 Compactification (physics)

For the concept of compactification in mathematics, see compactification (mathematics).
In physics, compactification means changing a theory with respect to one of its spacetime dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic.
Compactification plays an important part in thermal field theory where one compactifies time, in string theory where one compactifies the extra dimensions of the theory, and in two or onedimensional solid state physics, where one considers a system which is limited in one of the three usual spatial dimensions.
At the limit where the size of the compact dimension goes to zero, no fields depend on this extra dimension, and the theory is dimensionally reduced.
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Compactification in string theory
In string theory, compactification is a generalization of Kaluza–Klein theory. It tries to conciliate the gap between the conception of our universe based on its four observable dimensions with the ten, eleven, or twentysix dimensions theoretical equations lead to suppose the universe is made with. For this purpose it is assumed the extra dimensions are "wrapped" up on themselves, or "curled" up on Calabi–Yau spaces, or on orbifolds. Models in which the compact directions support fluxes are known as flux compactifications. The coupling constant of string theory, which determines the probability of strings to split and reconnect, can be described by a field called dilaton. This in turn can be described as the size of an extra (eleventh) dimension which is compact. In this way, the tendimensional type IIA string theory can be described as the compactification of Mtheory in eleven dimensions. Furthermore, different versions of string theory are related by different compactifications in a procedure known as Tduality.
The formulation of more precise versions of the meaning of compactification in this context has been promoted by discoveries such as the mysterious duality.
Flux compactification
A flux compactification is a particular way to deal with additional dimensions required by string theory. It assumes that the shape of the internal manifold is a Calabi–Yau manifold or generalized Calabi–Yau manifold which is equipped with nonzero values of fluxes, i.e. differential forms that generalize the concept of an electromagnetic field (see pform electrodynamics).
The hypothetical concept of the anthropic landscape in string theory follows from a large number of possibilities in which the integers that characterize the fluxes can be chosen without violating rules of string theory. The flux compactifications can be described as Ftheory vacua or type IIB string theory vacua with or without Dbranes.
See also
References
 Chapter 16 of Michael Green, John H. Schwarz and Edward Witten (1987) Superstring theory. Cambridge University Press. Vol. 2: Loop amplitudes, anomalies and phenomenology. ISBN 0521357535.
 Brian R. Greene, "String Theory on Calabi–Yau Manifolds". arXiv:hepth/9702155.
 Mariana Graña, "Flux compactifications in string theory: A comprehensive review", Physics Reports 423, 91–158 (2006). arXiv:hepth/0509003.
 Michael R. Douglas and Shamit Kachru "Flux compactification", Rev. Mod. Phys. 79, 733 (2007). arXiv:hepth/0610102.
 Ralph Blumenhagen, Boris Körs, Dieter Lüst, Stephan Stieberger, "Fourdimensional string compactifications with Dbranes, orientifolds and fluxes", Physics Reports 445, 1–193 (2007). arXiv:hepth/0610327.
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