T-duality
Encyclopedia
T-duality is a symmetry of quantum field theories with differing classical descriptions, of which the relationship between small and large distances in various string theories
is a special case. Discussion of the subject originated in a paper by T. S. Buscher and was further developed by Martin Rocek
and Erik Verlinde
. T-duality is not present in ordinary particle theory
, indicating that strings experience spacetime in a way that is fundamentally distinct than the way particles do. It relates different string theories
that were thought to be unrelated before T-duality was understood. T-duality preceded the Second Superstring Revolution.
predicts the existence of extra dimensions
in addition to the usual three space and single time dimensions. Different shapes and sizes of these extra dimensions result in different forces and particles appearing in the four-dimensional low energy physics and so universes with different shapes for the extra dimensions will have different physics. However, many of these geometries result in the same physics and this is the basis of T-duality.
For example, consider the case where one of the dimensions is a circle of radius . A particle or string can carry momentum
in this direction, and such a state is called a Kaluza-Klein mode. The momentum in this direction is quantized in that the momentum satisfies
As the radius becomes smaller, it costs more energy to excite one of these modes. On the other hand, as gets larger, the spacing between Kaluza-Klein states becomes smaller and in the limit of infinite radius, momentum is no longer quantized.
Unlike particles, a closed string can also wrap around the extra dimension. Such a state is called a winding mode. The energy to excite a winding mode is quantized proportional to the radius , so that as the radius gets smaller, the spacing between winding modes gets smaller, and in the limit of zero radius is no longer quantized while as the radius gets larger, it costs more energy to excite the winding modes. This is the opposite behavior of the Kaluza-Klein modes and suggests that the small and large radius behavior of the closed string is the same if we interchange winding modes and Kaluza-Klein modes. One can show that the physics at radius is the same as the physics at radius
. This relationship is an example of T-duality.
compactified on a circle of radius . The string may carry a net momentum in the compactified dimension. As is the case for particles, the momentum must be quantized in units of ,
where n is an integer. However, unlike the case for particles, a closed string may also wrap around the compactified dimension. The number of times that the closed string wraps around that dimension is called the winding number . The mass of a closed string is then
where and are the excitations for the left- and right-movers of the closed string, and is the slope parameter. This spectrum is invariant under the interchange
That is, the spectrum of the closed string is the same spectrum of a closed string in a background with radius . One can similarly show that the interactions of closed strings are also invariant under this interchange. This implies that the closed bosonic string compactified on a radius is equivalent to the theory with radius .
. T-duality interchanges the type II superstrings
with each other and also heterotic strings. For example, one might begin with a IIA string wrapped once around the direction in question. Under T-duality, it will be mapped to a IIB string which has momentum in that direction. A IIA string with a winding number of two (wrapped twice) will be mapped to a IIB string with two units of momentum, and so on.
, Shing-Tung Yau
, and Eric Zaslow have showed that mirror symmetry
can be understood as T-duality applied to three-dimensional toroidal fibres of the Calabi-Yau space.
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
is a special case. Discussion of the subject originated in a paper by T. S. Buscher and was further developed by Martin Rocek
Martin Rocek
Martin Rocek is a professor of theoretical physics at the State University of New York at Stony Brook and a member of the C. N. Yang Institute for Theoretical Physics. He received A.B. and Ph.D. degrees from Harvard University in 1975 and 1979...
and Erik Verlinde
Erik Verlinde
Erik Peter Verlinde is a Dutch theoretical physicist and string theorist. He is the identical twin brother of physicist Herman Verlinde. The Verlinde formula, which is important in conformal field theory and topological field theory, is named after him. His research deals with string theory,...
. T-duality is not present in ordinary particle theory
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, indicating that strings experience spacetime in a way that is fundamentally distinct than the way particles do. It relates different string theories
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
that were thought to be unrelated before T-duality was understood. T-duality preceded the Second Superstring Revolution.
