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Suppose you have a self-dual five form field strength $F_5=*F_5$. Kinetic term of this field strength is written as $$ \int F_5\wedge*F_5=\int F_5\wedge F_5=-\int F_5\wedge F_5 $$ where in second equality, I used $A\wedge B=(-1)^{pq}B\wedge A$ for $p$-form $A$ and $q$-form $B$. So you can conclude that $$ \int F_5\wedge*F_5=0\,. $$ This is the subtlety you ...



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