1 Introduction
The quantum K-theory was introduced by Givental [Reference Givental6] and Lee [Reference Lee9] decades ago. Recently, Givental shows that q-hypergeometric solutions represent K-theoretic Gromov-Witten invariants in the toric case [Reference Givental5] and Ruan-Zhang [Reference Ruan and Zhang14] introduce the level structures in quantum K-theory. There is a serendipitous discovery that some special toric spaces with certain level structures result in Mock theta functions. Nevertheless, beyond the toric case, much less is known.
The recent explosion of study of the quantum K-theory was from a fundamental relation between 3d supersymmetric gauge theories and quantum K-theory of the so-called Higgs branch discovered by the works of Nekrasov [Reference Nekrasov12], and Nekrasov and Shatashvili [Reference Nekrasov and Shatashvili10] [Reference Nekrasov and Shatashvili11], amongst many others. For the concrete case of massless theories with a nontrivial UV-IR flow, Jockers and Mayr [Reference Jockers and Mayr7] show a 3d gauge theory/quantum K-theory correspondence, connecting the BPS partition functions of specific $\mathcal {N}=2$ supersymmetric gauge theories to Givental’s permutation equivariant K-theory. In addition, Jockers et al. [Reference Jockers, Mayr, Ninad and Alexander8] and Ueda-Yoshida [Reference Ueda and Yoshida18] establish the correspondence between 3d gauge theory and the quantum K-theory of $Gr(r,n)$ independently. Now it is well-understood that the level structures introduced by Ruan-Zhang [Reference Ruan and Zhang14] are the key new feature for the so-called 3d $\mathcal {N}=2$ theory (Chern-Simons term).
One of the key features of gauge theory is Seiberg-duality, which has been studied in 2d by Bonelli et al. [Reference Bonelli, Sciarappa, Tanzini and Vasko1] and the first author. The 2d Seiberg-duality has a mathematical version known as mutation conjecture [Reference Ruan13]. As far as the authors know very little is known in the 3d $\mathcal {N}=2$ case. The results of this article hopefully will contribute some clarity. The simplest example of the mutation conjecture is the Grassmannian $Gr(r,V)$ versus dual Grassmannian $Gr(n-r,V^*)$ . However, it is unknown how to match the level structure. Without misunderstanding, we will use $Gr(r,n)$ and $Gr(n-r,n)$ to denote the Grassmannian and its dual, respectively. They are geometrically isomorphic. However, they encode very different combinatorial data. A long-standing problem is matching their combinatorial data directly. For example, the presentations of K-theoretic I-functions depend on their gauge theory representation/combinatorial data, and it is tough to see why the I-function of the Grassmannian equals the I-function of the dual Grassmannian. In this paper, we give the explicit formula of K-theoretic I-function of the Grassmannian with level structure by using abelian/nonabelian correspondence [Reference Wen20] as follows:
and
We want to remark here that the isomorphism between the Grassmannian and its dual would imply the equivalence of J-function when level l is $0$ . In fact, the I-function is known to be different from the J-function with negative levels.
In this paper, we use Theorem 1.2 to show the relations of the equivariant I-function between the Grassmannian $Gr(r,n)$ and that of the dual Grassmannian $Gr(n-r,n)$ with level structures; here we find an interval of levels within which two I-functions with levels are the same. On the boundary of that interval, two I-functions with levels are intertwining with each other. We call this phenomenon the level correspondence in Grassmann duality. The existence of a specific interval of level is very mysterious to us. We hope that our result will give some hints on formulating Seiberg-duality for a general target.
Theorem 1.1 Level correspondence
For the Grassmannian $Gr(r,n)$ and its dual Grassmannian $Gr(n-r,n)$ with standard $T=(\mathbb {C}^*)^n$ torus action, let $E_r$ , $E_{n-r}$ be the standard representation of $\operatorname {GL}(r,\mathbb {C})$ and $\operatorname {GL}(n-r,\mathbb {C})$ , respectively. Consider the following equivariant I-function:
Then we have the following relations between $I^{Gr(r,n),E_r,l}_{T,d}$ and $I_T^{Gr(n-r,n),E_{n-r}^\vee ,-l}$ in $K^{loc}_T(Gr(r,n))\otimes \mathbb {C}(q) \cong K^{loc}_T(Gr(n-r,n))\otimes \mathbb {C}(q)$ :
-
○ For $1-r\leq l\leq n-r-1$ , we have
$$ \begin{align*} I_{T,d}^{Gr(r,n),E_{r},l} = I_{T,d}^{Gr(n-r,n),E_{n-r}^\vee,-l}. \nonumber \end{align*} $$ -
○ For $l=n-r$ , we have
$$ \begin{align*} I_{T,d}^{Gr(r,n),E_{r},l}= \sum_{s=0}^d C_{s}( n-r,d) I_{T,d-s}^{Gr(n-r,n),E_{n-r}^\vee,-l}, \nonumber \end{align*} $$where $C_{s}(k,d) $ is defined as$$ \begin{align*} C_{s}(k,d)=\frac{(-1)^{k s}}{(q; q)_{s} q^{s(d-s+k)} \left(\bigwedge^{top}\mathcal{S}_{n-r} \right)^s }, \end{align*} $$and $\mathcal {S}_{n-r}$ is the tautological bundle of $Gr(n-r,n)$ . -
○ For $l=-r$ , we have
$$ \begin{align*} I_{T,d}^{Gr(n-r,n),E_{n-r}^\vee,-l}= \sum_{s=0}^d D_{s}( r, d) I_{T,d-s}^{Gr(r,n),E_{r},l}, \nonumber \end{align*} $$where$$ \begin{align*} D_{s}(r,d)=\frac{(-1)^{rs}}{ (q;q)_sq^{s(d-s)} \left(\bigwedge^{top}\mathcal{S}_{r} \right)^s}, \end{align*} $$and $\mathcal {S}_r$ is the tautological bundle of $Gr(r,n)$ .
