and the speed of light $c$. ${\bm A}$ and ${\bm\nabla}\times{\bm A}$ transform as vectors in real space, and therefore $S_{\text{c}}$ is invariant only under co-rotations of spin space and real space. It is this coupling of the magnetization to the fluctuating vector potential that allows one to consider the magnetization as having a particular direction in real space. The vector potential is governed by S\_A = \_V d[x]{} , \[eq:2.2c\] with $\rho$ any real number. The first term in Eq. (\[eq:2.2c\]) is the magnetic energy, and the second term with coupling constant $1/\rho$ is a gauge fixing term. One popular choice is $\rho = 0$, which enforces a Coulomb gauge, ${\bm\nabla}\cdot{\bm A} = 0$; another one is the Feynman gauge, $\rho=1$.[@Ryder_1985; @gauge_fixing_footnote] Either choice ensures a finite ${\bm A}$-propagator. In Coulomb gauge, it is A\_i([k]{}) A\_j(-[k]{})= 4T . \[eq:2.3\] The vector potential can now be integrated out exactly, which leads to an effective action in terms of ${\bm M}$ only. Alternatively, we can consider the magnetic induction ${\bm B} = {\bm\nabla}\times{\bm A}$ the fundamental field to be integrated out. In that case, the gauge fixing condition needs to be replaced by a constraint that enforces the Maxwell equation ${\bm\nabla}\cdot{\bm B} 