A mirror construction for the big equivariant quantum cohomology of toric manifolds

Abstract

We identify a certain universal Landau–Ginzburg model as a mirror of the big equivariant quantum cohomology of a (not necessarily compact or semipositive) toric manifold. The mirror map and the primitive form are constructed via Seidel elements and shift operators for equivariant quantum cohomology. Primitive forms in non-equivariant theory are identified up to automorphisms of the mirror.

Appendix 1: Formal geometry in infinite dimensions

For the sake of completeness, we prove a formal inverse function theorem and the existence of a flow of a vector field in infinite dimensions. The results here are straightforward generalizations of well-known results in finite dimensions, but we could not find a reference. Throughout the section, we assume that R is a linearly topologized ring containing $\mathbb {Q}$ and that R is complete and Hausdorff. We denote by $\{R_\nu \}$ a fundamental neighbourhood system of zero consisting of ideals of R.

Let $\mathbf {x}= \{x_1,x_2,x_3,\ldots \}$ be a countably infinite set of variables. A morphism $f :{\text {Spf}}(R[\![\mathbf {x}]\!]) \rightarrow {\text {Spf}}(R[\![\mathbf {x}]\!])$ of formal schemes over R (see Sect. 2.1 for $R[\![\mathbf {x}]\!]$) is given by a tuple $\{f^*(x_1),f^*(x_2), f^*(x_3),\ldots \}$ of elements in $R[\![\mathbf {x}]\!]$ such that $f^*(x_i)|_{\mathbf {x}=0}\in R_\mathrm{nilp}$ and $\lim _{n\rightarrow \infty } f^*(x_n) =0$, where $R_\mathrm{nilp}= \{x \in R : \lim _{n\rightarrow \infty } x^n = 0\}$. Consider the R-module

$$\begin{aligned} T := \left\{ (r_n)_{n=1}^\infty \in R^{\mathbb {N}} : \lim _{n\rightarrow \infty } r_n = 0 \right\} \cong (\mathbb {Q}^{\oplus \mathbb {N}})\mathbin {\widehat{\otimes }}R. \end{aligned}$$

The topology on T is defined by submodules $(\mathbb {Q}^{\oplus \mathbb {N}})\mathbin {\widehat{\otimes }}R_\nu$. A morphism f associates the (continuous) tangent map $df :T \rightarrow T$ defined by $df (e_i) = \sum _{j=1}^\infty \left. \frac{\partial f^*(x_j)}{\partial x_i}\right| _{\mathbf {x}=0} e_j$. The following gives two important classes of morphisms.

• for a continuous R-module homomorphism $A :T \rightarrow T$ with $A(e_i) = \sum _{j=1}^\infty a_{ji} e_j$, we have a linear map f given by $f^*(x_j) = \sum _{j=1}^\infty a_{ji} x_i$;

• for an element $(r_j)_{j=1}^\infty \in T$ with $r_j \in R_\mathrm{nilp}$, we have a translation map f given by $f^*(x_j) = x_j + r_j$.

Theorem 6.1

(formal inverse function theorem) Let $f :{\text {Spf}}(R[\![\mathbf {x}]\!]) \rightarrow {\text {Spf}}(R[\![\mathbf {x}]\!])$ be a morphism of formal schemes over R. If the tangent map $df :T \rightarrow T$ at $\mathbf {x}=0$ is an isomorphism, f is an isomorphism.

Proof

By composing with a linear map and a translation, we may assume that $f(0) = 0$ and the tangent map df is the identity. Then the truncation of $f^*$ given by $R[\![x_1,\ldots ,x_n]\!] \subset R[\![\mathbf {x}]\!] \xrightarrow {f^*} R[\![\mathbf {x}]\!] \twoheadrightarrow R[\![x_1,\ldots ,x_n]\!]$ is an isomorphism, by the inverse function theorem in finite dimensions (see [32, Appendix A]; the proof over a discrete ring works verbatim over R). It follows easily that $f^*$ is an isomorphism. $\square$

Next we discuss the integrability of a formal vector field. A formal vector field on ${\text {Spf}}(R[\![\mathbf {x}]\!])$ over R is a formal sum $V= \sum _{n=1}^\infty V_n(\mathbf {x}) \frac{\partial }{\partial x_n}$ with $V_n(\mathbf {x}) \in R[\![\mathbf {x}]\!]$ such that $\lim _{n \rightarrow \infty } V_n(\mathbf {x}) = 0$. We consider the flow $t\mapsto \mathbf {x}(t)=(x_n(t))_{n=1}^\infty$ satisfying

$$\begin{aligned} \frac{dx_n(t)}{dt} = V_n(\mathbf {x}(t)) \quad \text {with}\quad x_n(0) = x_n. \end{aligned}$$

(5.3)

Theorem 6.2

There exists a unique solution $\mathbf {x}(t) = (x_n(t))_{n=1}^\infty$ to the Eq. (5.3) which defines a morphism ${\text {Spf}}(R[\![\mathbf {x}]\!][\![t]\!]) \rightarrow {\text {Spf}}(R[\![\mathbf {x}]\!])$ of formal schemes. Let $I\subset R$ be an ideal such that, for any $\nu$, there exists $n\in \mathbb {N}$ such that $I^n \subset R_\nu$. If $V_n(\mathbf {x})\in I[\![\mathbf {x}]\!]$ for all n, then the substitution $t=1$ in the solution $\mathbf {x}(t)$ is well-defined and we obtain a time-one flow map ${\text {Spf}}(R[\![\mathbf {x}]\!]) \rightarrow {\text {Spf}}(R[\![\mathbf {x}]\!])$.

Proof

Note that V defines a well-defined continuous mapping $V :R[\![\mathbf {x}]\!] \rightarrow R[\![\mathbf {x}]\!]$. The flow is given by a continuous ring homomorphism $R[\![\mathbf {x}]\!] \rightarrow R[\![\mathbf {x}]\!][\![t]\!]$ defined by $\varphi \mapsto \exp (t V) \varphi = \sum _{k=0}^\infty \frac{1}{k!} t^k V^k(\varphi )$, where $V^k$ is the k-fold composition of V, see [34, Section 3C]. The former statement follows. To see the latter, it suffices to notice that $\lim _{k\rightarrow \infty } V^k(\varphi ) = 0$ uniformly for all $\varphi \in R[\![\mathbf {x}]\!]$ under the assumption. $\square$