A Note on Support in Triangulated Categories

Bergman, Aaron

doi:10.48550/arxiv.0804.3986

Cited by 3 publications

(4 citation statements)

References 14 publications

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“…If we consider the derived category of compactly supported coherent sheaves on X then this is equivalent to the derived category of finitedimensional quiver representations. See [17,18] for a further discussion of this. Let V be a finite dimensional representation of the the quiver Q (or a complex of representations with this as the only nonzero entry).…”

Section: The Flopmentioning

confidence: 99%

Probing Geometry with Stability Conditions

Aspinwall

2009

Preprint

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The notion that the geometry of spacetime is given by the moduli space of 0-branes is examined in four examples of Calabi-Yau threefolds. An important consideration when determining the moduli space of D-branes is the stability condition and this is key in our analysis. In the first two examples, the flop and the orbifold blowup, no surprises are found. Next, an exoflop is considered where the linear sigma model implies a P 1 external to the Calabi-Yau threefold is part of the geometry. The 0-brane probe sees no such external P 1 and furthermore exhibits a surprising discontinuity when following an extremal transition associated to the exoflop. Finally we consider a hybrid model of a Landau-Ginzburg fibration over a P 1 . Using the technology of matrix factorizations we find a D-brane probe whose moduli space is this P 1 but it is not a 0-brane and is not stable at the large radius limit of the Calabi-Yau manifold.

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Section: The Flopmentioning

confidence: 99%

Probing Geometry with Stability Conditions

Aspinwall

2009

Preprint

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“…So, following (19) we want to find elements δ ∈ D such that H k B Σ (S) δ = 0 for all k ≥ 2. Actually we will impose a slightly stronger condition to include k = 0 and 1: 4…”

Section: D(x) Generated By Line Bundles 41 Tilting Line Bundlesmentioning

confidence: 99%

“…There are related statements such as the equivalences concerning bounded derived categories of sheaves with compactly supported cohomology. See, for example,[18,19].…”

mentioning

confidence: 99%

D-Branes on Toric Calabi-Yau Varieties

Aspinwall

2008

Preprint

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We analyze B-type D-branes on noncompact toric Calabi-Yau spaces. A general program is presented to find a set of tilting line bundles that yields the associated quiver and its relations. In many cases, this set remains fixed as one moves between phases in the Kähler moduli space. This gives a particularly simple picture of how the derived category remains invariant across all phases. The combinatorial problems involving local cohomology used to determine the tilting set are also related to questions of Π-stability as one moves between phases. As a result, in some cases precisely those line bundles in the tilting set remain stable over the whole moduli space in some sense.

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“…This latter category is, in fact, a Calabi-Yau category of dimension three. It is proven in [23] that…”

Section: Equivalences Of Categoriesmentioning

confidence: 99%

Stability conditions and branes at singularities

Bergman

2008

J. High Energy Phys.

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I use Bridgeland's definition of a stability condition on a triangulated category to investigate the stability of D-branes on Calabi-Yau cones given by the canonical line bundle over a del Pezzo surface. In this context, I prove the existence of the decay of a D3-brane into a set of fractional branes. This is an important aspect of the derivation of quiver gauge theories from branes at singularities via the technique of equivalences of categories. Some important technical aspects of this equivalence are discussed. I also prove that the representations corresponding to skyscraper sheaves supported off the zero section are simple.

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A Note on Support in Triangulated Categories

Cited by 3 publications

References 14 publications

Probing Geometry with Stability Conditions

Probing Geometry with Stability Conditions

D-Branes on Toric Calabi-Yau Varieties

Stability conditions and branes at singularities

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