{\lambda }(S^\vee )$ at the ordinary markings. The pullback of the latter class along the stacky evaluation map has negative ${\mathbb {C}}^*$-weights and therefore the wall-crossing of weighted stability may change its invariants.