21 problems

Melted-volume conjecture.

Let the parameter space of hyperbolic monopoles contain a compact subset and a disjoint subset, and let the monopoles have sufficiently small mass. Small-mass asymptotic separation…

Let $\omega$ be the symmetric measure on the sphere at infinity $S^2$ of hyperbolic space ${\mathbb H}^3$. For $f\in C^0_c({\mathbb H}^3)$, define the action … where $x$ is regarde…

A hyperbolic monopole is a solution of the Bogomolny equation on hyperbolic space ${\mathbb H}^3$, with an asymptotic value given by its limiting data on the sphere at infinity…

Charge-existence conjecture. For a given $\mu$ there are monopoles with any collection of nonnegative magnetic charges $(m_1,\ldots,m_s)$. Given a choice of magnetic charges, there…

Austin–Braam holography conjecture. The holographic image of the monopole on the conformal boundary two-sphere determines the monopole on hyperbolic space.

Energy-density and cluster conjecture. For $m=0$ or $m=k$, the energy density is always confined to a finite neighbourhood of the plane $x_3=0$. For $0<m<k$, monopole clusters can…

Finite-time blow-up conjecture. Any invariant solution to the monopole equations on these three spaces without quadratic curvature decay must blow up in finite time.

Let $G_c$ be the compact form of a gauge group, and let $mu$ and $lambda$ be coweights with $lambda$ dominant. For dominant $nu$ satisfying $mu\leqnu\leqlambda$, let…

Cut-volume conjecture. The Kähler potential of a monowall moduli space is related to the cut volume $\mathcal{V}(R_f)$.

Let $X$ be a compact $G_2$-manifold, let $N$ be the singular set, and let $P$ be an $SO(3)$-bundle over $X\backslash N$ induced by restricting a bundle $\tilde{P}$ over $X$. Let…

Asymptotic expansion conjecture. The asymptotic expansion of $\operatorname{Tr} f(s,s;\mathbf{x})$ is determined to all orders by the conserved quantities…

Consider multiple monopoles in ${\cal N}=4$ super Yang–Mills theory. Sen's S-duality conjecture. The multiple monopoles form a unique bound state that builds the Montonen–Olive dua…

Consider the case $G=PSU(N)$ with $N_j$ singularities of fundamental weight $\omega_{N-j}$, so that … Let $m_{j,\alpha}$ be the positions of the singularities and define the monic…

Monopole-moduli gravitational-instanton conjecture. Precisely ${\cal M}_{2}^{0}$, being the gravitational instanton, gives the dominant contribution to the Euclidean path integral…

Large-charge Nahm-data convergence conjecture. For each solution $(t^i)$ corresponding to a magnetic bag, there is a sequence of solutions $(T^i_{(n)})$ such that, as…

Index–dimension conjecture. For general confined multimonopoles, the resulting index counts one quarter of the real dimension of the moduli space:

The Nahm equations provide the data for the ADHMN construction of BPS monopoles, whose Weyl equation yields the Higgs field and gauge potential through normalized solutions. In the…

Let $n$ and $m$ be the integers specifying the Ercolani–Sinha vector, and let the corresponding solution be considered on the interval $s\in(0,2)$. Unwanted-zero conjecture. The so…

Theta-function vanishing conjecture. For each such pair $(m,n)$, $H(y)$ vanishes $2(|n|-1)$ times on the interval $\lambda\in(0,2)$.

Uniqueness conjecture. For a symmetric monopole, the denominator of $Q_0(z)$ has $2(|n_1|-1)$ zeros, and consequently the tetrahedrally symmetric monopole is the only monopole in t…