K-theoretic Heisenberg algebras and permutation-equivariant Gromov–Witten theory

The definition of permutation-equivariant KGW invariants will be recalled later on (see Section 3.2). From the very beginning it is clear that we are dealing with representations of the symmetric groups. However, finding a good algebraic formalism to capture the entire information is a bit tricky. We refer to [5] for more details on the logic used by Givental. One of the surprises in the current paper (at least to the author) is that there is a slightly different way to organize the invariants which is equivalent to the choice made by Givental but proving the equivalence requires some non-trivial work.

Our logic is the following. Suppose that $X$ is a smooth projective variety. Let $\overline{\mathcal{M}}_{g,n+k}(X,d)$ be the moduli space of degree $d$ stable maps from a genus-$g$ nodal Riemann surface equipped with $n+k$ marked points. We let the symmetric group $S_{k}$ act on the moduli space by permuting the last $k$ marked points. We will also refer to the last $k$ marked points as permutable marked points. Let

and

be the evaluation maps at respectively the first $n$ and the last $k$ marked points. The permutation-equivariant KGW invariants contain information about the $S_{k}$-equivariant K-theoretic pushforward $\operatorname{ev}^{(k)}_{*}$. Therefore, it is natural to introduce the following graded vector space:

where $S_{n}$ is the symmetric group and $K_{S_{n}}(X^{n})$ denotes the $S_{n}$-equivariant topological K-ring with complex coefficients (see [17]). We will consider the case when $K^{1}(X)=0$ in order to make the construction more transperant. The case when $K^{1}(X)\neq 0$ is similar. The vector space $\mathcal{F}(X)$ was investigated before by Weiqiang Wang (see [19]). Partially motivated by the work of Grojnowski (see [9]) and Nakajima (see [16]) and following ideas of Segal, Wang proved that $\mathcal{F}(X)$ is a Fock space, i.e., it is an irreducible representation of a Heisenberg algebra. Let us outline Wang’s construction with some minor modification. The vector space $\mathcal{F}(X)$ is naturally a commutative graded algebra with multiplication defined by the induction operation (see Section 2.2)

On the other hand, each graded piece $K_{S_{n}}(X^{n})$ is a module over the representation ring $R(S_{n})$ of the symmetric group. It is well known (see [2], Proposition 39.4) that $R(S_{n})$ has a virtual representation $p_{n}$, that is, linear combination of irreducible representations with integer coefficients, such that, its character is

Note that $X^{n}\times p_{n}$ can be viewed as a virtual $S_{n}$-equivariant vector bundle on $X^{n}$. Given $E\in K_{S_{n}}(X^{n})$, we denote by $E\otimes p_{n}\in K_{S_{n}}(X^{n})$ the tensor product of $E$ and the trivial bundle $X^{n}\times p_{n}$. Let $\{\Phi_{\alpha}\}_{\alpha=1}^{N}$ be a basis of the topological K-ring $K(X)$. Let $\{\Phi^{\alpha}\}_{\alpha=1}^{N}$ be the basis dual to the above one with respect to the Euler pairing, that is, $(\Phi^{\alpha},\Phi_{\beta})=\delta_{\alpha,\beta}$ where $(E,F):=\chi(E\otimes F)$. Let $\nu_{n,\alpha}:=(\Phi^{\alpha})^{\boxtimes n}\otimes p_{n}$.

We refer to [19], Proposition 3 for the proof of Proposition 1.1.1. In fact, the result of Wang is more general. He considered the case when $X$ is a $G$-space where $G$ is a finite group. The corresponding Fock space $\mathcal{F}_{G}(X)$ could be viewed as the Fock space of the orbifold $[X/G]$. From this point of view, the work of Wang should have a generalization to the case when $X$ is an orbifold which is not necessarily a global quotient. The orbifold case will be important for the applications to permutation-equivariant orbifold KGW theory.

