K-theoretic Gromov–Witten invariants of line degrees on flag varieties

1. Introduction

In this paper, we study the n-pointed genus zero Gromov–Witten invariants of line degrees on a complex flag variety $X=G∕P$. Given an effective degree $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$, we let ${\overline{\mathcal{ℳ}}}_{0,n}(X,d)$ be the Kontsevich moduli space of n-pointed stable maps to X of degree d and genus zero. Given subvarieties ${\mathrm{\Omega }}_1,\dots ,{\mathrm{\Omega }}_n\subset X$ in general position, the cohomological Gromov–Witten invariant ${〈[{\mathrm{\Omega }}_1],\dots ,[{\mathrm{\Omega }}_n]〉}_d^X$ counts the number of parametrized curves ${\mathbb{P}}^1\to X$ of degree d with n marked points on the domain (up to projective transformation), such that the ith marked point is sent to ${\mathrm{\Omega }}_i$ for $1\le i\le n$, assuming that finitely many such curves exist. More generally, the $\mathrm{\text{K}}$-theoretic Gromov–Witten invariant ${〈[{\mathcal{𝒪}}_{{\mathrm{\Omega }}_1}],\dots ,[{\mathcal{𝒪}}_{{\mathrm{\Omega }}_n}]〉}_d^X$ is defined as the sheaf Euler characteristic $\chi ({\mathrm{\text{GW}}}_d,{\mathcal{𝒪}}_{{\mathrm{\text{GW}}}_d})$ of the Gromov–Witten subvariety ${\mathrm{\text{GW}}}_d\subset {\overline{\mathcal{ℳ}}}_{0,n}(X,d)$ of stable maps that send the ith marked point to ${\mathrm{\Omega }}_i$.

The degree $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$ is called a line degree if G acts transitively on the moduli space ${\overline{\mathcal{ℳ}}}_{0,0}(X,d)$ of zero-pointed stable maps of degree d. This happens when $d=[X_{s_{\alpha }}]$ is the class of a one-dimensional Schubert variety such that the defining simple root $\alpha$ satisfies combinatorial conditions given in Refs. 1,2,3 and discussed in Sec. 3. All one-dimensional Schubert classes are line degrees if G is simply-laced, if X is the variety $G∕B$ of complete flags, or if X is a cominuscule flag variety.

When d is a line degree, the 0- and 1-pointed moduli spaces $M_0={\overline{\mathcal{ℳ}}}_{0,0}(X,d)$ and $M_1={\overline{\mathcal{ℳ}}}_{0,1}(X,d)$ are flag varieties, $M_0=G∕P^{\prime }$ and $M_1=G∕(P\cap P^{\prime })$, and the natural projections $p:M_1\to X$ and $q:M_1\to M_0$ coincide with the evaluation map and the forgetful map. Our main result (Theorem 3.1) is a quantum equals classical formula stating that, for arbitrary (equivariant) $\mathrm{\text{K}}$-theory classes ${\mathcal{ℱ}}_1,\dots ,{\mathcal{ℱ}}_n\in {\mathrm{\text{K}}}_T(X)$, the associated Gromov–Witten invariant of line degree d is given by

where ${\chi }_{M_0}:{\mathrm{\text{K}}}_T(M_0)\to {\mathrm{\text{K}}}_T(\mathrm{\text{pt}})$ is the sheaf Euler characteristic map. The proof uses that ${\overline{\mathcal{ℳ}}}_{0,n}(X,d)$ has rational singularities and is birational to the n-fold product of $M_1$ over $M_0$. We also prove an n-pointed analogue of the Peterson comparison formula, stating that the Gromov–Witten invariant (1) coincides with a Gromov–Witten invariant of the variety of complete flags $G∕B$.

