Leibniz conformal bialgebras and the classical Leibniz conformal Yang-Baxter equation

We introduce the notion of Leibniz conformal bialgebras, presenting a bialgebra theory for Leibniz conformal algebras as well as the conformal analogues of Leibniz bialgebras. They are equivalently characterized in terms of matched pairs and conformal Manin triples of Leibniz conformal algebras. In the coboundary case, the classical Leibniz conformal Yang-Baxter equation is introduced, whose symmetric solutions give Leibniz conformal bialgebras. Moreover, such solutions are constructed from $\mathcal{O}$-operators on Leibniz conformal algebras and Leibniz-dendriform conformal algebras. On the other hand, the notion of Novikov bi-dialgebras is introduced, which correspond to a class of Leibniz conformal bialgebras, lifting the correspondence between Novikov dialgebras and a class of Leibniz conformal algebras to the context of bialgebras. In addition, we introduce the notion of classical duplicate Novikov Yang-Baxter equation whose symmetric solutions produce Novikov bi-dialgebras and thus Leibniz conformal bialgebras.