# T-duality

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T-duality is a symmetry of quantum field theories with differing classical descriptions,[1] of which the relationship between small and large distances in various string theories is a special case.[2] Discussion of the subject originated in a paper by T. H. Buscher and was further developed by Martin Rocek and Erik Verlinde. T-duality is not present in ordinary particle theory, indicating that strings would 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.[3]

## Qualitative description

String theory 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 R. 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

$p=\frac{n}{R}$

As the radius R becomes smaller, it costs more energy to excite one of these modes. On the other hand, as R 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 R, 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 R is the same as the physics at radius α'/R. This relationship is an example of T-duality.

## Bosonic string

To illustrate the ideas of T-duality, consider the bosonic string compactified on a circle of radius R. The string may carry a net momentum p in the compactified dimension. As is the case for particles, the momentum must be quantized in units of 1/R,

$p=\frac{n}{R}$

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 w. The mass of a closed string is then

$m^2=\frac{n^{2}}{R^{2}}+\frac{w^{2}R^{2}}{\alpha^{\prime 2}}+\frac{2}{\alpha'}\left(N+\tilde{N}-2\right)$

where N and Ñ are the excitations for the left- and right-movers of the closed string, and α' is the slope parameter. This spectrum is invariant under the interchange

$R\leftrightarrow \alpha'/R\quad n\leftrightarrow w$

That is, the spectrum of the closed string is the same spectrum of a closed string in a background with radius α'/R. 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 R is equivalent to the theory with radius α'/R.

## Superstrings

The idea of T-duality can be extended to more general backgrounds and even to superstring theories. 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.

## Open strings and D-branes

T-duality acting on D-branes changes their dimension by +1 or -1.

## Mirror symmetry

A three-dimensional projection of a six-dimensional Calabi–Yau manifold.

In string theory and algebraic geometry, the term "mirror symmetry" refers to a phenomenon involving complicated shapes called Calabi-Yau manifolds. These manifolds provide an interesting geometry on which strings can propagate, and the resulting theories may have applications in particle physics.[4] In the late 1980s, it was noticed that such a Calabi-Yau manifold does not uniquely determine the physics of the theory. Instead, one finds that there are two Calabi-Yau manifolds that give rise to the same physics.[5] These manifolds are said to be "mirror" to one another. This mirror duality is an important computational tool in string theory, and it has allowed mathematicians to solve difficult problems in enumerative geometry.[6]

A torus is the cartesian product of two circles.

One approach to understanding mirror symmetry is the SYZ conjecture, which was suggested by Andrew Strominger, Shing-Tung Yau, and Eric Zaslow in 1996.[7] According to the SYZ conjecture, mirror symmetry can be understood by dividing a complicated Calabi-Yau manifold into simpler pieces and considering the effects of T-duality on these pieces.[8]

The simplest example of a Calabi-Yau manifold is a torus (a surface shaped like a donut). Such a surface can be viewed as the product of two circles. This means that the torus can be viewed as the union of a collection of longitudinal circles (such as the red circle in the image). There is an auxiliary space which says how these circles are organized, and this space is itself a circle (the pink circle). This space is said to parametrize the longitudinal circles on the torus. In this case, mirror symmetry is equivalent to T-duality acting on the longitudinal circles, changing their radii from $R$ to $1/R$.

The SYZ conjecture generalizes this idea to the more complicated case of six-dimensional Calabi-Yau manifolds like the one illustrated above. As in the case of a torus, we can divide a six-dimensional Calabi-Yau manifold into simpler pieces, which in this case are 3-tori (three-dimensional objects which generalize the notion of a torus) parametrized by a 3-sphere (a three-dimensional generalization of a sphere).[9] T-duality can be extended from circles to the three-dimensional tori appearing in this decomposition, and the SYZ conjecture states that mirror symmetry is equivalent to the simultaneous application of T-duality to these three-dimensional tori.[10] In this way, the SYZ conjecture provides a geometric picture of how mirror symmetry acts on a Calabi-Yau manifold.

## Notes

1. ^
2. ^
3. ^ superstringtheory article Looking for extra dimensions by Patricia Schwarz
4. ^ Candelas et al. 1985
5. ^ Dixon 1988; Lerche, Vafa, and Warner 1989
6. ^ Zaslow 2008
7. ^ Strominger, Yau, and Zaslow 1996
8. ^ Yau and Nadis 2010, p.174
9. ^ More precisely, there is a 3-torus associated to every point on the three-sphere except at certain bad points, which correspond to singular tori. See Yau and Nadis 2010, pp.176--7.
10. ^ Yau and Nadis 2010, p.178

## References

• Becker, Katrin; Becker, Melanie; Schwarz, John (2007). String Theory and M-Theory: A Modern Introduction. Cambridge, UK: Cambridge University Press.
• Polchinski, J. (1998). String Theory. Cambridge, UK: Cambridge University Press.
• Yau, Shing-Tung; Nadis, Steve (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books. ISBN 978-0-465-02023-2.
• Zaslow, Eric (2008). "Mirror Symmetry". In Gowers, Timothy. The Princeton Companion to Mathematics