theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, compactification means changing a theory with respect to one of its

space-time In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional continuum (measurement), continu ...

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

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 Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values tha ...

. Compactification plays an important part in thermal field theory where one compactifies time, in

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

where one compactifies the

extra dimensions In physics, extra dimensions or extra-dimensional spaces are proposed as additional space or time dimensions beyond the (3 + 1) typical of observed spacetime — meaning 5-dimensional or higher. such as the first attempts based on the K ...

of the theory, and in two- or one-dimensional

solid state physics Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state p ...

, 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.

In string theory

In string theory, compactification is a generalization of

Kaluza–Klein theory In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to ...

Dean Rickles Dean Rickles (born July 17, 1977) is Professor of History and Philosophy of Modern Physics at the University of Sydney and a Director of the Sydney Centre for Time. Life Dean Rickles was born in Hull, Yorkshire. He briefly trained as a concert ...

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

are "wrapped" up on themselves, or "curled" up on Calabi–Yau spaces, or on

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. D ...

s. Models in which the compact directions support

flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phe ...

es are known as ''flux compactifications''. The

coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between tw ...

, which determines the probability of strings splitting and reconnecting, can be described by a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

called a

dilaton In particle physics, the hypothetical dilaton is a particle of a scalar field \varphi that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theory's compa ...

. 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 In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theories in ten dimensions. Both theories ...

can be described as the compactification of

M-theory In physics, M-theory is a theory that unifies all Consistency, consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1 ...

in eleven dimensions. Furthermore, different versions of string theory are related by different compactifications in a procedure known as

T-duality T-duality (short for target-space duality) in theoretical physics is an equivalence of two physical theories, which may be either quantum field theories or string theories. In the simplest example of this relationship, one of the theories descr ...

. 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

is a Calabi–Yau manifold or

generalized Calabi–Yau manifold In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures wer ...

which is equipped with non-zero values of fluxes, i.e.

differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...

s, that generalize the concept of an

electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...

(see

p-form electrodynamics In theoretical physics, -form electrodynamics is a generalization of Maxwell's theory of electromagnetism. Ordinary (via. one-form) Abelian electrodynamics We have a 1-form \mathbf, a gauge symmetry :\mathbf \rightarrow \mathbf + d\alpha , where ...

). The hypothetical concept of the

anthropic landscape In string theory, the string theory landscape (or landscape of vacua) is the collection of possible false vacua,The number of metastable vacua is not known exactly, but commonly quoted estimates are of the order 10500. See M. Douglas, "The stat ...

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 In theoretical physics, F-theory is a branch of string theory developed by Iranian-American physicist Cumrun Vafa. The new vacua described by F-theory were discovered by Vafa and allowed string theorists to construct new realistic vacua — in ...

vacua or

type IIB string theory In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theories in ten dimensions. Both theorie ...

vacua with or without

D-brane In string theory, D-branes, short for Dirichlet membrane, are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes are typically classified by their spatial dimensi ...

In string theory

Flux compactification

See also

References

Further reading