Editing Compactification (physics)

{{short description|Technique in theoretical physics}}

In [[theoretical physics]], '''compactification''' means changing a theory with respect to one of its [[Spacetime|space-time]] [[dimension]]s. 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 function|periodic]].

Compactification plays an important part in [[Thermal quantum field theory|thermal field theory]] where one compactifies time, in [[string theory]] where one compactifies the [[String theory#Extra dimensions|extra dimensions]] of the theory, and in two- or one-dimensional [[Solid-state physics|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 [[Dimensional reduction|dimensionally reduced]].

[[Image:Kaluza Klein compactification.svg|frame|The space {{math|''M'' × ''C''}} is compactified over the compact {{mvar|C}} and after Kaluza–Klein decomposition, we have an [[effective field theory]] over {{mvar|M}}.]]

==In string theory==

In string theory, compactification is a generalization of [[Kaluza–Klein theory]].<ref>[[Dean Rickles]] (2014). ''A Brief History of String Theory: From Dual Models to M-Theory.'' Springer, p. 89 n. 44.</ref> It tries to reconcile the gap between the conception of our universe based on its four observable dimensions with the ten, eleven, or twenty-six dimensions which theoretical equations lead us to suppose the universe is made with.

For this purpose it is assumed the [[String theory#Extra dimensions|extra dimensions]] are "wrapped" up on themselves, or "curled" up on [[Calabi–Yau manifold|Calabi–Yau spaces]], or on [[orbifold]]s. Models in which the compact directions support [[flux]]es are known as ''flux compactifications''. The [[coupling constant]] of [[string theory]], which determines the probability of strings splitting and reconnecting, can be described by a [[Field (physics)|field]] called a [[dilaton]]. This in turn can be described as the size of an extra (eleventh) dimension which is compact. In this way, the ten-dimensional [[String theory#Dualities|type IIA string theory]] can be described as the compactification of [[M-theory]] in eleven dimensions. Furthermore, [[String theory#Dualities|different versions of string theory]] are related by different compactifications in a procedure known as [[T-duality]].

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 complex structure|generalized Calabi–Yau manifold]] which is equipped with non-zero values of fluxes, i.e. [[differential form]]s, that generalize the concept of an [[electromagnetic field]] (see [[p-form electrodynamics]]).

The hypothetical concept of the [[String theory landscape|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 [[F-theory]] vacua or [[Type II string theory|type IIB string theory]] vacua with or without [[D-brane]]s.

==See also==
*[[Dimensional reduction]]

==References==
{{reflist}}

==Further reading==
* Chapter 16 of [[Michael Green (physicist)|Michael Green]], [[John H. Schwarz]] and [[Edward Witten]] (1987). ''Superstring theory''. Cambridge University Press. ''Vol. 2: Loop amplitudes, anomalies and phenomenology''. {{ISBN|0-521-35753-5}}.
* Brian R. Greene, "String Theory on Calabi–Yau Manifolds". {{arxiv|hep-th/9702155}}.
* Mariana Graña, "Flux compactifications in string theory: A comprehensive review", ''Physics Reports'' '''423''', 91–158 (2006). {{arxiv|hep-th/0509003}}.
* Michael R. Douglas and Shamit Kachru "Flux compactification", ''Rev. Mod. Phys.'' '''79''', 733 (2007). {{arxiv|hep-th/0610102}}.
* Ralph Blumenhagen, Boris Körs, Dieter Lüst, Stephan Stieberger, "Four-dimensional string compactifications with D-branes, orientifolds and fluxes", ''Physics Reports'' '''445''', 1–193 (2007). {{arxiv|hep-th/0610327}}.

[[Category:String theory]]