uad . \label{allRRfields}\end{aligned}$$ A single T-duality along the $a$ direction acts on the RR fields in the obvious way: if an $n$-form potential $C_n$ of one theory has no index parallel to $a$, it is mapped to the $(n+1)$-form $C_{n+1}$ of the other theory where the extra index is $a$, and vice versa. It is clear that one obtains all components of all the RR fields listed in eq. starting from any particular field and performing repeated T-dualities in all directions.

The $\alpha=-2$ branes arise from the ten-dimensional mixed-symmetry potentials [@stringsolitons; @dualdoubledgeometry] $${D_6 \ \ D_{7,1} \ \ D_{8,2} \ \ D_{9,3} \ \ D_{10,4} } \quad , \label{allDpotentials}$$ which are the same for the IIB and the IIA theory. The potential $D_6$ obviously corresponds to the NS5-brane, while the mixed-symmetry potential $D_{7,1}$ corresponds to the KK-monopole and the potential $D_{8,2}$ corresponds to the T-fold of [@deboer]. These latter two are solutions with one and two isometric directions respectively, and denoting with $a$ and $b$ these directions one obtains that the potentials corresponding to these solutions are the 6-forms $D_{6 \,  a ,a}$ and $D_{6 \, ab, ab}$. The KK-monopole solution is obtained by performing a T-duality on the NS5-brane solution along the direction $a$ transverse to the brane, while the T-fold solution is obtained by T-dualising the KK-monopole solution along the further isometry $b$. By generalising this, one finds that in terms of the $\alpha=-2$ potentials T-duality acts as follows: if the potential of one theory has no indices along $a$, then $T_a$ maps it to a potential of the other theory with $a$ added on both the first and the second set of indices, while if the potential of one theory has an index $a$ only along the first set of indices, it is mapped to the same potential of the other theory. It can easily be shown that these rules map the NS5-brane to all the other branes corresponding to the mixed-symmetry potentials in eq. by chains of T-dualities.

We now move to