Early results in this direction were proved for 3-pointed Gromov–Witten invariants of classical Grassmannians and cominuscule flag varieties.^{4,5,6,7,8,9,10} The comparison formula and a variant of (1) were proved for 3-pointed Gromov–Witten invariants in Ref. 11. Our results imply an analogous formula for n-pointed cohomological Gromov–Witten invariants, special cases of which were obtained in Refs. 12 and 13.

The (equivariant) big quantum $\mathrm{\text{K}}$-theory ring ${\mathrm{\text{QK}}}^{\mathrm{\text{big}}}(X)$ introduced in Refs. 14 and 15 is a power series deformation of the $\mathrm{\text{K}}$-theory ring $\mathrm{\text{K}}(X)$ that encodes the n-pointed $\mathrm{\text{K}}$-theoretic Gromov–Witten invariants of all degrees. Degrees of curves are encoded as powers $Q^d$ of Novikov variables, and additional variables $t_w$ dual to Schubert classes ${\mathcal{𝒪}}^w\in \mathrm{\text{K}}(X)$ encode insertions in Gromov–Witten invariants. The small quantum $\mathrm{\text{K}}$-theory ring $\mathrm{\text{QK}}(X)$ is recovered when all $t_w$ are specialized to 0.

Our formulas make it straightforward to compute the multiplicative structure of ${\mathrm{\text{QK}}}^{\mathrm{\text{big}}}(X)$ modulo the ideal $〈Q^e〉$ generated by degrees e larger than line degrees. We provide some examples of products in this ring in the last section. There are other approaches to computing n-pointed $\mathrm{\text{K}}$-theoretic Gromov–Witten invariants when $\mathrm{\text{K}}(X)$ is multiplicatively generated by line bundles; for example, some computations in type A have been obtained by using the J-function.^16,17,18

2. Preliminaries

2.1. Flag varieties

Let G be a complex reductive linear algebraic group, and fix a maximal torus T, a Borel subgroup B and a parabolic subgroup P, such that $T\subset B\subset P\subset G$. The opposite Borel subgroup $B^{-}\subset G$ is defined by $B^{-}\cap B=T$. Let $\mathrm{\Phi }$ be the associated root system, with basis of simple roots $\mathrm{\Delta }\subset {\mathrm{\Phi }}^{+}$. Let $W=N_G(T)∕T$ be the Weyl group of G, and $W_P=N_P(T)∕T$ the Weyl group of P. Then W is generated by the simple reflections $\{s_{\alpha }|\alpha \in \mathrm{\Delta }\}$, and $W_P$ is determined by (and determines) the set ${\mathrm{\Delta }}_P=\{\alpha \in \mathrm{\Delta }|s_{\alpha }\in W_P\}$.

Let $X=G∕P$ be the flag variety defined by P. Any Weyl group element $w\in W$ defines the Schubert varieties $X_w=\overline{BwP∕P}$ and $X^w=\overline{B^{-}wP∕P}$ in X. These varieties depend only on the coset $w{W}_P$ in $W∕W_P$. When w belongs to the subset $W^P\subset W$ of minimal representatives of the cosets in $W∕W_P$, we have $dim(X_w)=\mathrm{\text{codim}}(X^w,X)=\ell (w)$, where $\ell (w)$ denotes the Coxeter length of w.

2.2. Gromov–Witten invariants

For any projective T-variety Y, let ${\mathrm{\text{K}}}_T(Y)$ be the equivariant $\mathrm{\text{K}}$-theory ring, defined as the Grothendieck ring of T-equivariant algebraic vector bundles. This ring is an algebra over ${\mathrm{\text{K}}}_T(\mathrm{\text{pt}})$, the representation ring of T. Let ${\chi }_Y:{\mathrm{\text{K}}}_T(Y)\to {\mathrm{\text{K}}}_T(\mathrm{\text{pt}})$ be the push-forward map along the structure morphism. The equivariant $\mathrm{\text{K}}$-theory ${\mathrm{\text{K}}}_T(X)$ of the flag variety $X=G∕P$ is a free ${\mathrm{\text{K}}}_T(\mathrm{\text{pt}})$-module with basis $\{{\mathcal{𝒪}}^w|w\in W^P\}$, where ${\mathcal{𝒪}}^w=[{\mathcal{𝒪}}_{X^w}]\in {\mathrm{\text{K}}}_T(X)$ is the $\mathrm{\text{K}}$-theoretic Schubert class defined by the structure sheaf of $X^w$.

