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In physics, compactification means changing a theory with respect to one of its space-time 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 one-dimensional 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.
The space $$M \times C$$is compactified over the compact C and after Kaluza–Klein decomposition, we have an effective field theory over M.

Compactification in string theory

In string theory, compactification is a generalization of Kaluza–Klein theory.[1] 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 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 splitting and reconnecting, can be described by a 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 type IIA string theory can be described as the compactification of M-theory in eleven dimensions. Furthermore, 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 Calabi–Yau manifold which is equipped with non-zero values of fluxes, i.e. differential forms, that generalize the concept of an electromagnetic field (see p-form 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 F-theory vacua or type IIB string theory vacua with or without D-branes.

Dimensional reduction
Kaluza–Klein theory

Notes

Dean Rickles (2014). A Brief History of String Theory: From Dual Models to M-Theory. Springer, p. 89 n. 44.

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 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", Reviews of Modern Physics 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.

String theory
Background

Strings History of string theory
First superstring revolution Second superstring revolution String theory landscape

Calabi-Yau-alternate

Theory

String duality

T-duality S-duality U-duality Montonen–Olive duality

Particles and fields

Branes

Conformal field theory

Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model

Gauge theory

Anomalies Instantons Chern–Simons form Bogomol'nyi–Prasad–Sommerfield bound Exceptional Lie groups (G2, F4, E6, E7, E8) ADE classification Dirac string p-form electrodynamics

Geometry

Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold
Calabi–Yau manifold Hyperkähler manifold
K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten domain wall K-theory (physics) Twisted K-theory

Supersymmetry

Supergravity Superspace Lie superalgebra Lie supergroup

Holography

M-theory

String theorists

Aganagić Arkani-Hamed Atiyah Banks Berenstein Bousso Cleaver Curtright Dijkgraaf Distler Douglas Duff Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson Harvey Hořava Gibbons Kachru Kaku Kallosh Kaluza Kapustin Klebanov Knizhnik Kontsevich Klein Linde Maldacena Mandelstam Marolf Martinec Minwalla Moore Motl Mukhi Myers Nanopoulos Năstase Nekrasov Neveu Nielsen van Nieuwenhuizen Novikov Olive Ooguri Ovrut Polchinski Polyakov Rajaraman Ramond Randall Randjbar-Daemi Roček Rohm Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher Steinhardt Strominger Sundrum Susskind 't Hooft Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach

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