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 $Z:T_q=q^{\ast }T{\mathbb{P}}^1\to T_z=z^{\ast }T{\mathbb{P}}^1$, whose local coefficient in coordinate frames is $Z=z_q{\partial }_z\otimes dq$. Its divisor is $\mathrm{Div}(Z)=R_z-R_q$, and its local logarithmic derivative defines a rank-one logarithmic connection on $L=\mathrm{Hom}(T_q,T_z)$. 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 $H^{-1}\tilde{\omega }dz-H\omega dq=2id\log z_q=2i{Z}^{-1}dZ$, 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, ${\mathrm{Res}}_p\Theta =2i(r_z(p)-r_q(p))$, and globally ${\sum }_{p\in X}{\mathrm{Res}}_p\Theta =4i(\mathrm{deg}z-\mathrm{deg}q)$. 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.

arXiv comment19 pages