We think of $1\in\mathcal{F}(X)$ as the vacuum and of multiplication by $\nu_{n,\alpha}$ as the creation operations. It turns out that the operators of differentiation by $r\tfrac{\partial}{\partial\nu_{r,\alpha}}$ also have a natural K-theoretic interpretation. Namely, the following formula holds:

where $E\in K_{S_{n}}(X^{n})$, $\pi^{(n-r)}:X^{n}=X^{r}\times X^{n-r}\to X^{n-r}$ is the projection map, and $\operatorname{Res}^{S_{n}}_{S_{r}\times S_{n-r}}$ is the restriction functor (see Section 2.2). Here the K-theoretic pushforward $\pi^{(n-r)}_{*}$ yields a virtual $S_{n-r}$-bundle whose coefficients are representations of $S_{r}$ and hence after taking the trace $\operatorname{tr}_{(1,2,\dots,r)}$ we obtain an element in $K_{S_{n-r}}(X^{n-r})$. Let us point out that our construction of the Fock space is slightly different from the one in [19]. Namely, the multiplication operator of Wang involves the inverse of the Adam’s operations in $K(X)$ while the differential operator $r\partial_{\nu_{r,\alpha}}$ is represented by a contraction operation that involves a choice in the dual vector space $K(X)^{\vee}$. The proof of (1) could be obtained directly from Wang’s results but for the sake of completeness we give a self-contained proof (see Theorem 2.3.1).

We will refer to $\mathcal{F}(X)$ as the K-theoretic Fock space of $X$. We will think of $\nu=(\nu_{1},\nu_{2},\dots)$, where $\nu_{r}=(\nu_{r,a})_{1\leq a\leq N}$, as formal parameters. The KGW invariants of $X$ with values in $\mathcal{F}(X)$ are defined as follows:

where $L_{i}$ is the tautological line bundle formed by the cotangent lines at the $i$-th marked point, $\mathcal{O}_{g,n+k,d}$ is the virtual structure sheaf of the moduli space $\overline{\mathcal{M}}_{g,n+k}(X,d)$ (see [14]), and the pushforward is the $S_{k}$-equivariant $K$-theoretic pushforward, that is,

We will usually drop the virtual structure sheaf in the above notation. Also, let us introduce formal parameters $t_{i,a}$ and introduce the following formal Laurent series:

Then the KGW invariants with values in $\mathcal{F}(X)$ can be organized into a set of formal power series in $\mathbf{t}$ of the following form:

The permutation-equivariant KGW invariants of Givental (see [7]) will be recalled in Section 3.2. Using the Heisenberg relations, that is, formula (1), we will prove in Section 3.2 that the correlators (3) are related to the correlators of Givental by a simple substitution (see (23)). Let us point out that the above definition (3) does not contain the entire information of permutation-equivariant KGW theory. We did not allow descendants at the permutable marked points. Our construction can be extended naturally by introducing the K-theoretic Fock space $\mathcal{F}(X\times\mathbb{Z})$. We refer to Section 3.3 for more details. However, let us point out that in this paper the most general definition would not be needed.

The standard identities in KGW theory, after some minor modifications, extend to permutation-equivariant KGW theory (see [6]). More precisely, the correlators (3) satisfy the string equation (see Section 3.4) and if the genus is 0, then we also have the dilaton equation (see Section 3.5) and the WDVV equations (see Section 3.6). These identities allow us to introduce permutation-equivariant K-theoretic quantum differential equations and a corresponding fundamental solution called the $S$-matrix. Let $G$ be the $N\times N$ matrix with entries

where $(\Phi_{a},\Phi_{b})=\chi(\Phi_{a}\otimes\Phi_{b})$ is the Euler pairing. Let $G^{ab}(\nu)$ be the entries of the inverse matrix $G^{-1}$. The $S$-matrix is defined by the following identity:

Note that $S(\nu,q)$ is a formal power series in $\nu$ (and the Novikov variables $Q$) whose coefficients are in $\operatorname{End}(K(X))(q)$. We refer to Proposition 3.6.1 for a list of properties of the $S$-matrix. Let us define also the K-theoretic quantum cup product by

Let us recall also the so-called $J$-function:

Using the string equation, it is easy to prove that $J(\nu,q)=(1-q)S(\nu,q)^{-1}1$.

