2 Preliminaries on equivariant K theory

In this section, we recall some basic facts about the equivariant K-theory of a variety with a group action. For an introduction to equivariant K theory, and more details, see [Reference Chriss and GinzburgCG09].

Let X be a smooth projective variety with an action of a linear algebraic group G. The equivariant K theory ring $\operatorname {\mathrm {K}}_G(X)$ is the Grothendieck ring generated by symbols $[E]$ , where $E \to X$ is an G-equivariant vector bundle, modulo the relations $[E]=[E_1]+[E_2]$ for any short exact sequence $0 \to E_1 \to E \to E_2 \to 0$ of equivariant vector bundles. The additive ring structure is given by direct sum, and the multiplication is given by tensor products of vector bundles. Since X is smooth, any G-linearized coherent sheaf has a finite resolution by (equivariant) vector bundles, and the ring $\operatorname {\mathrm {K}}_G(X)$ coincides with the Grothendieck group of G-linearized coherent sheaves on X. In particular, any G-linearized coherent sheaf $\mathcal {F}$ on X determines a class $[\mathcal {F}] \in \operatorname {\mathrm {K}}_G(X)$ . An important special case is if $\Omega \subset X$ is a G-stable subscheme; then its structure sheaf determines a class $[{\mathcal {O}}_\Omega ] \in \operatorname {\mathrm {K}}_G(X)$ .

The ring $\operatorname {\mathrm {K}}_G(X)$ is an algebra over $\operatorname {\mathrm {K}}_G(pt) = \operatorname {\mathrm {Rep}}(G)$ , the representation ring of G. If $G=T$ is a complex torus, then this is the Laurent polynomial ring $\operatorname {\mathrm {K}}_T(pt) = {\mathbb {Z}}[T_1^{\pm 1}, \ldots , T_n^{\pm 1}]$ , where $T_i:=e^{t_i}$ are characters corresponding to a basis of the Lie algebra of T.

The (Hirzebruch) $\lambda _y$ class is defined by

$$\begin{align*}\lambda_y(E) := 1 + y [E] + y^2 [\wedge^2 E] + \ldots + y^e [\wedge^e E] \in \operatorname{\mathrm{K}}_G(X)[y] . \end{align*}$$

This class was introduced by Hirzebruch [Reference HirzebruchHir95] in order to help with the formalism of the Grothendieck-Riemann-Roch theorem. It may be thought as the K theoretic analogue of the (cohomological) Chern polynomial

$$\begin{align*}c_y(E)= 1+ c_1(E) y + \ldots + c_e(E) y^e\end{align*}$$

of the bundle E. The $\lambda _y$ class is multiplicative with respect to short exact sequences; that is. if

is such a sequence of vector bundle,s then

$$\begin{align*}\lambda_y(E_2) = \lambda_y(E_1) \cdot \lambda_y(E_3) ; \end{align*}$$

cf. [Reference HirzebruchHir95]. A particular case of this construction is when V is a (complex) vector space with an action of a complex torus T, and with weight decomposition $V = \oplus _i V_{\mu _i}$ , where each $\mu _i$ is a weight in the dual of the Lie algebra of T. The character of V is the element $\operatorname {\mathrm {ch}}_T(V):=\sum _i \dim V_{\mu _i} e^{\mu _i}$ , regarded in $\operatorname {\mathrm {K}}_T(pt)$ . The $\lambda _y$ class of V is the element $\lambda _y(V) =\sum _{i \ge 0} y^i ch(\wedge ^i V) \in \operatorname {\mathrm {K}}_T(pt)[y]$ . From the multiplicativity property of the $\lambda _y$ class, it follows that

$$\begin{align*}\lambda_y(V) = \prod_i (1+y e^{\mu_i})^{\dim V_{\mu_i}} ;\end{align*}$$

see [Reference HirzebruchHir95].

Since X is proper, the push-forward to a point equals the Euler characteristic, or, equivalently, the virtual representation,

$$\begin{align*}\chi(X, \mathcal{F})= \int_X [\mathcal{F}] := \sum_i (-1)^i \operatorname{\mathrm{ch}}_T H^i(X, \mathcal{F}) . \end{align*}$$

For $E,F$ equivariant vector bundles, this gives a pairing

$$\begin{align*}\langle - , - \rangle :\operatorname{\mathrm{K}}_G(X) \otimes \operatorname{\mathrm{K}}_G(X) \to \operatorname{\mathrm{K}}_G(pt); \quad \langle [E], [F] \rangle := \int_X E \otimes F = \chi(X, E \otimes F) . \end{align*}$$

3 Equivariant K theory of flag manifolds

In this paper, we will utilize the torus equivariant K-theory of the partial flag manifolds $\mathrm {Fl}(i_1, i_2, \ldots , i_k;n)$ , which parametrize partial flags $F_{i_1} \subset \ldots \subset F_{i_p} \subset {\mathbb {C}}^n$ , where $\dim F_i = i$ , to study the equivariant quantum K theory ring of Grassmann manifolds $\mathrm {Gr}(k;n)$ . The goal of this section is to review some of the basic features of the equivariant K rings of the partial flag manifolds. Of special importance is the calculations of K theoretic push forwards of Schur bundles, which will play a key role later in the paper.

3.1 Basic definitions

The flag manifold $\mathrm {Fl}(i_1, i_2, \ldots , i_k;n)$ is an algebraic variety homogeneous under the action of ${\mathrm {GL}_n:=\mathrm {GL}_n({\mathbb {C}})}$ . Let $T \subset \mathrm {GL}_n$ be the subgroup of diagonal matrices acting coordinate-wise on ${\mathbb {C}}^n$ . Denote by $T_i \in \operatorname {\mathrm {K}}_T(pt)$ the weights of this action.

Set W to be the symmetric group in n letters, and let $W_{i_1, \ldots , i_p} \le W$ be the subgroup generated by simple reflections $s_i = (i,i+1)$ where $i \notin \{ i_1, \ldots , i_p\}$ . Denote by $\ell :W \to \mathbb {N}$ the length function, and by $W^{i_1, \ldots , i_k}$ the set of minimal length representatives of $W/W_{i_1, \ldots , i_p}$ . This consists of permutations $w \in W$ which have descents at positions $i_1, \ldots , i_p$ , that is, $w(i_k+1)< \ldots <w(i_{k+1})$ , for $k=0, \ldots , p$ , with the convention that $i_0=1, i_{p+1} =n$ .

The torus fixed points $e_w \in \mathrm {Fl}(i_1, i_2, \ldots , i_k;n)$ are indexed by the permutations $w \in W^{i_1, \ldots , i_p}$ . For each such point, the Schubert cell $X_{w}^\circ \subset \mathrm {Fl}(i_1, i_2, \ldots , i_k;n)$ is the orbit $B^-. e_w$ of the Borel subgroup of lower triangular matrices, and the Schubert variety is the closure $X_w = \overline {X^{w,\circ }}$ . Our convention ensures that $X_w$ has codimension $\ell (w)$ . The Schubert varieties $X^w$ determine classes $\mathcal {O}_w:= [\mathcal {O}_{X_w}]$ in $\operatorname {\mathrm {K}}_T(X)$ . Similarly, the S-fixed points give classes denoted by $\iota _w:= [\mathcal {O}_{e_w}]$ . The equivariant K-theory $\operatorname {\mathrm {K}}_T(\mathrm {Fl}(i_1, \ldots , i_p;n))$ is a free module over $\operatorname {\mathrm {K}}_T(pt)$ with a basis given by Schubert classes $\{\mathcal {O}_w \}_{w \in W^{i_1, \ldots , i_p}}$ . For the Grassmannian $\mathrm {Gr}(k;n)$ , the Schubert varieties are indexed by partitions $\lambda = (\lambda _1, \ldots , \lambda _k)$ included in the $k \times (n-k) $ rectangle (i.e., $n-k \ge \lambda _1 \ge \ldots \ge \lambda _k \ge 0$ ). For Grassmanians, we denote the Schubert variety by $X_\lambda $ ; this has codimension $|\lambda | = \lambda _1 + \ldots + \lambda _k$ . The corresponding class in $\operatorname {\mathrm {K}}_T(\mathrm {Gr}(k;n))$ is denoted ${\mathcal {O}}_\lambda := [{\mathcal {O}}_{X_\lambda }]$ .

Let ${\mathbf {i}} = (1 < i_1< \ldots <i_p < n)$ and $\mathbf {j}$ obtained from $\mathbf {{i}}$ by removing some of the indices $i_s$ . Denote by $\mathrm {Fl}(\mathbf {i})$ and $\mathrm {Fl}(\mathbf {j})$ the respective flag manifolds and by $\pi _{\mathbf {i},\mathbf {j}}: \mathrm {Fl}(\mathbf {i}) \to \mathrm {Fl}(\mathbf { j})$ the natural projection. We will need the following well-known fact:

Lemma 3.1. Let $w \in W^{\mathbf {i}}$ . Then for any Schubert varieties $X_w \in \mathrm {Fl}(\mathbf {i})$ and $X_v \in \mathrm {Fl}(\mathbf {j})$ ,

$$\begin{align*}(\pi_{\mathbf{i},\mathbf{j}})_{*}\mathcal{O}_w = \mathcal{O}_{w'} \in K_T(\mathrm{Fl}(\mathbf{j})) , \textrm{ and } \pi_{\mathbf{i},\mathbf{j}}^{*}\mathcal{O}_v = \mathcal{O}_{v} \in K_T(\mathrm{Fl}(\mathbf{i})) ,\end{align*}$$

where $w' \in W^{\mathbf {j}}$ is the minimal length representative for $w \in W/W_{\mathbf {j}}$ .

Proof. The first equality follows from [Reference Brion and KumarBK05, Thm. 3.3.4(a)] and the second because $\pi _{\mathbf {i},\mathbf {j}}$ is a flat morphism.

4 Quantum K theory and ‘quantum=classical’

4.1 Definitions and notation

The quantum K ring was defined by Givental and Lee [Reference GiventalGiv00, Reference LeeLee04]. We recall the definition below, following [Reference GiventalGiv00]. The quantum K metric is a deformation of the usual K-theory pairing; we recall the definition of this metric for $X= \mathrm {Gr}(k;n)$ . Fix any basis $(a_\lambda )$ of $\operatorname {\mathrm {K}}_T(\mathrm {Gr}(k;n))$ over $\operatorname {\mathrm {K}}_T(pt)$ , where $\lambda $ varies over partitions in the $k \times (n-k)$ rectangle. The (small) quantum K metric is defined by

(9) $$ \begin{align} (a_\lambda , a_\mu)_{sm} = \sum_{d \ge 0} q^d \langle a_\lambda, a_\mu \rangle_d \quad \in \operatorname{\mathrm{K}}_T(pt)[\![q]\!] ,\end{align} $$

extended by $K_T(pt)[\![q]\!]$ -linearity. The elements $\langle a_\lambda , a_\mu \rangle _d \in \operatorname {\mathrm {K}}_T(pt)$ denote $2$ -point (genus $0$ , equivariant) K-theoretic Gromov-Witten (KGW) invariants if $d>0$ ; if $d=0$ , then $\langle a_\lambda , a_\mu \rangle _0= \langle a_\lambda , a_\mu \rangle $ are given by the ordinary K-theory pairing. The (n-point, genus $0$ ) KGW invariants $\langle a_1, \ldots , a_n \rangle _d \in \operatorname {\mathrm {K}}_T(pt)$ are defined by pulling back via the evaluation maps, then integrating, over the Kontsevich moduli space of stable maps $\overline {\mathcal {M}}_{0,n}(\mathrm {Gr}(k;n),d)$ . Instead of recalling the precise definition of the KGW invariants, in Theorem 4.5 below, we give a ‘quantum=classical’ statement calculating $2$ and $3$ -point KGW invariants of Grassmannians. Explicit combinatorial formulae for the $2$ -point KGW invariants for any homogeneous space may be found in [Reference Buch and MihalceaBM15, Reference Buch, Chung, Li and MihalceaBCLM20].

Example 4.1. Consider the projective plane ${\mathbb {P}}^2$ , and for simplicity, work non-equivariantly. Consider the Schubert basis ${\mathcal {O}}_0= [{\mathcal {O}}_{{\mathbb {P}}^2}], {\mathcal {O}}_1= [{\mathcal {O}}_{line}], {\mathcal {O}}_2=[{\mathcal {O}}_{pt}]$ . The classical K-theory pairing gives the matrix

$$\begin{align*}(\langle {\mathcal{O}}_i, {\mathcal{O}}_j \rangle) = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \end{align*}$$

For any $i,j \ge 0$ and $d>0$ , $\langle {\mathcal {O}}_i, {\mathcal {O}}_j \rangle _d =1$ . Then the quantum K metric is given by

$$\begin{align*}({\mathcal{O}}_i, {\mathcal{O}}_j )_{sm} = \langle {\mathcal{O}}_i, {\mathcal{O}}_j \rangle + q+q^2 + \ldots = \langle {\mathcal{O}}_i, {\mathcal{O}}_j \rangle + \frac{q}{1-q} .\end{align*}$$

Theorem 4.2 (Givental [Reference GiventalGiv00]).

Consider the $\operatorname {\mathrm {K}}_T[pt][\![q]\!]$ -module

$$\begin{align*}\operatorname{\mathrm{QK}}_T(\mathrm{Gr}(k;n)):= \operatorname{\mathrm{K}}_T(\mathrm{Gr}(k;n)) \otimes \operatorname{\mathrm{K}}_T(pt)[\![q]\!] . \end{align*}$$

Define the (small) equivariant quantum K product $\star $ on $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ by the equality

(10) $$ \begin{align}(a \star b, c)_{sm} = \sum_{d \ge 0} q^d \langle a,b,c \rangle_d ,\end{align} $$

for any $a, b, c \in \operatorname {\mathrm {K}}_T(\mathrm {Gr}(k;n))$ . Then $(\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n)), +,\star )$ is a commutative, associative $\operatorname {\mathrm {K}}_T(pt)[\![q]\!]$ -algebra. Furthermore, the small quantum K metric gives it a structure of a Frobenius algebra (i.e., $(a \star b, c)_{sm} = (a,b \star c)_{sm} $ ).

Remark 4.3. It was proved in [Reference Buch, Chaput, ‘Mihalcea and PerrinBCMP13] that the submodule $\operatorname {\mathrm {K}}_T(\mathrm {Gr}(k;n)) \otimes \operatorname {\mathrm {K}}_T(pt)[q]$ is stable under the QK product $\star $ . This means that the product of two Schubert classes in $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ has structure constants which are polynomials in q. Similar statements hold for any flag manifold [Reference Anderson, Chen and TsengACT22, Reference KatoKat18]. However, working over the ring of formal power series in q is needed for proving module finite generation of the claimed presentations. It is also natural if one studies quantizations of dual bundles, such as those from Theorem 6.1 below.

We record the following immediate consequence.

