Calabi–Yau Conifold Expansions

Cynk, Sławomir; Straten, Duco van

doi:10.1007/978-1-4614-6403-7_19

Cited by 3 publications

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References 25 publications

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“…A further interesting class of examples to look at are Calabi-Yaus that do not have a point of maximal unipotent monodromy as discussed for instance in [30][31][32]. This would translate into GLSMs that do not have geometric phases.…”

Section: Discussionmentioning

confidence: 99%

GLSM realizations of maps and intersections of Grassmannians and Pfaffians

Căldăraru

Knapp

Sharpe

2018

J. High Energ. Phys.

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In this paper we give gauged linear sigma model (GLSM) realizations of a number of geometries not previously presented in GLSMs. We begin by describing GLSM realizations of maps including Veronese and Segre embeddings, which can be applied to give GLSMs explicitly describing non-complete intersection constructions such as the intersection of one hypersurface with the image under some map of another. We also discuss GLSMs for intersections of Grassmannians and Pfaffians with one another, and with their images under various maps, which sometimes form exotic constructions of Calabi-Yaus, as well as GLSMs for other exotic Calabi-Yau constructions of Kanazawa. Much of this paper focuses on a specific set of examples of GLSMs for intersections of Grassmannians G(2, N ) with themselves after a linear rotation, including the Calabi-Yau case N = 5. One phase of the GLSM realizes an intersection of two Grassmannians, the other phase realizes an intersection of two Pfaffians. The GLSM has two nonabelian factors in its gauge group, and we consider dualities in those factors. In both the original GLSM and a double-dual, one geometric phase is realized perturbatively (as the critical locus of a superpotential), and the other via quantum effects. Dualizing on a single gauge group factor yields a model in which each geometry is realized through a simultaneous combination of perturbative and quantum effects.

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Section: Discussionmentioning

confidence: 99%

GLSM realizations of maps and intersections of Grassmannians and Pfaffians

Căldăraru

Knapp

Sharpe

2018

J. High Energ. Phys.

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Classification of double octic Calabi–Yau threefolds with h1,2 ≤ 1 defined by an arrangement of eight planes

Cynk

Kocel-Cynk

2018

Commun. Contemp. Math.

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In this paper, we propose a combinatorial approach to study Calabi–Yau threefolds constructed as a resolution of singularities of a double covering of [Formula: see text] branched along an arrangement of eight planes. We use this description to give a complete classification of arrangements of eight planes in [Formula: see text] defining Calabi–Yau threefolds modulo projective transformation with [Formula: see text] and to derive their geometric properties (Kummer surface fibrations, automorphisms, special elements in families).

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Calabi–Yau 3-folds of Borcea–Voisin type and elliptic fibrations

Cattaneo¹,

Garbagnati²

2016

Tohoku Math. J. (2)

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We consider Calabi-Yau 3-folds of Borcea-Voisin type, i.e. Calabi-Yau 3-folds obtained as crepant resolutions of a quotient (S × E)/(α S × α E ), where S is a K3 surface, E is an elliptic curve, α S ∈ Aut(S) and α E ∈ Aut(E) act on the period of S and E respectively with order n = 2, 3, 4, 6. The case n = 2 is very classical, the case n = 3 was recently studied by Rohde, the other cases are less known. First we construct explicitly a crepant resolution, X, of (S × E)/(α S × α E ) and we compute its Hodge numbers; some pairs of Hodge numbers we found are new. Then we discuss the presence of maximal automorphisms and of a point with maximal unipotent monodromy for the family of X. Finally, we describe the map En : X → S/α S whose generic fiber is isomorphic to E. 1 2 ANDREA CATTANEO AND ALICE GARBAGNATI more precise statement see Proposition 3.2). Hence, we consider Calabi-Yau 3folds constructed as resolution of a quotient (S × E)/(Z/nZ) for n = 2, 3, 4, 6 and we call them of Borcea-Voisin type. In case n = 2 one obtains the "classical" and well known Borcea-Voisin construction. A systematical analysis of the case n = 3 is presented in [R1] and [D2] and uses the classification of the non-symplectic automorphisms of K3 surfaces of order 3, described independently by Artebani and Sarti, [AS1], and by Taki, [T]. Sporadic examples of the case n = 4 are analyzed in [G], where some peculiar K3 surfaces with a non-symplectic automorphism of order 4 are constructed and the associated Calabi-Yau 3-folds are presented. The complete classification of the K3 surfaces with non-symplectic automorphisms of order 4 and 6 is still unknown, but a lot of it is understood, see [AS2] for n = 4 and [D1] for n = 6. Hence several families of Calabi-Yau 3-folds of Borcea-Voisin type obtained from quotients by automorphisms of order 4 and 6 can be described.Given a quotient (S × E)/(Z/nZ) as before, there could exist more then one crepant resolution. We construct explicitly one specific crepant resolution (see Sections 4.1, 5.1, 6.1, 7.1) and we call it of type X n . The properties of the fixed loci of α j S , j = 1, . . . n − 1, on S determine the Hodge numbers this 3-fold. We compute these for each admissible value of n (see Propositions 4.1, 5.1, 6.1, 7.1). Some of the Calabi-Yau 3-folds constructed have "new" Hodge numbers (here we refer to the database [J] of the known Calabi-Yau 3-folds).The 3-folds X of type X n admit an automorphism induced by α S × id. In certain cases (see Proposition 3.7) this automorphism is a maximal automorphism for the family, i.e. it deforms to an automorphism of the varieties which are deformations of X. Such an automorphism acts non-trivially on the period of X. In [R1], the maximal automorphisms which act non-trivially on the period are analyzed. In particular it is proved that if a family of Calabi-Yau 3-folds admits a maximal automorphism which acts on the period as the multiplication by an n-th root of unity, n = 2, then the family does not admit a point with maximal unipotent monodromy. In [R1] the fa...

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Calabi–Yau Conifold Expansions

Abstract: We describe examples of computations of Picard-Fuchs operators for families of Calabi-Yau manifolds based on the expansion of a period near a conifold point. We find examples of operators without a point of maximal unipotent monodromy, thus answering a question posed by J. Rohde.

Cited by 3 publications

References 25 publications

GLSM realizations of maps and intersections of Grassmannians and Pfaffians

GLSM realizations of maps and intersections of Grassmannians and Pfaffians

Classification of double octic Calabi–Yau threefolds with h1,2 ≤ 1 defined by an arrangement of eight planes

Calabi–Yau 3-folds of Borcea–Voisin type and elliptic fibrations

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