Sen's conjecture on $L^2$ harmonic forms on strongly centred monopole spaces

Let $\tilde{\mathcal{M}}^0_k$ be the universal cover of the quotient of the charge-$k$ monopole moduli space $\mathcal{N}_k$ by $\mathbb{R}^3\times S^1$, and let $\alpha_\zeta$ be the deck transformation corresponding to $\zeta\in\mathbb{Z}_k$. Denote by $\mathcal{H}^i(\tilde{\mathcal{M}}^0_k)$ the space of $L^2$ harmonic $i$-forms and set

$$\mathcal{H}^i_{k,\ell}=\{u\in\mathcal{H}^i(\tilde{\mathcal{M}}^0_k):\alpha_\zeta^*u=\zeta^\ell u\},\qquad \ell=0,1,\ldots,k-1.$$

Sen's conjecture. If $k$ and $\ell$ are coprime, then

$$\mathcal{H}^{2k-2}_{k,\ell}\cong\mathbb{C},\qquad \mathcal{H}^i_{k,\ell}=0\quad\text{for }i\neq 2k-2;$$

if $k$ and $\ell$ are not coprime, then

$$\mathcal{H}^i_{k,\ell}=0$$

for every $i$. The conjecture predicts the $L^2$ harmonic-form spectrum of strongly centred monopole moduli spaces; the paper proves the coprime case using asymptotic geometry and an argument of Segal and Selby, while the remaining cases require further analysis of the Hodge–de Rham operator.