Hyperconifold Transitions, Mirror Symmetry, and String Theory

Abstract

Multiply-connected Calabi-Yau threefolds are of particular interest for both string theorists and mathematicians. Recently it was pointed out that one of the generic degenerations of these spaces (occurring at codimension one in moduli space) is an isolated singularity which is a finite cyclic quotient of the conifold; these were called hyperconifolds. It was also shown that if the order of the quotient group is even, such singular varieties have projective crepant resolutions, which are therefore smooth Calabi-Yau manifolds. The resulting topological transitions were called hyperconifold transitions, and change the fundamental group as well as the Hodge numbers. Here Batyrev's construction of Calabi-Yau hypersurfaces in toric fourfolds is used to demonstrate that certain compact examples containing the remaining hyperconifolds - the Z_3 and Z_5 cases - also have Calabi-Yau resolutions. The mirrors of the resulting transitions are studied and it is found, surprisingly, that they are ordinary conifold transitions. These are the first examples of conifold transitions with mirrors which are more exotic extremal transitions. The new hyperconifold transitions are also used to construct a small number of new Calabi-Yau manifolds, with small Hodge numbers and fundamental group Z_3 or Z_5. Finally, it is demonstrated that a hyperconifold is a physically sensible background in Type IIB string theory. In analogy to the conifold case, non-perturbative dynamics smooth the physical moduli space, such that hyperconifold transitions correspond to non-singular processes in the full theory.

23 pages, PDFLaTeX. v2: Abstract and introduction slightly expanded, and examples of new manifolds added. Also added references and hyperref. v3: Minor corrections, including to relations on pg. 8

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Cited by in corpus (15)

• F-theory on Quotient Threefolds with (2,0) Discrete Superconformal Matter
• The string landscape and low energy supersymmetry
• Calabi-Yau Threefolds With Small Hodge Numbers
• The Expanding Zoo of Calabi-Yau Threefolds
• Hodge Numbers for CICYs with Symmetries of Order Divisible by 4
• F-Theory on Quotients of Elliptic Calabi-Yau Threefolds
• The 24-Cell and Calabi-Yau Threefolds with Hodge Numbers (1,1)
• Quantum black holes in Type-IIA String Theory
• Heterotic String Models on Smooth Calabi-Yau Threefolds
• Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories
• Classification and Properties of Hyperconifold Singularities and Transitions
• Two One-Parameter Special Geometries
• Deforming Geometric Transitions
• Geometric transitions and SYZ mirror symmetry
• A class of Calabi-Yau threefolds as manifolds of SU(2) structure