The length classification of threefold flops via noncommutative algebras

Karmazyn, Joseph

doi:10.48550/arxiv.1709.02720

Cited by 3 publications

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“…95-96], further developed in [35] and [37]. Concretely, we will use some technology that has recently been developed in [38], whereby one describes not the geometry directly, but a quiver that encompasses all relevant information. Such a quiver, known as the universal flopping algebra, has the property that its space of representations is literally a model for the geometry of interest (up to taking completions of local rings), with deformations and resolutions being regarded as Fayet-Illiopoulos terms.…”

Section: The Effective Theorymentioning

confidence: 99%

“…Let us now summarize our findings. The construction by Karmazyn [38] of the socalled universal flopping algebra of length provides us a CY n-fold, 7 from which we can take any three-dimensional slice (or appropriate covering of such a slice). Any such slice or cover will contain a U(1) vector multiplet, and matter with charge .…”

Section: Summary and Outlinementioning

confidence: 99%

“…This is the case that will ultimately produce threefolds with matter of charge two, starting from the D 4 surface. Here, we expose the technology recently developed in [38], where one constructs these families in terms of quivers. In section 4, we actually proceed to the construction of threefolds with charge-two matter.…”

Section: M-theorymentioning

confidence: 99%

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High electric charges in M-theory from quiver varieties

Collinucci

Fazzi

Morrison

et al. 2019

J. High Energ. Phys.

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M-theory on a Calabi-Yau threefold admitting a small resolution gives rise to an Abelian vector multiplet and a charged hypermultiplet. We introduce into this picture a procedure to construct threefolds that naturally host matter with electric charges up to six. These are built as families of Du Val ADE surfaces (or ALE spaces), and the possible charges correspond to the Dynkin labels of the adjoint of the ADE algebra. In the case of charge two, we give a new derivation of the answer originally obtained by Curto and Morrison, and explicitly relate this construction to the Morrison-Park geometry. We also give a procedure for constructing higher-charge cases, which can often be applied to F-theory models. arXiv:1906.02202v1 [hep-th] 5 Jun 20191 Introduction M-theory on certain singular spaces gives rise to non-Abelian gauge symmetries. More specifically, in Calabi-Yau (CY) geometries with non-isolated singularities that admit crepant resolutions (i.e. those that do not alter the CY condition), Dynkin diagrams show up naturally as a pattern traced out by families of small intersecting two-spheres in the resolved geometry (for non-isolated holomorphic two-spheres that come in a complex one-dimensional family). When this occurs, postulating the presence of light M2-branes wrapping such so-called vanishing spheres gives rise to light degrees of freedom that fill out the root system corresponding to the Dynkin diagram. This leads one to expect the effective field theory to contain a non-Abelian gauge multiplet in the limit where areas of the two-spheres approach zero and the geometry becomes singular [1,2]. (For rigid two-spheres, we get light matter multiplets instead, typically charged under the non-Abelian group.) This picture has been exploited for two decades in the field of geometric engineering in string theory and M-theory [3][4][5][6][7][8], as well as in some closely related Ftheory constructions [9][10][11]. In the latter, a direct interpretation of the singular space is not available in all but the simplest cases. However, two strategies connect F-theory to this phenomenon of non-Abelian degrees of freedom: Either one can take a particular limit of M-theory, such that the duality to type IIB compactified on a circle is manifest, and verify the correspondence. Alternatively, one can analyze the F-theory phenomena directly by considering which (p, q)-strings in the type IIB seven-brane background become light. (An important feature of F-theory is in fact the existence of seven-branes of "exotic" types, i.e. beyond the D7-branes and O7-planes of perturbative type IIB.)Much less clear is what to make of possible Abelian gauge symmetries. 1 These can also be related to singularities, but not as directly as in the case of simple Lie algebras. Since the inception of F-theory, it has been known that one way to understand U(1) gauge groups is by seeking sections (or in some cases, multi-sections) of the elliptic fibration of F-theory [11][12][13]. In M-theory, it suffices to think of divisors in the compactification ...

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Section: The Effective Theorymentioning

confidence: 99%

Section: Summary and Outlinementioning

confidence: 99%

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High electric charges in M-theory from quiver varieties

Collinucci

Fazzi

Morrison

et al. 2019

J. High Energ. Phys.

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“…The second author was supported by EPSRC grant EP/K021400/1. explicit examples of Type E flops [BW,K1]. It is also conjectured that such algebras classify smooth 3-fold flops [DW1,HT], and furthermore it is expected that they control divisor-to-curve contractions.…”

Section: Introductionmentioning

confidence: 99%

Contractions and deformations

Donovan¹,

Wemyss²

2019

American Journal of Mathematics

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Suppose that f is a projective birational morphism with at most onedimensional fibres between d-dimensional varieties X and Y , satisfying Rf * O X = O Y . Consider the locus L in Y over which f is not an isomorphism. Taking the schemetheoretic fibre C over any closed point of L, we construct algebras A fib and Acon which prorepresent the functors of commutative deformations of C, and noncommutative deformations of the reduced fibre, respectively. Our main theorem is that the algebras Acon recover L, and in general the commutative deformations of neither C nor the reduced fibre can do this. As the d = 3 special case, this proves the following contraction theorem: in a neighbourhood of the point, the morphism f contracts a curve without contracting a divisor if and only if the functor of noncommutative deformations of the reduced fibre is representable.

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“…It has become increasingly clear that various aspects of birational geometry should be enhanced into a mildly noncommutative setting. One example of this, namely associating noncommutative algebras to certain contractions, has recently yielded many new results, including: algorithmic ways to relate minimal models based on cluster theory [W], new invariants for flips and flops [DW1] linked to Gopakumar-Vafa invariants [T2], the braiding of flop functors [DW3], noncommutative versions of curve counting [T3], a full conjectural analytic classification of 3-fold flops [DW1,HT], and, in addition, the first new examples of 3-fold flops since 1983 [BW,K1].…”

mentioning

confidence: 99%

Noncommutative enhancements of contractions

Donovan

Wemyss

2019

Advances in Mathematics

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Given a contraction of a variety X to a base Y , we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial resolutions admitting a tilting bundle with trivial summand, and (2) all contractions with fibre dimension at most one. In all cases, this produces a global invariant. In the crepant setting, we then apply this to study derived autoequivalences of X. We work generally, dropping many of the usual restrictions, and so both extend and unify existing approaches. In full generality we construct a new endofunctor of the derived category of X by twisting over D, and then, under appropriate restrictions on singularities, give conditions for when it is an autoequivalence. We show that these conditions hold automatically when the non-isomorphism locus in Y has codimension 3 or more, which covers determinantal flops, and we also control the conditions when the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve contractions.

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The length classification of threefold flops via noncommutative algebras

Cited by 3 publications

References 0 publications

High electric charges in M-theory from quiver varieties

High electric charges in M-theory from quiver varieties

Contractions and deformations

Noncommutative enhancements of contractions

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