Stratified Hyperkähler Spaces From Semisimple Lie Algebras

Mayrand, Maxence

doi:10.1007/s00031-018-9501-x

Cited by 8 publications

(14 citation statements)

References 17 publications

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“…This follows immediately from the results of this paper together with the author's paper [29]. See also [28,Theorem 3.1] for the case where χ = 1.…”

Section: 2supporting

confidence: 73%

“…Then, if f is K-invariant, proper, and bounded below, there is still a moment map µ (not necessarily the same as above) such that µ −1 (0)/K ∼ = M //G (see e.g. [13,Lemma 6.1] or [28,Proposition 4.2]). The standard version can be recovered by taking f = 1 2 · 2 .…”

Section: Introductionmentioning

confidence: 99%

“…If χ = 1, the condition C[M ] ⊆ o(e f ) can be replaced by the weaker condition that f is proper and bounded below (see e.g. [13,Lemma 6.1] or [28,Proposition 4.2]). It is easy to see that if C[M ] ⊆ o(e f ) then f is proper and bounded below.…”

Section: Introductionmentioning

confidence: 99%

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Kempf–Ness type theorems and Nahm equations

Mayrand

2019

Journal of Geometry and Physics

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We prove a version of the affine Kempf-Ness theorem for nonalgebraic symplectic structures and shifted moment maps, and use it to describe hyperkähler quotients of T * G, where G is a complex reductive group.where k := Lie(K) and ·, · is the standard inner-product on C n . Recall that G is reductive so M //G is an affine variety. Moreover, if µ −1 (0)/K is smooth, then its reduced symplectic form is a Kähler form on M //G. This theorem admits many generalizations and variants; for instance, there are versions for projective manifolds [33, §2] Another important version -which is closer to the spirit of this paper -is when M is affine as above but we shift the moment map (1.1). More precisely, if χ : K → S 1 is a character, then ξ := i dχ ∈ k * is central (i.e. fixed by the coadjoint action), so we can consider the symplectic reduction µ −1 (ξ)/K. Then, King [22, §6] (see also [20]) showed that µ −1 (ξ)/K is homeomorphic to the twisted GIT quotient M // χ G, i.e. the GIT quotient of M by G with respect to the trivial line bundle M × C with the G-action g · (p, z) = (g · p, χ(g)z). In other words,such that u(g · p) = χ(g) n u(p) for all p ∈ M and g ∈ G. Recall that the quasi-projective variety M// χ G is a categorical quotient

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“…This follows immediately from the results of this paper together with the author's paper [29]. See also [28,Theorem 3.1] for the case where χ = 1.…”

Section: 2supporting

confidence: 73%

Section: Introductionmentioning

confidence: 99%

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Kempf–Ness type theorems and Nahm equations

Mayrand

2019

Journal of Geometry and Physics

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“…One can compute the Hasse diagram of a classical Higgs branch through the partial Higgs mechanism, or if a magnetic quiver is known, through quiver subtraction on the magnetic quiver [17]. Hasse diagrams for singular hyper-Kähler quotients were studied in [26]. For unitary quivers a procedure to produce the Higgs branch (quiver variety) Hasse diagram is given in [27].…”

Section: Jhep09(2020)159mentioning

confidence: 99%

Hasse diagrams for 3d $$ \mathcal{N} $$ = 4 quiver gauge theories — Inversion and the full moduli space

Grimminger

Hanany

2020

J. High Energ. Phys.

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We study Hasse diagrams of moduli spaces of 3d $$ \mathcal{N} $$ N = 4 quiver gauge theories. The goal of this work is twofold: 1) We introduce the notion of inverting a Hasse diagram and conjecture that the Coulomb branch and Higgs branch Hasse diagrams of certain theories are related through this operation. 2) We introduce a Hasse diagram to map out the entire moduli space of the theory, including the Coulomb, Higgs and mixed branches. For theories whose Higgs and Coulomb branch Hasse diagrams are related by inversion it is straight forward to generate the Hasse diagram of the entire moduli space. We apply inversion of the Higgs branch Hasse diagram in order to obtain the Coulomb branch Hasse diagram for bad theories and obtain results consistent with the literature. For theories whose Higgs and Coulomb branch Hasse diagrams are not related by inversion it is nevertheless possible to produce the Hasse diagram of the full moduli space using different methods. We give examples for Hasse diagrams of the entire moduli space of theories with enhanced Coulomb branches.

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“…Section 2 establishes some of our conventions regarding symplectic and hyperkähler geometry. Section 3 then uses [10], [22], and [27] to develop the hyperkähler-geometric features of T * (G/H) needed for the subsequent discussion of hyperkähler slices. This leads to Section 4, which reviews Bielawski's hyperkähler slice construction and reduces the non-emptiness of (T * (G/H) × (G × S reg ))/ / /K to the condition that h ⊥ contain a regular element.…”

mentioning

confidence: 99%

An Application of Spherical Geometry to Hyperkähler Slices

Crooks¹,

Pruijssen²

2020

Can. J. Math.-J. Can. Math.

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This work is concerned with Bielawski's hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice to the data of a complex semisimple Lie group G, a reductive subgroup H ⊆ G, and a Slodowy slice S ⊆ g := Lie(G), defining it to be the hyperkähler quotient of T * (G/H) × (G × S) by a maximal compact subgroup of G. This hyperkähler slice is empty in some of the most elementary cases (e.g. when S is regular and (G, H) = (SLn+1, GLn), n ≥ 3), prompting us to seek necessary and sufficient conditions for non-emptiness.We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when S = Sreg is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called a-regularity of (G, H). This a-regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of G/H. We also provide a classification of the a-regular pairs (G, H) in which H is a reductive spherical subgroup. Our arguments make essential use of Knop's results on moment map images and Losev's algorithm for computing Cartan spaces.2010 Mathematics Subject Classification. 20G20 (primary); 53C26, 14M17 (secondary).

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Stratified Hyperkähler Spaces From Semisimple Lie Algebras

Cited by 8 publications

References 17 publications

Kempf–Ness type theorems and Nahm equations

Kempf–Ness type theorems and Nahm equations

Hasse diagrams for 3d $$ \mathcal{N} $$ = 4 quiver gauge theories — Inversion and the full moduli space

An Application of Spherical Geometry to Hyperkähler Slices

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