The homology group ${\mathrm{\text{H}}}_2(X,\mathbb{Z})$ is a free abelian group generated by the classes $[X_{s_{\alpha }}]$ of the 1-dimensional Schubert varieties for $\alpha \in \mathrm{\Delta }\setminus {\mathrm{\Delta }}_P$. Given an effective degree $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$ and $n\in \mathbb{N}$, we let ${\overline{\mathcal{ℳ}}}_{0,n}(X,d)$ denote the Kontsevich moduli space of n-pointed stable maps to X of degree d and genus zero (see Ref. 19). This moduli space is nonempty when $d\ne 0$ or $n\ge 3$. In this case, the evaluation map ${\mathrm{\text{ev}}}_i:{\overline{\mathcal{ℳ}}}_{0,n}(X,d)\to X$, defined for $1\le i\le n$, sends a stable map to the image of the i-th marked point in its domain.

Given $\mathrm{\text{K}}$-theory classes ${\mathcal{ℱ}}_1,\dots ,{\mathcal{ℱ}}_n\in {\mathrm{\text{K}}}_T(X)$, the corresponding n-pointed (equivariant) K-theoretic Gromov–Witten invariant of degree d and genus zero is defined by

Similarly, the cohomological Gromov–Witten invariant given by ${\gamma }_1,\dots ,{\gamma }_n\in {\mathrm{\text{H}}}_T^{\ast }(X)$ is defined by Nonequivariant Gromov–Witten invariants are obtained by replacing T with the trivial group; these Gromov–Witten invariants are integers.

3. Gromov–Witten Invariants of Line Degrees

A nonzero homology class $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$ will be called a line degree if G acts transitively on the moduli space ${\overline{\mathcal{ℳ}}}_{0,0}(X,d)$ of 0-pointed stable maps to X of degree d and genus zero. Equivalently, $d=[X_{s_{\alpha }}]$ is the class of a one-dimensional Schubert variety defined by a simple root $\alpha \in \mathrm{\Delta }\setminus {\mathrm{\Delta }}_P$ such that $\alpha$ is a long root within its connected component of ${\mathrm{\Delta }}_P\cup \{\alpha \}$.^1,2,3 Here we identify the simple roots $\mathrm{\Delta }$ with the set of nodes in the Dynkin diagram of $\mathrm{\Phi }$, so that the connected component of $\alpha$ in ${\mathrm{\Delta }}_P\cup \{\alpha \}$ is an irreducible Dynkin diagram in itself. We further use the convention that all roots of a simply-laced root system are long. In particular, $[X_{s_{\alpha }}]$ is a line degree if the component of $\alpha$ in ${\mathrm{\Delta }}_P\cup \{\alpha \}$ is simply-laced, even if $\alpha$ is a short root of $\mathrm{\Phi }$. All one-dimensional Schubert classes are line degrees if $\mathrm{\Phi }$ is simply-laced, if X is the variety $G∕B$ of complete flags, or if X is a cominuscule flag variety. We note that the definition of line degree depends on the group G. For example, the projective space ${\mathbb{P}}^{2n-1}$ is a flag variety of both $\mathrm{\text{SL}}(2n)$ and $\mathrm{\text{Sp}}(2n)$, and $\mathrm{\text{SL}}(2n)$ acts transitively on the set of lines in ${\mathbb{P}}^{2n-1}$ whereas $\mathrm{\text{Sp}}(2n)$ does not.

