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1. Consequently, they go by slightly different names, depending mostly on context, such as Calabi-Yau manifolds or Calabi-Yau varieties.
2. A Calabi-Yau space is characterized by the existence of a nonvanishing harmonic spinor .
3. On a Calabi-Yau manifold , such a can be defined globally, and the Lie group is very important in the theory.
4. In fact, one of the many equivalent definitions, coming from Riemannian geometry, says that a Calabi-Yau manifold is a -dimensional manifold whose holonomy group reduces to .
5. Unfortunately, there are tens of thousands of possible Calabi-Yau manifolds for six dimensions, and string theory offers no reasonable means of determining which is the right one.
6. Calabi-Yau manifolds have become a topic of study in both mathematics and physics, dissolving the boundaries between the two subjects.
7. Calabi-Yau manifolds are complex manifolds, that is, they can be disassembled into patches which look like flat complex space.
8. Proving a conjecture of Eugenio Calabi, Shing-Tung Yau has shown that Calabi-Yau manifolds have a property which is very interesting to physics.
9. By Yau's theorem, not only is flat space a solution but so are Calabi-Yau manifolds.
10. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry.
11. On the other hand, their double covers are Calabi–Yau manifolds for both definitions (in fact, K3 surfaces).
12. Most definitions assert that Calabi–Yau manifolds are compact, but some allow them to be non-compact.
13. Some definitions put restrictions on the fundamental group of a Calabi–Yau manifold, such as demanding that it be finite or trivial.
14. By the conjecture of Calabi (1957) proved by Yau (1977; 1979), there exists on every Calabi-Yau manifold a Kähler metric with vanishing Ricci curvature.
15. While many beautiful properties of Calabi-Yau manifolds have been discovered, more questions have been raised and probed.
16. This proves the existence and provides a criterion for Kähler Calabi-Yau manifolds.
17. By convention, Calabi-Yau manifolds exclude those with infinite fundamental groups.
18. This is where all the interest into these Calabi-Yau manifolds in string theory comes from.
19. One also speaks of generalized Calabi-Yau spaces.
20. This is reflected notably in the mirror symmetry of the target Calabi-Yau manifolds.
21. For more see Calabi-Yau manifolds in SU-bordism theory.
22. Abstract We study aspects of the geometry and physics of type II string theory compactification on Calabi-Yau manifolds.
23. The emphasis is on non-perturbative phenomena which arise when the compactification manifold develops singularities, and the implications on quantum geometry of the the Calabi-Yau spaces.
24. If not, check out this Calabi-Yau manifold by the artist Bathsheba.
25. Abstract We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau.
26. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians.
27. Surveys in Dierential Geometry XIII A survey of Calabi-Yau manifolds Shing-Tung Yau Contents Introduction 1. 2.
28. The Ricci tensor of Calabi-Yau manifolds 2.2.
29. Examples of compact Calabi-Yau manifolds 2.6.
30. Noncompact Calabi-Yau manifolds 2.7.
31. Their size and six dimensions make Calabi-Yau spaces difficult to draw.
32. Calabi-Yau manifolds are complex geometrical spaces studied in mathematics and physics.
33. Different Calabi-Yau manifolds described by different quantum theories turn out to encode the same physics.
34. These shapes are known as Calabi-Yau manifolds — an example of which is depicted in this demonstration.
35. One of the main motivations for studying the collapsing of Ricci-flat Calabi-Yau manifolds comes from mirror symmetry.
36. A review of the necessary mathematics is undertaken fol- lowed by a review of the Calabi-Yau manifold and its role in physics.
37. The complete intersection Calabi-Yau manifold is explained and a machine learning approach is motivated.
38. Neural networks are then employed to learn the Hodge numbers of complex dimension 4 complete intersection Calabi-Yau manifolds.
39. In order for it to also produce a supersymmetric gauge theory with a realistic particle spectrum, the 6-manifold must be a Calabi-Yau manifold.
40. Calabi-Yau manifolds play an important role in physics owing to the part they play in passing from a 10-dimensional string theory vacuum to the observed world of four dimensions.
41. This speaker will comment on the mysterious modular behaviour of the zeta-function for the case that the Calabi-Yau manifold is singular.
42. We rst briey mention some of the work in Kahler Calabi-Yau manifolds that was inuenced by the discovery of mirror symmetry in the late 1980s.
43. We then discuss some of the mathemat- ical motivations behind the recent work on non-Kahler Calabi-Yau manifolds, which arise in string compactications with uxes.
44. After extending mirror symmetry to non-Kahler Calabi-Yau manifolds, we show how this leads to new cohomologies and invariants of non-Kahler symplectic manifolds.
45. As a prime example, mathematical research on Calabi-Yau spaces over the past two decades has been strongly motivated by string theory, and in particular, mirror symmetry.
46. Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent monodromy.
47. In the case that the group action has fixed points, it is often possible to resolve the resulting orbifold singularities in such a way as to again obtain a Calabi-Yau manifold.
48. Topologically, such a process is a surgery and yields a Calabi-Yau manifold topologically distinct from the original.
49. Only a little more difficult is the converse: under what conditions is a Calabi-Yau manifold, given that is?
50. In even dimensions, on the other hand, this simple calculation shows that there are no multiply-connected Calabi-Yau manifolds.
51. Calabi-Yau spaces are complex spaces with a vanishing first Chern class, or equivalently, with trivial canonical bundle (canonical class).

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Spacy 패턴 목록

• [{'LOWER': 'calabi'}, {'OP': '*'}, {'LOWER': 'yau'}, {'LEMMA': 'manifold'}]
• [{'LOWER': 'calabi'}, {'OP': '*'}, {'LOWER': 'yau'}, {'LEMMA': 'space'}]