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Tjotta, Erik N.

doi:10.1023/a:1017585802461

Cited by 12 publications

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References 20 publications

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“…We briefly review this construction for the special case of our Pfaffian varieties. For details see [19,43,23,41].…”

Section: Example: Ymentioning

confidence: 99%

On genus one fibered Calabi-Yau threefolds with 5-sections

Knapp,

Scheidegger,

Schimannek

2021

Preprint

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Elliptic and genus one fibered Calabi-Yau spaces play a prominent role in string theory and mathematics. In this article we discuss a class of genus one fibered Calabi-Yau threefolds with 5-sections from various perspectives. In algebraic geometry, such Calabi-Yaus can be constructed as complete intersections in Grassmannian fibrations and as Pfaffian varieties. These constructions naturally fit into the framework of homological projective duality and lead to dual pairs of Calabi-Yaus. From a physics perspective, these spaces can be realised as low-energy configurations ("phases") of gauged linear sigma models (GLSMs) with non-Abelian gauge groups, where the dual geometries arise as phases of the same GLSM. Using the modular bootstrap approach of topological string theory, one can compute all-genus Gopakumar-Vafa invariants of these Calabi-Yaus. We observe that homological projective duality acts as an element of Γ 0 (5) on the topological string partition function and the partition functions of dual geometries transform into each other. Moreover, we study the geometries from an M-/F-theory perspective. We compute the F-theory spectrum and show how the genus one-fibered Calabi-Yaus are connected to certain Calabi-Yaus in toric varieties via a series of Higgs transitions. Based on the F-theory physics, we conjecture that dual geometries are elements of the same Tate-Shafarevich group. Our analysis also leads to a classification of 5-section geometries, as well as the construction of F-theory models with charge 5 hypermultiplets.

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“…We briefly review this construction for the special case of our Pfaffian varieties. For details see [19,43,23,41].…”

Section: Example: Ymentioning

confidence: 99%

On genus one fibered Calabi-Yau threefolds with 5-sections

Knapp,

Scheidegger,

Schimannek

2021

Preprint

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Two-Sphere Partition Functions and Gromov–Witten Invariants

Jockers¹,

Kumar

Lapan

et al. 2014

Commun. Math. Phys.

136

315

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Many N = (2, 2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N = (2, 2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories -recently computed via localization by Benini et al. and Doroud et al. -yields the exact Kähler potential on the quantum Kähler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kähler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in α ′ . We compute these quantities for the quintic and for Rødland's Pfaffian Calabi-Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P 7 , recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.1 The work of Böhm [24, 25] provides a promising proposal, but we have been unable to implement it well enough to produce a mirror for this example.2 Local special Kähler manifolds are also often called projective special Kähler manifolds and are distinct from special Kähler manifolds -see, e.g., [32].3 This geometric structure gives rise to a Hodge filtration F 3 ⊂ F 2 ⊂ F 1 ⊂ F 0 of weight 3, with F 3 ≃ L, F 2 ≃ V, F 1 ≃ L ⊥ , and F 0 ≃ V C , where L ⊥ is the subspace of V C perpendicular to L with respect to the symplectic pairing ·, · -see, e.g., [33,34].The last expression involves the periods of Ω,B J Ω(ξ) , I, J = 0, . . . , h 2,1 , (2.6) with respect to a canonical symplectic basis (A I , B J ) of H 3 (Y ξ , Z) satisfying A I , B J = δ I J , A I , A J = B I , B J = 0 . (2.7) 2.3 The quantum Kähler moduli space M KählerThe main player of this note is the quantum-corrected Kähler moduli space M Kähler of a Calabi-Yau threefold Y , which is defined as the corresponding space of chiral-antichiral and antichiralchiral moduli of the underlying SCFT. It is also a local special Kähler manifold, parametrizing 15 We thank C. Vafa for pointing this out to us. 16 Note that in spacetime, some of these fields become quaternionic and are subject to further gs corrections.

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Geometric transitions between Calabi–Yau threefolds related to Kustin–Miller unprojections

Kapustka

2011

Journal of Geometry and Physics

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We study Kustin-Miller unprojections between Calabi-Yau threefolds, or more precisely the geometric transitions they induce. We use them to connect many families of Calabi-Yau threefolds with Picard number one to the web of Calabi-Yau complete intersections. This result enables us to find explicit description of a few known families of Calabi-Yau threefolds in terms of equations. Moreover, we find two new examples of Calabi-Yau threefolds with Picard group of rank one, which are described by Pfaffian equations in weighted projective spaces.shall call, to avoid confusion, a geometric bitransition. This class of bitransitions, thanks to its naturality, may be much better understood, and hence gives us a new variety of tools that may be used for explicit descriptions of examples of Calabi-Yau threefolds. The theory of unprojections has already proved its efficiency in classical algebraic geometry. Thanks to works of Reid and Papadakis [12,13,14,15,16], unprojections became a powerful tool to describe and construct new varieties. They are used in descriptions of singular K3 surfaces and Fano threefolds. In this paper we would like to show how these tools work in the context of Calabi-Yau threefolds. Differently from the approach to K3 surfaces or Fano 3 folds, we will not be interested in the singular varieties arising by unprojection, but in the families of smooth Calabi-Yau threefolds that might degenerate to such. The construction shall give rise to geometric bitransitions between these families. Throughout the paper we study examples of such given geometric bitransitions. In this way we find connections (consisting of an even number of geometric transitions) between many known families of Calabi-Yau threefolds with Picard number one. In particular we connect to the web the 5 families of Calabi-Yau threefolds introduced by Borcea [17] and described as complete intersections in homogeneous spaces.The main advantage of our constructions is that at least in low codimension they are well understood in terms of equations. Thanks to that, for some known examples of Calabi-Yau threefolds we find descriptions in terms of equations in weighted projective space. This fact enables us to understand much better the geometry of those examples of Calabi-Yau varieties, which were known only as smoothings of some degenerate varieties.Moreover, in two of the cases the construction leads us to new Calabi-Yau threefolds. They are described by the vanishing of Pfaffians of some 5 × 5 matrix with polynomial entries in some weighted projective spaces. These two families, together with a few others explicitly described in this paper, have conjectured Picard-Fuchs equations of their mirror (see [18]). We hope one might use the results of this paper to find these mirrors. There are at least two possibilities to proceed. The first is to use directly the constructed geometric transitions to find candidates for the mirrors. This idea is based on the conjecture by Morrisson (see [19]) which states that any geometric transition should admit a m...

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Untitled

Cited by 12 publications

References 20 publications

On genus one fibered Calabi-Yau threefolds with 5-sections

On genus one fibered Calabi-Yau threefolds with 5-sections

Two-Sphere Partition Functions and Gromov–Witten Invariants

Geometric transitions between Calabi–Yau threefolds related to Kustin–Miller unprojections

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