Appendix A Variants of the vortex equations

Instead of our variant of the vortex equations, we may also consider the following pair of equations on an oriented Riemannian \(2\)-manifold \((M,g)\) of negative Euler characteristic \[\tag{A.1} K_g-\delta_g\theta=-1+\ell e^{2f}|A|^2_g-\frac{1}{k}\Delta_g f \qquad\text{and} \qquad \overline{\partial} A=k\, \theta^{0,1}\otimes A.\] Here \(A\) is a differential of fractional degree \(1+\ell>1\), \(\theta \in \Omega^1\), \(f\in C^{\infty}\) and \(k\) is a real constant. Notice that we recover our vortex equations by choosing \(k=\ell\). We leave it as an exercise to the interested reader to check that for the choice \(c=2(\ell+1)\), the usual vortex equations (1.3) are equivalent to (A.1) when \(k=\ell+1\). Again, it is straightforward to verify that (A.1) are invariant under suitable gauge transformations. Namely, writing a gauge transformation as \(\tau=e^{w+\mathrm{i}\vartheta}\) for \(w,\vartheta\in C^{\infty}\), we obtain a solution \[\tau \cdot (A,\theta,f)=\left(e^{-(w+\mathrm{i}\vartheta)}A,\theta-\frac{1}{k}(\mathrm{d}w +\star_g\mathrm{d}\vartheta),f+w\right)\] to the above vortex equations from a solution \((A,\theta,f)\). As before, we obtain a thermostat on a suitable root \(SM^{1/n}\) of \(SM\), by defining \[\lambda=e^f a-\mathbb{V}\theta-\frac{1}{\ell}Hf.\] where we use notation as in 4. The thermostat is again invariant under real gauge transformations of the form \(\tau=e^w\), so that we can assume that \(f\) vanishes identically. Thus we have \[K_g-\delta_g\theta=-1+\ell|A|^2_g\qquad \text{and}\qquad \overline{\partial}A=k\, \theta^{0,1}\otimes A.\] Taking \(p=\theta+\mathbb{V}a/(1+\ell)\), we compute exactly as in the proof of Theorem A that \[\kappa_p=-1+(k-\ell)\operatorname{Re}\left((\theta+\mathrm{i}\mathbb{V}\theta)\boldsymbol A\right)\] where \(\boldsymbol{A}=\frac{\mathbb{V}a}{1+\ell}+\mathrm{i}a\). For the usual vortex equations with \(k=\ell+1\) we thus obtain \(\kappa_p=-1+\operatorname{Re}\left((\theta+\mathrm{i}\mathbb{V}\theta)\boldsymbol A\right)\). Moreover, for the usual vortex equations we have the bound \(|\boldsymbol A|^2\leqslant 1/\ell\), see [7]. Thus, we still obtain a dominated splitting provided \(|\theta+\mathrm{i}\mathbb{V}\theta|<\sqrt{\ell}\).