On Monopoles with Nonmaximal Symmetry Breaking

Ákos Nagy
(Joint work with Benoit Charbonneau.)

La Quinta, California

Friday, December 13, 2019.

Notation and assumptions

$P \rightarrow \mathbb{R}^3$ 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}

My object of interest is the following operator:

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

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

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

"THM" (Nahm):
$\mathcal{N}_{\nabla, \Phi, t}$ 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 for $\mathrm{SU} (2)$, by Nakajima, uses the original idea of Nahm (understanding harmonic spinors).

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!

Our result: doing it with spinors anyway!

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

Idea of the proof:

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.

What is left to do:

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.