Here we use q-Pochhammer symbol notation:
A key step in our proof is the following series of nontrivial q-Pochhammer symbol identities, which are of independent interest.
Theorem 1.2. Denoted by $[n]$ , the set of elements $\{ 1, \ldots ,n \}$ , let $\emptyset \neq I \subsetneq [n] $ be a subset of $[n]$ , $|I|$ be its cardinality and denoted by $I^\complement $ , the complementary set of I in $[n]$ . For constant positive integers d, n and l such that $1-|I|\leq l\leq n-|I|-1$ , let $A_d\left ( \vec {x} , I ,l\right )$ and $B_d\left ( \vec {x}, I,l\right )$ be two rational functions in $\vec {x}$ and q with an extra data l
where $\vec {d}_I$ is $|I|$ -tuple of non negative integers and $|\vec {d}_I|:=\sum _{i \in I} d_i$ . $x_i, i=1,$ … $,n$ are parameters. For convenience, we use the notation $x_{ij}:=x_{i}/x_{j}$ and $d_{ij}:=d_i-d_j$ . Then we have
Plan of the paper
This paper is arranged as follows. In Subsection 2.1, we prove Theorem 1.2 by constructing the rational function in equation (2.3) and then using the iterated residue method, which is useful in Nekrasov partition function [Reference Felder and Müller-Lennert4]. In the following Subsection 2.2, we provide two explicit examples to explain the proof and also provide a nontrivial identity by using Theorem 1.2. In Subsection 2.3, we expand the restriction to the boundary: that is, $l=-|I|$ and $l=n-|I|$ . In Section 3, we first revisit the K-theoretic quasi-map theory in which we review some basic definitions and theorems, especially the formula of equivariant I-function of the Grassmannian $Gr(r,n)$ . Finally, we apply Theorem 1.2 to obtain the level correspondence of the I-function in Grassmann duality.
2 The class of q-Pochhammer symbol identities
2.1 The proof of identities
Now we prove Theorem 1.2 for one case $I=\{1,\cdots ,r\}$ by constructing the following symmetric complex rational function $f(w_{1},\cdots ,w_{d})$ with parameters q and $x_1,\cdots ,x_n$ . We made the following assumptions for parameters:
Furthermore, there exists some $\rho>0$ such that
where $[n]:=\{1, \cdots , n \}$ and general situations follow from analytic continuation. Let $f(w_{1},\cdots ,w_{d})$ be as follows:
where U is a set of complex numbers all contained in open disk $|w|<\rho $ , at the moment $U = \{x_1,\cdots ,x_r\}$ , and g is a symmetric function of the form
From the condition in inequality (2.2) and the restriction of l, we know $g(\vec {w})$ is analytical in the polydiscs $\{(w_1,\cdots ,w_n):|w_i| \le \rho , \forall i \in [n]\}$ and g can only have possible zeros for some $w_j = 0$ .
We consider the following integration
where $(\hat {w}_{1},\cdots ,\hat {w}_{d})$ is any arrangement of $\{w_1,\cdots ,w_d\}$ and the integration contour $C_{\rho }$ for each variable $w_i$ is the circle centred at origin with radius $\rho $ and takes a counterclockwise direction. The condition in inequality (2.2) ensures that there isn’t a pole on the integration contour. By Fubini’s theorem, we could permute the order of integration variables; and since $f(w_{1},\cdots ,w_{d})$ is a symmetric function, we can change $(w_{1},\cdots , w_{d})$ to another order, such as $(\hat {w}_{1},\cdots ,\hat {w}_{d})$ .