Corollary 4.4. Let $a,b,c \in \operatorname {\mathrm {K}}_T(\mathrm {Gr}(k;n))[\![q]\!]$ . Then $a \star b = c$ if and only if for any $\kappa \in \operatorname {\mathrm {K}}_T(\mathrm {Gr}(k;n))$ ,

$$\begin{align*}\sum_{d \ge 0} q^d \langle a,b,\kappa \rangle_d = \sum_{d \ge 0} q^d \langle c , \kappa \rangle_d . \end{align*}$$

4.2 Quantum=classical

The goal of this section is to recall the ‘quantum = classical’ statement, which relates the ( $3$ -point, genus $0$ ) equivariant KGW invariants on Grassmannians to a ‘classical’ calculation in the equivariant K-theory of a two-step flag manifold. This statement was proved in [Reference Buch and MihalceaBM11], and it generalized results of Buch, Kresch and Tamvakis [Reference Buch, Kresch and TamvakisBKT03] from quantum cohomology. The proofs rely on the ‘kernel-span’ technique introduced by Buch [Reference BuchBuc03]. A Lie-theoretic approach, for large degrees d, and for the larger family of cominuscule Grassmannians, was obtained in [Reference Chaput and PerrinCP11]; see also [Reference Buch, Chaput, Mihalcea and PerrinBCMP18b] for an alternative way to ‘quantum=classical’ utilizing projected Richardson varieties.

We recall next the ‘quantum = classical’ result, proved in [Reference Buch and MihalceaBM11], and which will be used later in this paper. To start, form the following incidence diagram:

(11)

Here, all maps are the natural projections. Denote by $p_1: \mathrm {Fl}(k-d,k,k+d;n) \to \mathrm {Gr}(k;n)$ the composition $p_1:=p \circ p_1^{\prime }$ . If $d\ge k$ , then we set $Y_d:= \mathrm {Fl}(k+d;n)$ , and if $k+d \ge n$ , then we set $Y_d:= \mathrm {Gr}(k-d;n)$ . In particular, if $d \ge \min \{ k, n-k \}$ , then $Y_d$ is a single point.

Theorem 4.5. Let $a,b,c \in K_T(\mathrm {Gr}(k;n))$ and $d \ge 0$ a degree.

(a) The following equality holds in $K_T(pt)$ :

$$\begin{align*}\langle a, b, c \rangle_d = \int_{Y_d} (p_2)_{*}(p_1^{*}(a))\cdot (p_2)_{*}(p_1^{*}(b))\cdot (p_2)_{*}(p_1^{*}(c)) . \end{align*}$$

(b) Assume that $(p_2)_{*}(p_1^{*}(a)) = pr^{*}(a')$ for some $a' \in K_T(\mathrm {Gr}(k-d;n)$ . Then

$$\begin{align*}\langle a, b, c \rangle_d = \int_{\mathrm{Gr}(k-d;n)} a' \cdot (q)_{*}(p^{*}(b))\cdot (q)_{*}(p^{*}(c)) . \end{align*}$$

Observe that part (b) follows from (a) and the fact that the left diagram is a fibre square; for details, see [Reference Buch and MihalceaBM11]. We will often use the tautological sequence on $\mathrm {Gr}(k;n)$ :

$$\begin{align*}0 \to {\mathcal{S}}_k \to \mathbb{C}^n \to {\mathcal{Q}}_{n-k}:=\mathbb{C}^n/{\mathcal{S}}_k \to 0 . \end{align*}$$

To lighten notation, we will denote by the same letters the bundles on various flag manifolds from the diagram (11), but we will indicate in the subscript the rank of the bundle in question. To illustrate in a situation used below, observe that $\mathrm {Fl}(k-d,k;n)$ equals to the Grassmann bundle $\mathbb {G}(d, \mathbb {C}^n/{\mathcal {S}}_{k-d}) \to \mathrm {Gr}(k-d;n)$ , with tautological sequence

$$\begin{align*}0 \to {\mathcal{S}}_k/{\mathcal{S}}_{k-d} \to \mathbb{C}^n/{\mathcal{S}}_{k-d} \to \mathbb{C}^n/{\mathcal{S}}_{k} \simeq p^{*}(\mathbb{C}^n/{\mathcal{S}}_{k}) \to 0 ; \end{align*}$$

all these are bundles on $\mathrm {Fl}(k-d,k;n)$ , with ${\mathcal {S}}_k$ is pulled back from $\mathrm {Gr}(k;n)$ , and ${\mathcal {S}}_{k-d}$ is pulled back from $\mathrm {Gr}(k-d;n)$ .

5 Cohomological calculations from ‘quantum = classical’ diagrams

In this section, we calculate some push-forwards of tautological bundles, which will be utilized in the proof of our main result. For a vector bundle E of rank e, we denote by $\lambda (E)_{\ge s}$ , respectively $\lambda (E)_{\le s}$ and so on, the truncations $y^s [\wedge ^s E] + \ldots + y^e [\wedge ^e E]$ , respectively $1 + y [E] + \ldots + [\wedge ^{s} E]$ etc. We use the notation from the diagram (11).

Proof. Consider the exact sequence on $\mathrm {Fl}(k-d,k,k+d)$ given by

$$\begin{align*}0 \to {\mathcal{S}}_{k-d} \to S_k = p_1^{*}{\mathcal{S}} \to {\mathcal{S}}_k/{\mathcal{S}}_{k-d} \to 0 . \end{align*}$$

It follows that $\lambda _y({\mathcal {S}}_k) = \lambda _y({\mathcal {S}}_{k-d}) \cdot \lambda _y({\mathcal {S}}_k/{\mathcal {S}}_{k-d})$ . By the projection formula,

(15) $$ \begin{align} (p_2)_{*}(\lambda_y({\mathcal{S}}_k)) = \lambda_y({\mathcal{S}}_{k-d}) \cdot (p_2)_{*}(\lambda_y({\mathcal{S}}_k/{\mathcal{S}}_{k-d})) . \end{align} $$

Observe now that $p_2: \mathrm {Fl}(k-d,k,k+d;n) \to \mathrm {Fl}(k-d,k+d;n)$ may be identified with the Grassmann bundle $\mathbb {G}(d,{\mathcal {S}}_{k+d}/{\mathcal {S}}_{k-d})$ , with tautological subbundle ${\mathcal {S}}_k/{\mathcal {S}}_{k-d}$ . Then part (a) follows from the projection formula and Corollary 3.3, which shows that

$$\begin{align*}(p_2)_{*}(\lambda_y({\mathcal{S}}_k/{\mathcal{S}}_{k-d})) = [{\mathcal{O}}_{\mathrm{Fl}(k-d,k+d;n)}] .\end{align*}$$

For (b), consider the short exact sequence on $\mathrm {Fl}(k-d,k,k+d;n)$ ,

$$\begin{align*}0 \to {\mathcal{S}}_{k+d}/{\mathcal{S}}_k \to {\mathbb{C}}^n/{\mathcal{S}}_k = p_1^{*}{\mathcal{Q}}_{n-k} \to {\mathbb{C}}^n/{\mathcal{S}}_{k+d} \to 0 . \end{align*}$$

Then $\lambda _y(p_1^{*} {\mathcal {Q}}_{n-k}) = \lambda _y({\mathcal {S}}_{k+d}/{\mathcal {S}}_k) \cdot (p_2)^{*}\lambda _y({\mathbb {C}}^n/{\mathcal {S}}_{k+d})$ , and by the projection formula,

$$\begin{align*}(p_2)_{*}(\lambda_y(p_1^{*} {\mathcal{Q}}_{n-k})) = (p_2)_{*}(\lambda_y({\mathcal{S}}_{k+d}/{\mathcal{S}}_k)) \cdot \lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k+d}) . \end{align*}$$

The claim follows again by Corollary 3.3, using that ${\mathcal {S}}_{k+d}/{\mathcal {S}}_k$ is the tautological quotient bundle of the Grassmann bundle $\mathbb {G}(d,{\mathcal {S}}_{k+d}/{\mathcal {S}}_{k-d})$ ; thus,

$$\begin{align*}\pi_{*}\lambda_y({\mathcal{S}}_{k+d}/{\mathcal{S}}_k) = 1+ y[{\mathcal{S}}_{k+d}/{\mathcal{S}}_{k-d}] + \ldots + y^d \wedge^d [{\mathcal{S}}_{k+d}/{\mathcal{S}}_{k-d}] . \end{align*}$$

Part (c) follows similarly to (b), utilizing the exact sequence $0 \to ({\mathcal {S}}_k/{\mathcal {S}}_{k-d})^{*} \to {\mathcal {S}}_k^{*} = p_1^{*}({\mathcal {S}}_k^{*}) \to {\mathcal {S}}_{k-d}^{*} \to 0$ and Corollary 3.3.

We need the analogues of some of the previous equations in the equivariant K theory of $\mathrm {Gr}(k-d;n)$ .

Proof. Parts (a) and (b) follow similarly to Proposition 5.1 (a). For (a), we utilize the short exact sequence $0 \to {\mathcal {S}}_{k-d} \to S_k = p_1^{*}{\mathcal {S}}_k \to {\mathcal {S}}_k/{\mathcal {S}}_{k-d} \to 0$ on $\mathrm {Fl}(k-d, k;n)$ , and that the projection $q:\mathrm {Fl}(k-d,k;n) \to \mathrm {Gr}(k-d;n)$ is identified to the Grassmann bundle $\mathbb {G}(d;{\mathbb {C}}^n/{\mathcal {S}}_{k-d}) \to \mathrm {Gr}(k-d;n)$ with tautological subbundle ${\mathcal {S}}_k/{\mathcal {S}}_{k-d}$ . For (b), one takes the dual of this sequence. Then by Corollary 3.3,

$$\begin{align*}q_{*}(\lambda_y({\mathcal{S}}_{k}/{\mathcal{S}}_{k-d})) = [{\mathcal{O}}_{\mathrm{Gr}(k-d;n)}] \textrm{ and } q_{*}(\lambda_y(({\mathcal{S}}_{k}/{\mathcal{S}}_{k-d})^{*}))= \lambda_y(({\mathbb{C}}^n/{\mathcal{S}}_{k-d})^{*})_{\le d} .\end{align*}$$

Then (a) and (b) follow from this and the projection formula.

For part (c), observe that $\mathrm {Fl}(k-d,k;n) \to \mathrm {Gr}(k-d;n)$ is the Grassmann bundle $\mathbb {G}(d,{\mathbb {C}}^n/{\mathcal {S}}_{k-d})$ with tautological quotient bundle ${\mathbb {C}}^n/{\mathcal {S}}_k = p^{*}{\mathcal {Q}}_{n-k}$ . Then the claim follows again by Corollary 3.3.

Observe that $pr^{*}(\lambda _y({\mathcal {S}}_{k-d})) = \lambda _y({\mathcal {S}}_{k-d})$ in $\operatorname {\mathrm {K}}_T(Y_d)$ . Utilizing this, and combining Proposition 5.1, Corollary 5.2 and part (b) from Theorem 4.5, we obtain the following useful corollary:

Corollary 5.3. Fix arbitrary $b,c \in \operatorname {\mathrm {K}}_T(\mathrm {Gr}(k;n))$ and any degree $d \ge 0$ . Then the equivariant KGW invariant $\langle \lambda _y({\mathcal {S}}_k),b,c \rangle _d$ satisfies

$$\begin{align*}\langle \lambda_y({\mathcal{S}}_k),b,c \rangle_d = \int_{\mathrm{Gr}(k-d;n)} \lambda_y({\mathcal{S}}_{k-d}) \cdot q_{*}p^{*}(b) \cdot q_{*}p^{*}(c) . \end{align*}$$

In particular, the $2$ -point KGW invariant $\langle b,c \rangle _d$ satisfies

$$\begin{align*}\langle b,c \rangle_d = \int_{\mathrm{Gr}(k-d;n)} q_{*}p^{*}(b)\cdot q_{*}p^{*}(c) . \end{align*}$$

6 Quantum duals

The main goal of this section is to prove the next identity.

Theorem 6.1. The following holds in $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ :

(16) $$ \begin{align} (\lambda_y({\mathcal{S}}_k)-1) \star \det ({\mathcal{Q}}_{n-k}) = (1-q) ((\lambda_y({\mathcal{S}}_k) -1) \cdot \det ({\mathcal{Q}}_{n-k})) . \end{align} $$

Equivalently, for any $i>0$ ,

(17) $$ \begin{align} \wedge^i ({\mathcal{S}}_k) \star \det ({\mathcal{Q}}_{n-k}) = (1-q) \wedge^{k-i}({\mathcal{S}}_k^{*}) \cdot \det({\mathbb{C}}^n) . \end{align} $$

An equivalent formulation of the identity (16) is in terms of (equivariant) KGW invariants: for any degree $d \ge 0$ and any $a \in \operatorname {\mathrm {K}}_T(\mathrm {Gr}(k;n))$ ,

(18) $$ \begin{align} \langle \lambda_y({\mathcal{S}}_k) -1, \det ({\mathcal{Q}}_{n-k}) , a \rangle_d = \langle (\lambda_y({\mathcal{S}}_k) -1) \cdot \det ({\mathcal{Q}}_{n-k}), a \rangle_d - \langle (\lambda_y({\mathcal{S}}_k)-1) \cdot \det ({\mathcal{Q}}_{n-k}), a \rangle_{d-1} . \end{align} $$

By Corollary 5.3 and Corollary 5.2, the left-hand side of (18) may be calculated as

(19) $$ \begin{align} \langle \lambda_y({\mathcal{S}}_k)-1, \det ({\mathcal{Q}}_{n-k}) , a \rangle_d = \int_{\mathrm{Gr}(k-d;n)} (\lambda_y({\mathcal{S}}_{k-d})-1) \cdot \wedge^{n-k} ({\mathbb{C}}^n/{\mathcal{S}}_{k-d}) \cdot q_{*} p^{*}(a) . \end{align} $$

The next lemma calculates the push-forwards relevant for the right-hand side of (18).

