Tduality
String theory 

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Perturbative theory 
Nonperturbative results 
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Theorists

In theoretical physics, Tduality 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 describes strings propagating in an imaginary spacetime shaped like a circle of some radius R, while the other theory describes strings propagating on a spacetime shaped like a circle of radius 1/R. The two theories are equivalent in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, momentum in one description takes discrete values and is equal to the number of times the string winds around the circle in the dual description.
The idea of Tduality can be extended to more complicated theories, including superstring theories. The existence of these dualities implies that seemingly different superstring theories are actually physically equivalent. This led to the realization, in the mid1990s, that all of the five consistent superstring theories are just different limiting cases of a single elevendimensional theory called Mtheory.
In general, Tduality relates two theories with different spacetime geometries. In this way, Tduality suggests a possible scenario in which the classical notions of geometry break down in a theory of Planck scale physics.^{[1]} The geometric relationships suggested by Tduality are also important in pure mathematics. Indeed, according to the SYZ conjecture of Andrew Strominger, ShingTung Yau, and Eric Zaslow, Tduality is closely related to another duality called mirror symmetry, which has important applications in a branch of mathematics called enumerative algebraic geometry.
Contents

Overview 1
 Strings and duality 1.1
 Winding numbers 1.2
 Quantized momenta 1.3
 An equivalence of theories 1.4
 Superstrings 2
 Mirror symmetry 3
 See also 4
 Notes 5
 References 6
Overview
Strings and duality
Tduality is a particular example of a general notion of duality in physics. The term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be dual to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.
Like many of the dualities studied in theoretical physics, Tduality was discovered in the context of string theory.^{[2]} In string theory, particles are modeled not as zerodimensional points but as onedimensional extended objects called strings. The physics of strings can be studied in various numbers of dimensions. In addition to three familiar dimensions from everyday experience (up/down, left/right, forward/backward), string theories may include one or more compact dimensions which are curled up into circles.
A standard analogy for this is to consider multidimensional object such as a garden hose.^{[3]} If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling inside it would move in two dimensions. Such extra dimensions are important in Tduality, which relates a theory in which strings propagate on a circle of some radius R to a theory in which strings propagate on a circle of radius 1/R.
Winding numbers
In mathematics, the winding number of a curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The notion of winding number is important in the mathematical description of Tduality where it is used to measure the winding of strings around compact extra dimensions.
For example, the image below shows several examples of curves in the plane, illustrated in red. Each curve is assumed to be closed, meaning it has no endpoints, and is allowed to intersect itself. Each curve has an orientation given by the arrows in the picture. In each situation, there is a distinguished point in the plane, illustrated in black. The winding number of the curve around this distinguished point is equal to the total number of counterclockwise turns that the curve makes around this point.
\cdots  
−2  −1  0  
\cdots  
1  2  3 
When counting the total number of turns, counterclockwise turns count as positive, while clockwise turns counts as negative. For example, if the curve first circles the origin four times counterclockwise, and then circles the origin once clockwise, then the total winding number of the curve is three. According to this scheme, a curve that does not travel around the distinguished point at all has winding number zero, while a curve that travels clockwise around the point has negative winding number. Therefore, the winding number of a curve may be any integer. The pictures above show curves with winding numbers between −2 and 3:
Quantized momenta
The simplest theories in which Tduality arises are twodimensional sigma models with circular target spaces. These are simple quantum field theories that describe propagation of strings in an imaginary spacetime shaped like a circle. The strings can thus be modeled as curves in the plane that are confined to lie in a circle, say of radius R, about the origin. In what follows, the strings are assumed to be closed (that is, without endpoints).
Denote this circle by S_R^1. One can think of this circle as a copy of the real line with two points identified if they differ by a multiple of the circle's circumference 2\pi R. It follows that the state of a string at any given time can be represented as a function \varphi(\theta) of a single real parameter \theta. Such a function can be expanded in a Fourier series as
 \varphi(\theta)=mR\theta+x+\sum_{n\neq0}c_ne^{in\theta}.
Here m denotes the winding number of the string around the circle, and the constant mode x=c_0 of the Fourier series has been singled out. Since this expression represents the configuration of a string at a fixed time, all coefficients (x and the c_n) are also functions of time.
