(Joint work with Benoit Charbonneau.)

- $P \rightarrow \mathbb{R}^3$ is a smooth $\mathrm{SU} (N)$-principal bundle,
- $\nabla$ is a connection on $P$ and $\Phi$ is section for $\mathrm{ad} (P)$,
- and together they satisfy:

\begin{equation} F_\nabla = \ast \nabla \Phi, \qquad |F_\nabla| = |\nabla \Phi| \in L^2 (\mathbb{R}^3).\end{equation} - Furthermore:

\begin{equation} \Phi = \Phi_\infty - \frac{1}{2r} \kappa + O (r^{-2}), \quad \nabla \Phi = \frac{1}{2r^2} \kappa + O (r^{-3}).\end{equation}

$\mathcal{N}_{\nabla, \Phi, t} = D_\nabla + \Phi - i t \mathbb{I}: L_1^2 \rightarrow L^2$

$\mathcal{N}_{\nabla, \Phi, t}$ is Fredholm, if and only if $it \notin \mathrm{Spec} (\Phi_\infty)$.

$\mathrm{index} (\mathcal{N}_{\nabla, \Phi, t}) = i (\Phi_\infty, \kappa, t)$.

$\mathcal{N}_{\nabla, \Phi, t}$ has trivial kernel, and there is a family of operators \begin{equation}\mathcal{T} = (T_1 (t), T_2 (t), T_3 (t)),\end{equation} on the cokernel bundle of $\mathcal{N}_{\nabla, \Phi, t}$, such that they satisfy Nahm's equation, and encode $(\nabla, \Phi)$. The map sending $(\nabla, \Phi)$ to $\mathcal{T}$ is a bijection.

Proved in the case of *maximal symmetry breaking* by Hitchin, Nakajima, Hurtubise, and Murray.

The proof in the higher rank, maximal symmetry breaking case uses algebraic geometry (spectral curves) to circumvent the analysis of harmonic spinors.

Problem: These ideas do not work for **nonmaximal symmetry breaking**!

Today, I'll only talk about the asymptotic, \begin{equation}\mathrm{dist} (t, \mathrm{Spec}(\Phi_\infty)) \rightarrow 0,\end{equation} behavior of spinors.

- Initial decay estimates by (modified) Moser iteration.
- Write $\mathcal{N}_{\nabla, \Phi, t} = c(dr) \nabla_r + (\Phi_\infty - it \mathbb{I}) + \frac{1}{r} D_{\mathrm{model}} + O (r^{-2})$.
- $D_{\mathrm{model}} = D_{S^2} + \frac{1}{2} \kappa \mathbb{P}$.
- One can decompose a harmonic spinor according to the spectrum of $D_{\mathrm{model}}$.
- As $\mathrm{dist} (t, \mathrm{Spec}(\Phi_\infty)) \rightarrow 0$, one can find an approximate basis for harmonic spinors.

- This asymptotic basis extends Nakajima's construction.
- This is novel in the maximal symmetry breaking case too!
- One can compute the Nahm matrices in this basis, and find the generalized boundary conditions.
- Using this we defined the Nahm moduli corresponding for arbitrary symmetry breaking.
- Using this, we proved a conjecture of Singer and Murray.

- Finding new solutions: We have found novel new solutions with undergrads at Duke and Waterloo.

(Anuk Dayaprema, Christopher Lang, and Haoyang Yu) - Removing the asymptotic conditions on the monopoles and replace it with finite energy.
It was done by Stern for the maximal symmetry breaking cases (using ideas of Agmon).

We are currently working on the general case.