Suppose that $v(\mathbf{t},\nu_{2},\nu_{3},\dots)=\sum_{\alpha=1}^{N}v_{\alpha}(\mathbf{t},\nu_{2},\nu_{3},\dots)\Phi_{\alpha}$ is a formal power series in $\mathbf{t}=(t_{k,\alpha})$ and $\nu_{2},\nu_{3},\dots$ with coefficients in $K(X)$. We will think of $\nu_{2},\nu_{3},\dots$ as parameters and quite often we suppress them in the notation. For example, we will write simply $v(\mathbf{t})$ instead of $v(\mathbf{t},\nu_{2},\nu_{3},\dots)$. Furthermore, we identify $\nu_{1}:=v(\mathbf{t})$, that is, $\nu_{1,\alpha}:=v_{\alpha}(\mathbf{t})$ and write $S(v(\mathbf{t}),q):=S(\nu,q)|_{\nu_{1}=v(\mathbf{t})}$. The following system of differential equations is the K-theoretic version of Dubrovin’s principal hierarchy (see [3]):

where $\partial=\partial_{t_{0,1}}$ where $\Phi_{1}:=1\in K(X)$. According to Milanov–Tonita (see [15], Theorem 1), the fact that $S(\nu,q)$ is a solution to the quantum differential equations implies that the system of equations (7) is integrable, i.e., the system is compatible. The second goal of our paper is to construct a solution to (7) in terms of KGW invariants. Following Dubrovin, we will refer to this solution as the topological solution. The construction is the same as in [15], Theorem 2. Put

Note that the notation $t_{k,\alpha}$ here is slightly different from the corresponding notation in (3).

Suppose now that $X=\operatorname{pt}$. The Fock space in this case $\mathcal{F}(\operatorname{pt})=\bigoplus_{k=0}^{\infty}R(S_{k})\otimes\mathbb{C}=\mathbb{C}[\nu_{1},\nu_{2},\dots]$ is the representation ring of the symmetric group where $\nu_{n}=p_{n}$ is the virtual representation of $S_{n}$ introduced earlier (see Section 1.1).

I am thankful to Yukinobu Toda for a very useful discussion on sheaf cohomology and rational singularities. The idea to consider the K-theoretic Fock space of Wang came after a talk by Timothy Logvinenko on the MS seminar at Kavli IPMU. I am thankful to him for giving an inspiring talk. This work is supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan and by JSPS Kakenhi Grant Number JP22K03265.

Suppose that $\lambda=(\lambda_{1},\dots,\lambda_{r})$ is a partition of $n$, that is, a decreasing sequence of integers $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{r}>0$, such that, $\lambda_{1}+\cdots+\lambda_{r}=n$. Usually we write $\lambda\vdash n$ to denote that $\lambda$ is a partition of $n$. The number $\ell(\lambda):=r$ is called the length of $\lambda$. Put $\ell_{k}(\lambda):=\operatorname{card}\,\{i\ |\ \lambda_{i}=k\}$ where $k=1,2,\dots$. Given a partition $\lambda$ we define the permutation

