Holographic transport with topological term and entropy function

In this paper we consider the effect of a topological Maxwell term $W(\mathrm{\Phi })F_{\mu \nu }{\tilde{F}}^{\mu \nu }$ on holographic transport and thermodynamics in 2+1 dimensions, in the case with a dyonic black hole in the gravity dual. We find that for a constant $W$ the modifications to the thermodynamics are easily quantified, and transport is affected only for ${\sigma }_{xy}$. If one consider also the attractor mechanism, and writing the horizon transport in terms of charges, the transport coefficients are affected explicitly. We also introduce the case of case of radially dependent $W(z)$, in which case, however, analytical calculations become very involved. We also consider the implications of the two models for the S-duality of holographic transport coefficients.