Given a fixed line degree $d=[X_{s_{\alpha }}]$, we let $P^{\prime }\subset G$ be the parabolic subgroup defined by

that is, ${\mathrm{\Delta }}_{P^{\prime }}$ is obtained from ${\mathrm{\Delta }}_P\cup \{\alpha \}$ by removing the simple roots adjacent to $\alpha$ in the Dynkin diagram. In this case, the moduli spaces of 0- and 1-pointed stable maps to X of degree d and genus zero are the flag varieties $M_0={\overline{\mathcal{ℳ}}}_{0,0}(X,d)=G∕P^{\prime }$ and $M_1={\overline{\mathcal{ℳ}}}_{0,1}(X,d)=G∕(P\cap P^{\prime })$, and the natural projections $p:G∕(P\cap P^{\prime })\to X$ and $q:G∕(P\cap P^{\prime })\to G∕P^{\prime }$ coincide with the evaluation map and the forgetful map.^1,2,3

The curve of degree d in X corresponding to $y\in M_0$ is given by

Let ${\pi }_X:G∕B\to X$ be the projection. If $d=[X_{s_{\alpha }}]$ is a line degree of X, we also let d denote the unique line degree $[B{s}_{\alpha }B∕B]$ of $G∕B$ that is mapped to d by push-forward along ${\pi }_X$. This class $[B{s}_{\alpha }B∕B]$ is the Peterson lift of d, see Definition 1 in Ref. 20. In fact, one can check that any class in ${\mathrm{\text{H}}}_2(X;\mathbb{Z})$ is a line degree of X if and only if its Peterson lift is a line degree of $G∕B$.

Our main result is the following theorem. Part (B) and a variant of (A) were proved for 3-pointed Gromov–Witten invariants in Ref. 11.

Theorem 3.1. Let $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$ be a line degree of the flag variety X with associated projections $p:M_1\to X$ and $q:M_1\to M_0$, and let ${\mathcal{ℱ}}_1,\dots ,{\mathcal{ℱ}}_n\in {\mathrm{\text{K}}}_T(X)$ be $\mathrm{\text{K}}$-theory classes. The following identities hold in ${\mathrm{\text{K}}}_T(\mathrm{\text{pt}})$:

(A)	${〈{\mathcal{ℱ}}_1,\dots ,{\mathcal{ℱ}}_n〉}_d^X={\chi }_{G∕P^{\prime }}(q_{\ast }{p}^{\ast }{\mathcal{ℱ}}_1\cdot \cdots \cdot q_{\ast }{p}^{\ast }{\mathcal{ℱ}}_n)$.
(B)	${〈{\mathcal{ℱ}}_1,\dots ,{\mathcal{ℱ}}_n〉}_d^X={〈{\pi }_X^{\ast }{\mathcal{ℱ}}_1,\dots ,{\pi }_X^{\ast }{\mathcal{ℱ}}_n〉}_d^{G∕B}$.

Proof. To prove part (A), let $M_1^{(n)}=M_1{\times }_{M_0}\cdots {\times }_{M_0}{M}_1$ be the fiber product of n copies of $M_1$ over $M_0$, with projections $e_i:M_1^{(n)}\to M_1$ for $1\le i\le n$. Set $M_n={\overline{\mathcal{ℳ}}}_{0,n}(X,d)$, and let $\varphi :M_n\to M_1^{(n)}$ be the morphism defined by the n forgetful maps $M_n\to M_1$. We obtain the commutative diagram:

Since q is a locally trivial fibration with nonsingular base and fiber, it follows that $M_1^{(n)}$ is a nonsingular projective variety. Using that any morphism ${\mathbb{P}}^1\to X$ of line degree is an isomorphism onto its image, it follows that $\varphi$ is birational. The variety $M_n$ has rational singularities by Theorem 2(ii) in Ref. 19 and Proposition 5.15 in Ref. 21. We obtain ${\varphi }_{\ast }[{\mathcal{𝒪}}_{M_n}]=[{\mathcal{𝒪}}_{M_1^{(n)}}]$ in ${\mathrm{\text{K}}}_T(M_1^{(n)})$, and therefore where the last two equalities follow from the projection formula and Lemma 3.5 in Ref. 9.