Lemma 6.2. The following equality holds in $\operatorname {\mathrm {K}}_T(\mathrm {Gr}(k-d;n))$ :

$$\begin{align*}y^{n-k} q_{*}(p^{*}(\lambda_y(\mathcal{S}_k) \cdot \det ({\mathcal{Q}}_{n-k}) )) = \lambda_y(\mathcal{S}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d})_{\ge n-k} .\end{align*}$$

Proof. To start, observe that for any (equivariant) vector bundle E,

$$\begin{align*}\wedge^i E \otimes \det(E^{*}) = \wedge^{rk(E)-i} E^{*} \quad \textrm{ and } \quad y^{rk(E)} \lambda_{y^{-1}}(E^{*}) = \lambda_y(E) \cdot \det(E^{*}) . \end{align*}$$

Utilizing this and that $\det ({\mathbb {C}}^n/{\mathcal {S}}_k) = \det {\mathcal {S}}_k^{*} \otimes \det {\mathbb {C}}^n$ , we obtain that

(20) $$ \begin{align} \begin{aligned} & y^{n-k} q_{*}(p^{*}(\lambda_y(\mathcal{S}_k) \cdot \det ({\mathbb{C}}^n/\mathcal{S}_k))) \\ & = y^{n-k} q_{*}(p^{*}(\lambda_y(\mathcal{S}_k) \cdot \det \mathcal{S}_k^{*} \cdot \det {\mathbb{C}}^n))\\ & = y^{n-k} y^{k} q_{*}(\lambda_{y^{-1}}(\mathcal{S}_k^{*}) \cdot \det {\mathbb{C}}^n)) \\ & = y^n q_{*}(\lambda_{y^{-1}}(\mathcal{S}_{k-d}^{*}) \cdot \lambda_{y^{-1}}(\mathcal{S}_{k}/{\mathcal{S}}_{k-d})^{*}) \cdot \det {\mathbb{C}}^n \\ & = y^n \lambda_{y^{-1}}({\mathcal{S}}_{k-d}^{*}) \cdot q_{*}(\lambda_{y^{-1}}(\mathcal{S}_{k}/{\mathcal{S}}_{k-d})^{*}) \cdot \det {\mathbb{C}}^n . \end{aligned} \end{align} $$

By Corollary 3.3,

(21) $$ \begin{align} q_{*}(\lambda_y(({\mathcal{S}}_{k}/{\mathcal{S}}_{k-d})^{*}))= \lambda_y(({\mathbb{C}}^n/{\mathcal{S}}_{k-d})^{*})_{\le d} ;\end{align} $$

therefore,

(22) $$ \begin{align} \begin{aligned} y^d & q_{*}(\lambda_{y^{-1}}(\mathcal{S}_{k}/{\mathcal{S}}_{k-d})^{*}) = y^d (1+ y^{-1} ({\mathbb{C}}^n/{\mathcal{S}}_{k-d})^{*} + \ldots + y^{-d} \wedge^d ({\mathbb{C}}^n/{\mathcal{S}}_{k-d})^{*}) \\ & = \det ({\mathbb{C}}^n/{\mathcal{S}}_{k-d})^{*} \cdot ( y^d \wedge^{n-k+d} ({\mathbb{C}}^n/{\mathcal{S}}_{k-d}) \\ & \quad + y^{d-1} \wedge^{n-k+d-1}( {\mathbb{C}}^n/{\mathcal{S}}_{k-d}) + \ldots + \wedge^{n-k} ({\mathbb{C}}^n/{\mathcal{S}}_{k-d}) ) \\ & = y^{-(n-k)} \det ({\mathbb{C}}^n/{\mathcal{S}}_{k-d})^{*} \cdot \lambda_y( {\mathbb{C}}^n/{\mathcal{S}}_{k-d} )_{\ge n-k} . \end{aligned} \end{align} $$

Combining (20), (21) and (22) above, we obtain

$$\begin{align*}\begin{aligned} & y^{n-k} q_{*}(p^{*}(\lambda_y(\mathcal{S}_k) \cdot \det ({\mathbb{C}}^n/\mathcal{S}_k)) \\ & = y^n \lambda_{y^{-1}}({\mathcal{S}}_{k-d}^{*}) \cdot q_{*}(\lambda_{y^{-1}}(\mathcal{S}_{k}/{\mathcal{S}}_{k-d})^{*}) \cdot \det {\mathbb{C}}^n \\ & = y^{n-d} \lambda_{y^{-1}}({\mathcal{S}}_{k-d}^{*}) \cdot y^{-(n-k)} \det ({\mathbb{C}}^n/{\mathcal{S}}_{k-d})^{*} \cdot \lambda_y( {\mathbb{C}}^n/{\mathcal{S}}_{k-d} )_{\ge n-k} \cdot \det {\mathbb{C}}^n\\ & = y^{k-d} \lambda_{y^{-1}}({\mathcal{S}}_{k-d}^{*}) \cdot \det ({\mathcal{S}}_{k-d}) \cdot \lambda_y( {\mathbb{C}}^n/{\mathcal{S}}_{k-d} )_{\ge n-k} \\ & =\lambda_y({\mathcal{S}}_{k-d}) \cdot \lambda_y( {\mathbb{C}}^n/{\mathcal{S}}_{k-d} )_{\ge n-k} . \end{aligned} \end{align*}$$

The last expression is the one from the claim.

Consider the following projection maps:

(23)

We are now ready to prove Theorem 6.1.

Proof of Theorem 6.1.

A standard diagram chase and the equalities from Equation (19) and Lemma 6.2 imply that in order to prove Equation (18), it suffices to show that in $\operatorname {\mathrm {K}}_T(\mathrm {Fl}(k-d,k-d+1,k;n)$ ,

$$\begin{align*}\begin{aligned} y^{n-k} & p^{k-d}_{*}((\lambda_y({\mathcal{S}}_{k-d})-1) \cdot \wedge^{n-k}({\mathbb{C}}^n/{\mathcal{S}}_{k-d})) \\ & = p^{k-d}_{*}\Big(\lambda_y({\mathcal{S}}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d})_{\ge n-k} - \lambda_y({\mathcal{S}}_{k-d+1}) \cdot \lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d+1})_{\ge n-k}\Big) \\ & \quad - y^{n-k}\Big(\wedge^{n-k}({\mathbb{C}}^n/{\mathcal{S}}_{k-d}) - \wedge^{n-k} ({\mathbb{C}}^n/{\mathcal{S}}_{k-d+1})\Big) .\end{aligned} \end{align*}$$

After expanding and canceling the like terms, this amounts to showing that

(24) $$ \begin{align} \begin{aligned} p^{k-d}_{*}(\lambda_y({\mathcal{S}}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d})_{> n-k}) & = p^{k-d}_{*}(\lambda_y({\mathcal{S}}_{k-d+1})\cdot \lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d+1})_{\ge n-k}) \\& \quad - y^{n-k}p^{k-d}_{*}(\wedge^{n-k} ({\mathbb{C}}^n/{\mathcal{S}}_{k-d+1}) ) . \end{aligned} \end{align} $$

Since this identity will be used again in the proof of relations in the quantum K ring, we will prove it in the lemma below, thus finishing the proof of Theorem 6.1.

Proof. Consider the projection $p': \mathrm {Fl}(k-d,k-d+1,k;n) \to \mathrm {Fl}(k-d+1,k;n)$ ; this is the Grassmann bundle $\mathbb {G}(k-d;{\mathcal {S}}_{k-d+1})$ over $\mathrm {Fl}(k-d+1,k;n)$ with tautological subbundle ${\mathcal {S}}_k$ and quotient bundle $\mathcal {L} := {\mathcal {S}}_{k-d+1}/{\mathcal {S}}_{k-d}$ . In order to show the equality 24, it suffices to replace $p^{k-d}$ by $p'$ ; that is,

(25) $$ \begin{align} \begin{aligned} p^{\prime}_{*}(\lambda_y({\mathcal{S}}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d})_{> n-k}) &= p^{\prime}_{*}(\lambda_y({\mathcal{S}}_{k-d+1}) \cdot \lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d+1})_{\ge n-k}) \\& \quad - y^{n-k}p^{\prime}_{*}(\wedge^{n-k} ({\mathbb{C}}^n/{\mathcal{S}}_{k-d+1}) ) . \end{aligned} . \end{align} $$

To calculate the left-hand side, consider the short exact sequences

$$\begin{align*}0 \to \mathcal{S}_{k-d} \to \mathcal{S}_{k-d+1} \to \mathcal{L} \to 0 ; \quad 0 \to \mathcal{L} \to {\mathbb{C}}^{n}/\mathcal{S}_{k-d} \to {\mathbb{C}}^{n}/\mathcal{S}_{k-d+1} \to 0 . \end{align*}$$

From this, it follows that $\lambda _y({\mathbb {C}}^{n}/\mathcal {S}_{k-d}) = (1+y \mathcal {L}) \lambda _y({\mathbb {C}}^{n}/\mathcal {S}_{k-d+1})$ ; thus,

$$ \begin{align*} \lambda_y({\mathbb{C}}^{n}/\mathcal{S}_{k-d})_{>n-k} = \lambda_y({\mathbb{C}}^{n}/\mathcal{S}_{k-d+1})_{>n-k} + y \mathcal{L} \cdot \lambda_y({\mathbb{C}}^{n}/\mathcal{S}_{k-d+1})_{ \ge n-k} . \end{align*} $$

Observe that $p^{\prime }_{*}(\lambda _y(\mathcal {S}_{k-d})) =1$ , by Corollary 3.3. Using this and the projection formula, we calculate the left-hand side of Equation (25):

$$\begin{align*}\begin{aligned} & p^{\prime}_{*}(\lambda_y(\mathcal{S}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/\mathcal{S}_{k-d})_{> n-k}) \\ & = p^{\prime}_{*}(\lambda_y(\mathcal{S}_{k-d}) \cdot \lambda_y({\mathbb{C}}^{n}/\mathcal{S}_{k-d+1})_{>n-k}) \\ & \quad + p^{\prime}_{*}(\lambda_y(\mathcal{S}_{k-d}) \cdot (\lambda_y(\mathcal{L}) -1) \cdot \lambda_y({\mathbb{C}}^{n}/\mathcal{S}_{k-d+1})_{ \ge n-k} )\\ &= \lambda_y({\mathbb{C}}^{n}/\mathcal{S}_{k-d+1})_{>n-k}) + p^{\prime}_{*}((\lambda_y(\mathcal{S}_{k-d+1}) - \lambda_y(\mathcal{S}_{k-d})) \cdot \lambda_y({\mathbb{C}}^{n}/\mathcal{S}_{k-d+1})_{ \ge n-k}) \\ & = \lambda_y({\mathbb{C}}^{n}/\mathcal{S}_{k-d+1})_{>n-k}) + (\lambda_y(\mathcal{S}_{k-d+1}) - 1) \cdot \lambda_y({\mathbb{C}}^{n}/\mathcal{S}_{k-d+1})_{ \ge n-k} \\ & =\lambda_y(\mathcal{S}_{k-d+1}) \cdot \lambda_y({\mathbb{C}}^{n}/\mathcal{S}_{k-d+1})_{ \ge n-k} - y^{n-k} \wedge^{n-k} ({\mathbb{C}}^{n}/\mathcal{S}_{k-d+1}) . \end{aligned} \end{align*}$$

The last expression is the right-hand side of Equation (25), again by projection formula.

7 Relations in quantum K theory

Recall the tautological sequence $0 \to {\mathcal {S}}_k \to {\mathbb {C}}^n \to {\mathcal {Q}}_{n-k} \to 0$ on $X=\mathrm {Gr}(k;n)$ . The goal of this section is to prove the following theorem.

Theorem 7.1. The following equalities hold in $\operatorname {\mathrm {QK}}_T(X)$ :

(26) $$ \begin{align}\begin{aligned} \lambda_y({\mathcal{S}}_k) \star \lambda_y({\mathcal{Q}}_{n-k}) & = \lambda_y({\mathbb{C}}^n) - q y^{n-k} \bigl((\lambda_y({\mathcal{S}}_k) -1) \otimes \det {\mathcal{Q}}_{n-k}\bigr) \\ & = \lambda_y({\mathbb{C}}^n) - \frac{q}{1-q} y^{n-k} (\lambda_y({\mathcal{S}}_k) -1) \star \det {\mathcal{Q}}_{n-k}. \end{aligned} \end{align} $$

The second equality follows from Theorem 6.1; therefore, we focus on the first. Our strategy is to utilize again the ‘quantum=classical’ identity in order to show that for any $a \in \operatorname {\mathrm {K}}_T(\mathrm {Gr}(k;n))$ , and for any $d \ge 0$ ,

(27) $$ \begin{align} \langle \lambda_y({\mathcal{S}}_k), \lambda_y({\mathcal{Q}}_{n-k}), a\rangle_d = \lambda_y({\mathbb{C}}^n) \langle 1 , a \rangle_d - y^{n-k} \langle (\lambda_y({\mathcal{S}}_k) -1) \otimes \det ({\mathcal{Q}}_{n-k}), a \rangle_{d-1} . \end{align} $$

By Corollary 4.4 this will imply the desired identities. The KGW invariant from the left-hand side may be calculated using the next lemma.

Lemma 7.2. The following equality holds in $\operatorname {\mathrm {K}}_T(\mathrm {Gr}(k-d;n))$ :

$$\begin{align*}\begin{aligned} & q_{*} p^{*}(\lambda_y({\mathcal{S}}_k)) \cdot q_{*} p^{*}(\lambda_y({\mathcal{Q}}_{n-k})) = \lambda_y({\mathbb{C}}^n) - \lambda_y({\mathcal{S}}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d})_{> n-k} .\end{aligned} \end{align*}$$

Proof. This follows from Corollary 5.2, parts (a) and (c), working in the Grassmannian $\mathrm {Gr}(k-d;n)$ with tautological sequence $0 \to {\mathcal {S}}_{k-d} \to {\mathbb {C}}^n \to {\mathbb {C}}^n/{\mathcal {S}}_{k-d} \to 0$ :

$$\begin{align*} q_{*} p^{*}(\lambda_y({\mathcal{S}}_k)) \cdot q_{*} p^{*}(\lambda_y({\mathcal{Q}}_{n-k})) & = \lambda_y({\mathcal{S}}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d})_{\le n-k} \\ & =\lambda_y({\mathcal{S}}_{k-d}) \cdot (\lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d}) -\lambda_y({\mathbb{C}}^n/{\mathcal{S}}_{k-d}) _{> n-k}) \\ & =\lambda_y({\mathbb{C}}^n) - \lambda_y(\mathcal{S}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/\mathcal{S}_{k-d})_{> n-k} . \\[-35pt] \end{align*}$$

We are now ready to prove Theorem 7.1.

Proof of Theorem 7.1.

We need to verify Equation (27), and for that, we utilize the ‘quantum= classical’ statement in Theorem 4.5. From Lemma 7.2, the left-hand side of Equation (27) equals

$$\begin{align*}\int_{\mathrm{Gr}(k-d;n)} (\lambda_y({\mathbb{C}}^n) - \lambda_y(\mathcal{S}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/\mathcal{S}_{k-d})_{> n-k}) \cdot q_{*} p^{*}(a) . \end{align*}$$

From Lemma 6.2, the right-hand side of (27) is equal to

$$\begin{align*}\begin{aligned} & \int_{\mathrm{Gr}(k-d;n)} \lambda_y({\mathbb{C}}^n) \cdot q_{*} p^{*}(a) - \\ & \int_{\mathrm{Gr}(k-d+1;n)} \bigl(\lambda_y(\mathcal{S}_{k-d+1}) \cdot \lambda_y({\mathbb{C}}^n/\mathcal{S}_{k-d+1})_{\ge n-k} + y^{n-k} \wedge^{n-k} ({\mathbb{C}}^n/\mathcal{S}_{k-d+1})\bigr) \cdot \bar{q}_{*} \bar{p}^{*}(a) . \end{aligned} \end{align*}$$

Here, $\bar {p}: \mathrm {Fl}(k-d+1,k;n) \to \mathrm {Gr}(k;n)$ and $\bar {q}: \mathrm {Fl}(k-d+1,k;n) \to \mathrm {Gr}(k-d+1;n)$ are the natural projections. After cancelling the like terms, this amounts to proving the equality

(28) $$ \begin{align} \begin{aligned} & \int_{\mathrm{Gr}(k-d;n)} (\lambda_y(\mathcal{S}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/\mathcal{S}_{k-d})_{> n-k}) \cdot q_{*} p^{*}(a) \\& \quad = \int_{\mathrm{Gr}(k-d+1;n)} \bigl(\lambda_y(\mathcal{S}_{k-d+1}) \cdot \lambda_y({\mathbb{C}}^n/\mathcal{S}_{k-d+1})_{\ge n-k} - y^{n-k} \wedge^{n-k} ({\mathbb{C}}^n/\mathcal{S}_{k-d+1})\bigr) \cdot q_{*} p^{*}(a) . \end{aligned} \end{align} $$

Recall the projection $p^{k-d}: \mathrm {Fl}(k-d,k-d+1,k;n) \to \mathrm {Gr}(k;n)$ . As in the proof of Theorem 6.1, by diagram chasing and projection formula, one shows the equality (28) by demonstrating the following:

$$\begin{align*}\begin{aligned} & p^{k-d}_{*}(\lambda_y(\mathcal{S}_{k-d}) \cdot \lambda_y({\mathbb{C}}^n/\mathcal{S}_{k-d})_{> n-k}) \\ & = p^{k-d}_{*}\bigl(\lambda_y(\mathcal{S}_{k-d+1}) \cdot \lambda_y({\mathbb{C}}^n/\mathcal{S}_{k-d+1})_{\ge n-k} - y^{n-k} \wedge^{n-k} ({\mathbb{C}}^n/\mathcal{S}_{k-d+1})\bigr) . \end{aligned} \end{align*}$$

This is Equation (24), proved in Lemma 6.3.

