The Calabi – Yau space ( Calabi – Yau manifold ) is a compact complex manifold with a Kähler metric for which the Ricci tensor vanishes. In superstring theory, it is sometimes assumed that additional dimensions of space-time take the form of a 6-dimensional Calabi-Yau manifold, which led to the idea of mirror symmetry . The name was coined in 1985 [1] , in honor of Eugenio Calabi , who first suggested [2] [3] that such dimensions could exist, and Yau Shintun , who in 1978 proved [4] .
Integrated -dimensional space Calabi - Yau is -dimensional Riemannian manifold with a ricci-flat metric and additional symplectic structure.
Content
Examples and classification
In the one-dimensional case, any Calabi-Yau space is a torus , which is considered as an elliptic curve .
All two-dimensional Calabi-Yau spaces are tori and the so-called K3-surfaces . Classification in higher dimensions is not completed, including in the important three-dimensional case.
Use in String Theory
String theory uses three-dimensional (with a real dimension of 6) Calabi-Yau manifolds that act as a compactification layer of space-time , so that every point in four-dimensional space-time corresponds to a Calabi-Yau space.
More than 470 million three-dimensional Calabi – Yau spaces are known [5] , which satisfy the requirements for additional dimensions arising from string theory.
One of the main problems of string theory (given the current state of development) is such a selection from the indicated satisfactory subset of three-dimensional Calabi - Yau spaces, which would provide the most adequate justification for the number and composition of families of all known particles. The phenomenon of freedom of choice of Calabi - Yau spaces and the emergence of a huge number of false vacuums in string theory in this regard is known as the landscape problem of string theory . Moreover, if theoretical developments in this area lead to the identification of a single Calabi – Yau space that meets all the requirements for additional measurements, this will become a very weighty argument in favor of the truth of string theory [6] .
Notes
- ↑ Candelas, Philip; Horowitz, Gary; Strominger, Andrew & Witten, Edward (1985), " Vacuum configurations for superstrings ", Nuclear Physics B T. 258: 46–74 , DOI 10.1016 / 0550-3213 (85) 90602-9
- ↑ Calabi, Eugenio (1954), "The space of Kähler metrics", Proc. Internat. Congress Math. Amsterdam , p. 206–207
- ↑ Calabi, Eugenio (1957), "On Kähler manifolds with vanishing canonical class", Algebraic geometry and topology. A symposium in honor of S. Lefschetz , Princeton University Press , p. 78—89, MR : 0085583
- ↑ Yau, Shing Tung (1978), " On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I ", Communications on Pure and Applied Mathematics Vol. 31 (3): 339-411, MR : 480 350 , ISSN 0010-3640 , DOI 10.1002 / cpa.3160310304
- ↑ Shintan Yau , Steve Nadis. String theory and the hidden dimensions of the universe. - St. Petersburg: Publishing House "Peter", 2016. - 400 p. - ISBN 978-5-496-00247-9 .
- ↑ B. Green The Elegant Universe. Superstrings, hidden dimensions and the quest for the ultimate theory . Per. from English, total ed. V.O. Malyshenko, - M .: EditorialURSS, 2004 .-- 288 p. - ISBN 5-354-00161-7 .
Literature
- Tian, Gang & Yau, Shing-Tung (1990), " Complete Kähler manifolds with zero Ricci curvature, I ", Amer. Math. Soc. T. 3 (3): 579–609 , DOI 10.2307 / 1990928
- Tian, Gang & Yau, Shing-Tung (1991), " Complete Kähler manifolds with zero Ricci curvature, II ", Invent. Math. T. 106 (1): 27-60 , DOI 10.1007 / BF01243902