Residue mirror symmetry for Grassmannian quasimaps and gauged linear sigma models

Let GG act on a GG-space VV, and let WVW\subset V be the zero locus of GG-semi-invariant polynomials fif_i, for i=1,,ri=1,\ldots,r. For PC[t]WP\in\mathbb{C}[\mathfrak{t}]^{\mathcal{W}}, write PWGITG\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 PC[t]WP\in\mathbb{C}[\mathfrak{t}]^{\mathcal{W}} one has, up to an overall sign,

cor{P}=±PWGITG.\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.

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Primary source

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

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