For part (B), let $P_{\alpha }\subset G$ be the minimal parabolic subgroup given by ${\mathrm{\Delta }}_{P_{\alpha }}=\{\alpha \}$, where $\alpha \in \mathrm{\Delta }\setminus {\mathrm{\Delta }}_P$ is defined by $d=[X_{s_{\alpha }}]$. We obtain a commutative diagram

where $\pi$, ${\pi }_{\alpha }$ and $q_{\alpha }$ are the natural projections of flag varieties. Using that the square is Cartesian and ${\pi }_X=p\circ \pi$, we obtain where the last equality follows from part (A) applied to $G∕B$, noting that we have ${\overline{\mathcal{ℳ}}}_{0,0}(G∕B,d)=G∕P_{\alpha }$ and ${\overline{\mathcal{ℳ}}}_{0,1}(G∕B,d)=G∕B$. □

Theorem 3.1 implies the analogous identities for cohomological Gromov–Witten invariants in the following corollary by the equivariant Hirzebruch formula,²² see Subsec. 4.1 in Ref. 9. A similar statement is proved in Ref. 13 when P is maximal and used to study semisimplicity of big quantum cohomology; when X is a Grassmannian, the statement is proved and applied in Ref. 12. Part (B) for 3-pointed Gromov–Witten invariants is a special case of Peterson’s comparison formula, proved in Woodward’s paper (Ref. 20).

Corollary 3.1. Let $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$ be a line degree of X, and let ${\gamma }_1,\dots ,{\gamma }_n\in {\mathrm{\text{H}}}_T^{\ast }(X)$ be cohomology classes. The following identities hold in ${\mathrm{\text{H}}}_T^{\ast }(\mathrm{\text{pt}})$:

(A)	${〈{\gamma }_1,\dots ,{\gamma }_n〉}_d^X={\int }_{G∕P^{\prime }}{q}_{\ast }{p}^{\ast }{\gamma }_1\cdot \cdots \cdot q_{\ast }{p}^{\ast }{\gamma }_n$.
(B)	${〈{\gamma }_1,\dots ,{\gamma }_n〉}_d^X={〈{\pi }_X^{\ast }{\gamma }_1,\dots ,{\pi }_X^{\ast }{\gamma }_n〉}_d^{G∕B}$.

Remark 3.1. Theorem 3.1 and Corollary 3.1 imply that nonequivariant Gromov–Witten invariants of line degree are enumerative for Schubert classes in the following sense. Let ${\mathrm{\Omega }}_1,\dots ,{\mathrm{\Omega }}_n\subset X$ be Schubert varieties in general position and let

be the subvariety of $M_0=G∕P^{\prime }$ parametrizing lines meeting ${\mathrm{\Omega }}_1,\dots ,{\mathrm{\Omega }}_n$. Then This follows from Theorem 3.1(A) since the projection $q:p^{-1}({\mathrm{\Omega }}_i)\to q(p^{-1}({\mathrm{\Omega }}_i))$ of the Schubert variety $p^{-1}({\mathrm{\Omega }}_i)$ is cohomologically trivial.²³ The left-hand side of (2) is the sheaf Euler characteristic $\chi ({\mathrm{\text{GW}}}_d,{\mathcal{𝒪}}_{{\mathrm{\text{GW}}}_d})$ of the Gromov–Witten variety ${\mathrm{\text{GW}}}_d={\bigcap }_{i=1}^n{\mathrm{\text{ev}}}_i^{-1}({\mathrm{\Omega }}_i)\subset {\overline{\mathcal{ℳ}}}_{0,n}(X,d)$ of stable maps that send the ith marked point to ${\mathrm{\Omega }}_i$ (see Subsec. 4.1 in Ref. 9).

