T-duality

T-duality 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, T-duality was discovered in the context of string theory.[3] In string theory, particles are modeled not as zero-dimensional points but as one-dimensional 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.[4] 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 T-duality, which relates a theory in which strings propagate on a circle of some radius ${\displaystyle R}$ to a theory in which strings propagate on a circle of radius ${\displaystyle 1/R}$.

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 T-duality 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.

${\displaystyle \cdots }$
	−2	−1	0
				${\displaystyle \cdots }$
	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:

The simplest theories in which T-duality arises are two-dimensional 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 ${\displaystyle R}$, about the origin. In what follows, the strings are assumed to be closed (that is, without endpoints).

Denote this circle by ${\displaystyle S_{R}^{1}}$. 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 ${\displaystyle 2\pi R}$. It follows that the state of a string at any given time can be represented as a function ${\displaystyle \varphi (\theta )}$ of a single real parameter ${\displaystyle \theta }$. Such a function can be expanded in a Fourier series as

${\displaystyle \varphi (\theta )=mR\theta +x+\sum _{n\neq 0}c_{n}e^{in\theta }}$.

Here ${\displaystyle m}$ denotes the winding number of the string around the circle, and the constant mode ${\displaystyle x=c_{0}}$ of the Fourier series has been singled out. Since this expression represents the configuration of a string at a fixed time, all coefficients (${\displaystyle x}$ and the ${\displaystyle c_{n}}$) are also functions of time.

Let ${\displaystyle {\dot {x}}}$ denote the time derivative of the constant mode ${\displaystyle x}$. 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 ${\displaystyle {\dot {x}}=n/R}$ for some integer ${\displaystyle n}$. In more physical language, one says that the momentum spectrum is quantized.

In the situation described above, the total energy, or Hamiltonian, of the string is given by the expression

${\displaystyle H=(mR)^{2}+{\dot {x}}^{2}+\sum _{n}|{\dot {c}}_{n}|^{2}+n^{2}|c_{n}|^{2}}$.

Since the momenta of the theory are quantized, the first two terms in this formula are ${\displaystyle (mR)^{2}+(n/R)^{2}}$, and this expression is unchanged when one simultaneously replaces the radius ${\displaystyle R}$ by ${\displaystyle 1/R}$ and exchanges the winding number ${\displaystyle m}$ and the integer ${\displaystyle n}$. The summation in the expression for ${\displaystyle H}$ 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 ${\displaystyle R}$, while the other describes string propagating in a circle of radius ${\displaystyle 1/R}$ with momentum and winding numbers interchanged. This equivalence of theories is the simplest manifestation of T-duality.