Spherical subcategories in algebraic geometry

Hochenegger, Andreas; Kalck, Martin; Ploog, David

doi:10.1002/mana.201400232

Cited by 21 publications

(31 citation statements)

References 27 publications

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“…A particular class of auto-equivalences of a triangulated category C is given by the Thomas-Seidel twists associated to spherical objects [53,54]; they have the advantage of having a very explicit form. In many cases the spherical twists generate the full group of autoequivalences 22 Aut(C ), and hence we may expect them to produce a 'substantial' part of the S-duality group.…”

Section: Thomas-seidel Twists In a Triangle Categorymentioning

confidence: 99%

Homological $S$-duality in 4d $\mathcal{N}=2$ QFTs

Caorsi

Cecotti

2018

Advances in Theoretical and Mathematical Physics

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The S-duality group S(F ) of a 4d N = 2 supersymmetric theory F is identified with the group of triangle equivalences of its cluster category C (F ) modulo the subgroup acting trivially on the physical quantities. S(F ) is a discrete group commensurable to a subgroup of the Siegel modular group Sp(2g, Z) (g being the dimension of the Coulomb branch). This identification reduces the determination of the S-duality group of a given N = 2 theory to a problem in homological algebra. In this paper we describe the techniques which make the computation straightforward for a large class of N = 2 QFTs. The group S(F ) is naturally presented as a generalized braid group.The S-duality groups are often larger than expected. In some models the enhancement of S-duality is quite spectacular. For instance, a QFT with a huge S-duality group is the Lagrangian SCFT with gauge group SO(8) × SO(5) 3 × SO(3) 6 and half-hypermultiplets in the bi-and tri-spinor representations.

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Section: Thomas-seidel Twists In a Triangle Categorymentioning

confidence: 99%

Homological $S$-duality in 4d $\mathcal{N}=2$ QFTs

Caorsi

Cecotti

2018

Advances in Theoretical and Mathematical Physics

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“…P 1 [k]-objects are well-known as spherical objects; see [45]. Without the Calabi-Yau property they are called spherelike objects and studied in [21,22] by Kalck, Ploog and the first author. P n [2]-objects are known as P n -objects and studied in [25] by Huybrechts and Thomas.…”

Section: Definition and Basic Examplesmentioning

confidence: 99%

Formality of -objects

Hochenegger¹,

Krug²

2019

Compositio Math.

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We show that a P-object and simple configurations of Pobjects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.

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“…In [HKP16] the authors show that if E is a d-spherelike object, there exists a unique maximal triangulated subcategory of T in which E becomes d-spherical. In the following we will imitate this construction for a larger class of objects.…”

Section: 2mentioning

confidence: 99%

“…However, a strongly crepant categorical resolution inside D b ( Y ) is unique, as we show in Proposition 4.9. Our concept of weakly crepant neighbourhoods was motivated by the idea that some non-CY objects possess 'CY neighbourhoods" (a construction akin to the spherical subcategories of spherelike objects in [HKP16]), i.e. full subcategories in which they become Calabi-Yau.…”

Section: Introductionmentioning

confidence: 99%

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Derived categories of resolutions of cyclic quotient singularities

Krug

Ploog

Sosna

2017

The Quarterly Journal of Mathematics

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For a cyclic group G acting on a smooth variety X with only one character occurring in the G-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold [X/G] and the blow-up resolution Y → X/G.Some results generalise known facts about X = A n with diagonal G-action, while other results are new also in this basic case. In particular, if the codimension of the fixed point locus equals |G|, we study the induced tensor products under the equivalence D b ( Y ) ∼ = D b ([X/G]) and give a 'flop-flop=twist' type formula. We also introduce candidates for general constructions of categorical crepant resolutions inside the derived category of a given geometric resolution of singularities and test these candidates on cyclic quotient singularities.

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Spherical subcategories in algebraic geometry

Cited by 21 publications

References 27 publications

Homological $S$-duality in 4d $\mathcal{N}=2$ QFTs

Homological $S$-duality in 4d $\mathcal{N}=2$ QFTs

Formality of -objects

Derived categories of resolutions of cyclic quotient singularities

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