8 The QK Whitney presentation

In this section, we prove that the relations found in Theorem 7.1 give a presentation of the algebra $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ . The proof strategy is similar to that employed in [Reference Bernd and TianST97, Reference Fulton and PandharipandeFP97] from quantum cohomology: one first proves that in the classical limit, these generate the full ideal of relations, and then one uses Nakayama-type arguments to upgrade to the quantum situation. Unlike the quantum cohomology ring, the quantum K theory is not a graded ring. One may still use the usual Nakayama lemma for modules over completed rings to prove that the same phenomenon holds. The necessary statements are collected in the Appendix A below.

As usual, we consider the Grassmannian $\mathrm {Gr}(k;n)$ equipped with the tautological sequence $0 \to {\mathcal {S}} \to {\mathbb {C}}^n \to {\mathcal {Q}} \to 0$ , but from now, we omit the ranks in the subscripts. Let $X=(X_1, \ldots , X_k)$ and $\tilde {X}=(\tilde {X}_1, \ldots , \tilde {X}_{n-k})$ denote formal variables. The elementary symmetric polynomials ${e_i(X)= e_i(X_1, \ldots , X_k)}$ and $e_j(\tilde {X})= e_j(\tilde {X}_1, \ldots , \tilde {X}_{n-k})$ are algebraically independent. Geometrically, in $\mathrm {K}_T(\mathrm {Gr}(k;n))$ ,

$$\begin{align*}\lambda_y({\mathcal{S}}) = \prod_{i=1}^k (1+ y X_i) ; \quad \lambda_y({\mathcal{Q}}) = \prod_{j=1}^{n-k} (1+ y \tilde{X}_j) . \end{align*}$$

Define by $I\subset \mathrm {K}_T(pt)[e_1(X), \ldots , e_k(X), e_1(\tilde {X}), \ldots , e_{n-k}(\tilde {X})]$ the ideal determined by the Whitney relations in $\mathrm {K}_T(\mathrm {Gr}(k;n))$ ; that is, I is generated by the coefficients of the powers of y in

(29) $$ \begin{align} \prod_{i=1}^k (1+ yX_i) \prod_{i=1}^{n-k} (1+y \tilde{X}_i) = \prod_{i=1}^{n} (1+y T_i) . \end{align} $$

A non-equivariant variant of the following proposition appears in [Reference LascouxLas90, §7].

Proposition 8.1. There is an isomorphism of $\mathrm {K}_T(pt)$ -algebras

$$\begin{align*}\Psi: \mathrm{K}_T(pt)[e_1(X), \ldots, e_k(X), e_1(\tilde{X}), \ldots, e_{n-k}(\tilde{X})]/I \to \mathrm{K}_T(\mathrm{Gr}(k;n)) , \end{align*}$$

sending $e_i(X) \mapsto \wedge ^i {\mathcal {S}}$ and $e_j(\tilde {X}) \mapsto \wedge ^j {\mathcal {Q}}$ .

Proof. Denote the ring on the left by A. Since $\lambda _y({\mathcal {S}}) \cdot \lambda _y({\mathcal {Q}}) = \lambda _y({\mathbb {C}}^n)$ , the homomorphism $\Psi : A \to \mathrm {K}_T(\mathrm {Gr}(k;n))$ is well defined. Consider the polynomial ring

$$\begin{align*}A' := {\mathbb{Z}}[T_1, \ldots, T_n][e_1(X), \ldots, e_k(X), e_1(\tilde{X}), \ldots, e_{n-k}(\tilde{X})] . \end{align*}$$

Note that I is also an ideal in $A'$ . Observe that $A'/I$ is a free ${\mathbb {Z}}[T_1, \ldots , T_n]$ -module of rank ${n \choose k}$ . (There are several proofs. For instance, (temporarily) identify ${\mathbb {Z}}[T_1, \ldots , T_n]$ to $H^{*}_T(pt)$ . Then from the Whitney relations $c^T({\mathcal {S}}) \cdot c^T({\mathcal {Q}})= c^T({\mathbb {C}}^n)$ in $H^{*}_T(\mathrm {Gr}(k;n))$ , there is an isomorphism of $H^{*}_T(pt)$ -algebras $A'/I\simeq H^{*}_T(\mathrm {Gr}(k;n))$ , sending $e_i(X) \mapsto c_i^T({\mathcal {S}})$ and $e_j(\tilde {X}) \mapsto c_j^T({\mathcal {Q}})$ .) If we regard the Laurent polynomial ring $\mathrm {K}_T(pt)= {\mathbb {Z}}[T_1^{\pm 1}, \ldots , T_n^{\pm 1}]$ as a ${\mathbb {Z}}[T_1, \ldots , T_n]$ -algebra (under the natural inclusion of polynomials into Laurent polynomials), we obtain that

$$\begin{align*}A = (A'/I) \otimes_{{\mathbb{Z}}[T_1, \ldots, T_n]} \mathrm{K}_T(pt) \end{align*}$$

is a free $\mathrm {K}_T(pt)$ -module of rank ${n \choose k}$ . We utilize this to show that $\Psi $ induces an isomorphism between the associated graded rings. To calculate $\mathrm {gr}(A)$ , we make the change of variables $z_i = 1- X_i$ ( $1 \le i \le k$ ), $\tilde {z}_j = 1 - \tilde {X}_j$ ( $1 \le j \le n- k$ ), and $\zeta _s = 1- T_s$ ( $1 \le s \le n$ ). Each of these variables has degree $1$ . Under this change, A becomes

$$\begin{align*}\frac{\operatorname{\mathrm{K}}_T(pt)[e_1(z), \ldots, e_k(z);e_1(\tilde{z}), \ldots, e_{n-k}(\tilde{z})]}{ \langle \sum_{i+j =\ell} e_i(z_1, \ldots, z_k) e_j(\tilde{z}_1, \ldots, \tilde{z}_{n-k}) - e_\ell(\zeta_1, \ldots, \zeta_n) \rangle_{1 \le \ell \le n}} ;\end{align*}$$

see also (56) below (with $q=0$ ). The variables $z_i = 1 - X_i$ are sent to the $\operatorname {\mathrm {K}}$ -theoretic Chern roots of ${\mathcal {S}}^{*}$ , and similarly, $\tilde {z}_j$ to the $\operatorname {\mathrm {K}}$ -theoretic Chern roots of ${\mathcal {Q}}^{*}$ . We deduce that the initial term of $\Psi (e_i(z))$ , and of $\Psi (e_j(\tilde {z}))$ , equals to $c_i^T({\mathcal {S}}^{*})$ , respectively $c_j^T({\mathcal {Q}}^{*})$ in $H^{*}_T(\mathrm {Gr}(k;n))$ . Thus, when taking the associated graded rings, $\Psi $ recovers the usual presentation of the equivariant cohomology ring. This implies that $\Psi $ is injective.

The surjectivity follows from the theory of factorial Grothendieck polynomials. More precisely, from [Reference BuchBuc02, Thm. 2.1] or [Reference OetjenOet21, Thm. 1.2], see also [Reference Fulton and LascouxFL94, Reference McNamaraMcN06, Reference Ikeda and NaruseIN13], for each partition $\lambda $ , the equivariant Schubert class ${\mathcal {O}}_\lambda $ is a symmetric polynomial in the $\operatorname {\mathrm {K}}$ -theoretic Chern roots $1-X_1, \ldots , 1-X_k$ of ${\mathcal {S}}$ with coefficients in $\operatorname {\mathrm {K}}_T(pt)$ . By (58) below, this is a $\operatorname {\mathrm {K}}_T(pt)$ -linear combination of the (images of) $e_i(X)$ . Then $\Psi $ is also surjective, and this finishes the proof.

Recall from Theorem 7.1 that in $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ ,

(30) $$ \begin{align} \lambda_y({\mathcal{S}}) \star \lambda_y({\mathcal{Q}}) = \lambda_y({\mathbb{C}}^n) - \frac{q}{1-q} y^{n-k} (\lambda_y({\mathcal{S}}) -1) \star \det {\mathcal{Q}} . \end{align} $$

Motivated by this, define the ideal

$$\begin{align*}I_q \subset \mathrm{K}_T[pt][\![q]\!][e_1(X), \ldots, e_k(X), e_1(\tilde{X}), \ldots, e_{n-k}(\tilde{X})]\end{align*}$$

generated by polynomials obtained by equating the powers of y in the equality:

(31) $$ \begin{align} \begin{aligned} \prod_{i=1}^k & (1+y X_i) \times \prod_{j=1}^{n-k} (1+y \tilde{X}_i) \\ = & \prod_{i=1}^n (1+y T_i) -\frac{q}{1-q} y^{n-k} \tilde{X}_1 \cdot \ldots \cdot \tilde{X}_{n-k} \bigl( \prod_{i=1}^k (1+y X_i) -1 \bigr) . \end{aligned} \end{align} $$

Theorem 8.2. There is an isomorphism of $\mathrm {K}_T[pt][\![q]\!]$ -algebras

$$\begin{align*}\Psi: \mathrm{K}_T[pt][\![q]\!][e_1(X), \ldots, e_k(X), e_1(\tilde{X}), \ldots, e_{n-k}(\tilde{X})]/I_q \to \operatorname{\mathrm{QK}}_T(\mathrm{Gr}(k;n)) , \end{align*}$$

sending $e_i(X) \mapsto \wedge ^i {\mathcal {S}}$ and $e_j(\tilde {X}) \mapsto \wedge ^j {\mathcal {Q}}$ .

Proof. There exists a ring homomorphism

$$\begin{align*}\widetilde{\Psi}:\mathrm{K}_T[pt][\![q]\!][e_1(X), \ldots, e_k(X), e_1(\tilde{X}), \ldots, e_{n-k}(\tilde{X})]\to \operatorname{\mathrm{QK}}_T(\mathrm{Gr}(k;n)) , \end{align*}$$

sending $e_i(X) \mapsto \wedge ^i {\mathcal {S}}$ and $e_j(\tilde {X}) \mapsto \wedge ^j {\mathcal {Q}}$ . It follows from Equation (30) that $\widetilde {\Psi }(I_q) = 0$ ; therefore, this induces the homomorphism $\Psi $ from the claim. In order to prove that $\Psi $ is an isomorphism, we will use the Nakayama-type result from Proposition A.3, applied to the case when

$$\begin{align*}M = \mathrm{K}_T[pt][\![q]\!][e_1(X), \ldots, e_k(X), e_1(\tilde{X}), \ldots, e_{n-k}(\tilde{X})]/I_q \end{align*}$$

is the claimed presentation, regarded as a module over the $\langle q \rangle $ -adically complete ring $R=\mathrm {K}_T[pt][\![q]\!]$ , and the free R-module $N=\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ . Proposition 8.1 implies that if one takes the quotient by $\langle q \rangle $ , $\Psi $ becomes an isomorphism. The fact that M is a finite R-module follows from [Reference EisenbudEis95, Ex. 7.4, p. 203] (cf. Remark A.4), applied to the ideal $\mathfrak {m}=\langle q \rangle $ , and then noting that M is finite over

$$\begin{align*}S=\mathrm{K}_T[pt][\![q]\!][e_1(X), \ldots, e_k(X), e_1(\tilde{X}), \ldots, e_{n-k}(\tilde{X})] . \end{align*}$$

Note that R and S are Noetherian rings (e.g., by [Reference Atiyah and MacdonaldAM69, Thm. 10.26]) and that $\langle q \rangle $ is included in the Jacobson radical of R because $1-a q$ is invertible in R, for any $a \in \operatorname {\mathrm {K}}_T(pt)$ .Footnote ²

9 Physics, Wilson lines and Jacobian ring presentations

In this section, we will outline the approach giving the Jacobian relations (42) derived from a holomorphic function W called the (twisted) superpotential in the physics literature. These relations will be utilized in the next section to obtain the Coulomb branch presentation of $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ . See e.g. [Reference Jockers and MayrJM20, Reference Jockers and MayrJM19, Reference Jockers, Mayr, Ninad and TablerJMNT20, Reference Gu, Mihalcea, Sharpe and ZouGMSZ22, Reference Ueda and YoshidaUY20] for references.

Briefly, in the special case that a space can be realized as $V // G$ for V a complex vector space V and G a reductive algebraic group, the quantum K theory of $V//G$ arises from a three-dimensional ‘supersymmetric gauge theory’, which is ultimately defined by G, a representation $\rho $ defining the G action on V, and a matrix of real numbers $\tilde {k}$ (of the same rank as G), whose values we will give momentarily. The three-dimensional theory lives on a three-manifold, which is taken to be $\Sigma \times S^1$ for a Riemann surface $\Sigma $ , partly as a result of which the theory can be described as a two-dimensional theory on $\Sigma $ . Correlation functions in the two-dimensional theory include ‘Wilson lines’ (see, for example, [Reference Closset and KimCK16, section 2])

(32) $$ \begin{align} \mathrm{Tr}_{\rho} P \exp\left( \int_{S^1} A \right) \end{align} $$

(for $\rho $ a representation of G, A a connection on a principal G bundle, P a path-ordering symbol) on $S^1$ over a fixed point in $\Sigma $ , and those Wilson lines correspond to K theory elements on $V//G$ . More precisely, we will identify them with Schur functors on certain vector bundles on $V//G$ associated to representations of G; see Remark 9.3 below. Quantum K theory of spaces described as critical loci of holomorphic functions on noncompact symplectic quotients can also be described physically, but in this section, we focus on the simpler case of spaces that are themselves symplectic quotients of vector spaces. Given $V//G$ , the quantum K theory relations arise as derivatives of a holomorphic function known as the superpotential and conventionally denoted W; cf. Equation (36) below.

For simplicity, we specialize to the case that $G = \mathrm {GL}_k$ for a positive integer k. Then, define $\tilde {k}_{1}$ , $\tilde {k}_{2} \in {\mathbb R}$ as follows. Write $\rho $ as a sum of irreducible representations

(33) $$ \begin{align} \rho \: = \: \rho_1 \oplus \cdots \oplus \rho_{\ell}. \end{align} $$

Define

(34) $$ \begin{align} \tilde{k}_{1} \: = \: -\frac{1}{2} \sum_{\gamma=1}^{\ell} \left( \mathrm{Cas}_1(\rho_{\gamma}) \right)^2, \: \: \: \tilde{k}_{2} \: = \: k \: - \: \frac{1}{2} \sum_{\gamma=1}^{\ell} \frac{\dim \rho_{\gamma}}{\dim GL(k)} \mathrm{Cas}_2(\rho_{\gamma}), \end{align} $$

where Cas denotes eigenvalues of Casimir operators as in, for example, [Reference IachelloIac06, chapter 7].

