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## Hyperconifold Transitions, Mirror Symmetry, and String Theory,” 1102.1428. * Temporary entry

Citations: | 9 - 3 self |

### Citations

1192 |
Introduction to Toric Varieties,
- FULTON
- 1993
(Show Context)
Citation Context ...sibilities are shown in Figure 1. To answer this question we will examine the exceptional set of the resolution. Inspection of Figure 1, and the ‘star construction’ of toric geometry, as described in =-=[27]-=-, tell us that in the first case, the exceptional set consists of two copies of the Hirzebruch surface F1, intersecting along a P1, while in the second it consists of two disjoint surfaces, each isomo... |

474 | The homogeneous coordinate ring of a toric variety,”
- Cox
- 1995
(Show Context)
Citation Context ...milies of Calabi-Yau hypersurfaces are then mirror to each other. 2.1 Homogeneous coordinates It is very convenient to use homogeneous coordinates for the ambient toric space, as introduced by Cox in =-=[24]-=-. Let Σ be a fan for a toric variety Z. Then we can construct Z from Σ as follows. Suppose Σ contains d one-dimensional cones, which are rays, and let vρ be the first lattice vector on the ρ’th ray. W... |

467 | Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties,
- Batyrev
- 1994
(Show Context)
Citation Context ...quotient variety develops a hyperconifold singularity.3 2 Toric geometry and the Batyrev construction Here we will briefly review Batyrev’s construction of Calabi-Yau hypersurfaces in toric varieties =-=[2]-=-. This will serve mainly to establish notation, as several conventions have been used in the literature. We will specialise to the case of Calabi-Yau threefolds in toric fourfolds. Let N be a lattice,... |

333 |
Electric - magnetic duality, monopole condensation, and confinement in supersymmetric Yang-Mills theory,"
- Seiberg, Witten
- 1994
(Show Context)
Citation Context ..., and so should be included in the low-energy theory. If instead it is integrated out, it exactly reproduces the classical singularity of the moduli space, via a divergent one-loop contribution to F1 =-=[35]-=-, F1 ∼ const. + 1 2pii Z1 logZ1 . (9) If this is substituted into Equation (7), it is easily seen that the moduli space metric becomes singular at Z1 = 0. However, this is now seen to be merely an art... |

270 | Massless black holes and conifolds in string theory,
- Strominger
- 1995
(Show Context)
Citation Context ...onifold singularities represent singularities of the worldsheet theory. However, in nonperturbative Type IIB string theory, the conifold singularity is resolved by the effects of light D-brane states =-=[32]-=-. Furthermore, when it is mathematically possible to carry out a conifold transition to a new Calabi-Yau manifold, this manifests in the physics 16 as a new branch of the low-energy moduli space [10].... |

166 |
de la Ossa, Moduli space of Calabi–Yau manifolds, Nucl. Phys. B355
- Candelas, X
- 1991
(Show Context)
Citation Context ...es {AI , BI}I=1,...,h2,1(X̃)+1. The complex structure moduli space of X̃ admits complex homogeneous coordinates ZI , and holomorphic ‘functions’ FI defined in terms of the holomorphic three-form Ω by =-=[34]-=- ZI = ∫ AI Ω , FI = ∫ BI Ω . The moduli space metric is Kähler, with Kähler potential K = − log [ i ( Z I FI − ZIF I )] . (7) The low-energy dynamics of the complex structure moduli fields is that o... |

162 | Black hole condensation and the unification of string vacua,
- Greene, Morrison, et al.
- 1995
(Show Context)
Citation Context ...i space, at least in Type IIB string theory, is perfectly smooth through a point corresponding to a hyperconifold transition. The story is very similar to that of a conifold transition, worked out in =-=[10]-=-. The results of [1] and the present paper therefore have significant implications for the connectedness of the moduli space of Calabi-Yau threefolds, and the associated string vacua. Soon after Reid ... |

