Anisotropic shear viscosity of a strongly coupled non-Abelian plasma from magnetic branes

Recent estimates for the electromagnetic fields produced in the early stages of noncentral ultrarelativistic heavy ion collisions indicate the presence of magnetic fields B~O(0.1-15m_π²), where m_π is the pion mass. It is then of special interest to study the effects of strong (Abelian) magnetic fields on the transport coefficients of strongly coupled non-Abelian plasmas, such as the quark-gluon plasma formed in heavy ion collisions. In this paper we study the anisotropy in the shear viscosity induced by an external magnetic field in a strongly coupled N=4 super Yang-Mills (SYM) plasma. Due to the spatial anisotropy created by the magnetic field, the most general viscosity tensor of a magnetized plasma has five shear viscosity coefficients and two bulk viscosities. We use the holographic correspondence to evaluate two of the shear viscosities, η_⊥≡ηxyxy (perpendicular to the magnetic field) and η_∥≡ηxzxz=ηyzyz (parallel to the field). When B≠0 the shear viscosity perpendicular to the field saturates the viscosity bound η_⊥/s=1/(4π), while in the direction parallel to the field the bound is violated since η_∥/s<1/(4π). However, the violation of the bound in the case of strongly coupled SYM is minimal even for the largest value of B that can be reached in heavy ion collisions.