According to ref.1 (§A.3), the naive definition of Chern-Simons $$ S[A]=k\int_M \mathrm{CS}[A]\tag{A.17} $$ is ill-defined, because $A$ may have "Dirac string singularities". The solution is to extend $M$ to the bulk, such that $M=\partial X$, and define $$ S[A]=k\int_X c(F)\tag{A.18} $$ with $c$ the Chern form of $F=\mathrm dA$. It is argued that, provided $k$ is properly quantised, this integral is independent of $X$ and of the extension of $A$.
But how does this procedure fix potential "Dirac string singularities"? We are integrating over all $A$, and therefore we will have singular configurations regardless of whether we use $\mathrm A.17$ or $\mathrm A.18$. How does extending $A$ into the bulk help remove these singular configurations?
References.
- N. Seiberg, E. Witten, Gapped Boundary Phases of Topological Insulators via Weak Coupling, https://arxiv.org/abs/1602.04251.