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Abstract We consider a pair of quiver varieties $$(X;X^{\prime})$$ related by 3D mirror symmetry, where $$X =T^*{Gr}(k,n)$$ is the cotangent bundle of the Grassmannian of $$k$$-planes of $$n$$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $$X \times X^{\prime} $$ (the mother function) whose restrictions to $$X$$ and $$X^{\prime} $$ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $$X$$ and $$X^{\prime}$$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.
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Pursuing Quantum Difference Equations II: 3D mirror symmetry
Abstract Let $$\textsf {X}$$ and $$\textsf {X}^{!}$$ be a pair of symplectic varieties dual with respect to 3D mirror symmetry. The $$K$$-theoretic limit of the elliptic duality interface is an equivariant $$K$$-theory class $$\mathfrak {m} \in K(\textsf {X}\times \textsf {X}^{!})$$. We show that this class provides correspondences $$ \begin{align*} & \Phi_{\mathfrak{m}}: K(\textsf{X}) \leftrightarrows K(\textsf{X}^{!}) \end{align*}$$mapping the $$K$$-theoretic stable envelopes to the $$K$$-theoretic stable envelopes. This construction allows us to relate various representation theoretic objects of $$K(\textsf {X})$$, such as action of quantum groups, quantum dynamical Weyl groups, $$R$$-matrices, etc., to those for $$K(\textsf {X}^{!})$$. In particular, we relate the wall $$R$$-matrices of $$\textsf {X}$$ to the $$R$$-matrices of the dual variety $$\textsf {X}^{!}$$. As an example, we apply our results to $$\textsf {X}=\textrm {Hilb}^{n}({{\mathbb {C}}}^2)$$—the Hilbert scheme of $$n$$ points in the complex plane. In this case, we arrive at the conjectures of Gorsky and Negut from [10].
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- Award ID(s):
- 2054527
- PAR ID:
- 10433249
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2023
- Issue:
- 15
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- p. 13290-13331
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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