Suppose we have the following evaluating sequence for some $S_1 \le d$ by induction,
which are all simple poles inside $|w|<\rho $ . Then we have
Now, integrating variable $\hat {w}_{S_1}$ , we pick up the residue as $\hat {w}_{S_1} = q^{-k_1}u_1$ for some $0\le k_1< S_1$ and $u_1 \in U=\{x_1,\cdots ,x_r\}$ because the condition in equation (2.2), $|\hat {w}_{S_1}|< \rho $ , implies that $k_1=0$ . Evaluating $\hat {w}_{S_1} = u_1$ , we obtain
where
All elements of $\tilde {U}$ are still in the open disk $|w|<\rho $ , and
So we just write $\tilde {f} := \underset {\hat {w}_{S_1} = u}{\operatorname {Res}} \,\underset {\hat {w}_{S_1-1} = q \hat {w}_{S_1}}{\operatorname {Res}} \cdots \underset {\hat {w}_2 = q \hat {w}_3}{\operatorname {Res}} \, \underset {\hat {w}_1 = q \hat {w}_2}{\operatorname {Res}}f$ in the same pattern as in the original form in equation (2.4). One could check that setting $S_1 = 1$ in equation (2.7) is valid.
If one takes the following evaluation sequence of simple poles by induction
we get
which agrees with equation (2.7), since $g(\vec {w})$ is a symmetric function. That is to say, we get the same results from two different evaluation sequences
As with the evaluation process for the sequence in equation (2.6), we now pick up the residues of $\tilde {f}$ in the following sequence:
Suppose $\hat {w}_{S_1+S_2}=u_2$ . We have two cases here: $u_2 \ne q^{S_1} u_1$ or $u_2 = q^{S_1} u_1$ . With a little computation, we obtain the following.
Case 1: $u_2 \ne q^{S_1} u_1$ ,
Case 2: $u_2 = q^{S_1} u_1$ ,
To summarise all of the above, together with equations (2.13), (2.15) and (2.16), we know that the iterated residue does not depend on the order of the poles we pick but depends on the final set of poles we choose. So we can integrate all variables for the integrand of the form as in equation (2.4) with one less variable each time.
When there is only one variable left
we still update the set U to $U\backslash \{u\} \cup \{qu\}$ after choosing a pole at $\hat {w} = u \in U$ . Using the same argument to get equations (2.15) and (2.16) after picking up poles for all $w_i, i \in [d]$ , the result only depends on the final set U, which is of the form
where $d_1 + \cdots + d_r = d$ , which means for each sequence, the final result can be indexed by a r-tuple partition of d.
Suppose there is a sequence with final set $\{q^{d_1} x_1, \cdots , q^{d_r}x_r\}$ . Then we can compute the result with the following sequence:
Note that we can do permutations on the left side, so for each partition $|\vec {d}| = d$ , we have $d!$ possible evaluation sequences.
In all, we obtain the following lemma to compute $E_d$ .
Lemma 2.1. We can write E as
where
here
and the order in which to take the limit is from $w_1$ to $w_d$ .
We now evaluate one specific configuration of these simple pole residues for given $\vec d$ by changing variables:
Notations: From now on, we frequently use the following notations:
Then
Now we pick up the simple pole terms and evaluate the function with $z_{i,j}=1$ . Note that
where the order in which to take the limit is from $z_{i,1}$ to $z_{i,d_i}$ . So this specific configuration of residues is
and the factor with only $x_{ij}$ for $i=1,\ldots ,r$ and $j=r+1,\ldots ,n$ is
The factor that does not involve any $x_{ij}$ is
And define $P_d$ as
By simple induction, it is easy to show that
and
The factor left is
The above equations prove that the summand in equation (2.20) corresponding to a given $\vec d$ equals to one summand in $A_{d}\left (\vec {x},I,l\right )$ . Thus we have
Now calculate the integration in a clockwise direction:
The assumption with l ensures that when integrating in any order, for each variable w, the residue at infinity is 0. By definition, we can calculate this integration by taking the sum of residues outside the circle $|w_i|=\rho $ .
The iterated residues, in this case, are similar to the previous counterclockwise direction. Arguments similar to those in equation (2.1) show
where
here
and the order in which to take the limit is from $w_1$ to $w_d$ .
We now do the following, changing variables and calculating the residues:
Similarly,
Note that
where the order in which to take the limits is from $z_{i,1}$ to $z_{i,d_i}$ . So the residues for one specific configuration of residues of type $\vec {d'}$ are
After almost the same computation as for $E_{\vec {d}}$ , we can simplify the above equation to
which proves
Since the residue at infinity is zero, using the Cauchy Residue Theorem d times,
we arrive at equations (2.24), (2.25) and (2.26) of the following theorem stated in the introduction.
Theorem 2.2. Denoted by $[n]$ , the set of elements $\{ 1, \ldots ,n \}$ , let $ \emptyset \neq I \subsetneq [n] $ be a subset of $[n]$ , $|I|$ be its cardinality, and denoted by $I^\complement $ , the complementary set of I in $[n]$ . Then for constant positive integers d, n and integer l with restriction $1-|I|\leq l\leq n-|I|-1$ , let $A_d\left ( \vec {x} , I ,l\right )$ and $B_d\left ( \vec {x}, I,l\right )$ be two rational functions in $\vec {x}$ and q with an extra data l:
where $\vec {d}_I$ is an $|I|$ -tuple of nonnegative integers and $|\vec {d}_I|:=\sum _{i \in I} d_i$ . $x_i, i=1,$ … $,n$ are parameters. For convenience, we use the notation $x_{ij}:=x_{i}/x_{j}$ and $d_{ij}:=d_i-d_j$ . Then we have
2.2 Examples
In the following two examples, we show how the proof of Theorem 2.2 works.
