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

Dinkins, Hunter

doi:10.48550/arxiv.2011.08603

Cited by 7 publications

(16 citation statements)

References 16 publications

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“…The cotangent bundle of the full flag variety is known to be self dual with respect to 3d mirror symmetry, see [25], [7], and [10]. We denote by X !…”

Section: Dual Varietymentioning

confidence: 99%

“…In this paper, we study this problem for the special case when X is the cotangent bundle of the full flag variety. This variety is known to be self-dual with respect to 3d mirror symmetry, see [25] and [7]. We denote the 3d mirror dual copy of this variety by X !…”

Section: Introductionmentioning

confidence: 99%

“…are isomorphic as varieties, the 3d mirror symmetry relationship requires that various tori associated to X and X ! are related in non-trivial ways, see (7) below. Hence it will be necessary to distinguish them.…”

Section: Introductionmentioning

confidence: 99%

“…Using results from [7], we identify in Theorem 2 the power series expansion of χ Stab s,X,K C,T 1/2 X (p) around a certain point with an object we call the index vertex of X ! .…”

Section: Introductionmentioning

confidence: 99%

“…More specifically, we expect that for any two varieties related by 3d mirror symmetry satisfying the conditions listed at the beginning of the introduction, the index vertex and the K-theoretic stable envelopes should be related in the same way as described here. Indeed, the proof of Theorem 2 shows that the relationship between the K-theoretic stable envelopes and the index vertex is really a consequence of a more general conjecture regarding elliptic stable envelopes and vertex functions of 3d mirror dual varieties, see [9], [7], [28], [14] and the introduction of [1]. While a general construction of 3d mirror dual pairs is not presently known, the construction of Coloumb branches in [17] and [2] provides a large class of varieties 3d mirror dual to quiver varieties, and more generally Higgs branches of 3d N = 4 gauge theories.…”

Section: Introductionmentioning

confidence: 99%

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Euler characteristic of stable envelopes

Dinkins¹,

Smirnov²

2021

Preprint

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In this paper we prove a formula relating the equivariant Euler characteristic of K-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the index vertex is the power series expansion of a rational function. This result is a consequence of the 3d mirror self-symmetry of the variety considered here. In general, one expects an analogous result to hold for any two varieties related by 3d mirror symmetry.

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“…The cotangent bundle of the full flag variety is known to be self dual with respect to 3d mirror symmetry, see [25], [7], and [10]. We denote by X !…”

Section: Dual Varietymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Using results from [7], we identify in Theorem 2 the power series expansion of χ Stab s,X,K C,T 1/2 X (p) around a certain point with an object we call the index vertex of X ! .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Euler characteristic of stable envelopes

Dinkins¹,

Smirnov²

2021

Preprint

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On supersymmetric interface defects, brane parallel transport, order-disorder transition and homological mirror symmetry

Galakhov

2022

J. High Energ. Phys.

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We concentrate on a treatment of a Higgs-Coulomb duality as an absence of manifest phase transition between ordered and disordered phases of 2d $$ \mathcal{N} $$ N = (2, 2) theories. We consider these examples of QFTs in the Schrödinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated categories of D-brane boundary conditions. As a result of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are equivalent, this resembles an analog of homological mirror symmetry. Following construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one is able to introduce interface defects in these theories and associate them to D-brane parallel transport functors. We concentrate on surveying simple examples, analytic when possible calculations, numerical estimates and simple physical picture behind curtains of geometric objects. Categorification of hypergeometric series analytic continuation is derived as an Atiyah flop of the conifold. Finally we arrive to an interpretation of the braid group action on the derived category of coherent sheaves on cotangent bundles to flag varieties as a categorification of Berry connection on the Fayet-Illiopolous parameter space of a sigma-model with a quiver variety target space.

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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.

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

Cited by 7 publications

References 16 publications

Euler characteristic of stable envelopes

Euler characteristic of stable envelopes

On supersymmetric interface defects, brane parallel transport, order-disorder transition and homological mirror symmetry

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

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