Qualitative Description
String theoryString theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
predicts the existence of extra dimensions
Extra dimensions
Several speculative physical theories have introduced extra dimensions of space for various reasons:*Kaluza-Klein theory introduces extra dimensions to explain the fundamental forces other than gravity ....
in addition to the usual three space and single time dimensions. Different shapes and sizes of these extra dimensions result in different forces and particles appearing in the four-dimensional low energy physics and so universes with different shapes for the extra dimensions will have different physics. However, many of these geometries result in the same physics and this is the basis of T-duality.
For example, consider the case where one of the dimensions is a circle of radius . A particle or string can carry momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
in this direction, and such a state is called a Kaluza-Klein mode. The momentum in this direction is quantized in that the momentum satisfies
As the radius becomes smaller, it costs more energy to excite one of these modes. On the other hand, as gets larger, the spacing between Kaluza-Klein states becomes smaller and in the limit of infinite radius, momentum is no longer quantized.
Unlike particles, a closed string can also wrap around the extra dimension. Such a state is called a winding mode. The energy to excite a winding mode is quantized proportional to the radius , so that as the radius gets smaller, the spacing between winding modes gets smaller, and in the limit of zero radius is no longer quantized while as the radius gets larger, it costs more energy to excite the winding modes. This is the opposite behavior of the Kaluza-Klein modes and suggests that the small and large radius behavior of the closed string is the same if we interchange winding modes and Kaluza-Klein modes. One can show that the physics at radius is the same as the physics at radius
. This relationship is an example of T-duality.
Bosonic String
To illustrate the ideas of T-duality, consider the bosonic stringBosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s.In the early 1970s, supersymmetry was discovered in the context of string theory, and a new version of string theory called superstring theory became the real focus...
compactified on a circle of radius . The string may carry a net momentum in the compactified dimension. As is the case for particles, the momentum must be quantized in units of ,
where n is an integer. However, unlike the case for particles, a closed string may also wrap around the compactified dimension. The number of times that the closed string wraps around that dimension is called the winding number . The mass of a closed string is then
where and are the excitations for the left- and right-movers of the closed string, and is the slope parameter. This spectrum is invariant under the interchange
That is, the spectrum of the closed string is the same spectrum of a closed string in a background with radius . One can similarly show that the interactions of closed strings are also invariant under this interchange. This implies that the closed bosonic string compactified on a radius is equivalent to the theory with radius .
Superstrings
The idea of T-duality can be extended to more general backgrounds and even to superstring theoriesSuperstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
. T-duality interchanges the type II superstrings
Type II string theory
In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings. These account for two of the five consistent superstring theories in ten dimensions. Both theories have the maximal amount of supersymmetry — namely 32 supercharges...
with each other and also heterotic strings. For example, one might begin with a IIA string wrapped once around the direction in question. Under T-duality, it will be mapped to a IIB string which has momentum in that direction. A IIA string with a winding number of two (wrapped twice) will be mapped to a IIB string with two units of momentum, and so on.
Open Strings and D-branes
T-duality acting on D-branes changes their dimension by +1 or -1.Mirror Symmetry
Andrew StromingerAndrew Strominger
Andrew Eben Strominger is an American theoretical physicist who works on string theory and son of Jack L. Strominger. He is currently a professor at Harvard University and a senior fellow at the Society of Fellows...
, Shing-Tung Yau
Shing-Tung Yau
Shing-Tung Yau is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province....
, and Eric Zaslow have showed that mirror symmetry
Mirror symmetry
In physics and mathematics, mirror symmetry is a relation that can exist between two Calabi-Yau manifolds. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden...
can be understood as T-duality applied to three-dimensional toroidal fibres of the Calabi-Yau space.
See also
- S-dualityS-dualityIn theoretical physics, S-duality is an equivalence of two quantum field theories or string theories. An S-duality transformation maps states and vacua with coupling constant g in one theory to states and vacua with coupling constant 1/g in the dual theory...
- U-dualityU-dualityIn physics, U-duality is a symmetry of string theory or M-theory combining S-duality and T-duality transformations. The term is most often met in the context of the "U-duality group" of M-theory as defined on a particular background space . This is the union of all the S- and T-dualities...
- Mirror symmetryMirror symmetryIn physics and mathematics, mirror symmetry is a relation that can exist between two Calabi-Yau manifolds. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden...
- Dualities in string theory