This action reads $$S=-\frac{1}{4g_{D9}^2}\int d^{10}x F_{MN} F^{MN}-\frac{1}{4g_{D5}^2}\int d^{6}x F'_{MN} F'^{MN}- \int d^6 x \left[ D_{\mu} \chi^{\dagger} D^{\mu} \chi + \frac{g_{D5}^2}{2}\sum\limits_{A=1}^{3}(\chi_i^{\dagger}\sigma_{ij}^A \chi_j)^2\right]$$

My question is about the $\chi$. The field content of the theory is : 2 vector multiplets (one from the $5-5$ strings and one from the $9-9$ strings) and 3 hypermultiplets (one from the $5-5$ strings, one from the $9-9$ strings, one half from the $5-9$ and another half from the $9-5$).

In the action, the first two integrals are the kinetic terms for the vector multiplets. If I understand correctly, $\chi$ is a doublet describing the $5-9$ and $9-5$ hypermultiplet scalars. The first term in the square brakets is the kinetic term for those, but then the second term looks like the potential for the $5-5$ hypermultiplet (see eq. B.7.3) !

So here is my question : what is $\chi$ really, and how is the action above obtained ?

Edit : my guess would be that $\chi$ is a generic notation for the scalars of all the hypermultiplets. But then there would be something like another index, because I assume that the range of $i$ is $1,2$ if $\sigma_{ij}^A$ are the usual Pauli matrices.



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