Bogusław Broda, Grzegorz Duniec, Giorgi Khimshiashvili

The Non-Abelian Stokes Theorem in Low Dimensions

abstract:
We discuss some aspects of the non-Abelian version of Stokes theorem in two and three dimensions. In particular, the existence of isotopy for practical implementation of the non-Abelian Stokes theorem for topologically nontrivial knots, possibly with self-intersections, in three-dimensional Euclidean space is proved. A generalization to the case of a more general 3-manifold $\mathcal{M}^{3}$ is proposed and some results concerned with the issues of links and stability are presented. A 2-dimensional version of the non-Abelian Stokes theorem is also established and its generalization to the case of a real analytic surface with isolated singularities is formulated in the setting of square-integrable forms.

Mathematics Subject Classification: 26B20, 35Q60, 57M25, 81T13.

Key words and phrases: Stokes theorem, product integration, Seifert surface, gauge theory, real analytical surface, isolated singularity.