Asymptotically cylindrical Calabi-Yau manifolds

Haskins, Mark E.; Hein, Hans-Joachim; Nordström, Johannes

doi:10.48550/arxiv.1212.6929

Cited by 11 publications

(38 citation statements)

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“…This result is analogous to [14, Theorem 3.1], but Li's proof is different from the proof in [14]. The aim of this appendix is to clarify the relation between the conical and the cylindrical case.…”

Section: Appendix a Li's Compactification Theoremmentioning

confidence: 61%

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Quasi-asymptotically conical Calabi–Yau manifolds

2019

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We construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). We do so by first providing a natural compactification of QAC-spaces by manifolds with fibred corners and by giving a definition of QAC-metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on QAC-spaces developed by the second author and Mazzeo, we can in many instances obtain Kähler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. This allows us to obtain QAC Calabi-Yau metrics in the Kähler classes of these metrics by solving a corresponding complex Monge-Ampère equation.

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Section: Appendix a Li's Compactification Theoremmentioning

confidence: 61%

“…It seems possible that these cases can also be treated using the holomorphic isometries of C instead of Mori theory 4. Thus, unlike the construction of X as a complex orbifold in[22] (compare Appendix A), our proof that X is Kähler is quite different in spirit from the treatment of the asymptotically cylindrical case in[14].…”

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confidence: 81%

Quasi-asymptotically conical Calabi–Yau manifolds

2019

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“…We therefore seek a similar result in the Calabi-Yau case. We note that by work beginning with Tian-Yau and continued by Kovalev and Haskins-Hein-Nordström [15,27,38], asymptotically cylindrical Calabi-Yau manifolds do indeed exist. It is known that they can be glued by deformation in the sense of complex algebraic geometry.…”

Section: Introductionmentioning

confidence: 77%

“…Conversely to Kovalev, they did not explicitly consider deformations of the Kähler form, but observed (Lemma 9.1) that if the first Betti number of the compactifying orbifold is zero, then Kähler classes remain Kähler under such deformations of complex structure. Combining this with [15,Theorem D], which says in particular that for any Kähler class we can find an asymptotically cylindrical Calabi-Yau metric, we essentially expect that every Kähler class on the orbifold gives a deformation of the metric corresponding to this complex deformation.…”

Section: G2mentioning

confidence: 84%

Gluing and deformation of asymptotically cylindrical Calabi–Yau manifolds in complex dimension three

Talbot¹

2019

Differential Geometry and its Applications

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We study gluings of asymptotically cylindrical special Lagrangian submanifolds in asymptotically cylindrical Calabi-Yau manifolds. We prove both that there is a well-defined gluing map, and, after reviewing the deformation theory for special Lagrangians, prove that this gluing map defines a local diffeomorphism from matching pairs of deformations of asymptotically cylindrical special Lagrangians to deformations of special Lagrangians. We also give some examples of asymptotically cylindrical special Lagrangian submanifolds to which these results apply.

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“…It is natural to ask how these gluing operations equipped with Higgs bundles lift to a local Spin(7) geometry, and more globally, to possibly compact Spin(7) spaces. As noted in [2], these sorts of operations can often be interpreted as a generalization of the standard connected sums construction of references [88,89] (see also [1] for a proposed extension to Spin(7) spaces). Related to this, it would be interesting to better understand the possible constraints which come from coupling our 3d theories to gravity.…”

Section: Discussionmentioning

confidence: 99%

Reflections on the Matter of 3d $\mathcal{N} = 1$ Vacua and Local $Spin(7)$ Compactifications

Cvetič,

Heckman,

Torres

et al. 2021

Preprint

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We use Higgs bundles to study the 3d N = 1 vacua obtained from M-theory compactified on a local Spin(7) space given as a four-manifold M 4 of ADE singularities with further generic enhancements in the singularity type along one-dimensional subspaces. There can be strong quantum corrections to the massless degrees of freedom in the low energy effective field theory, but topologically robust quantities such as "parity" anomalies are still calculable. We show how geometric reflections of the compactification space descend to 3d reflections and discrete symmetries. The "parity" anomalies of the effective field theory descend from topological data of the compactification. The geometric perspective also allows us to track various perturbative and non-perturbative corrections to the 3d effective field theory. We also provide some explicit constructions of well-known 3d theories, including those which arise as edge modes of 4d topological insulators, and 3d N = 1 analogs of grand unified theories. An additional result of our analysis is that we are able to track the spectrum of extended objects and their transformations under higher-form symmetries.

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Asymptotically cylindrical Calabi-Yau manifolds

Cited by 11 publications

References 0 publications

Quasi-asymptotically conical Calabi–Yau manifolds

Quasi-asymptotically conical Calabi–Yau manifolds

Gluing and deformation of asymptotically cylindrical Calabi–Yau manifolds in complex dimension three

Reflections on the Matter of 3d $\mathcal{N} = 1$ Vacua and Local $Spin(7)$ Compactifications

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