Residue mirror symmetry for Grassmannian quasimaps and gauged linear sigma models

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Let GG act on a GG-space VV, and let W⊂VW\subset V be the zero locus of GG-semi-invariant polynomials fif_i, for i=1,…,ri=1,\ldots,r. For P∈C[t]WP\in\mathbb{C}[\mathfrak{t}]^{\mathcal{W}}, write ⟨P⟩WGITG\langle P\rangle_{W\mathbin{\mathrm{GIT}}G} for the generating function of quasimap invariants, and let cor⁡{P}\operatorname{cor}\{P\} denote the corresponding correlation function of the A-twisted gauged linear sigma model with potential ∑ifipi\sum_i f_i p_i on V×ArV\times\mathbb{A}^r, where pip_i has R-charge 22. Residue mirror symmetry conjecture. Provided the conditions in Sections C1 and C2 hold, for every P∈C[t]WP\in\mathbb{C}[\mathfrak{t}]^{\mathcal{W}} one has, up to an overall sign,

cor⁡{P}=±⟨P⟩WGITG.\operatorname{cor}\{P\}=\pm\langle P\rangle_{W\mathbin{\mathrm{GIT}}G}.

This predicts equality between quasimap invariants and A-twisted GLSM correlation functions, but the supplied text does not state whether the claim has been proved or remains open.

References

Primary source

Bumsig Kim, Jeongseok Oh, Kazushi Ueda and Yutaka Yoshida, “Residue mirror symmetry for Grassmannians”, arXiv:1607.08317 (2019).

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