Vortices in the effective field theory of Fractional Quantum Hall Effect I am a grad student tasked with explaining what vortices are in the Zhang Hansson and Kivelson effective field theory of fractional quantum hall effect. I am explaining them to a final year bachelors student who is doing a project in our group and that student doesn't know much geometry and topology. The simple, intuitive explanation that I gave is that they are just fields that locally behave like vortices that you find in hydrodynamics. By that what I meant was that these two vortices have certain properties that are analogous such as being able to define the notion of winding number. Is there a more rigorous and intuitive explanation for it?
 A: yes there is.  first explain vortices in a simpler setting then move on to FQHE.
First, we start with two-dimensional spins   interacting with nearest neighbours located in three-dimensional Euclidean cubic lattice. We can parametrize a spin orientation $\vec{s}_n$ at a lattice site $n$ as, \begin{equation}
 \vec{s}_n=(\cos\theta_n, \sin\theta_n),\label{eq:s}
\end{equation}
where $\theta_n\in[0,2\pi)$ is the angle between spin and a fixed axis of reference.
The action is ,
\begin{equation}
 S=-\frac{1}{T}\sum_{n\mu}\vec{s}_n\cdot\vec{s}_{n+\hat{\mu}}=-\frac{1}{T}\sum_{n\mu}\cos(\Delta_\mu\theta_n),\label{eq:xy}
\end{equation}
such model is usually called the ``XY Model''  where $T$ is the temperature.
Now, let's consider the continuum limit, for $T\ll1$. In this low temperature regime, we would not expect too much difference  between  the orientations of adjacent spins, so we have
\begin{equation}
-\frac{1}{T}\sum_{n\mu}\cos(\Delta_\mu\theta_n)\to\frac{1}{T}\sum_{n\mu}\bigg(\frac{1}{2}(\Delta_\mu\theta_n)^2\bigg)\to\frac{1}{2T}\int d^3x(\partial_\mu\theta_n)^2.\label{eq:lowtempapxwrong}
\end{equation}
However this approximation is not correct. As a result  of the compactness of $\theta$ field, two adjacent spins can  have almost same orientation but corresponding  angle between them can be in order of $2\pi$.  Such example is shown in 1. A corresponding field configuration can be,
\begin{align}
 \theta_n&=2\pi-\epsilon{\notag},\\
 \theta_{n+\delta}&=\epsilon,
\end{align}
such that $\theta_{n_0+\delta}-\theta_{n_0}=2\epsilon-2\pi$ is not small even though $\vec{s}_n$ and $\vec{s}_{n_0+\delta}$ are very closely aligned. However the approximation we have just made, highly suppresses such field configurations. These kind of multi-valued configurations are vortex line configurations \cite{Polyakov:1987ez}. The object we call \emph{vortex line},      is the singular line shaped region of a given vortex configuration.
So this is the basic explanation of three dimensional vortex loops but it applies to 2D vortices as well. after this we can go back to FQHE. The full action of 2D FQHE system is given as, \begin{align}
 S_{\phi}=\int d^3x\Bigg[\bar{\phi}i(\partial_0+iqA_0)\phi&-\frac{1}{2m}\bigg|\Big(\partial_i+i(qA_i+a_i[\bar{\phi}\phi])\Big)\phi\bigg|^2+\mu|\phi|^2\Bigg]\notag\\ &-\frac{1}{2}\int d^3x\int d^3y\delta\rho(x)V(x-y)\delta\rho(y),
\end{align}
plus Chern-simons terms. Now we can apply a simple  integral transformation as, $\phi=\sqrt{\rho(x)}e^{i\theta(x)}$ and $\bar{\phi}=\sqrt{\rho(x)}e^{-i\theta(x)}$ where $\theta(x)=\theta(t,\vec{x})$ is angular field with $\theta(x)\in[-\pi,\pi)$ it is the phase at spatial coordinate $\vec{x}$ and at time $t$. As we have seen in  just now such an angular field can have smooth, single-valued configurations, as well as multi-valued singular configurations,  which we call vortices.   However, in the previous case, these singularities were three dimensional loops since we were considering a three dimensional Euclidean lattice. However, in this case we consider a system with two spatial dimensions. Thus the singularities are point-like. We then write  $\theta=\theta_s+\theta_{\rm v}$ as smooth single valued part $\theta_s$ and as singular  multi-valued part $\theta_{\rm v}$, as an example one of the possible configurations that multi-valued part can have is ,  $\theta_{\rm v}=\arctan(\frac{x_2}{x_1})$ which is shown in 2.

