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.
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Notes
Small means that the parameter \(\tau \) is restricted to lie in \(H^2\); big means that the parameter space is the whole cohomology group.
This is equivalent to \(X_\Sigma \) being a GIT quotient of a vector space. We do not assume that \(X_\Sigma \) is projective or \(c_1(X_\Sigma )\) is semipositive.
Recall the convention on the T-action on \(H^0(X_\Sigma ,\mathcal {O})\) at the beginning of Sect. 2.3.
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Acknowledgments
I thank Eduardo González and Si Li for very helpful discussions. Joint work [25] with Eduardo González was a starting point of the present research. Si Li suggested me to consider deformations of F(x) by infinitely many monomials. This research is supported by JSPS Kakenhi Grant Numbers 16K05127, 16H06337, 25400069, 26610008, 23224002, 25400104.
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Communicated by Denis Auroux.
Appendix 1: Formal geometry in infinite dimensions
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
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.
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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 \(f^*(x_j) = \sum _{j=1}^\infty a_{ji} x_i\);
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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
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. \(\square \)
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Iritani, H. A mirror construction for the big equivariant quantum cohomology of toric manifolds. Math. Ann. 368, 279–316 (2017). https://doi.org/10.1007/s00208-016-1437-7
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DOI: https://doi.org/10.1007/s00208-016-1437-7