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→ℙ1. Their differentials determine a canonical meromorphic relative differential morphism Z:Tq=q*Tℙ1→ Tz=z*Tℙ1, whose local coefficient in coordinate frames is Z=zq ∂z⊗ dq. Its divisor is Div(Z)=Rz−Rq, and its local logarithmic derivative defines a rank-one logarithmic connection on L=Hom(Tq,Tz). 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−1ω̃ dz−Hω dq =2i dlog zq =2i Z−1dZ, 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, RespΘ=2i(rz(p)−rq(p)), and globally ∑p∈ XRespΘ=4i(degz−degq). 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.
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BibTeX
@misc{nuramatov2026relative,
title = {{The Relative Differential Morphism and the Logarithmic Connection of Dual Conformal Nets}},
author = {A. G. Nuramatov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26803},
eprint = {2609.26803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26803},
}
RIS
TY - UNPB
TI - The Relative Differential Morphism and the Logarithmic Connection of Dual Conformal Nets
AU - A. G. Nuramatov
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.26803
UR - https://arxiv.org/abs/2609.26803
AB - Let X be a compact connected Riemann surface equipped with two nonconstant meromorphic projections q,z:X→ℙ^1. Their differentials determine a canonical meromorphic relative differential morphism Z:T_q=q^*Tℙ^1→ T_z=z^*Tℙ^1, whose local coefficient in coordinate frames is Z=z_q ∂_z⊗ dq. Its divisor is Div(Z)=R_z−R_q, and its local logarithmic derivative defines a rank-one logarithmic connection on L=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^−1ω̃ dz−Hω dq =2i dlog z_q =2i Z^−1dZ, 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, Res_pΘ=2i(r_z(p)−r_q(p)), and globally ∑_p∈ XRes_pΘ=4i(degz−degq). 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.
ER -