Note that $\{\sigma(\lambda)\}_{\lambda\vdash n}$ is a complete set of representatives for the conjugacy classes of $S_{n}$ and that the conjugacy class $C_{\lambda}:=\{g\sigma(\lambda)g^{-1}\ |\ g\in S_{n}\}$ consists of $n!z_{\lambda}^{-1}$ elements where $z_{\lambda}:=\prod_{k}k^{\ell_{k}(\lambda)}\ell_{k}(\lambda)!$ is the number of elements in $S_{n}$ that commute with $\sigma(\lambda)$. Finally, let us denote by $Y_{i}(\lambda)$ ($1\leq i\leq\ell(\lambda)$) the sequences $Y_{1}(\lambda):=\{1,2,\dots,\lambda_{1}\}$, $Y_{2}(\lambda):=\{\lambda_{1}+1,\lambda_{1}+2,\dots,\lambda_{1}+\lambda_{2}\}$, etc. The subgroup of $S_{n}$ consisting of permutations that leave the sets $Y_{i}(\lambda)$ invariant for all $i$ will be denoted by $S_{\lambda}$, that is,

The subgroups $S_{\lambda}$ ($\lambda\vdash n$) are known as the Young subgroups of $S_{n}$.

Suppose now that $\lambda$ and $\mu$ are partitions of $n$. We would like to recall the classification of the double coset classes $S_{\mu}\backslash S_{n}/S_{\lambda}$ (see [20], Appendix 3, Section A3.2). For a given parmutation $\sigma\in S_{n}$, let us define

Note that $\gamma_{ij}$ are non-negative integers satisfying

The proof follows immediately from the definition (see [20], Appendix 3, Section A3.2). Suppose that we fix a matrix $\gamma=(\gamma_{ij})$ of size $r\times s$ satisfyning the above conditions (8). Let us construct a permutation $\sigma\in S_{n}$ such that the corresponding matrix is $\gamma$. Split each set $Y_{i}(\lambda)$ into subsets $Y_{i1}(\lambda),\dots,Y_{is}(\lambda)$, such that, $Y_{i1}(\lambda)$ consists of the first $\gamma_{i1}$ elements of $Y_{i}(\lambda)$, $Y_{i2}(\lambda)$ – of the next $\gamma_{i2}$ elements, etc. We can do this because the number of elements in $Y_{i}(\lambda)$ is $\lambda_{i}=\gamma_{i1}+\gamma_{i2}+\cdots+\gamma_{is}.$ Similarly, let us split each set $Y_{j}(\mu)$ into subsets $Y_{1j}(\mu),\dots,Y_{rj}(\mu)$ consisting of respectively $\gamma_{1j},\dots\gamma_{rj}$ elements. We define $\sigma$ to be the permutation that maps the elements of $Y_{ij}(\lambda)$ to $Y_{ij}(\mu)$ and for the sake of definitness, let us require that $\sigma$ preserves the order of the elements. Note that $\sigma(Y_{i}(\lambda))\cap Y_{j}(\mu)=Y_{ij}(\mu)$ consists of $\gamma_{ij}$ elements.

We proved the following proposition.

Note that in Proposition 2.1.1 the condition that $\lambda$ and $\mu$ are partitions can be relaxed, that is, we do not have to require that $\lambda=(\lambda_{1},\dots,\lambda_{r})$ and $\mu=(\mu_{1},\dots,\mu_{s})$ are decreasing sequences.

We will assume that the reader is familiar with the notion of a $G$-space and $G$-vector bundle. For some background, we refer to [17]. Suppose that $G$ is a finite group, $H$ a subgroup of $G$, and $X$ is a $G$-space. For every $H$-bundle on $X$ we define a $G$-bundle $\operatorname{Ind}_{H}^{G}(E)$ on $X$ whose points are the set theoretic maps $f:G\to E$ satisfying the following two conditions

1. (i)

   The composition $\pi\circ f:G\to X$ is $G$-equivariant, that is, $\pi(f(g))=g\pi(f(1))$ for all $g\in G$ where $\pi:E\to X$ is the projection map.

2. (ii)

   The map $f$ is $H$-equivariant, that is, $f(hg)=hf(g)$ for all $h\in H$ and $g\in G$.

