The conjectural cohomological stable-envelope formula for bow varieties

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Let \DD\DD be a brane diagram and let f∈\Ch(\DD)\Tf\in\Ch(\DD)^{\T} be a fixed point. For each tangent-weight term αx,y,kxyhk\alpha_{x,y,k}\frac{x}{y}\mathbf h^k, call it ff-small when its restriction satisfies \LocfK(xy)=uiuj\Loc^K_f\left(\frac{x}{y}\right)=\frac{u_i}{u_j} with i<ji<j. Define

W~f=e(∑f-smallαx,y,kxyhk),Wf=1NfSym⁡(W~f),\tilde W_f=e\left(\sum_{f\text{-small}}\alpha_{x,y,k}\frac{x}{y}\mathbf h^k\right),\qquad W_f=\frac{1}{N_f}\operatorname{Sym}(\tilde W_f),

where Sym⁡\operatorname{Sym} is the symmetrizing operator and Nf=∏U D5∏i=1sdiU!N_f=\prod_{U\text{ D5}}\prod_{i=1}^s d_i^U!. Cohomological stable-envelope conjecture. Cohomological stable envelopes exist for bow varieties, and the stable envelope for ff is represented by WfW_f. The conjecture further asserts that the substitution defining each cohomological restriction of WfW_f has a limit that exists and is a polynomial, so that WfW_f defines an element of ⨁f′H\T∗(f′)\bigoplus_{f'}H^*_{\T}(f'). The formula proposes an explicit construction of stable envelopes from tangent weights and fixed-point combinatorics; the supplied passage gives no resolution or partial proof, so its status is open.

References

Primary source

R. Rimanyi and Y. Shou, “Bow varieties—geometry, combinatorics, characteristic classes”, arXiv:2012.07814 (2020).

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