Then, the quantum K theory ring relations are given physically as the critical locus equations of a function W known as the superpotential, which is given as [Reference Closset and KimCK16, equ’n (2.33)], [Reference Gu, Mihalcea, Sharpe and ZouGMSZ22, equ’n (2.1)]

(35) $$ \begin{align} W & = \frac{\tilde{k}_{2}}{2} \sum_{a=1}^{k} \left(\ln X_{a} \right)^2 \: + \: \frac{\tilde{k}_{1} - \tilde{k}_{2}}{2 k} \left( \sum_{a=1}^{k} \ln X_{a} \right)^2\nonumber \\& \quad + \left( \ln(-1)^{k-1} q \right) \sum_{a=1}^{k} \ln X_{a}\nonumber \\& \quad + \sum_{\alpha} \left[ \mathrm{Li}_2\left( \exp (\tilde{\rho}_{\alpha}, \ln X) \right) \: + \: \frac{1}{4} \left( \tilde{\rho}_{\alpha}, \ln X \right)^2 \right], \end{align} $$

where we use $\ln X$ to denote a vector with components $( \ln X_a )$ . Here, $(X_{a})$ is a point in $({\mathfrak R} \otimes _{\mathbb Z} {\mathbb C} - \Delta )/{\mathcal W} \cong ({\mathbb C}^{k} - \Delta )/{\mathcal W}$ for ${\mathfrak R}$ the root lattice of $\mathrm {GL}_k$ ,

(36) $$ \begin{align} \Delta \: = \: \coprod_{a < b} \{ X_a = X_b \} \, \coprod_a \{X_a = 1 \} \, \coprod_a \{X_a = 0 \}, \end{align} $$

${\mathcal W} = S_k$ the Weyl group of $U(k)$ , $(\cdot ,\cdot )$ denotes a natural pairing between root and weight lattice vectors, and $\{ \tilde {\rho }_{\alpha } \}$ are the weight vectors of the representation $\rho $ . It can be shown that the superpotential is invariant under the action of the Weyl group.

For the case of a Grassmannian $\mathrm {Gr}(k;n)$ , described as the GIT quotient $V // \mathrm {GL}_k$ for ${V = \mathrm {Hom}({\mathbb {C}}^k, {\mathbb {C}}^n)}$ , with $\rho $ given by a sum of n copies of the fundamental representation,

(37) $$ \begin{align} \tilde{k}_1 \: = \: -n/2, \: \: \: \tilde{k}_2 \: = \: k - n/2, \end{align} $$

and the superpotential specializes to

(38) $$ \begin{align} W & = \frac{k}{2} \sum_{a=1}^k \left( \ln X_a \right)^2 \: - \: \frac{1}{2} \left( \sum_{a=1}^k \ln X_a \right)^2\nonumber \\& \quad + \ln\left( (-1)^{k-1} q \right) \sum_{a=1}^k \ln X_a \: + \: n \sum_{a=1}^k \mathrm{Li}_2 \left( X_a \right). \end{align} $$

Remark 9.3. A Wilson line in representation $\phi $ of $U(k)$ is the Chern character of the Schur functor $\mathfrak {S}_{\phi } {\mathcal S}$ for ${\mathcal S}$ the universal subbundle, where the $X_a$ are exponentials of Chern roots. We give a few simple examples below:

We claim that the quantum K theory ring is determined by an analogue of the Jacobian ring of W (involving exponentials of derivatives rather than just derivatives). For the moment, we compute the relations generated by W, and then later we will observe that the resulting ring is the quantum K theory ring of $\mathrm {Gr}(k;n)$ as presented by [Reference Gorbounov and KorffGK17].

Returning to the Grassmannian $\mathrm {Gr}(k;n)$ , using the possibly obscure fact that

(39) $$ \begin{align} y \frac{\partial}{\partial y} \mathrm{Li}_2(y) \: = \: - \ln(1 - y), \end{align} $$

we find for each $1 \le a \le k$ that

(40) $$ \begin{align} \exp\left( \frac{\partial W}{\partial \ln X_a} \right) \: = \: 1 \end{align} $$

implies that

(41) $$ \begin{align} (-1)^{k-1} q \left( X_a \right)^k \: = \: \left( \prod_{b=1}^k X_b \right) \left(1 - X_a \right)^n. \end{align} $$

There is also an equivariant version of these identities. Let $T_i \in \operatorname {\mathrm {K}}_T(pt)$ denote equivariant parameters. These appear in the pertinent physical theories as exponentials of ‘twisted masses’. Concretely, in cases with twisted masses, the superpotential (38) for $\mathrm {Gr}(k;n)$ generalizes to [Reference Ueda and YoshidaUY20]

$$ \begin{align*} W & = \frac{k}{2} \sum_{a=1}^k \left( \ln X_a \right)^2 \: - \: \frac{1}{2} \left( \sum_{a=1}^k \ln X_a \right)^2 \\& \quad + \ \ln\left( (-1)^{k-1} q \right) \sum_{a=1}^k \ln X_a \: + \: \sum_{i=1}^n \sum_{a=1}^k \mathrm{Li}_2 \left( X_a T_i^{-1} \right). \end{align*} $$

Simplifying

(42) $$ \begin{align} \exp\left( \frac{\partial W}{\partial \ln X_a} \right) \: = \: 1 \end{align} $$

for each $1 \le a \le k$ , we find

(43) $$ \begin{align} (-1)^{k-1} q (X_a)^k \prod_{j=1}^{n} T_j = \left(\prod_{b=1}^k X_b\right) \cdot \prod_{i=1}^n (T_i - X_a) . \end{align} $$

In the next section, we will symmetrize these relations to obtain the Coulomb branch presentation of $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ . This will be proved to be isomorphic to the $\operatorname {\mathrm {QK}}$ Whitney ring from Theorem 8.2. We also note that the equations (43) are the same as the Bethe Ansatz equations from [Reference Gorbounov and KorffGK17, equ’n (4.17)]. In loc.cit., the authors utilize an approach based on algebraic properties of integrable systems and of equivariant localization, to obtain a distinct presentation of $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ . See also §12.1.2 below for a comparison.

Observe that in the non-equivariant specialization (i.e., when $T_i = 1$ ), the equations (43) specialize to (41).

Example 9.6. In the case of $\mathrm {Gr}(2;5)$ , the superpotential is given by

(44) $$ \begin{align} W & = \frac{1}{2} \left( \ln X_1 \right)^2 \: + \: \frac{1}{2} \left( \ln X_2 \right)^2 \: - \: \left( \ln X_1 \right) \left( \ln X_2 \right) \nonumber \\& \quad + \ln\left( - q \right) \sum_{a=1}^2 \ln X_a \: + \: \sum_{i=1}^5 \sum_{a=1}^2 \mathrm{Li}_2 \left( X_a T_i^{-1} \right), \end{align} $$

and the chiral ring relations (43) are, for $a \in \{1, 2 \}$ ,

(45) $$ \begin{align} \prod_{i=1}^5 \left( T_i - X_a \right) \: = \: (-q) \frac{ X_a^2 }{ X_1 X_2} \prod_{j=1}^5 T_j. \end{align} $$

In the nonequivariant case, we take $T_i = 1$ , then the chiral ring relations become

(46) $$ \begin{align} -q X_1 \: = \: X_2 (1-X_1)^5, \: \: \: -q X_2 \: = \: X_1 (1-X_2)^5, \end{align} $$

in agreement with (41). We show in the next section how the symmetrization of this leads the quantum $\operatorname {\mathrm {K}}$ relations; see also Appendix B below.

10 Coulomb branch and quantum Whitney presentations

The goal of this section is to obtain the Coulomb branch presentation we denoted by $\widehat {\operatorname {\mathrm {QK}}}_T(\mathrm {Gr}(k;n))$ , and predicted by physics. We will prove in Theorem 10.2 that this is equivalent to the presentation of $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ from Theorem 8.2.

The idea of obtaining the Coulomb branch presentation was already used in the authors’ previous work [Reference Gu, Mihalcea, Sharpe and ZouGMSZ22]. First, one rewrites the ‘vacuum equations’ (43) in terms of the ‘shifted variables’ from (47) below. Since $\mathrm {Gr}(k;n) = Hom({\mathbb {C}}^k, {\mathbb {C}}^n)//\mathrm {GL}_k$ , any presentation has to satisfy a symmetry with respect to $S_k$ , the Weyl group of $\mathrm {GL}_k$ . While the ideal generated by equations (43) is symmetric under permutations in $S_k$ , the individual generators are not. To rectify this, we write down a ‘characteristic polynomial’ (cf. Equation (52) below) where all the coefficients satisfy the required symmetry. Then we utilize the Vieta equations for this polynomial to obtain a set of polynomial equations. These are $S_k \times S_{n-k}$ symmetric, and they give a presentation of the equivariant quantum K ring. To start, define

(47) $$ \begin{align} \zeta_i \: = \: 1 - T_i, \: \: \: z_a \: = \: 1-X_a \quad (1 \le i \le n ; \quad 1 \le a \le k) , \end{align} $$

so that for any a, equation (43) becomes

(48) $$ \begin{align} \left( \prod_{i=1}^n (z_a - \zeta_i ) \right) \left( \prod_{b \neq a} ( 1- z_b) \right) \: + \: (-1)^k q (1-z_a)^{k-1} \prod_{i=1}^n (1-\zeta_i) \: = \: 0 . \end{align} $$

A key observation is that we may rewrite this in the form

(49) $$ \begin{align} (z_a)^n \: + \: \sum_{i=0}^{n-1} (-1)^{n-i} (z_a)^i \hat{g}_{n-i}(z,\zeta,q) \: = \: 0 . \end{align} $$

where the $\hat {g}_i(z,\zeta ,q)$ are symmetric polynomials in $z_1, \ldots , z_k$ and $\zeta _1, \ldots , \zeta _n$ . To do this, it suffices to take $a=1$ . The only nontrivial part is the symmetry in the variables z. This follows from repeated application of the identity

$$\begin{align*}e_j(z_2, \ldots, z_k) = e_j(z_1, \ldots, z_k) - z_1 e_{j-1}(z_2, \ldots , z_k) \end{align*}$$

to the factor $\prod _{j=2}(1-z_j)= \sum _{i=0}^{k-1} (-1)^i e_i(z_2, \ldots , z_k)$ , then collecting the resulting powers of $z_1$ .

To state the formula for the polynomial $\hat {g}_\ell (z, \zeta ,q)$ , we fix some notation. Set

$$\begin{align*}c^z= \prod_{i=1}^k (1-z_i) = \sum_{i \ge 0} (-1)^i e_i(z) ; ~ c_{\le j}^z = \sum_{i = 0}^j (-1)^i e_i(z) ;~ c_{\ge j}^z = (-1)^j( c(z)- c_{\le j-1}(z)) .\end{align*}$$

Informally, these are truncations of the Chern polynomial in $z=(z_1, \ldots , z_k)$ . One defines similarly $c^\zeta , c_{\le j}^\zeta ,c_{\ge j}^\zeta $ ; these are polynomials in $\zeta =(\zeta _1,\ldots , \zeta _n)$ . Set

(50) $$ \begin{align} c^{\prime}_{\ge \ell}(z,\zeta)= e_\ell(\zeta) + e_{\ell-1}(\zeta) c_{\ge 2}^z+ e_{\ell-2}(\zeta) c_{\ge 3}^z+ \ldots + e_{\ell - k+1}(\zeta) c_{\ge k}^z . \end{align} $$

If clear from the context, we will drop the variables $z,\zeta $ from the notation. (As usual, $e_i(z) =e_i(\zeta ) = 0$ for $i < 0$ and $e_0(z) = e_0(\zeta ) =1$ .) Define the matrices

$$\begin{align*}E= \begin{pmatrix} -1 & 0 & \ldots & 0 \\ -e_1 & -1 & \ldots & 0 \\ \vdots & \vdots & \ddots & 0 \\ -e_{k-1} & -e_{k-2} & \ldots & -1 \end{pmatrix} ; \end{align*}$$

$$\begin{align*}C^\zeta_{\ge n-k+2}= \begin{pmatrix} c_{\ge n-k+2}^\zeta \\ \vdots \\ c_{\ge n}^\zeta \\ 0 \end{pmatrix} ; \quad C^{z,\zeta}_{\ge n-k+1}= \begin{pmatrix} c^{\prime}_{\ge n-k+1} \\ c^{\prime}_{\ge n-k+2} \\ \vdots \\ c^{\prime}_{\ge n} \end{pmatrix} .\end{align*}$$

Besides their usefulness in the lemma below, these matrices will appear naturally in §11 below, in relation to an equivariant generalization of Grothendieck polynomials.

Lemma 10.1. The polynomial coefficients $\hat {g}_{\ell }(z,\zeta ,q)$ from (49) are given by

(51) $$ \begin{align} \begin{cases} c^{\prime}_{\ge \ell}(z,\zeta) & \textrm{ if } 1 \leq \ell \leq n-k \\ c^{\prime}_{\ge \ell}(z,\zeta) + \bigl(E\cdot C^\zeta_{\ge n-k+2}\bigr)_\ell + (-1)^{n+k} q \binom{k-1}{n-\ell} c^\zeta & \textrm{ if } n-k+1 \le \ell \le n . \end{cases} \end{align} $$

Here, $M_\ell $ denotes the $\ell -n+k$ -th component of the k-component column matrix M.

Define a ‘characteristic polynomial’ $f(\xi , z, \zeta ,q)$ by

(52) $$ \begin{align} f(\xi,z,\zeta,q) \: = \: \xi^n \: + \: \sum_{i=0}^{n-1} (-1)^{n-i} \xi^i \hat{g}_{n-i}(z,\zeta,q) . \end{align} $$

From (49), we deduce that $f(\xi ,z,\zeta ,q) = 0$ whenever $\xi = z_a$ for some a. Since f is a degree n polynomial in $\xi $ , the equation $f(\xi ,z,\zeta ,q) = 0$ has n roots in some appropriate field extension, which by construction include $z_1, \ldots , z_k$ . Let $\{z, \hat {z} \} = \{z_1, \cdots , z_k; \hat {z}_{k+1}, \cdots , \hat {z}_n \}$ denote the n roots of (52). From Vieta’s formula,

(53) $$ \begin{align} \sum_{i+j = \ell} e_i(z) e_j(\hat{z}) \: = \: \hat{g}_{\ell}(z,\zeta,q) . \end{align} $$

This determines the ‘Coulomb branch ring’:

(54) $$ \begin{align} \widehat{\operatorname{\mathrm{QK}}}_T(\mathrm{Gr}(k;n))= \operatorname{\mathrm{K}}_T(pt)[\![q]\!][e_1(z), \cdots, e_k(z), e_1(\hat{z}), \cdots, e_{n-k}(\hat{z})] / \hat{J}, \end{align} $$

where $\hat {J}$ is the ideal generated by the polynomials

(55) $$ \begin{align} \sum_{i+j=\ell} e_i(z) e_j(\hat{z}) - \hat{g}_{\ell}(z,\zeta,q) ; \quad 1 \leq \ell \leq n . \end{align} $$

Our goal is to demonstrate that the relations above are equivalent to those from Theorem 8.2 arising from the $\operatorname {\mathrm {QK}}$ -Whitney relations. To this aim, apply the same change of variables (47) to the relations (31), denoting in addition $\tilde {z}_i = 1 - \tilde {X}_i$ . Then the presentation in Theorem 8.2 can be written as

(56) $$ \begin{align} \widetilde{\operatorname{\mathrm{QK}}}_T(\mathrm{Gr}(k;n))= \operatorname{\mathrm{K}}_T(pt)[\![q]\!][e_1(z),\ldots,e_k(z),e_1(\tilde{z}),\ldots,e_{n-k}(\tilde{z} )]/{\tilde{J}}, \end{align} $$

where the ideal $\tilde {J}$ is generated by $\sum _{i+j=\ell } e_{i}(z) e_{j}(\tilde {z}) \: - \: \tilde {g}_{\ell } (z,\zeta ,q)$ for

(57) $$ \begin{align} \tilde{g}_{\ell} (z,\zeta,q) \: = \: e_{\ell}(\zeta) \: - \: \frac{q}{1-q}\sum_{s=n-k+1}^{\ell}(-1)^s \binom{n-s}{\ell-s} \Delta_{s+k-n}, \end{align} $$

for $1 \leq \ell \leq n$ , and $\Delta _{i} = e_{i}(1-z) e_{n-k}(1-\tilde {z})$ . Note that $\tilde {g}$ also depends on $\tilde {z}$ , although this is not included in the notation. (In fact, in the proof of Theorem 10.2, we will eliminate the dependence on $\tilde {z}$ ; see Theorem 11.8.) To get a more explicit formula, observe that

(58) $$ \begin{align} e_{i}(1-x_1, \ldots, 1-x_n) = \sum_{s=0}^i (-1)^s {n-s \choose i-s} e_s(x_1, \ldots, x_n) .\end{align} $$

An easy algebra manipulation based on this formula shows that for $1 \le \ell \le n$ ,

$$\begin{align*}\sum_{s=n-k+1}^{\ell}(-1)^s \binom{n-s}{\ell-s} e_{s+k-n}(1-z) = (-1)^{n-k+1}\Big({k \choose \ell +k-n}- e_{\ell+k-n}(z)\Big) ;\end{align*}$$

therefore, (57) may be rewritten as

(59) $$ \begin{align} \tilde{g}_{\ell} (z,\zeta,q) \: = \: e_{\ell}(\zeta) + (-1)^{n-k} \frac{q}{1-q} e_{n-k}(1-\tilde{z}) \Big({k \choose \ell +k-n}- e_{\ell+k-n}(z) \Big) . \end{align} $$

Note that the ring in (c) was proved in Theorem 8.2 to be isomorphic to the ‘geometric’ ring $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ . We already proved that the isomorphism of the rings in (b) and (c) follows from the change of variables

$$ \begin{align*} z_i \: = \: 1 - X_i \quad (1 \le i \le k) ; \quad \tilde{z}_j \: = \: 1 - \tilde{X}_j \quad (1 \le j \le n-k) . \end{align*} $$

The isomorphism between (a) and (b) is proved in the next section. In the process, we will reformulate these presentations in terms of (equivariant) Grothendieck polynomials and (equivariant) complete homogeneous symmetric functions; see Theorem 11.8 below. An example for $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(2;5))$ is given in Appendix B.