Example 2.3 d=1
For the case l=0, d=1, r=2, n=3, equation (2.3) becomes the following simple form:
Consider the integration in equation (2.5). Then there are simple poles of type $(1,0)$ and $(0,1)$ in the counter $C_{\rho }$ :
-
○ type (1,0): $ w = x_1 $ ,
-
○ type (0,1): $ w = x_2 $ .
Then the residue for each type is as follows:
-
○ type ${(1,0)}$ :
$$ \begin{align*} E_{(1,0)}=\underset{\hat{w} = x_1}{\operatorname{Res}}f = \frac{1}{(1-q)(1-x_{21})(1-qx_{13})},\\[-15pt] \end{align*} $$ -
○ type ${(0,1)}$ :
$$ \begin{align*} E_{(0,1)}=\underset{\hat{w} = x_2}{\operatorname{Res}}f= \frac{1}{(1-q)(1-x_{12})(1-qx_{23})}.\\[-15pt] \end{align*} $$
And there is only one simple pole $w=q^{-1}x_3 $ in the counter $C^{\prime }_{\rho }$ , so
-
○ type $1$ :
$$ \begin{align*} E^{\prime}_{1}= \underset{\hat{w} = q^{-1}x_3}{\operatorname{Res}}f = \frac{-1}{(1-q)(1-qx_{13})(1-qx_{23})}.\\[-15pt] \end{align*} $$
It is easy to obtain
which agrees with equation (2.26).
Example 2.4 d=2
For the case l=0, d=2, r=2, n=3, equation (2.3) becomes the following simple form:
Consider the integration in equation (2.5). Then there are simple poles of type $(2,0)$ , $(1,1)$ and $(0,2)$ in the counter $C_{\rho _i}$ :
-
○ type (2,0): $\{ w_1, w_2 \} = \{ x_1, x_1 q \} $ ,
-
○ type (1,1): $\{ w_1, w_2 \} = \{ x_1, x_2 \} $ ,
-
○ type (0,2): $\{ w_1, w_2 \} = \{ x_2, x_2 q \} $ .
Then the residue for each type is as follows:
-
○ type ${(2,0)}$ :
$$ \begin{align*} 2!E_{(2,0)} &= \underset{\hat{w}_{2}= qx_1}{\operatorname{Res}}\underset{\hat{w}_{1} = x_1}{\operatorname{Res}}f+\underset{\hat{w}_{2}= x_1}{\operatorname{Res}}\underset{\hat{w}_{1} = qw_2}{\operatorname{Res}}f \\ &=\frac{1}{2(1-q)^2} \frac{1}{(1+q)(1-q x_{13})(1-q^2x_{13})(1-x_{21})(1-q^{-1}x_{21})} \\ &\quad + \frac{1}{2(1-q)^2} \frac{1}{(1+q)(1-q^2x_{13})(1-qx_{13})(1-q^{-1}x_{21})(1-x_{21})} \\ &= \frac{1}{(1-q)^2} \frac{1}{(1+q)(1-q^2 x_{13})(1-q x_{13})(1-q^{-1} x_{21})(1- x_{21})}, \end{align*} $$ -
○ type ${(1,1)}$ :
$$ \begin{align*} 2!E_{(1,1)} &= \underset{\hat{w}_{2} = x_2}{\operatorname{Res}}\underset{\hat{w}_{1} = x_1}{\operatorname{Res}}f +\underset{\hat{w}_{2} = x_1}{\operatorname{Res}}\underset{\hat{w}_{1} = x_2}{\operatorname{Res}}f \\ &= \frac{1}{2(1-q)^2} \frac{1}{(1-qx_{12})(1-qx_{21})(1-qx_{13})(1-qx_{23})} \\ &\quad + \frac{1}{2(1-q)^2} \frac{1}{(1-qx_{21})(1-qx_{12})(1-qx_{23})(1-qx_{13})} \\ &= \frac{1}{(1-q)^2} \frac{1}{(1-q x_{21})(1-q x_{12})(1-q x_{23})(1-q x_{13})}, \end{align*} $$ -
○ type ${(0,2)}$ :
$$ \begin{align*} 2!E_{(0,2)} &= \underset{\hat{w}_{2} = qx_2}{\operatorname{Res}}\underset{\hat{w}_{1} = x_2}{\operatorname{Res}}f+\underset{\hat{w}_{2} = x_2}{\operatorname{Res}}\underset{\hat{w}_{1} = qw_2}{\operatorname{Res}}f \\ &= \frac{1}{2(1-q)^2} \frac{1}{(1+q)(1-x_{12})(1-qx_{23})(1-q^{-1}x_{12})(1-q^2x_{23})} \\ &\quad + \frac{1}{2(1-q)^2}\frac{1}{(1+q)(1-q^{-1}x_{12})(1-q^2x_{23})(1-x_{12})(1-qx_{23})} \\ &= \frac{1}{(1-q)^2}\frac{1}{(1+q)(1-q^{-1} x_{12})(1-q^2 x_{23})(1- x_{12})(1-q x_{23})}. \end{align*} $$
Consider the integration in equation (2.23). Then there are simple poles of type 2 in the counter $C^{\prime }_{\rho _i}$ :
-
○ type 2: $\{ w_1, w_2 \} = \{ q^{-1}x_3, q^{-2}x_3 \} $ .
