Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional $$ \mathcal{N}=4 $$ theories

Hanany, Amihay; Sperling, Marcus

doi:10.1007/jhep08(2018)189

Cited by 12 publications

(14 citation statements)

References 60 publications

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“…In a Lagrangian field theory (like these LSQs) , turning on an FI parameter decreases the Coulomb branch dimension by 1. Since there are rank(G) (= number of gauge nodes in the quiver) FI parameters available to turn on, this indicates that we can only flow to an IR SCFT , we note here that the height of a nilpotent orbit has also played an important role in recent works [81,111]. In unpublished work, A. Hanany and G. Ferlito [112] have also considered the behaviour of Dynkin quivers at various heights.…”

Section: Little String Quiversmentioning

confidence: 89%

Masses, Sheets and Rigid SCFTs

Balasubramanian¹,

Distler²

2018

Preprint

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We study mass deformations of certain three dimensional N 4 Superconformal Field Theories (SCFTs) that have come to be called T ρ [G] theories. These are associated to tame defects of the six dimensional (0, 2) SCFT X[j] for j A, D, E. We describe these deformations using a refined version of the theory of sheets, a subject of interest in Geometric Representation Theory. In mathematical terms, we parameterize local mass-like deformations of the tamely ramified Hitchin integrable system and identify the subset of the deformations that do admit an interpretation as a mass deformation for the theories under consideration. We point out the existence of non-trivial Rigid SCFTs among these theories. We classify the Rigid theories within this set of SCFTs and give a description of their Higgs and Coulomb branches. We then study the implications for the endpoints of RG flows triggered by mass deformations in these 3d N 4 theories. Finally, we discuss connections with the recently proposed idea of Symplectic Duality and describe some conjectures about its action.

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Section: Little String Quiversmentioning

confidence: 89%

Masses, Sheets and Rigid SCFTs

Balasubramanian¹,

Distler²

2018

Preprint

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“…It follows from early mathematical works [42,43] (and also from [44]) that, for a wealth of ADE quivers, H [Q] is the closure of a nilpotent orbit of the Lie algebra g into which Q • is shaped [5]. The singularity structure of closures of nilpotent orbits is then reinterpreted as the phase diagram of the gauge theory, a claim tested and successfully reproduced in a vast class of examples [45][46][47][48][49].…”

Section: Moduli Spaces Of Vacua Of 3d N = 4 Theoriesmentioning

confidence: 99%

Crystal bases and three-dimensional 𝒩 = 4 Coulomb branches

Santilli

Tierz

2022

J. High Energ. Phys.

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We establish and develop a correspondence between certain crystal bases (Kashiwara crystals) and the Coulomb branch of three-dimensional 𝒩 = 4 gauge theories. The result holds for simply-laced, non-simply laced and affine quivers. Two equivalent derivations are given in the non-simply laced case, either by application of the axiomatic rules or by folding a simply-laced quiver. We also study the effect of turning on real masses and the ensuing simplification of the crystal. We present a multitude of explicit examples of the equivalence. Finally, we put forward a correspondence between infinite crystals and Hilbert spaces of theories with isolated vacua.

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“…The enrichment of quivers §4.3.4 via [KKW15] provides a new link to mutations and is possibly related to [HS18].…”

Section: Collection Of Master Equations For Operad-type Examplesmentioning

confidence: 99%

Feynman categories and representation theory

Kaufmann¹

2021

Representations of Algebras, Geometry And Physics

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We give a presentation of Feynman categories from a representation-theoretical viewpoint.Feynman categories are a special type of monoidal categories and their representations are monoidal functors. They can be viewed as a far reaching generalization of groups, algebras and modules. Taking a new algebraic approach, we provide more examples and more details for several key constructions. This leads to new applications and results.The text is intended to be a self-contained basis for a crossover of more elevated constructions and results in the fields of representation theory and Feynman categories, whose applications so far include number theory, geometry, topology and physics.Organization of the text. The text is designed to be as self-contained as possible and is aimed at a diverse audience.We start in §1 with collecting classic results for groups and quiver representations, but reformulated in categorical language. This presentation might be of independent interest as a primer.The next paragraph, §2, contains the definition of Feynman category introduced as a special type of monoidal category. The representations are then given by strong monoidal functors. The development is parallel to that of §1. The presentation of indexed Feynman categories is new. The section ends with examples which are based on finite sets. Here we provide new details. The representations of these are various kinds of algebras. The group(oid) representations are also included as a basic example. Further examples are provided by graphical Feynman categories. The theory of graphs we use is detailed in Appendix A.We then turn to various constructions for Feynman categories in §3. These yield Feynman categories whose representations are lax-monoidal or simply functors. At this level the finite set based Feynman categories have FI-modules and (co)-(semi)-simplicial objects as objects. The next operation is that of decoration. It yields the graphical Feynman categories that encode operadic-types, see Table 7. The next construction is the plus construction. Here we give a detailed exposition of the condensed presentation in [KW17, §3.6], providing several explicit calculations. The new precision yields gcp-version of the plus-construction, which is a generalization of hyper version contained in [KW17, §3.7]. The relationship to indexing is also made more explicit here then previously. A detailed graphical based analysis is given in Appendix B, where we also give a careful discussion of levels.In §4 we tackle the enriched version. This is technically the most demanding and contains many new details. The bar/co-bar transformation and a dual transformation, aka. Feynman transform along with the master equations are discussed in §5. Traditionally the bar/co-bar adjunction can be used to define resolutions. For this one needs a model structure in general. The relevant details are reviewed in Appendix D. The W-construction is reviewed in §6. Here we also reconstruct the associahedra in their cubical decompositions.We end the paper with an outlook, §7...

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Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional $$ \mathcal{N}=4 $$ theories

Cited by 12 publications

References 60 publications

Masses, Sheets and Rigid SCFTs

Masses, Sheets and Rigid SCFTs

Crystal bases and three-dimensional 𝒩 = 4 Coulomb branches

Feynman categories and representation theory

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