11 An isomorphism between the Whitney and Coulomb branch presentations

The goal of this section is to prove Theorem 11.12, thus finishing the proof of Theorem 10.2. We utilize the notation from §10.

11.1 Grothendieck polynomials

We start by recording some algebraic identities about the Grothendieck polynomials $G_j(z)=G_j(z_1, \ldots , z_k)$ , indexed by single row partitions. As usual, $z=(z_1, \ldots , z_k)$ , and $e_i=e_i(z), h_i=h_i(z)$ denote the elementary symmetric function, respectively the complete homogeneous symmetric function. It was proved in [Reference LenartLen00, p.80] that

(60) $$ \begin{align} G_j(z) = \sum_{a,b \ge 0 ; a+b \le k} (-1)^b h_{j+a} e_b = h_j+(h_{j+1} - h_j e_1) + (h_{j+2} - h_{j+1} e_1+ h_j e_2) + \ldots . \end{align} $$

An equivalent formulation proved in [Reference LenartLen00, Thm. 2.2]) is

(61) $$ \begin{align} G_j(z) = h_j(z) - \sum_{a=2}^{k} (-1)^{a} s_{(j,1^{a-1})}(z) , \end{align} $$

where $s_\mu (z)$ denotes the Schur polynomial, and $(j,1^{a-1})$ is the partition $(j, 1, \ldots , 1)$ with $a-1$ $1$ ’s. We refer to [Reference LenartLen00] for more about these polynomials. We record a Cauchy-type identity for Grothendieck polynomials. It is likely well known, but we could not find a reference.

Lemma 11.1. If $\ell \ge 1$ and $z=(z_1, \ldots , z_k)$ , then

(62) $$ \begin{align} \sum_{i,j \ge 0 ; i+j=\ell} (-1)^j e_i(z) G_j(z) = e_{\ell+1}(z) - e_{\ell+2}(z) + \ldots . \end{align} $$

In particular, for $\ell \ge k$ , $\sum _{i,j \ge 0 ; i+j=\ell } (-1)^j e_i(z) G_j(z) = 0$ .

Proof. From (61), we obtain

$$ \begin{align*} \sum_{i+j=\ell} (-1)^j e_i(z) G_j(z) &\:=\: e_\ell(z) + \sum_{j=1}^{\ell} \sum_{a=1}^{k}(-1)^{j+a-1} e_{\ell-j} (z) s_{(j,1^{a-1})}(z) \\ &\:=\: e_\ell(z) + \sum_{a=1}^k (-1)^{a-1} \left[\sum_{j=1}^{\ell} (-1)^{j} e_{\ell-j}(z) s_{(j,1^{a-1})}(z) \right]. \end{align*} $$

To prove the lemma, it suffices to show that

$$\begin{align*}\sum_{j=1}^{\ell} (-1)^{j} e_{\ell-j}(z) s_{(j,1^{a-1})}(z) = - e_{\ell + a- 1}(z) . \end{align*}$$

The case when $a=1$ follows from $\sum _{i+j=\ell } (-1)^j e_i(z) h_j(z) =0$ (i.e., the usual Cauchy formula) for $\ell \ge 1$ . For $a>1$ , we utilize the Pieri formula to multiply a Schur function by $e_i(z)$ (cf. e.g., [Reference FultonFul97]). To start, only Schur functions $s_\mu $ with $\mu _1 \le \ell +1$ and $1 \le \mu _2 \le 2$ may appear. Furthermore, if $\mu =(\mu _1, \ldots , \mu _s)$ is such a partition with $1<\mu _1\le \ell +1$ , then the Schur function $s_\mu $ appears in the left-hand side twice, in the expansion of the terms

$$\begin{align*}(-1)^{\mu_1}e_{\ell-\mu_1}(z) s_{\mu_1,1^{a-1}}(z) + (-1)^{\mu_1-1}e_{\ell-\mu_1+1}(z) s_{\mu_1-1,1^{a-1}}(z) . \end{align*}$$

Since the coefficients of $s_\mu $ appearing in the Pieri rule equal to $1$ , and since the terms above have opposite signs, it follows that the corresponding $s_\mu $ ’s cancel. We are left with the situation when $\mu _1 =1$ . Then necessarily $j=1$ (i.e., we consider the terms $- e_{\ell -1}(z) e_{a}(z)$ ). Again by Pieri formula, the only multiple of $s_\mu $ with $\mu _1=1$ is $- e_{\ell +a-1}(z)$ . This finishes the proof.

Define the column matrices:

$$\begin{align*}H= \begin{pmatrix} (-1)^{n-k+1} h_{n-k+1} \\ (-1)^{n-k+2} h_{n-k+2} \\ \vdots \\ (-1)^{n} h_{n} \end{pmatrix} ; \quad G= \begin{pmatrix} (-1)^{n-k+1} G_{n-k+1} \\ (-1)^{n-k+2} G_{n-k+2} \\ \vdots \\ (-1)^{n} G_{n} \end{pmatrix} .\end{align*}$$

Define also the $k \times k$ matrix

$$ \begin{align*} { A \: = \: \begin{pmatrix} c_{ \le k-1} &- c_{ \le k-2} &\cdots &\cdots &\cdots &\cdots &(-1)^{k-1} \\ (-1)^{k-2}c_{ \ge k} &c_{\le k-2} &\cdots &\cdots &\cdots &\cdots &(-1)^{k-2} \\ \vdots & \ddots &\ddots &\ddots &\cdots &\cdots &\vdots \\ (-1)^{k-i}c_{ \ge k} &\cdots & (-1)^{k-i} c_{ \ge k-i+2} & c_{ \le k-i} &-c_{\le k-i-1} &\cdots &(-1)^{k-i} \\ \vdots & \ddots &\ddots &\ddots &\ddots &\cdots &\vdots \\ -c_{\ge k} &\cdots &\cdots &\cdots &\cdots &c_{\le 1} &-1 \\ c_{ \ge k} &\cdots &\cdots &\cdots &\cdots &c_{\ge 2} & 1 \\ \end{pmatrix}. } \end{align*} $$

Lemma 11.2. The following equality holds: $G = A H$ .

Proof. This follows from (60) together with the Cauchy formula $\sum _{a+b=j} (-1)^a h_a e_b =0$ for $j \ge 1$ .

11.2 Equivariant deformations

Motivated by Propositions 11.6 and 11.7 below, define the polynomials $h^{\prime }_j,G^{\prime }_j \in {\mathbb {C}}[z,\zeta ]$ by

$$\begin{align*}h^{\prime}_j(z,\zeta) = \sum_{a+b=j} (-1)^{a} e_a(\zeta) h_b(z) ; \quad G_j^{\prime} (z,\zeta)=\sum_{a+b=j} (-1)^{a} e_a(\zeta) G_b(z) . \end{align*}$$

Note that both are symmetric polynomials in variables z and $\zeta $ . The polynomial $h^{\prime }_j(z,\zeta )$ is a specialization of the factorial complete homogeneous polynomial (see, for example, [Reference MihalceaMih08]), but the polynomial $G^{\prime }_j(z,\zeta )$ is far from being the factorial Grothendieck polynomial [Reference McNamaraMcN06], which is much more complicated.Footnote ⁴

Lemma 11.3. The following Cauchy-type formulae hold:

1. (a) If $\ell \ge 1$ , then

   $$\begin{align*}\sum_{a+b=\ell} (-1)^a h_a^{\prime}(z,\zeta) e_b(z) = e_\ell(\zeta) . \end{align*}$$

2. (b) If $\ell \ge 1$ , then

   $$\begin{align*}\sum_{a+b=\ell} (-1)^a G_a^{\prime}(z,\zeta) e_b(z) = c^{\prime}_{\ge \ell}(z,\zeta) ,\end{align*}$$
   where $c^{\prime }_{\ge \ell }(z,\zeta )$ is defined in (50).

Proof. For (a), we collect the coefficients of $e_i(\zeta )$ from both sides. On the left-hand side this coefficient equals to

$$\begin{align*}\sum_{i \le a} (-1)^{a} e_{\ell-a}(z) (-1)^{i} h_{a-i}(z) = \sum_{i \le a} (-1)^{a-i} e_{\ell-a} (z) h_{a-i}(z) . \end{align*}$$

By Cauchy formula, this equals to $0$ if $i <\ell $ , and it equals to $1$ if $i=\ell $ , thus proving part (a). We apply the same strategy for (b). The coefficient of $e_\ell (\zeta )$ on the left-hand side equals to $1$ , and a calculation similar to (a) shows that if $i < \ell $ , the coefficient of $e_i(\zeta )$ is

$$\begin{align*}\sum_{i \le a} (-1)^{a} e_{\ell-a} (z) (-1)^i G_{a-i}(z) = e_{\ell-i+1} (z) - e_{\ell-i+2} (z) + \ldots = c_{\ge \ell - i+1}^z . \end{align*}$$

Here, the first equality follows from the Cauchy-type formula (62). This finishes the proof of (b).

Our next task is to write down the analogue of Lemma 11.2 relating the polynomials $G^{\prime }_\ell $ and $h^{\prime }_\ell $ . To this aim, in analogy to (60), define the polynomial

(63) $$ \begin{align} G_j^{\dagger}(z,\zeta) = \sum (-1)^b h^{\prime}_{j+a}(z,\zeta) e_b(z) = h^{\prime}_j+(h^{\prime}_{j+1} - h^{\prime}_j e_1) + (h^{\prime}_{j+2} - h^{\prime}_{j+1} e_1+ h^{\prime}_j e_2) + \ldots , \end{align} $$

where the sum is over $a,b \ge 0$ and $a+b \le k$ .

Lemma 11.4. For any $\ell \ge 1$ , the following holds:

$$\begin{align*}G_\ell^{\dagger}(z,\zeta) - G^{\prime}_\ell(z,\zeta) = (-1)^{\ell+1} (e_{\ell+1}(\zeta) - e_{\ell+2}(\zeta) + \ldots + (-1)^{k-1} e_{\ell+k}(\zeta)) .\end{align*}$$

Proof. This follows from a direct calculation, by identifying the coefficients of $e_i(\zeta )$ in both sides. We leave the details to the reader.

Define the column matrices:

$$\begin{align*}H'= \begin{pmatrix} (-1)^{n-k+1} h^{\prime}_{n-k+1} \\ (-1)^{n-k+2} h_{n-k+2} \\ \vdots \\ (-1)^{n} h^{\prime}_{n} \end{pmatrix} ; \quad G'= \begin{pmatrix} (-1)^{n-k+1} G^{\prime}_{n-k+1} \\ (-1)^{n-k+2} G^{\prime}_{n-k+2} \\ \vdots \\ (-1)^{n} G^{\prime}_{n} \end{pmatrix} .\end{align*}$$

Corollary 11.5. The following holds: $G'=AH'+C^\zeta _{\ge n-k+2}$ .

Proof. Let $G^\dagger $ be the column matrix with entries $(-1)^{n-k+i} G^\dagger _{n-k+i}(z,\zeta )$ . As in the proof of Lemma 11.2, $G^\dagger = A H'$ . Then the claim follows from the formula in Lemma 11.4.

11.3 Elimination of variables

Our strategy is to utilize the Vieta relations in order to eliminate the variables $\hat {z}_i, \tilde {z}_i$ from the quantum rings $\widehat {\operatorname {\mathrm {QK}}}_T(\mathrm {Gr}(k;n))$ and $\widetilde {\operatorname {\mathrm {QK}}}_T(\mathrm {Gr}(k;n))$ respectively, and then prove that the resulting rings are isomorphic. It turns out that the elimination process naturally leads to the study of variations of the Grothendieck, respectively the complete homogeneous polynomials.

Proposition 11.6. Consider the Coulomb branch ring $\widehat {\operatorname {\mathrm {QK}}}_T(\mathrm {Gr}(k;n))$ from (54). Then the following holds for $1 \le \ell \le n-k$ :

$$\begin{align*}e_\ell(\hat{z}) \equiv \sum_{i+j=\ell} (-1)^j e_i(\zeta) G_j(z) = (-1)^{\ell} G^{\prime}_\ell(z,\zeta) \quad\mod \hat{J} . \end{align*}$$

Proof. We proceed by induction on $\ell \ge 1$ . If $\ell =1$ , the claim follows from the definition of $g_1(z,\zeta ,q)$ and the fact that $G_1(z) = \sum _{i=1}^k (-1)^i e_i(z)$ from Equation (61). By induction and the relations (53), we obtain that for $\ell \ge 2$ ,

(64) $$ \begin{align} \begin{aligned} e_\ell(\hat{z}) & \equiv g_\ell(z,\zeta,q) - \sum_{s=0}^{\ell-1} e_{\ell -s}(z) \left(\sum_{i+j=s} (-1)^j e_i(\zeta) G_j(z) \right) \\ & = g_\ell(z,\zeta,q) - \sum_{s=0}^{\ell-1} e_{\ell -s}(z) (-1)^s G^{\prime}_s (z,\zeta) \\ & = g_\ell(z,\zeta,q) + (-1)^{\ell} G^{\prime}_\ell(z,\zeta) - c^{\prime}_{\ge \ell} (z,\zeta) \\ & = (-1)^{\ell} G^{\prime}_\ell(z,\zeta) . \end{aligned} \end{align} $$

Here, the third equality follows from Lemma 11.3(b), and the fourth follows from the definition of $\hat {g}_\ell $ from Lemma 10.1.