Then the residue for each type 2 is as follows:
-
○ type $ {2}$ :
$$ \begin{align*} (-1)^22!E^{\prime}_{2} &= \underset{\hat{w}_{2} = q^{-2}x_3}{\operatorname{Res}}\underset{\hat{w}_{1} = q^{-1}x_3}{\operatorname{Res}}f+\underset{\hat{w}_{2} = q^{-1}x_3}{\operatorname{Res}}\underset{\hat{w}_{1} = q^{-1}w_2}{\operatorname{Res}}f \\ &=\frac{1}{(1+q)(1-q)^2(1-q^2 x_{13})(1-q x_{13})(1-q^2 x_{23})(1-q x_{23})}. \end{align*} $$
By a little computation, we have
Example 2.5. From Proposition 1.2, if we take $n=3$ , $l=0$ and $I=[2]$ , we know that $A_d(\vec {x},[2],0)=B_d(\vec {x},[3]\backslash [2],0 )$ . By the following computation, there is a phenomenon that we can extract from $A_d(\vec {x},[2],0)$ to get $B_d(\vec {x},[3]\backslash [2],0 )$ times another factor when $d=1,2$ : that is, $A_d(\vec {x},[2],0)=B_d(\vec {x},[3]\backslash [2],0 ) \times G(\vec {x}), \ d=1,2 $ . Thus we can conclude that $G(\vec {x})=1$ . Furthermore, this is a general phenomenon for all d; see the following Corollary 2.1.
By definition, $\vec {x}=\{x_1,x_2,x_3 \}$ , so
For $d=1$ – that is, $(d_1,d_2)=(1,0)$ or $(0,1)$ – we have
For $d=2$ – that is, $(d_1,d_2)=(2,0),(1,1)$ or $(0,2)$ – similarly we have
More generally, we have the following corollary.
Corollary 2.1.
Proof. Set $l=0$ , $r=2,n=3$ in equation (2.26). We have
Since we know $A_d(\vec {x},I ,0)$ equals $B_d(\vec {x},I^{\complement } ,0)$ , we get the conclusion.
2.3 Boundary cases
For $l=-|I|$ , $l=n-|I|$ , equation (2.26) no longer holds, since the residue at infinity is nonzero, but we can compare the behaviour of some special limit in equation (2.26) to obtain following results.
Corollary 2.2.
-
○ For $l=n-|I|$ , we have
(2.29) $$ \begin{align} A_d\left( \vec{x},I,l\right)= \sum_{s=0}^d C_{s}( \vec{x},I^\complement,d) B_{d-s}\left( \vec{x},q,I^\complement,-l\right), {} \end{align} $$where $C_{s}( \vec {x},I,d) $ is defined as$$ \begin{align*} C_{s}( \vec{x},I,d)=\frac{(-1)^{|I|\cdot s}\prod_{i\in I^\complement}x_i^{s}}{(q; q)_{s} q^{s(d-s+|I|)}}. \end{align*} $$ -
○ For $l=-|I|$ , we have
(2.30) $$ \begin{align} B_d\left( \vec{x},I^\complement,-l\right)= \sum_{s=0}^d D_{s}( \vec{x},I,d) A_{d-s}\left( \vec{x},q,I,l\right), {} \end{align} $$where$$ \begin{align*} D_{s}( \vec{x},I,d)=\frac{(-1)^{|I|\cdot s}\prod_{i \in I} x^{-s}_{i}}{ (q;q)_sq^{s(d-s)} }. \end{align*} $$
Proof. Consider $[n+1]$ , $I \subsetneq [n+1] $ , $\{n+1\} \notin I $ , $l=n-|I|$ in equation (2.26). Then we have
It is easy to see that taking $\lim _{ x_{n+1} \rightarrow \infty }$ in equation (2.31), we obtain
Now let’s take limit $ \lim _{x_{n+1} \rightarrow \infty } $ in equation (2.32):
The limits of the last two terms in equation (2.33) equal 1, and by a little computation, we obtain that equation (2.34) equals
Then we obtain
We obtain the conclusion.