When the Schubert varieties satisfy the condition

we also have that is, the cohomological Gromov–Witten invariant ${〈[{\mathrm{\Omega }}_1],\dots ,[{\mathrm{\Omega }}_n]〉}_d^X$ is the number of lines in X of degree d meeting ${\mathrm{\Omega }}_1,\dots ,{\mathrm{\Omega }}_n$.

Example 3.1. Let $\gamma \in {\mathrm{\text{H}}}^{\ast }({\mathbb{P}}^3)$ be the class of a line. Then Remark 3.1 implies that ${〈\gamma ,\gamma ,\gamma ,\gamma 〉}_1^{{\mathbb{P}}^3}$ counts the number of lines that meet four given lines in general position in ${\mathbb{P}}^3$, and this number can be computed as a classical intersection number on $M_0=\mathrm{\text{Gr}}(2,4)$.

4. Applications to Big Quantum K-Theory

4.1. Definitions

Set $\mathrm{\Gamma }={\mathrm{\text{K}}}_T(\mathrm{\text{pt}})\otimes \mathbb{Q}$. Given a fixed flag variety $X=G∕P$, we let

be the ring of formal power series over $\mathrm{\Gamma }$, in Novikov variables $Q_{\alpha }$ dual to the Schubert basis of ${\mathrm{\text{H}}}_2(X,\mathbb{Z})$, and formal variables $t_w$ dual to the Schubert basis of ${\mathrm{\text{K}}}_T(X)$. The big equivariant quantum K-theory ring of X is a $\mathrm{\Gamma }〚Q,t〛$-algebra defined by as a module. We proceed to define the multiplicative structure on ${\mathrm{\text{QK}}}_T^{\mathrm{\text{big}}}(X)$ following Refs. 14 and 15.

For $d={\sum }_{\alpha \in \mathrm{\Delta }\setminus {\mathrm{\Delta }}_P}{d}_{\alpha }[X_{s_{\alpha }}]\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$, we write

and for any function $h:W^P\to \mathbb{N}$, we define Given $\mathrm{\text{K}}$-theory classes ${\mathcal{ℱ}}_1,{\mathcal{ℱ}}_2,{\mathcal{ℱ}}_3\in {\mathrm{\text{K}}}_T(X)$, we let ${〈{\mathcal{ℱ}}_1,{\mathcal{ℱ}}_2,{\mathcal{ℱ}}_3,{\mathcal{𝒪}}^h〉}_d^X$ denote the $(|h|+3)$-pointed Gromov–Witten invariant of degree d with $h(w)$ insertions of ${\mathcal{𝒪}}^w$, for $w\in W^P$, in addition to the first three insertions. We then define with the sum over all effective degrees $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$ and functions $h:W^P\to \mathbb{N}$. We extend this by linearity to a symmetric 3-form on the $\mathrm{\Gamma }〚Q,t〛$-module ${\mathrm{\text{QK}}}_T^{\mathrm{\text{big}}}(X)$. The quantum metric on ${\mathrm{\text{QK}}}_T^{\mathrm{\text{big}}}(X)$ is then defined by $(({\mathcal{ℱ}}_1,{\mathcal{ℱ}}_2))=(({\mathcal{ℱ}}_1,{\mathcal{ℱ}}_2,1))$, and the quantum product ${\mathcal{ℱ}}_1\star {\mathcal{ℱ}}_2\in {\mathrm{\text{QK}}}_T^{\mathrm{\text{big}}}(X)$ is the unique class defined by for all ${\mathcal{ℱ}}_3\in K_T(X)$. The small quantum $\mathrm{\text{K}}$-theory ring is the quotient ${\mathrm{\text{QK}}}_T(X)={\mathrm{\text{QK}}}_T^{\mathrm{\text{big}}}(X)∕〈t〉$ by the ideal generated by $t_w$ for $w\in W^P$.

