3d Mirror Symmetry and Elliptic Stable Envelopes

Abstract

We consider a pair of quiver varieties (X;X') 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' (the Mother function) whose restrictions to X and X' are the elliptic stable envelopes of those varieties. This implies, that the restriction matrices of the elliptic stable envelopes for X and X' are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.

arXiv comment50 pages, 5 figures

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Cited by in corpus (14)

• Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety
• 3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve
• Intersecting Defects and Supergroup Gauge Theory
• 3d Mirror Symmetry and Quantum $K$-theory of Hypertoric Varieties
• Quiver ${\text{W}}_{{\epsilon }_1,{\epsilon }_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories
• Symplectic Duality of $T^{\ast }Gr(k,n)$
• Bow varieties---geometry, combinatorics, characteristic classes
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• Quantum $K$-theory of toric varieties, level structures, and 3d mirror symmetry
• Elliptic canonical bases for toric hyper-Kahler manifolds
• Quasimaps to zero-dimensional $A_{\infty }$-quiver varieties
• Shuffle products for elliptic stable envelopes of Nakajima varieties
• Twisted Fock module of toroidal algebra via DAHA and vertex operators
• Quantum differential and difference equations for ${\mathrm{Hilb}}^n({\mathbb{C}}^2)$