Wrapped Fukaya–matrix factorisation mirror symmetry for equivariant Milnor fibres

Let w\mathbf{w} be an invertible polynomial with admissible symmetry group ΓΓw\Gamma\subseteq\Gamma_{\mathbf{w}}, and let Γˇ\widecheck{\Gamma} be the corresponding dual group. Let Vˇ=wˇ1(1)\widecheck{V}=\widecheck{\mathbf{w}}^{-1}(1) be the completed Milnor fibre of the mirror polynomial, viewed with the action of Γˇ\widecheck{\Gamma}, and let W([Vˇ/Γˇ])\mathcal{W}([\widecheck{V}/\widecheck{\Gamma}]) denote the wrapped Fukaya category of the quotient stack. Wrapped Fukaya–matrix factorisation mirror symmetry. There is a quasi-equivalence

W([Vˇ/Γˇ])mf(An+1,Γ,w+x0x1xn).\mathcal{W}([\widecheck{V}/\widecheck{\Gamma}])\simeq \mathrm{mf}(\mathbb{A}^{n+1},\Gamma,\mathbf{w}+x_0x_1\dots x_n).

This is an expected extension of homological mirror symmetry to equivariant Milnor fibres and quotient stacks. The source notes that an appropriate wrapped Fukaya-category definition in this setting is still needed, so the statement remains open.

Sources & referencesView supporting material

Primary source

Matthew Habermann, “Homological mirror symmetry for nodal stacky curves”, arXiv:2101.12178 (2023).

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