2020
DOI: 10.48550/arxiv.2008.05516
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Symplectic Duality of $T^*Gr(k,n)$

Hunter Dinkins

Abstract: In this paper, we explore a consequence of symplectic duality (also known as 3d mirror symmetry) in the setting of enumerative geometry. The theory of quasimaps allows one to associate hypergeometric functions called vertex functions to quiver varieties. In this paper, we prove a formula which relates the vertex functions of T * Gr(k, n) and its symplectic dual. In the course of the proof, we study a family of q-difference operators which act diagonally on Macdonald polynomials. Our results may be interpreted … Show more

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Cited by 3 publications
(5 citation statements)
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“…The work in this paper has applications to half-indices or partition function of supersymmetric boundary conditions on three-manifolds M 3 with boundary ∂M 3 = E τ [16][17][18][19]. In combination with our previous work [20], this can be used to shed light on the role of elliptic stable envelopes in implementing mirror symmetry of vertex functions [5,21,22]. We will return to this topic in a future publication [23].…”
Section: Enriched Neumannmentioning
confidence: 89%
See 1 more Smart Citation
“…The work in this paper has applications to half-indices or partition function of supersymmetric boundary conditions on three-manifolds M 3 with boundary ∂M 3 = E τ [16][17][18][19]. In combination with our previous work [20], this can be used to shed light on the role of elliptic stable envelopes in implementing mirror symmetry of vertex functions [5,21,22]. We will return to this topic in a future publication [23].…”
Section: Enriched Neumannmentioning
confidence: 89%
“…Finally, we mention that the difference between exceptional Dirichlet and enriched Neumann boundary conditions is also detected by the general half index [16][17][18] counting quarter-BPS boundary local operators. We will explore this topic, connecting with the mathematical literature on vertex functions [67] and exploring their mirror symmetry properties [21,22,68] in a future work [23].…”
Section: Higgs Branch Supportmentioning
confidence: 99%
“…The work in this paper has applications to half-indices or partition function of supersymmetric boundary conditions on three-manifolds M 3 with boundary ∂ M 3 = E τ [16][17][18][19]. In combination with our previous work [20], this can be used to shed light on the role of elliptic stable envelopes in implementing mirror symmetry of vertex functions [5,73,74]. We will return to this topic in a future publication [21].…”
Section: Enriched Neumannmentioning
confidence: 89%
“…Finally, we mention that the difference between exceptional Dirichlet and enriched Neumann boundary conditions is also detected by the general half index [16][17][18] counting quarter-BPS boundary local operators. We will explore this topic, connecting with the mathematical literature on vertex functions [72] and exploring their mirror symmetry properties [73][74][75] in a future work [21].…”
Section: Higgs Branch Supportmentioning
confidence: 99%
“…This result has been proven in the case where X is a hypertoric variety in [20], as well as for the vertex function at a particular fixed point point for the cotangent bundle of the Grassmannian in [5]. As far as we are aware, this has not been proven for any other examples.…”
Section: Introductionmentioning
confidence: 94%