The Relative Differential Morphism and the Logarithmic Connection of Dual Conformal Nets

Abstract

Let $X$ be a compact connected Riemann surface equipped with two nonconstant meromorphic projections $q,z:X\to\mathbb{P}^1$. Their differentials determine a canonical meromorphic relative differential morphism $Z:T_q=q^*T\mathbb{P}^1\to T_z=z^*T\mathbb{P}^1$, whose local coefficient in coordinate frames is $Z=z_q\,\partial_z\otimes dq$. Its divisor is $\operatorname{Div}(Z)=R_z-R_q$, and its local logarithmic derivative defines a rank-one logarithmic connection on $L=\operatorname{Hom}(T_q,T_z)$. The general theory of logarithmic connections is classical. The main point of the present paper is that, in the dual conformal-net setting, the Levi--Civita curvature data of the two nets realize this logarithmic connection by the symmetric formula $H^{-1}\widetildeω\,dz-Hω\,dq =2i\,d\log z_q =2i\,Z^{-1}dZ$, where the last expression is understood through local trivializations of $L$. Thus the two dual conformal connection differentials occur with opposite signs, and their difference is the logarithmic Maurer--Cartan form of the relative differential morphism. Its residues recover the relative ramification multiplicities, $\operatorname{Res}_pΘ=2i(r_z(p)-r_q(p))$, and globally $\sum_{p\in X}\operatorname{Res}_pΘ=4i(\operatorname{deg}z-\operatorname{deg}q)$. Thus local conformal connection data determine both relative ramification and the difference of the degrees of the two projections. We also discuss the dual morphism, the square-root issue, and a hyperelliptic example.