Proposition 11.7. Consider the quantum $\operatorname {\mathrm {K}}$ Whitney ring $\widetilde {\operatorname {\mathrm {QK}}}_T(\mathrm {Gr}(k;n))$ from (56). Then the following holds for $1 \le \ell \le n-k$ :

$$\begin{align*}{e_\ell(\tilde{z})} \equiv \sum_{i+j=\ell} (-1)^j e_i(\zeta) h_j(z) = (-1)^{\ell} h^{\prime}_\ell(z,\zeta) \quad\mod \tilde{J} . \end{align*}$$

Proof. The proof is similar to that for Proposition 11.6, utilizing the Cauchy formula in Lemma 11.3(a), and that in this case, $\tilde {g}_\ell (z,\zeta ) = e_\ell (\zeta )$ by Equation (59).

Recall the notation from (50) and the surrounding paragraphs. Define the column matrices

(65) $$ \begin{align} \tilde{R}= \begin{pmatrix} \tilde{g}_{n-k+1} \\ \tilde{g}_{n-k+2} \\ \vdots \\ \tilde{g}_{n} \end{pmatrix} ; \quad \hat{R}= \begin{pmatrix} \hat{g}_{n-k+1} \\ \hat{g}_{n-k+2} \\ \vdots \\ \hat{g}_{n} \end{pmatrix} ; \quad E_{n-k+1}^\zeta = \begin{pmatrix} e_{n-k+1}(\zeta) \\ \vdots \\ \vdots \\ e_n(\zeta) \end{pmatrix}. \end{align} $$

Theorem 11.8. (a) The ring $\widetilde {\operatorname {\mathrm {QK}}}(\mathrm {Gr}(k;n))$ is isomorphic to $\operatorname {\mathrm {K}}_T(pt)[\![q]\!][e_1, \ldots , e_k]/\widetilde {I}$ , where the ideal $\widetilde {I}$ is defined by $\tilde {R}=E H'+ E_{n-k+1}^\zeta $ ; that is,

$$\begin{align*}\begin{pmatrix} \tilde{g}_{n-k+1} \\ \tilde{g}_{n-k+2} \\ \vdots \\ \tilde{g}_{n} \end{pmatrix} = \begin{pmatrix} -1 & 0 & \ldots & 0 \\ -e_1 & -1 & \ldots & 0 \\ \vdots & \vdots & \ddots & 0 \\ -e_{k-1} & -e_{k-2} & \ldots & -1 \end{pmatrix} \begin{pmatrix} (-1)^{n-k+1} h^{\prime}_{n-k+1} \\ (-1)^{n-k+2} h^{\prime}_{n-k+2} \\ \vdots \\ (-1)^{n} h^{\prime}_{n} \end{pmatrix} + \begin{pmatrix} e_{n-k+1}(\zeta) \\ \vdots \\ \vdots \\ e_n(\zeta) \end{pmatrix}.\end{align*}$$

(b) The ring $\widehat {\operatorname {\mathrm {QK}}}(\mathrm {Gr}(k;n))$ is isomorphic to $\operatorname {\mathrm {K}}_T(pt)[\![q]\!][e_1, \ldots , e_k]/\widehat {I}$ , where the ideal $\widehat {I}_q$ is defined by $\hat {R}=E G'+C_{\ge n-k+1}^{z,\zeta } $ ; that is,

$$\begin{align*}\begin{pmatrix} \hat{g}_{n-k+1} \\ \hat{g}_{n-k+2} \\ \vdots \\ \hat{g}_{n} \end{pmatrix} = \begin{pmatrix} -1 & 0 & \ldots & 0 \\ -e_1 & -1 & \ldots & 0 \\ \vdots & \vdots & \ddots & 0 \\ -e_{k-1} & -e_{k-2} & \ldots & -1 \end{pmatrix} \begin{pmatrix} (-1)^{n-k+1} G^{\prime}_{n-k+1} \\ (-1)^{n-k+2} G^{\prime}_{n-k+2} \\ \vdots \\ (-1)^{n} G^{\prime}_{n} \end{pmatrix} + \begin{pmatrix} c^{\prime}_{\ge n-k+1} \\ \vdots \\ \vdots \\ c^{\prime}_{\ge n} \end{pmatrix}. \end{align*}$$

Proof. The isomorphisms are obtained by eliminating the variables $\tilde {z}_i$ , respectively $\hat {z}_i$ ( $1 \le i \le n-k$ ) from the first $n-k$ relations in $\widehat {\operatorname {\mathrm {QK}}}_T(\mathrm {Gr}(k;n))$ respectively $\widetilde {\operatorname {\mathrm {QK}}}_T(\mathrm {Gr}(k;n))$ (cf. (54) respectively (56)). We indicate the main steps to obtain these formulae.

Consider first $\widetilde {\operatorname {\mathrm {QK}}}_T(\mathrm {Gr}(k;n))$ . By Proposition 11.7, $e_j(\tilde {z})\equiv (-1)^j h^{\prime }_j(z,\zeta )$ modulo $\widetilde {I}$ ; we utilize this to eliminate the first $n-k$ relations from $\tilde {J}$ to show that $\widetilde {I}$ is generated by

$$\begin{align*}\sum_{a+b=\ell ; b \le n-k} (-1)^b e_a(z) h^{\prime}_b (z,\zeta)- \tilde{g}_\ell(z,\zeta) , \quad n-k+1 \le \ell \le n . \end{align*}$$

From the Cauchy formula in Lemma 11.3(a), it follows that for $n-k+1 \le \ell \le n$ ,

$$\begin{align*}\sum_{a+b=\ell ; b \le n-k} (-1)^b e_a h^{\prime}_b = e_\ell(\zeta)-\sum_{b=n-k+1}^{\ell} (-1)^{b} e_{\ell - b} h^{\prime}_{b} \equiv \tilde{g}_\ell \quad \mod \widetilde{I} . \end{align*}$$

Writing these expressions in matrix form proves the claim for $\widetilde {I}$ .

The same proof works for $ \widehat {I}$ , after utilizing that $e_j(\hat {z})\equiv (-1)^j G^{\prime }_j(z,\zeta )$ modulo $\widehat {I}$ by Proposition 11.6, and the Cauchy-type formula from Lemma 11.3(b); the matrix $C_{\ge n-k+1}^{z,\zeta }$ accounts for the right-hand side of the Cauchy formula.

11.4 Proof of Theorem 10.2

We need the following algebraic identities.

Lemma 11.9. (a) Let $A'$ be the antidiagonal transpose of A from (11.1). Then

$$\begin{align*}A' \begin{pmatrix} {k \choose 1}- e_{1}(z) \\ {k \choose 2}- e_{2}(z) \\ \vdots \\ {k \choose k}- e_{k}(z) \end{pmatrix} = \prod_{i=1}^{k}(1-z_i) \begin{pmatrix} {k-1 \choose k-1} \\ {k-1 \choose k-2}\\ \vdots \\ {k-1 \choose 0} \end{pmatrix} . \end{align*}$$

(b) $\hat {R} - C_{\ge n-k+1}^{z,\zeta } - E\cdot C^\zeta _{\ge n-k+2} = (-1)^{n+k} q \prod _{i=1}^n (1-\zeta _i) \begin {pmatrix} {k-1 \choose k-1} \\ {k-1 \choose k-2}\\ \vdots \\ {k-1 \choose 0} \end {pmatrix} $ .

Proof. Part (a) follows, for example, from induction on i; we leave the details to the reader. Part (b) is equivalent to Lemma 10.1.

Next, we state the key result which relates the two presentations.

Lemma 11.10. The following equality holds:

$$\begin{align*}\hat{R} - C_{\ge n-k+1}^{z,\zeta} - E\cdot C^\zeta_{\ge n-k+2} \equiv A' (\tilde{R} - E_{n-k+1}^\zeta) \quad\mod \widetilde{I} , \end{align*}$$

where $A'$ is the antidiagonal transpose of A.

Proof. From part (b) of Lemma 11.9, we need to show that

$$\begin{align*}A' (\tilde{R} - E_{n-k+1}^\zeta) \equiv (-1)^{n+k} q \prod_{i=1}^n (1-\zeta_i) \begin{pmatrix} {k-1 \choose k-1} \\ {k-1 \choose k-2}\\ \vdots \\ {k-1 \choose 0} \end{pmatrix} . \end{align*}$$

From the definition of $\tilde {g}_\ell $ from (59), it follows that for $\ell \ge n-k+1$ ,

$$\begin{align*}\tilde{g}_\ell - e_{\ell}(\zeta) \equiv (-1)^{n-k} \frac{q}{1-q} e_{n-k}(1-\tilde{z}) \Big({k \choose \ell +k-n}- e_{\ell+k-n}(z) \Big) . \end{align*}$$

By Lemma 11.9(a), for $n-k+1 \le \ell \le n$ ,

$$\begin{align*}\begin{aligned} A' (\tilde{R} - E_{n-k+1}^\zeta) & \equiv (-1)^{n-k} \frac{q}{1-q} e_{n-k}(1-\tilde{z}) A' \cdot \Big({k \choose \ell +k-n}- e_{\ell+k-n}(z) \Big)_{\ell+k-n} \\ & = (-1)^{n-k} \frac{q}{1-q} e_{n-k}(1-\tilde{z}) e_k (1-z) \begin{pmatrix} {k-1 \choose k-1} \\ {k-1 \choose k-2}\\ \vdots \\ {k-1 \choose 0} \end{pmatrix} . \end{aligned} \end{align*}$$

If one writes $e_{n-k}(1-\tilde {z}) e_k (1-z)$ in the presentation from Theorem 10.2(c), it follows that in $\widetilde {QK}_T(\mathrm {Gr}(k;n))$ ,

$$\begin{align*}e_{n-k}(1-\tilde{z}) e_k (1-z) = e_{n-k}(\tilde{X}) e_k(X)\equiv (1-q) \prod_{i=1}^n (1- \zeta_i) \quad\mod \widetilde{I} . \end{align*}$$

Then the claim follows by combining the previous equalities.

Corollary 11.11. The ideal $\widehat {I} \subset \widetilde {I}$ .

Proof. We need to show that $\hat {R}-EG'-C_{\ge n-k+1}^{z,\zeta } \equiv 0$ modulo $\widetilde {I}$ . From definitions, Lemma 11.10 and Corollary 11.5, we have

$$\begin{align*}\begin{aligned} \hat{R}-EG'-C_{\ge n-k+1}^{z,\zeta} & = \hat{R}-E(AH'+C_{\ge n-k+2}^\zeta)-C_{\ge n-k+1}^{z,\zeta} \\ & = \hat{R} - E C_{\ge n-k+2}^\zeta - C_{\ge n-k+1}^{z,\zeta} - EAH' \\ & \equiv A' (\tilde{R} - E_{n-k+1}^\zeta) - EAH' \\ & \equiv A E H' - EAH' . \end{aligned} \end{align*}$$

Then the claim follows because $EA = A' E$ as matrices with polynomial coefficients.

The next theorem finishes the proof of Theorem 10.2.

Theorem 11.12. There is $\operatorname {\mathrm {K}}_T(pt)[\![q]\!]$ -algebra isomorphism

$$\begin{align*}\operatorname{\mathrm{K}}_T(pt)[\![q]\!][e_1, \ldots, e_k]/\widehat{I} \to \operatorname{\mathrm{K}}_T(pt)[\![q]\!][e_1, \ldots, e_k]/\widetilde{I} \end{align*}$$

sending $e_i(z) \mapsto e_i(z)$ .

Proof. By Corollary 11.11, there is a surjective ring homomorphism of $\operatorname {\mathrm {K}}_T(pt)[\![q]\!]$ -algebras

$$\begin{align*}\Phi: \operatorname{\mathrm{K}}_T(pt)[\![q]\!][e_1(z), \ldots, e_k(z)]/\widehat{I} \to \operatorname{\mathrm{K}}_T(pt)[\![q]\!][e_1(z), \ldots, e_k(z)]/\widetilde{I} ,\end{align*}$$

sending $e_i(z) \mapsto e_i(z)$ . To prove $\Phi $ is an isomorphism, we follow the same approach as in the proof of Theorem 8.2, relying on Proposition A.3. We need to check that the Coulomb branch presentation $\operatorname {\mathrm {K}}_T(pt)[\![q]\!][e_1, \ldots , e_k]/\widehat {I}$ is a finite module over $\operatorname {\mathrm {K}}_T(pt)[\![q]\!]$ , and that we obtain an isomorphism after taking the quotient by $\langle q \rangle $ . To start, note that from the definition of the ideal $\hat {J}$ from (55), the variables $z_1, \ldots , z_k$ are solutions of the Bethe Ansatz equations (48). Then the same happens for the variables $z_1, \ldots , z_k$ appearing in $\operatorname {\mathrm {K}}_T(pt)[\![q]\!][e_1, \ldots , e_k]/\widehat {I}$ . From this, and from the special case $q=0$ of [Reference Gorbounov and KorffGK17, Lemma 4.7], we obtain that the quotient $\operatorname {\mathrm {K}}_T(pt)[\![q]\!][e_1, \ldots , e_k]/(\langle q \rangle + \widehat {I})$ is $\operatorname {\mathrm {K}}_T(pt)$ -module-generated by the factorial Grothendieck polynomials $G_\lambda (z;\xi )$ from [Reference McNamaraMcN06], where $\lambda $ is included in the $k \times (n-k)$ rectangle. (This result uses a determinantal formula for factorial Grothendieck polynomials from [Reference Gorbounov and KorffGK17, eq. 2.14], in turn attributed to [Reference Ikeda and NaruseIN13].) Since $\Phi $ is surjective, Proposition 8.1 and Lemma A.2 imply that the images $\Phi (G_\lambda (z;\xi ))$ modulo q form a basis in $\operatorname {\mathrm {K}}_T(\mathrm {Gr}(k;n))\simeq \operatorname {\mathrm {K}}_T(pt)[\![q]\!][e_1, \ldots , e_k]/(\langle q \rangle + \widetilde {I})$ . This implies that $\Phi $ must be an isomorphism modulo $\langle q \rangle $ . Then the same argument as in Theorem 8.2, based on [Reference EisenbudEis95, Ex. 7.4, p. 203] (cf. Remark A.4), implies that $\operatorname {\mathrm {K}}_T(pt)[\![q]\!][e_1, \ldots , e_k]/\widehat {I}$ is a finite module; thus, $\Phi $ is an isomorphism, by Proposition A.3.

12 Examples: Non-equivariant and quantum cohomology specializations

In this section, we illustrate the non-equivariant specializations of the Coulomb $\operatorname {\mathrm {QK}}$ Whitney presentations of $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(k;n))$ (i.e., when $\zeta _i=0$ for $1 \le i \le n$ ). The second part is dedicated to the quantum cohomology limit. In particular, we show that both presentations specialize to Witten’s presentation of $\operatorname {\mathrm {QH}}^{*}_T(\mathrm {Gr}(k;n))$ .

