Euler fluid in 2+1 dimensions as a gauge theory and an action for the Euler fluid in any dimension

In this paper we parallel the construction of Tong of a gauge theory for shallow water, by writing a gauge theory for the Euler fluid in $2+1$ dimensions. We then extend it to a Euler fluid coupled to an electromagnetic background. We argue that the gauge theory formulation provides a topological argument for the quantization of $2+1$ dimensional Euler Hopfion solution. In the process, we find a (nongauge) action for the Euler fluid that can be extended to any dimension, including the physical $3+1$ dimensions. We also discuss several aspects of the Arnold-Beltrami-Childress flow.