Symplectic Duality of $T^*Gr(k,n)$

Dinkins, Hunter

doi:10.48550/arxiv.2008.05516

Cited by 3 publications

(5 citation statements)

References 18 publications

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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%

“…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%

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3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve

Bullimore¹,

Zhang²

2021

Preprint

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This paper studies 3d N = 4 supersymmetric gauge theories on an elliptic curve, with the aim to provide a physical realisation of recent constructions in equivariant elliptic cohomology of symplectic resolutions. We first study the Berry connection for supersymmetric ground states in the presence of mass parameters and flat connections for flavour symmetries, which results in a natural construction of the equivariant elliptic cohomology variety of the Higgs branch. We then investigate supersymmetric boundary conditions and show from an analysis of boundary 't Hooft anomalies that their boundary amplitudes represent equivariant elliptic cohomology classes. We analyse two distinguished classes of boundary conditions known as exceptional Dirichlet and enriched Neumann, which are exchanged under mirror symmetry. We show that the boundary amplitudes of the latter reproduce elliptic stable envelopes introduced by Aganagic-Okounkov, and relate boundary amplitudes of the mirror symmetry interface to the mother function in equivariant elliptic cohomology. Finally, we consider correlation functions of Janus interfaces for varying mass parameters, recovering the chamber R-matrices of elliptic integrable systems.

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Section: Enriched Neumannmentioning

confidence: 89%

Section: Higgs Branch Supportmentioning

confidence: 99%

3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve

Bullimore¹,

Zhang²

2021

Preprint

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show abstract

“…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%

3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve

Bullimore

Zhang

2022

SciPost Phys.

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This paper studies 3d3d\mathcal{N}=4𝒩=4 supersymmetric gauge theories on an elliptic curve, with the aim to provide a physical realisation of recent constructions in equivariant elliptic cohomology of symplectic resolutions. We first study the Berry connection for supersymmetric ground states in the presence of mass parameters and flat connections for flavour symmetries, which results in a natural construction of the equivariant elliptic cohomology variety of the Higgs branch. We then investigate supersymmetric boundary conditions and show from an analysis of boundary ’t Hooft anomalies that their boundary amplitudes represent equivariant elliptic cohomology classes. We analyse two distinguished classes of boundary conditions known as exceptional Dirichlet and enriched Neumann, which are exchanged under mirror symmetry. We show that the boundary amplitudes of the latter reproduce elliptic stable envelopes introduced by Aganagic-Okounkov, and relate boundary amplitudes of the mirror symmetry interface to the mother function in equivariant elliptic cohomology. Finally, we consider correlation functions of Janus interfaces for varying mass parameters, recovering the chamber R-matrices of elliptic integrable systems.

show abstract

“…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%

3d mirror symmetry of the cotangent bundle of the full flag variety

Dinkins¹

2020

Preprint

Self Cite

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Aganagic and Okounkov proved that the elliptic stable envelope provides the pole cancellation matrix for the enumerative invariants of quiver varieties known as vertex functions. This transforms a basis of a system of q-difference equations holomorphic in z with poles in a to a basis of solutions holomorphic in a with poles in z. The resulting functions are expected to be the vertex functions of the 3d mirror dual variety. In this paper, we prove that for the cotangent bundle of the full flag variety, the functions obtained in this way recover the vertex functions for the same variety under an exchange of the parameters a ↔ z. As a corollary of this, we deduce the expected 3d mirror relationship for the elliptic stable envelope.

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Symplectic Duality of $T^*Gr(k,n)$

Cited by 3 publications

References 18 publications

3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve

3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve

3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve

3d mirror symmetry of the cotangent bundle of the full flag variety

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