 Reduction of the eleven-dimensional Chern-Simons term

ID: reduction-of-the-eleven-dimensional-chern-simons-term

Put

A_{3} = C_{3} + B_{2} \land d y

and

F_{4} = G_{4} + H_{3} \land d y

, where

G_{4} = d C_{3}

. The circle component of

A_{3} \land F_{4} \land F_{4}

[B_{2} \land G_{4} \land G_{4} + 2 C_{3} \land G_{4} \land H_{3}] \land d y

. Since

d (C_{3} \land B_{2} \land G_{4}) = B_{2} \land G_{4} \land G_{4} - C_{3} \land H_{3} \land G_{4}

, its integral equals three copies of

B_{2} \land G_{4} \land G_{4}

- 1/2

after circle integration. Boundary terms and global potential patching must be retained when the spacetime has boundary or nontrivial flux topology.

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