Reduction of the eleven-dimensional Chern-Simons term ($-\frac{1}{6}\int A_3{F}_4{F}_4⟶-\frac{1}{2}\int B_{2}d{C}_{3}d{C}_3$)

Put $A_3=C_3+B_2\wedge dy$ and $F_4=G_4+H_3\wedge dy$, where $G_4=d{C}_3$. The circle component of $A_3\wedge F_4\wedge F_4$ is $[B_2\wedge G_4\wedge G_4+2C_3\wedge G_4\wedge H_3]\wedge dy$. Since $d(C_3\wedge B_2\wedge G_4)=B_2\wedge G_4\wedge G_4-C_3\wedge H_3\wedge G_4$, its integral equals three copies of $B_2\wedge G_4\wedge G_4$ up to a boundary term. The coefficient therefore becomes $-1/2$ after circle integration. Boundary terms and global potential patching must be retained when the spacetime has boundary or nontrivial flux topology.