In fact this duality is present between anomalous chiral fluids in four spacetime dimensions and spinning fluids in six spacetime dimensions. However, since we want to highlight the string theory origin of these chiral fluids we consider spinning fluids in ten spacetime dimensions.

There is also another equation that in general arises from ${\nabla }_{\mu }{\mathbf{T}}^{\mu \nu }=0$ describing the transverse dynamics of the brane but it is not relevant for the purposes of this letter so we present it elsewhere.

We note that in other formulations of hydrodynamics of spin [32, 33, 34] there is a Belinfante-Rosenfeld ambiguity in the definition of stress tensor and spin current which does not appear in the formulation presented here because the stress tensor $T^{ab}$ is symmetric.

This idea is similar to the one implemented in [52].

One point of departure with theories with intrinsic spin is that in this formulation with transverse spin, coupling to the sources ${\gamma }_{ab}$ and ${{\omega }_a}^{ij}$ directly in (5) and not to the tangent ${\partial }_a{X}^{\mu }$ and normal $n_i^{\mu }$ vectors, does not lead to a relation between the spin chemical potential ${\mu }^{ij}$ and the thermal vorticity ${\partial }_{[a}{u}_{b]}$ of the fluid in equilibrium.