Similarly, consider $A_d\left ( \vec {x}\cup x_{n+1} ,\tilde {I},l\right )$ and $B_d\left ( \vec {x} \cup x_{n+1},\tilde {I} ^\complement ,-l\right )$ . For $\tilde {I}= I \cup \{n+1\}$ and $l=-|I|$ , from equation (2.26), we have
It is easy to see that after taking $\operatorname{lim}_{x_{n+1} \rightarrow 0} $ in equation (2.36), we obtain
First, rewrite equation (2.35) as follows:
Now let’s take limit $ \lim _{x_{n+1} \rightarrow 0 } $ in the above formula. We obtain
3 K-theoretic I-function with level structure
3.1 Definitions
Let X be a GIT quotient $V/\!/_\theta G$ , where V is a vector space and ${G}$ is a connected reductive complex Lie group. Let ${\mathcal {Q}}^{\epsilon }_{g,n}(X,\beta )$ be the moduli stack of $\epsilon $ -stable quasimaps [Reference Ciocan-Fontanine and Kim3] parametrising data $(C,p_1,$ … $,p_n,\mathcal {P},s)$ , where C is an n-pointed genus g Riemann surface, $\mathcal {P}$ is a principal G-bundle over C, s is a section and $\beta \in \mathrm {Hom}(\mathrm {Pic}^G(V))$ . There are natural maps
given by evaluation at the ith marked point: that is,
There are line bundles
which are called universal cotangent line bundles. The fibre of $L_i$ over the point $(C^{\epsilon },p_1,$ … $,p_n,\mathcal {P},s)$ is the cotangent line to C at the point $p_i$ .
The permutation-equivariant K-theoretic quasimap invariants with level structures [Reference Ruan and Zhang14] are holomorphic Euler characteristics over ${\mathcal {Q}}^{\epsilon }_{g,n}(X,d)$ of the sheaves
where $\mathcal {O}^{v i r}_{g,n,d}$ is called the virtual structure sheaf [Reference Lee9]. $\mathbf {t}(q)$ is defined as follows:
$\{\phi _\alpha \}$ is a basis in $K^0(X)\otimes Q$ and $t_{k,\alpha }$ are formal variables. The last term in equation (3.1) is the level l determinant line bundle over $\mathcal {Q}^{\epsilon }_{g,n}(X,d)$ defined as
where $\pi $ is the forgetful map from the universal curve: that is,
The bundle $\mathcal {P}$ is the universal principal bundle over the universal curve, and R is a G-representation.
Similarly, we can define a quasimap graph space $ \mathcal { QG }^{\epsilon } _ { 0 , n } ( X , \beta ) $ , which parametrises quasimaps with parametrised component $\mathbb {P}^1$ , so there is a natural $\mathbb {C}^*$ -action on the quasimap graph space. It is denoted by $\mathrm {F}_{0,\beta }$ , the special fixed loci in $ ( \mathcal { QG }^{\epsilon } _ { 0 , n } ( X , \beta ) )^{\mathbb {C}^*}$ , and denoted by q, the weight of the cotangent bundle at $0 :=[1,0] $ of $\mathbb {P}^1$ ; for details, see [Reference Ciocan-Fontanine and Kim3].
Definition 3.1 [Reference Ruan and Zhang14]
The permutation-equivariant K-theoretic $\mathcal {J}^{\hspace{1.5pt}R,l,\epsilon }$ -function of $V // {G}$ of level l is defined as
where $\{ \phi _\alpha \}$ is a basis of $K^0(V/\!/{G})$ and $\{ \phi ^\alpha \}$ is the dual basis with respect to twisted pairing $( \ \ , \ \ )^{R,l}$ : that is,
Definition 3.2 [Reference Ruan and Zhang14]
When taking $\epsilon $ small enough, denoted by $\epsilon =0^{+}$ , we call $\mathcal {J}^{R,l,0^{+}}(0)$ the small I-function of level l: that is,
3.2 Level correspondence in Grassmann duality
Let V be $r \times n$ matrixes $M_{r \times n}$ , G be the general linear group $GL_r$ and $ \theta $ be the $\mathbf {det}: GL_r \rightarrow \mathbb {C}^*$ . Then we have
There is a natural $T=(\mathbb {C}^*)^n$ -action on $\mathbb {C}^n$ with weights $\mathbb {C}^n=\Lambda _1 + \cdots + \Lambda _n $ . Then deducing an action on $Gr(r,n)$ by $T \cdot A = AT$ , $A \in M_{r\times n}$ . Using general abelian/nonabelian correspondence in [Reference Wen20] for $Gr(r,n)$ , we have
where $\vec {d}=\{d_1 \leq d_2 \leq \cdots \leq d_r \}$ such that $d_1 = d_2 = \cdots = d_{r_1} < d_{r_1+1}= \cdots = d_{r_1+r_2} < d_{r_1+\cdots +r_h}\cdots = d_{r_1+\cdots +r_h+r_{h+1}}$ : that is, $r_1+\cdots +r_{h+1}=r$ . $\omega $ is the Weyl group acting on $L_i$ to change the index, $\{ L_i \}^r_{i=1}$ come from the filtration of tautological bundle $\mathcal {S}_r$ of $Gr(r,n)$ . We could rewrite the equivariant I-function in the following way
Suppose $\omega $ changes $i_1$ to $i_2$ and $j_1$ to $j_2$ . Then one of the factors changes from
to
Since $\omega \in S_{r} / S_{r_{1}} \times \cdots \times S_{r_{h+1}}$ , we have $d_{i_1} \neq d_{i_2}, d_{j_1} \neq d_{j_2}$ . In equation (3.2), we have an order of partition $\vec {d}$ ; one can see from equation (3.3) to equation (3.4) that $\omega $ -action is just $\{ d_i \}$ rearranged without changing the form. There is a unique $\omega \in S_{r} /\left (S_{r_{1}} \times \ldots \times S_{r_{h+1}}\right )$ whose inverse $\omega ^{-1}$ arranges $\left (d_{1}, \ldots , d_{r}\right )$ in nondecreasing order $d_{1} \leq d_{2} \leq \ldots \leq d_{r}$ . Then we have
Note that in [Reference Taipale15], the author claimed a version of the mirror theorem with a different I-function.
