# Action with self-dual field strength

It is said that writing down an action in presence of a self-dual field strength is subtle and not known till date. The familiar example people give is that of type IIB super-gravity which has a self-dual 5-form field strength. Can someone elaborate on what exactly the subtlety is.

## 1 Answer

Suppose you have a self-dual five-form field strength $F_5=*F_5$. The kinetic term of this field strength is proportional to $$\int F_5\wedge*F_5=\int F_5\wedge F_5=-\int F_5\wedge F_5$$ where I used $A\wedge B=(-1)^{pq}\, B\wedge A$ for $p$-form $A$ and $q$-form $B$ in the second equality. So you can conclude that $$\int F_5\wedge*F_5=0\,.$$ This is the subtlety you mentioned. People usually impose self-dual condition on $F_5$ after obtaining equations of motion. But this is not a satisfactory resolution, because we want to obtain self-dual condition as a part of equations of motion.

(Edit) The above answer was my naive understanding. For more detailed explanation, see page 313 of Becker, Becker and Schwarz.