12.1 Non-equivariant presentations

We abuse notation and denote by the same symbols the non-equivariant specializations of relations, ideals, etc. For $\ell \ge 1$ , the polynomials $\tilde {g}_\ell , \hat {g}_\ell $ are given by

$$\begin{align*}\tilde{g}_\ell(z,q) = (-1)^{n-k} \frac{q}{1-q} e_{n-k}(1-\tilde{z})\Big({k \choose \ell +k-n}- e_{\ell+k-n}(z) \Big) ; \end{align*}$$

$$\begin{align*}\hat{g}_\ell(z,q) = c_{\ge \ell +1}^z + (-1)^{n+k} q {k-1 \choose n- \ell} . \end{align*}$$

In particular, if $1 \le \ell \le n-k$ , $\tilde {g}_\ell =0$ and $\hat {g}_\ell $ does not depend on q. The non-equivariant versions of propositions 11.7 and 11.6 state that for any $1 \le j \le n-k$ , the following identities hold:

(66) $$ \begin{align} e_j(\tilde{z}) = (-1)^j h_j(z) \quad\mod \tilde{J} ; \quad e_j(\hat{z}) = (-1)^j G_j(z) \quad\mod \hat{J} . \end{align} $$

Remark 12.1. It is well known that the Grothendieck polynomials $G_j(z)$ represent Schubert classes ${\mathcal {O}}_j \in K(\mathrm {Gr}(k;n))$ [Reference LascouxLas90]. Nevertheless, the geometric interpretation $e_j(\hat {z}) = (-1)^j {\mathcal {O}}_j$ fails in the equivariant case, already for $j=1$ . The Schubert divisor class may be calculated from the exact sequence:

$$\begin{align*}0 \to \wedge^k {\mathcal{S}} \otimes {\mathbb{C}}_{-t_1 - \ldots - t_k} \to {\mathcal{O}}_{\mathrm{Gr}(k;n)} \to {\mathcal{O}}_1 \to 0 . \end{align*}$$

Then, from the geometric interpretation of the variables $z_i,\zeta _i$ ,

(67) $$ \begin{align} {\mathcal{O}}_1 = 1 - \frac{(1-z_1) \cdot \ldots \cdot (1-z_k)} {(1-\zeta_1) \cdot \ldots \cdot (1-\zeta_k)} \quad \in \operatorname{\mathrm{K}}_T(\mathrm{Gr}(k;n)) .\end{align} $$

However, from Proposition 11.6,

$$\begin{align*}e_1(\hat{z}) = - G_1^{\prime}(z,\zeta) = - G_1(z) + e_1(\zeta) , \end{align*}$$

which is different from the expression in Equation (67).

Corollary 12.2. The rings $\widetilde {\operatorname {\mathrm {QK}}}(\mathrm {Gr}(k;n))$ and $\widehat {\operatorname {\mathrm {QK}}}(\mathrm {Gr}(k;n))$ are isomorphic to

$$\begin{align*}\widetilde{\operatorname{\mathrm{QK}}}(\mathrm{Gr}(k;n)) = {\mathbb{Z}}[\![q]\!][e_1, \ldots, e_k]/\widetilde{I} ; \quad \widehat{\operatorname{\mathrm{QK}}}(\mathrm{Gr}(k;n)) = {\mathbb{Z}}[\![q]\!][e_1, \ldots, e_k]/\widehat{I} , \end{align*}$$

where the ideal $\widetilde {I}$ is defined by $\tilde {R}=E H$ ; that is,

$$\begin{align*}\begin{pmatrix} \tilde{g}_{n-k+1} \\ \tilde{g}_{n-k+2} \\ \vdots \\ \tilde{g}_{n} \end{pmatrix} = \begin{pmatrix} -1 & 0 & \ldots & 0 \\ -e_1 & -1 & \ldots & 0 \\ \vdots & \vdots & \ddots & 0 \\ -e_{k-1} & -e_{k-2} & \ldots & -1 \end{pmatrix} \begin{pmatrix} (-1)^{n-k+1} h_{n-k+1} \\ (-1)^{n-k+2} h_{n-k+2} \\ \vdots \\ (-1)^{n} h_{n} \end{pmatrix}, \end{align*}$$

and the ideal $\widehat {I}$ is defined by $\hat {R}=E G+C^z_{\ge n-k+2}$ ; that is,

$$\begin{align*}\begin{pmatrix} \hat{g}_{n-k+1} \\ \hat{g}_{n-k+2} \\ \vdots \\ \hat{g}_{n} \end{pmatrix} = \begin{pmatrix} -1 & 0 & \ldots & 0 \\ -e_1 & -1 & \ldots & 0 \\ \vdots & \vdots & \ddots & 0 \\ -e_{k-1} & -e_{k-2} & \ldots & -1 \end{pmatrix} \begin{pmatrix} (-1)^{n-k+1} G_{n-k+1} \\ (-1)^{n-k+2} G_{n-k+2} \\ \vdots \\ (-1)^{n} G_{n} \end{pmatrix} + \begin{pmatrix} c_{\ge n-k+2}^z \\ \vdots \\ c_{\ge n}^z\\ 0 \end{pmatrix}. \end{align*}$$

12.1.1 Projective spaces

To illustrate both presentations, we take two ‘opposite’ examples: for the projective space $\mathrm {Gr}(1;n)$ and its dual $\mathrm {Gr}(n-1;n)$ . These are isomorphic manifolds, and their quantum $\operatorname {\mathrm {K}}$ rings are also isomorphic. But the isomorphism is highly nontrivial; this is expected, given that the Grassmannians are realized as different GIT quotients. A worked-out example for $\operatorname {\mathrm {QK}}_T(\mathrm {Gr}(2;5)$ is included in Appendix B.

If $k=1$ , then $G_n(z) = h_n(z) = z_1^n$ . The presentations from Corollary 12.2 are

$$\begin{align*}\widehat{\operatorname{\mathrm{QK}}}(\mathrm{Gr}(1;n)) = {\mathbb{Z}}[\![q]\!][z]/\langle z^{n} - q \rangle = \widetilde{\operatorname{\mathrm{QK}}}(\mathrm{Gr}(1;n)) . \end{align*}$$

Consider now $k=n-1$ . From (61), it follows that the Grothendieck polynomials $G_j$ are significantly more complicated than the polynomials $h_j$ . To illustrate, consider $\operatorname {\mathrm {QK}}(\mathrm {Gr}(3;4))$ . The $\operatorname {\mathrm {QK}}$ Whitney presentation is

$$\begin{align*}\widetilde{\operatorname{\mathrm{QK}}}(\mathrm{Gr}(3;4)) = \frac{{\mathbb{Z}}[\![q]\!][e_1(z_1,z_2,z_3),e_2(z_1,z_2,z_3),e_3(z_1,z_2,z_3)]}{\langle \tilde{g}_2 + h_2, \tilde{g}_3 + e_1 h_2 - h_3, \tilde{g}_4 + e_2 h_2 - e_1 h_3 + h_4 \rangle }, \end{align*}$$

where

$$\begin{align*}\tilde{g}_2 = \frac{q}{q-1}(1+e_1) (3-e_1) ; \quad \tilde{g}_3 = \frac{q}{q-1}(1+e_1)(3-e_2) ; \quad \tilde{g}_4 = \frac{q}{q-1}(1+e_1)(1-e_3) . \end{align*}$$

The Coulomb presentation is

$$\begin{align*}\widehat{\operatorname{\mathrm{QK}}}(\mathrm{Gr}(3;4)) = \frac{{\mathbb{Z}}[\![q]\!][e_1(z_1,z_2,z_3),e_2(z_1,z_2,z_3),e_3(z_1,z_2,z_3)]}{\langle \hat{g}_2 + G_2, \hat{g}_3 + e_1 G_2 - G_3- e_3, \hat{g}_4 + e_2 G_2 - e_1 G_3 + G_4 \rangle } ,\end{align*}$$

where $\hat {g}_2 = e_3 - q, {\hat {g}_3} = - 2q, \hat {g}_4 = - q$ and

$$\begin{align*}\begin{aligned} G_2 = h_2 - s_{2,1}+ s_{2,1,1} ; \quad G_3 = h_3 - s_{3,1}+ s_{3,1,1} ; \quad G_4 = h_4 - s_{4,1}+ s_{4,1,1} . \end{aligned} \end{align*}$$

12.1.2 Relation to the Gorbounov and Korff presentation

In this section, we recall the description of the non-equivariant quantum $\operatorname {\mathrm {K}}$ -theory ring $\operatorname {\mathrm {QK}}(\mathrm {Gr}(k;n))$ from Gorbounov and Korff’s paper [Reference Gorbounov and KorffGK17], based on the Bethe Ansatz equations. Then we indicate an isomorphism between this and our non-equivariant Coulomb presentation $\widehat {\operatorname {\mathrm {QK}}}(\mathrm {Gr}(k;n))$ .

For a variable $\xi $ , denote by $\ominus \xi = \frac {-\xi }{1-\xi }$ . Consider a sequence of indeterminates $E_1, \ldots , E_k, H_1, \ldots , H_{n-k}$ , and extend this sequence by requiring $E_0 = H_0 = 1$ and $H_{r} =0$ for $r> n-k$ and $E_{r}=0$ for $r> k$ . Consider the generating series

$$\begin{align*}H(\xi) = \sum_{r=0}^{n-k}(H_r - H_{r+1}) \xi^{n-k-r} ; \quad E(\xi)= \sum_{r=0}^{k}(E_r - E_{r+1}) \xi^{k-r} . \end{align*}$$

The following is stated in [Reference Gorbounov and KorffGK17, Theorem 1.1].

Theorem 12.3 (Gorbounov-Korff).

The non-equivariant quantum K theory ring of $\mathrm {Gr}(k;n)$ is generated by $H_1, \ldots , H_{n-k}$ , $E_1, \ldots , E_k$ with relations given by the coefficients of $\xi $ in the expansion of

(68) $$ \begin{align} H(\xi)E(\ominus \xi)\:=\: \left( \prod_{i=1}^k \ominus \xi \right) \xi^{n-k} (1-H_1) + q . \end{align} $$

Denote by $\operatorname {\mathrm {QK}}^{GK}(\mathrm {Gr}(k;n))$ the Gorbounov and Korff’s presentation. A direct computation gives that the coefficient of $\xi ^{\ell }$ in (68) is equal to

(69) $$ \begin{align} (-1)^k(H_{\ell} - H_{\ell+1}) + \sum_{j=1}^{\ell} (H_{\ell-j} - H_{\ell+1-j}) &\left[(-1)^{k-j}\sum_{s=j}^k\binom{s-1}{s-j}E_s\right]\notag \\ &\qquad\qquad\:=\: \left\{\begin{array}{cl} 0 & 1\leq \ell \leq n-k-1, \\ (-1)^{n-\ell}q \binom{k}{n-\ell} & n-k\leq \ell \leq n . \end{array}\right. \end{align} $$

We provide next an algebra isomorphism between the (non-equivariant) Coulomb branch presentation $\widehat {\operatorname {\mathrm {QK}}}(\mathrm {Gr}(k;n))$ and the presentation $\operatorname {\mathrm {QK}}^{GK}(\mathrm {Gr}(k;n))$ .

Proposition 12.4. Consider the map $\Xi : \operatorname {\mathrm {QK}}^{GK}(\mathrm {Gr}(k;n)) \to \widehat {\operatorname {\mathrm {QK}}}(\mathrm {Gr}(k;n))$ defined by

$$\begin{align*}\Xi (H_j) = G_j(z) \textrm{ for } 1 \le j \le n-k ; \quad \Xi(E_j) = G_{1^j}(z) \textrm{ for } 1 \le j \le k . \end{align*}$$

Then $\Xi $ is an algebra isomorphism.

Proof. The main argument is showing that $\Xi $ is well defined. To calculate the image of the left-hand side of (69), one utilizes the polynomial identity $e_\ell (z) = \sum _{j=\ell }^k\binom {j-1}{j-\ell }G_{1^j}(z)$ (see, for example, [Reference LenartLen00, Thm. 2.2]), together with the (non-equivariant) Cauchy identity from Lemma 11.3(b). We leave the details to the reader.

12.2 Quantum cohomology as a limit

Next, we discuss how the quantum K theory reduces to the quantum cohomology. Mathematically, this is achieved by taking leading terms or, equivalently, taking the associated graded rings. In physics this corresponds to ‘taking a two-dimensional limit’, which we recall next; see [Reference Gu, Mihalcea, Sharpe and ZouGMSZ22]. If we assume the theory is defined on a 3-manifold $S^1 \times \Sigma $ for some Riemann surface $\Sigma $ , where $S^1$ has diameter L, then

(70) $$ \begin{align} \begin{aligned} q &= L^n q_{2d} , \quad X_a = \exp\left( L \sigma_a \right) = 1 + L \sigma_a + \frac{L^2}{2} {\sigma_a}^2 + \cdots, \\ T_i &= \exp\left( L t_i \right) = 1 + L t_i + \frac{L^2}{2} {t_i}^2 + \cdots \end{aligned} \end{align} $$

Here, $q_{2d}$ is the quantum parameter in the $2d$ theory and $t_i$ are elements in $H^{*}_T(pt)$ (in physics terminology, the twisted masses). The term $-L t_i$ corresponds to the cohomological parameter $t_i \in H^{*}_T(pt)$ . We have

$$\begin{align*}z_a = 1 - X_a = -L \sigma_a - \frac{L^2}{2} {\sigma_a}^2 - \ldots ; \quad \zeta_i = 1-T_i = -L t_i - \frac{L^2}{2} {t_i}^2 - \ldots . \end{align*}$$

In the two-dimensional limit, $L \rightarrow 0$ and the leading terms will dominate. We now take the $2d$ limit in Equation (43) arising from the Coulomb branch. After taking the leading terms, (43) becomes

$$ \begin{align*} \prod_{i=1}^n\left(\sigma_a - t_i \right) = (-1)^{k-1} q_{2d}, \quad a = 1, \dots, k. \end{align*} $$

These are the generators which determine the chiral rings of the $2d$ gauge linear sigma model (GLSM) for $\mathrm {Gr}(k;n)$ when twisted masses $t_i$ are turned on. The $2d$ limits of the polynomials $\hat {g}_{\ell }(z,\zeta ,q)$ are $\hat {g}_{\ell }(z,\zeta ,q) \rightarrow L^{\ell } g_{\ell }^{2d}(\sigma ,m,q_{2d})$ , with $\hat {g}_{\ell }^{2d}(z,t,q_{2d})$ given by

$$ \begin{align*} \hat{g}_{\ell}^{2d}(\sigma,t,q_{2d}) = e_{\ell}(t)+ (-1)^{n+k} q_{2d} \delta_{\ell, n},\quad \ell = 1,\ldots,n, \end{align*} $$

where $\delta _{\ell , n}$ is the Kronecker delta. Then the $2d$ limit of the Coulomb branch presentation in (54) is

(71) $$ \begin{align} \sum_{i+j=\ell} e_{i}(\sigma) e_{j}(\hat{\sigma}) = \hat{g}_{\ell}^{2d}(\sigma,t,q_{2d}),\quad \ell=1,\ldots,n . \end{align} $$

We may identify $\{-\sigma _a\}$ with Chern roots of ${\mathcal {S}}$ and $\{-\hat {\sigma }_a\}$ with Chern roots of ${\mathcal {Q}}$ . This recovers the equivariant quantum Whitney relations from (1):

$$ \begin{align*} c^T({\mathcal{S}}) \star c^T({\mathcal{Q}}) = c^T({\mathbb{C}}^n) + (-1)^{k} q_{2d}. \end{align*} $$

As we observed earlier, the $2d$ limits of the Coulomb branch and the Whitney presentations from (54) and (56) coincide. Indeed, under the change of variables from (70) and after taking leading terms, we obtain

$$ \begin{align*} \tilde{g}_{\ell}^{2d}(\sigma,t,q_{2d}) = e_{\ell}(t) + (-1)^{n-k} q_{2d} \delta_{\ell, n} = {g}_{\ell}^{2d}(\sigma,t,q_{2d}), \quad \ell = 1,\ldots,n. \end{align*} $$