85 |
Local mirror symmetry: calculations and interpretations,
- Chiang, Klemm, et al.
- 1999
(Show Context)
Citation Context ... conjecture, that the mirror process to any ZN -hyperconifold transition is a conifold transition in which the intermediate variety has N nodes. It is probably possible to use the local techniques of =-=[5, 6]-=- to prove this [7]. The mirror conifold transitions have another interesting feature. Batyrev and Kreuzer showed that within the class of Calabi-Yau hypersurfaces in toric fourfolds, mirror symmetry e... |

69 |
PALP: A Package for Analyzing Lattice Polytopes with Applications to Toric Geometry,”
- Kreuzer, Skarke
- 2004
(Show Context)
Citation Context ...e will turn to examples of hyperconifold transitions between CalabiYau hypersurfaces in toric fourfolds. The required analysis of reflexive polytopes was greatly assisted by the software package PALP =-=[25]-=-. 3.1 The Z3 quotient of the bicubic The family of ‘bicubic’ manifolds X2,83 are hypersurfaces in P2×P2, cut out by a single polynomial of bidegree (3, 3). Products of projective spaces are toric vari... |

55 |
Complete intersection Calabi-Yau manifolds (II).
- Candelas, Lutken, et al.
- 1988
(Show Context)
Citation Context ...ns between them. Most known multiply-connected Calabi-Yau threefolds are obtained as free quotients of complete intersections in products of projective spaces. A few examples were discovered long ago =-=[14, 15, 16]-=-, and recently a more systematic search has been performed, leading to a complete enumeration of the manifolds which can be constructed this way [17, 18, 19]. A smaller number of examples occur as hyp... |

52 | Complete Classification of Reflexive Polyhedra
- Kreuzer, Skarke
- 2002
(Show Context)
Citation Context ...ent work has several objectives. We work mainly within the class of CalabiYau hypersurfaces in toric fourfolds, first described systematically by Batyrev [2] and then enumerated by Kreuzer and Skarke =-=[3]-=-. The formalism is reviewed in Section 2, 2 and then used in Section 3 to demonstrate that Z3- and Z5-hyperconifold transitions do connect compact Calabi-Yau manifolds. Perhaps more interestingly, it ... |

48 |
Imagined Communities
- Acquisti, Gross
- 2006
(Show Context)
Citation Context ... mirror process to any ZN -hyperconifold transition is a conifold transition in which the intermediate variety has N nodes. It is probably possible to use the local techniques of [5, 6] to prove this =-=[7]-=-. The mirror conifold transitions have another interesting feature. Batyrev and Kreuzer showed that within the class of Calabi-Yau hypersurfaces in toric fourfolds, mirror symmetry exchanges the funda... |

33 | Integral cohomology and mirror symmetry for Calabi-Yau 3-folds,” in Mirror Symmetry.
- Batyrev, Kreuzer
- 2006
(Show Context)
Citation Context ...Calabi-Yau hypersurfaces in toric fourfolds, mirror symmetry exchanges the fundamental group (which in these cases can only be Z2,Z3 or Z5) with the Brauer group, which is the torsion part of H3(X,Z) =-=[8]-=-. Since the hyperconifold transitions studied here destroy the fundamental group, their mirror conifold transitions should destroy the Brauer group. This is not a new phenomenon (see for example [9]),... |

30 |
Examples of special Lagrangian fibrations. In Symplectic geometry and mirror symmetry
- Gross
- 2001
(Show Context)
Citation Context ... conjecture, that the mirror process to any ZN -hyperconifold transition is a conifold transition in which the intermediate variety has N nodes. It is probably possible to use the local techniques of =-=[5, 6]-=- to prove this [7]. The mirror conifold transitions have another interesting feature. Batyrev and Kreuzer showed that within the class of Calabi-Yau hypersurfaces in toric fourfolds, mirror symmetry e... |