Remark 4.1. Let 0 denote the identity element of W, so that $t_0$ is dual to $1\in {\mathrm{\text{K}}}_T(X)$. The product ${\mathcal{ℱ}}_1\star {\mathcal{ℱ}}_2\in {\mathrm{\text{QK}}}_T^{\mathrm{\text{big}}}(X)$ is known to be independent of $t_0$ for ${\mathcal{ℱ}}_1,{\mathcal{ℱ}}_2\in {\mathrm{\text{K}}}_T(X)$; this follows from Ref. 14 (see also Proposition 2.10 in Ref. 18). In fact, since the general fibers of the forgetful map ${\overline{\mathcal{ℳ}}}_{0,n+1}(X,d)\to {\overline{\mathcal{ℳ}}}_{0,n}(X,d)$ are isomorphic to ${\mathbb{P}}^1$, we have ${〈1,{\mathcal{𝒪}}^h〉}_d^X={〈{\mathcal{𝒪}}^h〉}_d^X$ for all effective $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$ and $h:W^P\to \mathbb{N}$ with $|h|\ge 3$, and therefore $(({\mathcal{ℱ}}_1,{\mathcal{ℱ}}_2,{\mathcal{ℱ}}_3))=e^{t_0}(({\mathcal{ℱ}}_1,{\mathcal{ℱ}}_2,{\mathcal{ℱ}}_3)){|}_{t_0=0}$ for all ${\mathcal{ℱ}}_1,{\mathcal{ℱ}}_2,{\mathcal{ℱ}}_3\in {\mathrm{\text{K}}}_T(X)$. We obtain

It follows that ${\mathcal{ℱ}}_1\star {\mathcal{ℱ}}_2\in {\mathrm{\text{QK}}}_T^{\mathrm{\text{big}}}(X)$ is the unique class that is independent of $t_0$ and satisfies for all ${\mathcal{ℱ}}_3\in {\mathrm{\text{K}}}_T(X)$. Notice that the product ${\mathcal{ℱ}}_1\star {\mathcal{ℱ}}_2$ can be constructed from the quantum potential $\mathcal{𝒢}(Q,t)=((1,1,1))\in \mathrm{\Gamma }〚Q,t〛$ specialized at $t_0=0$, by observing that $(({\mathcal{𝒪}}^u,{\mathcal{𝒪}}^v,{\mathcal{𝒪}}^w)){|}_{t_0=0}={\partial }_{t_u}{\partial }_{t_v}{\partial }_{t_w}{\mathcal{𝒢}}_0$ holds for $u,v,w\in W^P\setminus \{0\}$.

Remark 4.2. The product in the equivariant big quantum cohomology ring

is defined by where ${〈[X^u],[X^v],[X_w],{[X]}^h〉}_d^X$ is the cohomological Gromov–Witten invariant with $h(w)$ insertions of $[X^w]$ in addition to the first three insertions.

4.2. Examples

In this section we apply Theorem 3.1 to compute some examples of big quantum $\mathrm{\text{K}}$-theory products modulo powers of the Novikov variables of degrees larger than line degrees. In each case, there is a unique line degree, and only one Novikov variable which will be denoted by Q. Congruence $\equiv$ is always modulo $Q^2$. Our examples are compatible with a positivity property of big quantum K-theory that we plan to discuss elsewhere.

