The necessary and sufficient condition for the existence of the harmonic function $h$ is

where $\mathbb{T}_{\sigma}$ denotes the complement of the union of small balls of radius $\sigma$ centred at the punctures. Thus if (4.1) is satisfied, a harmonic function $h$ with the singular behaviour (4.2) does indeed exists and is unique up to the addition of a constant.

By Lefschetz–Poincaré duality $H_{2}(\mathbb{T}^{\ast})\simeq H^{1}_{c}(\mathbb{T}^{\ast})$. The latter group sits in a long exact sequence

where $\mathbb{Z}^{2n+8}$ is generated by the $2n+8$ punctures. Thus $H_{2}(\mathbb{T}^{\ast})$ is $(2n+10)$–dimensional and maps onto $H_{2}(\mathbb{T})$ with kernel spanned by the classes of $2n+8$ spheres centred at the punctures. Note that the sum of these $2n+8$ homology classes vanishes.

Because of (4.2), $2n+7$ of the $2n+10$ integrality constraints on $\tfrac{i}{2\pi}\ast dh$ to represent the first Chern class of a line bundle are automatically satisfied since we chose $2m_{j}-4,k_{i}\in\mathbb{Z}$. The remaining $3$ constraints can be reinterpreted in terms of the position of the punctures following the arguments in the proof of [9, Proposition 3.5]:

Since the points $q_{j}$ belong to the half-lattice $\tfrac{1}{2}\Lambda$ this condition is automatically satisfied.

We have therefore proved the existence of a principal $U(1)$ bundle $P\rightarrow\mathbb{T}^{\ast}$ endowed with a connection $\theta$ with curvature $\ast dh$. Since $\mathbb{T}$ is not simply connected $\theta$ is uniquely determined up to a flat connection, i.e. a point of the dual torus $\hat{\mathbb{T}}$. This concludes the proof of (i) and (ii).

By uniqueness up to the addition of a constant the harmonic function $h$ is $\tau$–invariant and therefore can be thought of as defined on $\mathbb{T}^{\ast}/\tau$. Since $\tau^{\ast}(\ast dh)=-\ast dh$, we can lift $\tau$ (uniquely up to gauge transformations) to an involution $\tilde{\tau}$ of the circle bundle $P$ by requiring that $\tilde{\tau}$ acts simultaneously as $\tau$ on $\mathbb{T}^{\ast}$ and as the standard involution on the circle fibres. ∎