30 |
A rank 2 vector bundle on P4 with 15,000 symmetries,
- Horrocks, Mumford
- 1973
(Show Context)
Citation Context ... algebraic torus T4 = ( C∗ )4 via the map N 3 (n1, n2, n3, n4) 7→ {(λn1 , λn2 , λn3 , λn4) | λ ∈ C∗} , 2There are also certain exceptional cases, such as the quotients of the Horrocks-Mumford quintic =-=[21]-=- and the Gross-Popescu manifolds [22, 23], but these are not discussed here. 3It is possible for worse singularities to occur instead, because the quadratic terms in the analogue of Equation (2) may a... |

23 | A three-generation Calabi-Yau manifold with small Hodge numbers,” Fortschritte der Physik,
- Braun, Candelas, et al.
- 2010
(Show Context)
Citation Context ...s. A few examples were discovered long ago [14, 15, 16], and recently a more systematic search has been performed, leading to a complete enumeration of the manifolds which can be constructed this way =-=[17, 18, 19]-=-. A smaller number of examples occur as hypersurfaces in toric fourfolds [8], or as free non-toric quotients of such hypersurfaces [20], which is a largely unexplored class.2 The cyclic fundamental gr... |

19 |
Compact three dimensional Kähler manifolds with zero Ricci curvature
- Yau
- 1985
(Show Context)
Citation Context ...8 are certainly new manifolds, since no existing manifolds have the same Hodge numbers and fundamental group. X6,9, on the other hand, could well be the same as Yau’s famous three-generation manifold =-=[28, 29]-=-. This suspicion is strengthened by the fact that their covering spaces have the same Hodge numbers. 3.2 The Z5 quotient of the quintic A smooth quintic hypersurface in P4 is a Calabi-Yau manifold, wi... |

18 | Through the looking glass
- Morrison
- 1995
(Show Context)
Citation Context ..., it can also be used to study the mirror processes to these transitions, which turn out to be ordinary conifold transitions. They therefore provide a counter-example to an old conjecture of Morrison =-=[4]-=- that the mirror of a conifold transition is another conifold transition. The examples herein show that, while this is a very tempting conjecture, it is not true in general. They also motivate a modes... |

17 | Triadophilia:
- Candelas, Ossa, et al.
- 2008
(Show Context)
Citation Context ...termined by ∆ miss the orbifold points, and therefore give a family of smooth, multiply connected Calabi-Yau threefolds, with Hodge numbers (h1,1, h2,1) = (2, 29). This is well known; see for example =-=[26]-=-. We are now interested in specialising to the case where the Calabi-Yau hypersurface intersects one of these singularities, and therefore has a Z3-hyperconifold singularity. Let us focus on the fixed... |

15 |
New manifolds for superstring compactification
- Strominger, Witten
- 1985
(Show Context)
Citation Context ...ns between them. Most known multiply-connected Calabi-Yau threefolds are obtained as free quotients of complete intersections in products of projective spaces. A few examples were discovered long ago =-=[14, 15, 16]-=-, and recently a more systematic search has been performed, leading to a complete enumeration of the manifolds which can be constructed this way [17, 18, 19]. A smaller number of examples occur as hyp... |

15 | On free quotients of complete intersection Calabi-Yau manifolds,”
- Braun
- 2011
(Show Context)
Citation Context ...s. A few examples were discovered long ago [14, 15, 16], and recently a more systematic search has been performed, leading to a complete enumeration of the manifolds which can be constructed this way =-=[17, 18, 19]-=-. A smaller number of examples occur as hypersurfaces in toric fourfolds [8], or as free non-toric quotients of such hypersurfaces [20], which is a largely unexplored class.2 The cyclic fundamental gr... |

14 |
Connecting Moduli Spaces of Calabi-Yau Threefolds, Commun.Math.Phys
- Green, Hübsch
(Show Context)
Citation Context ...ated string vacua. Soon after Reid suggested the idea that all threefolds with c1 = 0 may be connected by conifold transitions [11], this was shown to be true for almost all known Calabi-Yau examples =-=[12, 13]-=-. But conifold transitions cannot change the fundamental group, so this cannot be the whole story. Hyperconifold transitions then fill an important gap, since they still involve relatively mild singul... |