Example 4.1. Let $X={\mathbb{P}}^1=\mathrm{\text{SL}}(2)∕B$. The only line degree in ${\mathrm{\text{H}}}_2({\mathbb{P}}^1,\mathbb{Z})$ gives $M_1={\mathbb{P}}^1$ and $M_0=\{\mathrm{\text{pt}}\}$. Let $T={ℂ}^{\ast }$act on ${\mathbb{P}}^1$ and let $P\in {\mathbb{P}}^1$ be a T-fixed point. Then ${\mathrm{\text{K}}}_T({\mathbb{P}}^1)$ has basis $\{1,{\mathcal{𝒪}}^1\}$, where ${\mathcal{𝒪}}^1=[{\mathcal{𝒪}}_P]$. Set $a=1-{\mathcal{𝒪}}^1{|}_P\in {\mathrm{\text{K}}}_T(\mathrm{\text{pt}})$. Using that ${({\mathcal{𝒪}}^1)}^n={(1-a)}^{n-1}{\mathcal{𝒪}}^1\in {\mathrm{\text{K}}}_T({\mathbb{P}}^1)$ for $n\ge 1$, we compute the specialized potential (modulo $Q^2$) as

where $t=t_1$ is dual to ${\mathcal{𝒪}}^1$. If we write ${\mathcal{𝒪}}^1\star {\mathcal{𝒪}}^1=c_0+c_1{\mathcal{𝒪}}^1$ with $c_0,c_1\in \mathrm{\Gamma }〚Q,t〛$, then the equations for $\mathcal{ℱ}\in \{1,{\mathcal{𝒪}}^1\}$, are equivalent to for $i\in \{0,1\}$. By solving for $c_0$ and $c_1$ modulo $Q^2$, we arrive at

Example 4.2. Let $X={\mathbb{P}}^2$. Then $M_1=\mathrm{\text{Fl}}(3)$ is a complete flag variety and $M_0={{\mathbb{P}}^2}^{\ast }$ is the dual projective plane. The Schubert basis is $\{1,{\mathcal{𝒪}}^1,{\mathcal{𝒪}}^2\}$, where ${\mathcal{𝒪}}^1=[{\mathcal{𝒪}}_L]$ is the class of a line and ${\mathcal{𝒪}}^2=[{\mathcal{𝒪}}_P]$ is the class of a point in ${\mathbb{P}}^2$.

Working nonequivariantly for simplicity, we have

from which we obtain (see also Subsec. 4.3 in Ref. 18) :

Example 4.3. Let $X=\mathrm{\text{Gr}}(2,4)$ be the Grassmannian of 2-planes in ${ℂ}^4$. Here, $M_0=\mathrm{\text{Fl}}(1,3;4)$ is a point-hyperplane incidence variety and $M_1=\mathrm{\text{Fl}}(4)$ is a complete flag variety. The Schubert basis is $\{1,{\mathcal{𝒪}}^1,{\mathcal{𝒪}}^{1,1},{\mathcal{𝒪}}^2,{\mathcal{𝒪}}^{2,1},{\mathcal{𝒪}}^{2,2}\}$, where ${\mathcal{𝒪}}^{\lambda }$ is the Schubert class indexed by the partition $\lambda$. Working nonequivariantly, we obtain (by a computation in Maple) :

and

Acknowledgments

This project was initiated at the ICERM^a Women in Algebraic Geometry workshop in July 2020 and the ICERM Combinatorial Algebraic Geometry program in Spring 2021. We thank A. Gibney, L. Heller, E. Kalashnikov, H. Larson as well as P. E. Chaput, L. C. Mihalcea and N. Perrin for inspiring collaborations on related projects. We also thank an anonymous referee for a careful reading of our paper and for several helpful suggestions. A.B. was partially supported by NSF Grants DMS-2152316 as well as DMS-1929284 while in residence at ICERM during the Spring of 2021. L.C. was partially supported by NSF Grant DMS-2101861.

ORCID

Anders S. Buch  https://orcid.org/0000-0001-6139-2392

Linda Chen  https://orcid.org/0000-0002-8606-1634

Weihong Xu  https://orcid.org/0000-0003-0990-5327

Notes

^a Institute for Computational and Experimental Research in Mathematics in Providence, RI.