Double field theory in

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 ...

refers to formalisms that capture the

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 ...

property of

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 ...

as a manifest

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

of a field theory.

Background

In double field theory, the T-duality transformation of exchanging

momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...

and

winding An electromagnetic coil is an electrical conductor such as a wire in the shape of a coil ( spiral or helix). Electromagnetic coils are used in electrical engineering, in applications where electric currents interact with magnetic fields, in ...

modes of closed strings on

toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...

al backgrounds translates to a generalized coordinate transformation on a doubled

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

, where one set of its coordinates is dual to momentum modes and the second set of coordinates is interpreted as dual to winding modes of the closed string. Whether the second set of coordinates has physical meaning depends on how the level-matching condition of closed strings is implemented in the theory: either through the weak constraint or the strong constraint. In strongly constrained double field theory, which was introduced by Warren Siegel in 1993, the strong constraint ensures the dependency of the fields on only one set of the doubled coordinates; it describes the massless fields of closed string theory, i.e. the

graviton In theories of quantum gravity, the graviton is the hypothetical elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with re ...

, Kalb Ramond B-field, and

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 ...

, but does not include any winding modes, and serves as a T-duality invariant reformulation of

supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...

. Weakly constrained double field theory, introduced by

Chris Hull Christopher Michael Hull (born 1957) One or more of the preceding sentences incorporates text from the royalsociety.org website where: is a professor of theoretical physics at Imperial College London. Hull is known for his work on string theory, ...

and

Barton Zwiebach Barton Zwiebach (born ''Barton Zwiebach Cantor'', October 4, 1954) is a Peruvian string theorist and professor at the Massachusetts Institute of Technology. Work Zwiebach studied electrical engineering at the Universidad Nacional de Ingenierí ...

in 2009, allows for the fields to depend on the whole doubled

and encodes genuine momentum and winding modes of the string. Double field theory has been a setting for studying various string theoretical properties such as: consistent Kaluza-Klein truncations of higher-dimensional supergravity to lower-dimensional theories, generalized fluxes, and alpha-prime corrections of string theory in the context of

cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...

and

black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...

References

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