Let \dot{x} denote the time derivative of the constant mode x. This represents a type of momentum in the theory. One can show, using the fact that the strings considered here are closed, that this momentum can only take on discrete values of the form \dot{x}=n/R for some integer n. In more physical language, one says that the momentum spectrum is quantized.
An equivalence of theories
In the situation described above, the total energy, or Hamiltonian, of the string is given by the expression
 H=(mR)^2+\dot{x}^2+\sum_n\dot{c}_n^2+n^2c_n^2.
Since the momenta of the theory are quantized, the first two terms in this formula are (mR)^2+(n/R)^2, and this expression is unchanged when one simultaneously replaces the radius R by 1/R and exchanges the winding number m and the integer n. The summation in the expression for H is similarly unaffected by these changes, so the total energy is unchanged. In fact, this equivalence of Hamiltonians descends to an equivalence of two quantum mechanical theories: One of these theories describes strings propagating on a circle of radius R, while the other describes string propagating in a circle of radius 1/R with momentum and winding numbers interchanged. This equivalence of theories is the simplest manifestation of Tduality.
Superstrings
Up until the mid 1990s, physicists working on string theory believed there were five distinct versions of the theory: type I, type IIA, type IIB, and the two flavors of heterotic string theory (SO(32) and E_{8}×E_{8}). The different theories allow different types of strings, and the particles that arise at low energies exhibit different symmetries.
In the mid 1990s, physicists noticed that these five string theories are actually related by highly nontrivial dualities. One of these dualities is Tduality. For example, it was shown that type IIA string theory is equivalent to type IIB string theory via Tduality and also that the two versions of heterotic string theory are related by Tduality.
The existence of these dualities showed that the five string theories were in fact not all distinct theories. In 1995, at the string theory conference at University of Southern California, Edward Witten made the surprising suggestion that all five of these theories were just different limits of a single theory now known as Mtheory.^{[4]} Witten's proposal was based on the observation that different superstring theories are linked by dualities and the fact that type IIA and E_{8}×E_{8} heterotic string theories are closely related to a gravitational theory called elevendimensional supergravity. His announcement led to a flurry of work now known as the second superstring revolution.
Mirror symmetry
In string theory and algebraic geometry, the term "mirror symmetry" refers to a phenomenon involving complicated shapes called CalabiYau manifolds. These manifolds provide an interesting geometry on which strings can propagate, and the resulting theories may have applications in particle physics.^{[5]} In the late 1980s, it was noticed that such a CalabiYau manifold does not uniquely determine the physics of the theory. Instead, one finds that there are two CalabiYau manifolds that give rise to the same physics.^{[6]} 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.^{[7]}
One approach to understanding mirror symmetry is the SYZ conjecture, which was suggested by Andrew Strominger, ShingTung Yau, and Eric Zaslow in 1996.^{[8]} According to the SYZ conjecture, mirror symmetry can be understood by dividing a complicated CalabiYau manifold into simpler pieces and considering the effects of Tduality on these pieces.^{[9]}
The simplest example of a CalabiYau manifold is a 3tori (threedimensional objects which generalize the notion of a torus) parametrized by a 3sphere (a threedimensional generalization of a sphere).^{[10]} Tduality can be extended from circles to the threedimensional tori appearing in this decomposition, and the SYZ conjecture states that mirror symmetry is equivalent to the simultaneous application of Tduality to these threedimensional tori.^{[11]} In this way, the SYZ conjecture provides a geometric picture of how mirror symmetry acts on a CalabiYau manifold.
See also
Notes
 ^ Seiberg 2006
 ^ Other dualities that arise in string theory are Sduality, Uduality, mirror symmetry, and the AdS/CFT correspondence.
 ^ This analogy is used for example in Greene 2000, p.186
 ^ Witten 1995
 ^ Candelas et al. 1985
 ^ Dixon 1988; Lerche, Vafa, and Warner 1989
 ^ Zaslow 2008
 ^ Strominger, Yau, and Zaslow 1996
 ^ Yau and Nadis 2010, p.174
 ^ More precisely, there is a 3torus associated to every point on the threesphere except at certain bad points, which correspond to singular tori. See Yau and Nadis 2010, pp.176–7.
 ^ Yau and Nadis 2010, p.178
References
 Candelas, Philip; Horowitz, Gary; Strominger, Andrew; Witten, Edward (1985). "Vacuum configurations for superstrings". Nuclear Physics B 258: 46–74.
 Dixon, Lance (1988). "Some worldsheet properties of superstring compactifications, on orbifolds and otherwise". ICTP Ser. Theoret. Phys. 4: 67–126.
 Greene, Brian (2000). The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. Random House.
 Lerche, Wolfgang; Vafa, Cumrun; Warner, Nicholas (1989). "Chiral rings in N = 2 superconformal theories". Nuclear Physics B 324 (2): 427–474.
 Seiberg, Nathan (2006). "Emergent Spacetime".
 Strominger, Andrew; Yau, ShingTung; Zaslow, Eric (1996). "Mirror symmetry is Tduality". Nuclear Physics B 479 (1): 243–259.
 Witten, Edward (March 13–18, 1995). "Some problems of strong and weak coupling". Proceedings of Strings '95: Future Perspectives in String Theory. World Scientific.
 Witten, Edward (1995). "String theory dynamics in various dimensions". Nuclear Physics B 443 (1): 85–126.
 Yau, ShingTung; Nadis, Steve (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books.
 Zaslow, Eric (2008). "Mirror Symmetry". In Gowers, Timothy.