See [6] for more details.

This condition holds for the known solutions in [4, 8] with $M_8$ an $S^4$ fibration over $B_4$. One can relax this condition and obtain off-shell expressions for the central charge that will depend on this flux. It would be interesting to know whether or not any such solutions exist.

Take any curve connecting two fixed points $x_a$, $x_{a^{\prime }}$ in $B_4$, together with the $S^4$ bundle over that curve. This is a five-dimensional submanifold of $M_8$ with boundaries $S_a^4$ and $S_{a^{\prime }}^4$ (the spheres over the fixed points), showing the latter are homologous.

Recall that this is given by $a=\frac{256}{7}(b_1{b}_2{)}^2{N}^3$, subject to $b_1+b_2=1$. This off-shell expression only assumes the existence of a $U(1{)}^2\subset SO(5)$ $R$-symmetry and after extremizing $a$ over the constrained $b_i$, one finds the on-shell weights $b_1=b_2=\frac{1}{2}$ and on-shell central charge $a=\frac{16}{7}{N}^3$, which is the correct result for the $d=6$, $\mathcal{N}=(0,2)$ SCFT with its $SO(5)$ $R$-symmetry, in the large $N$ limit. Further discussion in the context of equivariant localization can be found in Appendix B of [6].