12 |
Phase transitions among (many of) Calabi-Yau compactifications
- Green, Hubsch
- 1988
(Show Context)
Citation Context ...ated string vacua. Soon after Reid suggested the idea that all threefolds with c1 = 0 may be connected by conifold transitions [11], this was shown to be true for almost all known Calabi-Yau examples =-=[12, 13]-=-. But conifold transitions cannot change the fundamental group, so this cannot be the whole story. Hyperconifold transitions then fill an important gap, since they still involve relatively mild singul... |

11 | A Calabi-Yau threefold with Brauer group (Z/8Z)2,” math.AG/0512182 - Gross, Pavanelli |

11 |
Three generation superstring model. 1. Compactification and discrete symmetries. Nucl. Phys
- Greene, Kirklin, et al.
- 1986
(Show Context)
Citation Context ...8 are certainly new manifolds, since no existing manifolds have the same Hodge numbers and fundamental group. X6,9, on the other hand, could well be the same as Yau’s famous three-generation manifold =-=[28, 29]-=-. This suspicion is strengthened by the fact that their covering spaces have the same Hodge numbers. 3.2 The Z5 quotient of the quintic A smooth quintic hypersurface in P4 is a Calabi-Yau manifold, wi... |

10 | Quotients of the conifold in compact Calabi-Yau threefolds, and new topological transitions,” arXiv:0911.0708 [hep-th
- Davies
(Show Context)
Citation Context .... . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Hyperconifolds and their resolutions . . . . . . . . . . . . . . . . . . . . 18 1 Introduction and discussion This paper is a follow-up to =-=[1]-=-, in which a class of threefold singularities and associated topological transitions were studied. These are isolated Calabi-Yau threefold singularities which are quotients of the conifold by a finite... |

9 |
The moduli space of 3-folds with K=0 may nevertheless be irreducible
- Reid
- 1987
(Show Context)
Citation Context ...onnectedness of the moduli space of Calabi-Yau threefolds, and the associated string vacua. Soon after Reid suggested the idea that all threefolds with c1 = 0 may be connected by conifold transitions =-=[11]-=-, this was shown to be true for almost all known Calabi-Yau examples [12, 13]. But conifold transitions cannot change the fundamental group, so this cannot be the whole story. Hyperconifold transition... |

6 |
Calabi-Yau Threefolds and Moduli
- Gross, Popescu
(Show Context)
Citation Context ... map N 3 (n1, n2, n3, n4) 7→ {(λn1 , λn2 , λn3 , λn4) | λ ∈ C∗} , 2There are also certain exceptional cases, such as the quotients of the Horrocks-Mumford quintic [21] and the Gross-Popescu manifolds =-=[22, 23]-=-, but these are not discussed here. 3It is possible for worse singularities to occur instead, because the quadratic terms in the analogue of Equation (2) may always be degenerate. This does not seem t... |

3 |
Completing the Web of Z3
- Candelas, Constantin
(Show Context)
Citation Context ...s. A few examples were discovered long ago [14, 15, 16], and recently a more systematic search has been performed, leading to a complete enumeration of the manifolds which can be constructed this way =-=[17, 18, 19]-=-. A smaller number of examples occur as hypersurfaces in toric fourfolds [8], or as free non-toric quotients of such hypersurfaces [20], which is a largely unexplored class.2 The cyclic fundamental gr... |

1 |
Branes and fundamental groups,” Adv.Theor.Math.Phys
- Gopakumar, Vafa
- 1998
(Show Context)
Citation Context ...] that a similar story should hold for hyperconifolds, and we will now show that this is indeed the case. The following argument is closely modelled on that of [10, 32], and also uses the insights of =-=[33]-=- about D-branes wrapped on multiply-connected cycles. Much of what follows is well known, but is included in order to give a relatively self-contained account. 4.1 The conifold For expository reasons,... |