FixPointSubspace

Denote by \(X_n\) the phase space for \(n\) vortices, so \(X_n\simeq \mathbb{C}^n\setminus \Delta\).

\(\mathrm{Fix}(\mathbb{Z}_n,X_{kn})\)

The fixed points consist of \(k\) rings with \(n\) vortices in each = that is \(n\)-rings. It is a symplectic subspace, with an action of \(\mathsf{SO}(2)\), or more generally \(\mathsf{SO}(2)\times \mathfrak{S}\), where \(\mathfrak{S}\) is a subgroup of the permutation group \(S_k\) depending on the vortex strengths of each ring (vortex strength within a ring must be constant).

One \(n\)-ring

Write \(\zeta=\exp(2\pi i/n)\).

Let \(z_0=\rho e^{i\theta}\in\mathbf{C}^*\) be one of the vortices in the ring. The other points are \(z_j=z_0\zeta^j\) (for \(j=1,\dots n-1\)).

Hamiltonian is

\[H = -\frac1{4\pi} \sum_{j<k} \log|z_j-z_k|.\]

Now, \(|z_j-z_k| = \rho|\zeta^j-\zeta^k| = \rho|\zeta^{k-j}-1|\).

Lemma \(\prod_{j<k} |\zeta^j-\zeta^k| = n^{n/2}\).

Thus the Hamiltonian is equal to

\[H = -\frac{n(n-1)}{8\pi} \log\rho - \frac{n}{8\pi}\log n.\] The second term is a constant so irrelevant (just curious).

This must count as the Robin function for the single \(n\)-ring.

Interaction of 2 \(n\)-rings

Let \(z_1,z_2\) be points, one in each ring. The other points in the rings are then \(z_1\zeta^j\) and \(z_2\zeta^j\) (for \(j=1,\dots,n-1\))

The interaction Hamiltonian is then \[H_{\textrm{int}} = -\frac{\kappa_1\kappa_2 }{4\pi} \sum_{j,k=1}^n \log |z_1\zeta^j - z_2\zeta^k|^2.\] Now \(z_1\zeta^j - z_2\zeta^k = \zeta^j(z_1-z_2\zeta^{k-j})\), whence \[\prod_{j,k} |z_1\zeta^j - z_2\zeta^k|^2 = \prod_{j,k} |z_1-z_2 \zeta^k|^2 = \left[\prod_k|z_1-z_2 \zeta^k|^2\right]^n.\]

Now let \(P(z) = \prod_{k=1}^n(z-z_2 \zeta^k)\). Then \(P(z) = z^n-z_2^n\). Therefore,

\[H_{\textrm{int}} = -\frac{\kappa_1\kappa_2 n}{4\pi} \log |z_1^n - z_2^n|^2.\]