If we consider the standard representation of $GL_r$ , denoted by $E_r$ , then the associated bundle $\mathcal {P} \times _G R|_{F_{0,\beta }}$ can be identified with $\oplus _{i=1}^{r}L_i \otimes \mathcal {O}_{\mathbb {P}^1}(-d_i) $
Similarly, if we take a dual standard representation, denoted by $E_r^{\vee }$ , then
So the equivariant I-function of $Gr(r,n)$ with a level structure is as follows:
and
Remark. For the dual Grassmannian $Gr(n-r,n)$ , the $(\mathbb {C}^*)^n$ -action on $\mathbb {C}^n$ is the dual action, so the weights are $\mathbb {C}^n=\Lambda _1^{-1}+ \cdots + \Lambda _n^{-1}$ . The deduced action on $Gr(n-r,n)$ is as follows: $T \cdot B = BT$ , $B \in M_{n-r \times n}$ . So the corresponding equivariant I-function is as follows
and
where $\tilde {L}_i$ for $i=1,\ldots ,n-r$ come from the filtration of tautological bundle $\mathcal {S}_{n-r}$ over $Gr(n-r,n)$ .
Let T act on the Grassmannian $Gr(r,n)$ as before. Then there are ${n \choose r} $ fixed points: that is, denoted by $\{e_1, \ldots , e_n \}$ , the basis of $\mathbb {C}^n$ . Then the subspace V spanned by $\{e_{i_1}, \ldots , e_{i_r} \}$ is a T-fixed point for $\{i_1, \cdots , i_r \} \subset [n] $ . Let
be the map induced from the close embedding $\mathfrak {l}: Gr(r,n)^T \hookrightarrow Gr(r,n)$ . The kernel and cokernel are $K_T(pt)$ -modules and have some support in the torus T. From a very general localisation theorem of Thomason [Reference Thomason16], we know
where the union is over finitely many nontrivial characters $\mu $ . The same is true of $\text { ker } \mathfrak {l}_{*}$ , but since
has no such torsion, this forces $\text { ker } \mathfrak {l}_{*}=0$ , so after inverting finitely many coefficients of the form $t^{\mu }-1$ , we obtain an isomorphism: that is,
We denote $K^{loc}_{T}(-)$ by
where $\mathcal {R} \cong \mathbb {Q}(t_1,\ldots ,t_n)$ and $\{t_i\}$ are the charaters of torus T.
Similarly, $T=\left (\mathbb {C}^{*}\right )^{n}$ -action on $Gr(n-r,n)$ also has $\binom {n}{n-r} = \binom {n}{r}$ isolated fixed points, which are indexed by $(n-r)$ -element subsets of $[n]$ , so identification of $Gr(r,n)^{T}$ with $Gr(n-r,n)^{T}$ gives an $\mathcal { R }$ -module isomorphism of $K^{loc}_{T} ( Gr(r,n) )$ with $K^{loc}_{T} ( Gr(n-r,n) )$ . Indeed, suppose W is a subspace of dimension r in a vector space V of dimension n. Then we have a natural short exact sequence
Taking the dual of this short exact sequence yields an inclusion of $(V / W)^{*}$ in $V^{*}$ with quotient $W^{*}$
so $ \psi : W \mapsto (V/W)^{*}$ gives a cannocial equivariant isomorphism $Gr(r, V) \cong Gr(n-r, V^{*})$ , where the action of $T=\left (\mathbb {C}^{*}\right )^{n}$ on $V^{*}$ is induced from the action of T on V. Thus, $\psi $ gives the canonical identification of fixed points
where I is a set of $[n]$ with $|I|=r$ and $\{e^i\}_{i = 1} ^{n}$ is the dual basis of $\{e_i\}_{i = 1} ^{n}$ . Now we can state the following Level correspondence in Grassmann duality.
