Why is it interesting?

Winding numbers are crucial to understand the QCD vacuum and important related effects like instantons in quantum field theory.

Layman

Student

Source: page 80 Selected Topics in Gauge Theories by Walter Dittrich, Martin Reuter

To understand the notion "winding number", we therefore consider $U(1)$ transformations in a toy model with just one spatial dimension that is curled up to a circle. The $U(1)$ transformations depend on the spatial coordinate, which simply means we have a map from each point in space to an $U(1)$ element.

In other words, we consider maps from the circle $S^1$ to the group $U(1)$. Points on the circle can be parameterized by an angle $\phi$ that runs from $0$ to $2\pi$ and therefore, we can write possible maps as follows:

$$ S^1 \to U(1) : g(\phi)= e^{i\alpha(\phi)} \, . $$

The thing is now, that the set of all possible $g(\phi)$ is divided into various topological sectors, which can be labelled by an integer $n$. This can be understood as follows:

The map from the circle $S^1$ to $U(1)$ needs not to be one-to-one. The degree to which a given map is not one-to-one is the winding number. For example, when the map is two-to-one, the winding number is 2. A map from the circle onto elements of $U(1)$ is

$$ S^1 \to U(1) : f_n(\phi)= e^{in\phi} \, . $$

This map eats elements of the circle $S^1$ and spits out an $U(1)$. Now, depending on the value of $n$ in the exponent we get for multiple elements of the circle the same $U(1)$ element. Formulated differently, this means that depending on $n$ our map $f_n(\phi)$ maps several points on the circle onto the same $U(1)$ element.

For example, if $n=2$, we have $$ f_2(\phi)= e^{i2\phi} .$$ Therefore $$ f_2(\pi/2)= e^{i \pi} = -1 $$ and also $$ f_2(3\pi/2)= e^{i3 \pi} = e^{i2 \pi} e^{i1 \pi} = -1 .$$

Therefore, as promised, for $n=2$ the map is two-to-one, because $\phi=\pi/2$ and $\phi= 3\pi/2$ are mapped onto the same $U(1)$ element. Equally, for $n=3$, we get for $\phi=\pi/3$, $\phi=\pi$ and $\phi= 5\pi/3$ the same $U(1)$ element $f_3(\pi/3)=f_3(\pi)=f_3(5\pi/3)=-1$.

In this sense, the map $f_n(\phi)$ determines how often $U(1)$ is wrapped around the circle and this justifies the name "winding number" for the number $n$.

As a side remark: The elements of $U(1)$ also lie on a circle in the complex plane. ($U(1)$ is the group of the unit complex numbers). Thus, in this sense, $f_n(\phi)$ is a map from $S^1 \to S^1$.

A clever way to extract the winding number for an arbitrary map $ S^1 \to U(1)$ is to compute the following integral

$$ \int_0^{2\pi} d\phi \frac{f_n'(\phi)}{f_n(\phi)} = 2\pi i n, $$ where $f_n'(\phi)$ is the derivative of $f_n(\phi)$. Such tricks are useful for more complicated structures where the winding number isn't that obvious.

Researcher

Examples

Example1

Example2:

FAQ

Is the winding number always an integer?

No!

However, there is no reason for c1 to take any particular value. c1 is not necessarily an integer for a pure abelian gauge field in R2 . This is because α is not necessarily single-valued, nor does it need to increase by an integer multiple of 2π around the circle at infinity. However, we shall see later, in our discussion of vortices, that the coupling of the gauge potential to a scalar field φ does bring in further restrictions, and then c1 must be an integer.

page 62 in Topological Solions by Manton and Sutcliff

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