On the hydrodynamics of (2 + 1)-dimensional strongly coupled relativistic theories in external magnetic fields

https://arxiv.org/abs/2209.11589

I was invited to produce a review covering theoretical progress in the development of (2+1)-dimensional hydrodynamics in an external magnetic field. In the review I discuss why external magnetic fields inevitably break  translation and boost invariance. I detail the nature of thermodynamic equilibrium in the presence of this field and translation breaking scalars. Special attention is paid to ensuring that the Ward identities are satisfied. This had previously been a problem in the fluid/gravity literature going back to the early days of applied holography. In particular, if the magnetic field is treated as order one in derivatives, it is not possible to satisfy the current-current Ward identities except at leading order in the magnetic field. 

The resolution to this paradox is that one can never treat the magnetic field as order one in derivatives in a holographic model. It always enters into the thermodynamics. Consequently, a crucial piece of a transport coefficient, linear in the magnetic field had been errorneously dropped from previous works.  Unfortunately, this mistake continues to happen to this day.  

Finally, in this paper we show how one can obtain conductivities that respect the Ward identities in the presence of translation breaking scalars and an external magnetic field.