Theorem 3.1 Level correspondence
For the Grassmannian $Gr(r,n)$ and its dual Grassmannian $Gr(n-r,n)$ with standard $T=(\mathbb {C}^*)^n$ torus action, let $E_r$ , $E_{n-r}$ be the standard representation of $\operatorname {GL}(r,\mathbb {C})$ and $\operatorname {GL}(n-r,\mathbb {C})$ , respectively. Consider the following equivariant I-function:
Then we have the following relations between $ I^{Gr(r,n),E_r,l}_{T,d} $ and $ I^{Gr(n-r,n),E^\vee _{n-r},-l}_{T,d} $ in $ K^{loc}_T(Gr(r,n)) \otimes \mathbb {C}(q) $ (which equals $ K^{loc}_T(Gr(n-r,n)) \otimes \mathbb {C}(q) $ ):
-
○ For $1-r\leq l\leq n-r-1$ , we have
$$ \begin{align*} I_{T,d}^{Gr(r,n),E_{r},l} = I_{T,d}^{Gr(n-r,n),E_{n-r}^\vee,-l}. \nonumber \end{align*} $$ -
○ For $l=n-r$ , we have
$$ \begin{align*} I_{T,d}^{Gr(r,n),E_{r},l}= \sum_{s=0}^d C_{s}( n-r,d) I_{T,d-s}^{Gr(n-r,n),E_{n-r}^\vee,-l}, \nonumber \end{align*} $$where $C_{s}(k,d) $ is defined as$$ \begin{align*} C_{s}(k,d)=\frac{(-1)^{k s}}{(q; q)_{s} q^{s(d-s+k)} \left(\bigwedge^{top}\mathcal{S}_{n-r} \right)^s}, \end{align*} $$and $\mathcal {S}_{n-r}$ is the tautological bundle of $Gr(n-r,n)$ . -
○ For $l=-r$ , we have
$$ \begin{align*} I_{T,d}^{Gr(n-r,n),E_{n-r}^\vee,-l}= \sum_{s=0}^d D_{s}( r, d) I_{T,d-s}^{Gr(r,n),E_{r},l}, \nonumber \end{align*} $$where$$ \begin{align*} D_{s}(r,d)=\frac{(-1)^{rs}}{ (q;q)_sq^{s(d-s)}\left( \bigwedge^{top}\mathcal{S}_{r} \right)^s}, \end{align*} $$and $\mathcal {S}_r$ is the tautological bundle of $Gr(r,n)$ .
Proof. From the discussion above, we prove the above identity by comparing $i^*_I I^{E_r, l}_{T}$ and $i^*_{I^\complement } I^{E_{n-r}^\vee , -l}_{T}$ ; here $i_{I}$ and $i_{I^\complement }$ are inclusion maps from the corresponding fixed points: that is, we compare two I-functions by restricting them to corresponding fixed points. Let $I= (j_1, \cdots , j _r)$ be the subset of $[n]=\{1, \ldots , n \}$ , with $|I|=r$ . Denote $v_1, v_2,\cdots , v_r$ , the fibre coordinates in the fibre of $\mathcal {S}$ at fixed point $<e_j>_{j \in I}$ , $\forall ( t_1, \cdots , t_n ) \in \left (\mathbb {C}^{*}\right )^{n}$ , with weights $\mathbb {C}^n= \Lambda _1 + \cdots + \Lambda _n $ and
So the weights of $i^*_{I}\mathcal {S}_r$ are $\{\Lambda _i \}_{i \in I}$ and the weights of $i^*_{I^\complement }\mathcal {S}_{n-r}$ are $\{\Lambda ^{-1}_i \}_{i \in I^\complement }$ . Since the I-function is symmetric with respect to $\{L_i\}$ , we can take any choice of weights
and
Using notation $\Lambda _{ij}=\Lambda _i \Lambda _j^{-1}$ and the following Lemma 3.2, we obtain
and
Comparing equations (3.8) and (3.9) with equations (2.24) and (2.25), we obtain the conclusion.
Lemma 3.2. Let I be the subset of $[n]=\{1, \ldots , n \}$ . We have
Proof. It is sufficient to consider one term. If $d_i \geq d_j $ , then
If $d_i \leq d_j$ , then
Acknowledegments
We would like to thank Prof. Yongbin Ruan for suggesting this problem and for valuable discussions. Thanks are also due to Prof. Yutaka Yoshida for showing us his formula of the I-function of the Grassmannian, Ming Zhang for discussion about level structures and Zijun Zhou for helpful discussions. We also appreciate the hospitality and support of IASM (Institute For Advanced Study In Mathematics) of Zhejiang University. The second author is supported by a KIAS Individual Grant (MG083901) at Korea Institute for Advanced Study.
Conflicts of interest
None.