Non-abelian reciprocity laws on algebraic surfaces

We prove and generalize both Kato–Parshin reciprocity laws on an algebraic surface over a field of characteristic zero, using iterated integrals. We generalize the known reciprocity laws to reciprocity laws on generating series of iterated integrals. That gives us a reciprocity law for each coefficient of the generating series. The terms of degree 2 give us analogues of sum of the residues is zero. Degree 3 terms give us Kato-Parshin reciprocity laws. Higher degree terms give new reciprocity laws.