Quantizations of conical symplectic resolutions I: local and global structure

Braden, Tom; Proudfoot, Nicholas; Webster, Ben

doi:10.48550/arxiv.1208.3863

Cited by 56 publications

(167 citation statements)

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“…The action of S := C × on T * (C n ) by s (x, y) = (s −1 x, s −1 y) gives actions on M and M 0 such that ν is equivariant; we have s • ω = s 2 ω where ω is the symplectic form on M. The S action on C[M 0 ] gives M 0 the structure of an affine cone; it has nonnegative weights with zero weight space consisting of constant functions. Thus, when M is smooth, M ν − → M 0 is a conical symplectic resolution in the sense of [BPW16]. In general, the map M ν − → M 0 with the S actions is invariant under equivalences of rational arrangements V.…”

Section: Hypertoric Varieties and Algebrasmentioning

confidence: 99%

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

Lauda,

Licata,

Manion

2020

Preprint

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We give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement, using the algebras defined in [BLP + 11] from the equivariant cohomology of toric varieties. We prove this conjecture for cyclic arrangements by showing that these algebras are isomorphic to algebras appearing in work of Ozsváth-Szabó [OSz18] in bordered Heegaard Floer homology [LOT18]. The proof of our conjecture in the cyclic case extends work of Karp-Williams [KW19] on sign variation and the combinatorics of the m = 1 amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of gl(1|1).

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Section: Hypertoric Varieties and Algebrasmentioning

confidence: 99%

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

Lauda,

Licata,

Manion

2020

Preprint

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“…From the perspective of the Coulomb branch, the W(F) action has a different interpretation. In the framework of [114], the family of Coulomb branches (parameterized by the value of the complex mass parameters) obtained by mass deformations can be identified with a canonical Poisson deformation associated to every Symplectic resolution . The Flavour symmetry group F does not act on the Coulomb branch.…”

Section: Connections To Symplectic Dualitymentioning

confidence: 99%

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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“…There is an N-algebra attached to (X, λ) and the choice of a very ample line bundle L on X. To construct it, let η ∈ H 2 (X, C) be the first Chern class of L. Then, for any pair of integers k, m, [BPW,Proposition 5.2] shows that there is a unique sheaf of C * -equivariant (Q λ+kη X , Q λ+mη X )bimodules kη B mη (λ) which quantizes the line bundle L k−m . Set kη S mη (λ) = Γ C * ( kη B mη (λ)[h −1/m ]) ∈ (U λ+kη , U λ+mη )-bimod.…”

Section: Equidimensionalitymentioning

confidence: 99%

“…Then the desired N-algebra is S(λ, η) = k≥m≥0 kη S mη (λ), with multiplication induced from tensor products (see [BPW,Definition 5.5]). 9.5.…”

Section: Equidimensionalitymentioning

confidence: 99%

The Auslander–Gorenstein property forZ-algebras

Gordon

Stafford

2014

Journal of Algebra

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We provide a framework for part of the homological theory of Z-algebras and their generalizations, directed towards analogues of the Auslander-Gorenstein condition and the associated double Ext spectral sequence that are useful for enveloping algebras of Lie algebras and related rings. As an application, we prove the equidimensionality of the characteristic variety of an irreducible representation of the Z-algebra, and for related representations over quantum symplectic resolutions. In the special case of Cherednik algebras of type A, this answers a question raised by the authors. S = 0≤q≤p∈Z S p,q with matrix-style multiplication and where each non-zero S q,q is a noetherian, unital algebra and the S p,q are finitely generated modules over both S p,p and S q,q .These algebras are important in noncommutative algebraic geometry -see, for example, [SV] or [VdB]. They are also useful in geometric representation theory, including in the study of rational Cherednik algebras [GS1, GS2], deformed preprojective algebras [Bo, Mu], finite W -algebras [Gi] and deformations of conical symplectic singularities [BPW]. In applications, the Z-algebra provides an effective way to relate the representation theory of the given algebra to the geometry of a resolution of singularities for its associated graded ring: indeed, the Z-algebra can be regarded as a quantization of that resolution. In the above examples, the Z-algebras quantize Hilbert schemes of points on the plane, minimal resolutions of Kleinian singularities, resolutions of Slodowy slices and, most generally, symplectic resolutions of conical symplectic singularities, respectively.What is missing, and what is provided in this paper, is a suitable homological machine to relate the commutative and noncommutative theories. Our aim is show how the Auslander-Gorenstein condition and the related double Ext spectral sequence, that are so useful for enveloping algebras and related rings, can be generalised to work for Z-algebras. As an application we generalize Gabber's equidimensionality result to Z-algebras.1.2. To explain our results in more detail, we need some notation. Write S 0,0 -mod for the category of noetherian left S 0,0 -modules, S-grmod for the category of noetherian left S-modules and S-qgr for the quotient category of S-grmod modulo the bounded modules. Let π S : S-grmod → S-qgr be the natural projection. If the S p,q are (S p,p , S q,q )-progenerators, we say that S is a Morita Z-algebra. In this case, for any q ≥ 0, there is an equivalence of categories S q,q -mod ∼ −→ S-qgr given by Φ : M → π S (S * ,q ⊗ Sq,q M ), where S * ,q = p≥q S p,q .We assume that there is a filtration F on S such that the associated graded algebra gr F S = gr F S p.q = ∆(R). Here, ∆(R) = ∆(R) p,q is the Z-algebra associated to some finitely generated commutative graded algebra R = R n by defining ∆(R) p,q = R p−q for all p ≥ q. There are natural equivalences ∆(R)-qgr ≃ R-qgr ≃ Coh(X), for X = Proj(R). If M ∈ S 00 -mod has a good filtration F , then M = Φ(M ) is naturally filtered an...

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Quantizations of conical symplectic resolutions I: local and global structure

Cited by 56 publications

References 57 publications

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

Masses, Sheets and Rigid SCFTs

The Auslander–Gorenstein property forZ-algebras

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