Note that when $X$ is a pont we have: $E$ is a representation of $H$ and $\operatorname{Ind}_{H}^{G}(E)$ is the induced representation. The structure of a $G$-bundle on $\operatorname{Ind}_{H}^{G}(E)$ is defined as follows. First, the $G$-action on $\operatorname{Ind}_{H}^{G}(E)$ is defined by $(gf)(g^{\prime}):=f(g^{\prime}g)$. The structure projection $p:\operatorname{Ind}_{H}^{G}(E)\to X$ is defined by $f\mapsto\pi(f(1))$. It is strightforward to check that $p$ is a $G$-equivariant map. Suppose that $f:G\to E$ is in the fiber $p^{-1}(x)$. Then $\pi(f(g))=g\pi(f(1))=gp(f)=gx$, that is, $f(g)\in E_{gx}$ where $E_{y}$ denotes the fiber of $E$ at $y$. Using the linear strucure on the fibers of $E$ we get that $p^{-1}(x)$ is naturally a linear vector space: $(f_{1}+f_{2})(g):=f_{1}(g)+f_{2}(g)$ and $(cf)(g):=cf(g)$. Let $g_{1},g_{2},\dots,g_{s}$ be a complete set of representatives of the right coset classes in $H\backslash G$. Since $f(hg)=hf(g)$ we see that the map $f$ is uniquely determined by its values $f(g_{i})\in E_{g_{i}x}$. This proves that $p^{-1}(x)$ is a finite dimensional vector space of dimension $\operatorname{rk}(E)|G:H|$ where $|G:H|$ is the index of $H$ in $G$. Moreover, there is a set-theoretic isomorphism

where $i=i(g)$ is the unique index, such that, $g\in Hg_{i}$ (note that $gg_{i}^{-1}\in H$ acts on $E$) and $\widetilde{g}_{i}$ is the map defined by the following Cartesian square:

where slightly abusing the notation we denote by $g_{i}:X\to X$ the map defined by $x\mapsto g_{i}x$. Using the map (9), we equip $\operatorname{Ind}_{H}^{G}(E)$ with the structure of a vector bundle over $X$. It is easy to check that the $G$-action on $\operatorname{Ind}_{H}^{G}(E)$ defined above is continuous and therefore $\operatorname{Ind}_{H}^{G}(E)$ is a topological $G$-vector bundle on $X$.

We also have a restriction functor. Namely, if $X$ is a $G$-space, $H\leq G$ is a subgroup, and $E$ is a $G$-vector bundle, then $E$ is also an $H$-vector bundle which will be denoted by $\operatorname{Res}^{G}_{H}(E)$. The Frobenius reciprocity rules take the following form.

If $X$ is a point, Proposition 2.2.1 is well known in the representation theory of finite groups (see [2], Section 34). The proofs in general remains the same. The following proposition is Lemma 7 in [19].

In the case when $X$ is a point, the statement is well known (see [18], Proposition 22). Serre’s argument works in the general case too.

In order to define the anihilation operations, let us first recall the notion of a trace of a vector bundle. Suppose that $E$ is a complex vector bundle on some compact topological space $Y$ and that $g:E\to E$ is a finite order automorphism acting trivially on the base $Y$. To be more precise, we have $g(y)=y$ for all $y\in Y$ and the induced map $g_{y}:E_{y}\to E_{y}$ is a finite order linear map. In particular, the maps $g_{y}$ are diagonalizable and their eigenvalues are independent of $y$. The trace of $g$ on $E$ is a virtual vector bundle on $Y$ defined by

where only finitely many terms in the above sum are non-zero because $\lambda$ must be an eigenvalue of $g$ and $E_{\lambda}$ is a vector bundle because the eigenvalues of $g_{y}$ and their multiplicities are independent of $y$. The following lemma is well known (see [1], Section 2).

Using the Chern character map, we get that the Adam’s operation $\psi^{m}:K(X)\to K(X)$ is an isomorphism where recall that we work with the K-ring with complex coefficients. Therefore, for every $F\in K(X)$, there exists an $E\in K(X)$, such that, $\psi^{m}(E)=F$. We will need also the formula for the trace of an induced vector bundle.