Holographic Transport with Topological Term and Entropy Function

Abstract

In this paper, we consider the effect of a topological Maxwell term $W(\Phi )F_{\mu \nu }\tilde{F}^{\mu \nu }$ 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 considers 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 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.

Appendices

Solutions for Fluctuations in the Case of W(z)

$$\begin{aligned} \begin{aligned} \mathcal {W}(u) =&-\frac{i}{\gamma _x} \int _0^u -i \left( \frac{4 \left( h^2+q^2\right) W(z) (\gamma _y q z-\alpha _y)}{q} \right) \, dz \\&-u \frac{\gamma _x h^3 \left( u^2 \left( 2 q^2 (u-1) (4 u-3)-7 u+6\right) +4\right) }{\gamma _x h \left( h^2+q^2-3\right) } \\&-u \frac{2 h^2 \left( \gamma _y q^3 (3-2 u)+\left( 2 q^2-3\right) Z_0 (\gamma _y q u-2 \alpha _y)+3 \gamma _y q (u-1)\right) }{\gamma _x h \left( h^2+q^2-3\right) } \\&-u \frac{h^4 (\gamma _y q (2 u (Z_0-1)+3)-4 \alpha _y Z_0)+\gamma _x h^5 (u-1) u^2 (4 u-3)}{\gamma _x h \left( h^2+q^2-3\right) } \\&-u \frac{\gamma _x h \left( q^2 \left( u^2 \left( q^2 (u-1) (4 u-3)-7 u+6\right) +4\right) +3 u^2\right) }{\gamma _x h \left( h^2+q^2-3\right) } \\&-u \frac{q \left( q^2-3\right) \left( 3 \gamma _y+\gamma _y q^2 (2 u (Z_0-1)+3)-4 \alpha _y q Z_0\right) }{\gamma _x h \left( h^2+q^2-3\right) }. \end{aligned} \end{aligned}$$

(A.1)

$$\begin{aligned} \begin{aligned} \mathcal {A}^0_x(u)&= i\frac{ (h^3 z^2 (4 z-3) (\gamma _x q z-\alpha _x)+h^2 q (\gamma _y q z-\alpha _y)}{h (z-1) \left( h^2+q^2-3\right) \left( z \left( z \left( z \left( h^2+q^2\right) -1\right) -1\right) -1\right) }\\&+ i\frac{hz^2 \left( q^2 (4 z-3)-3\right) (\gamma _x q z-\alpha _x)+q \left( q^2-3\right) (\gamma _y q z-\alpha _y))}{h (z-1) \left( h^2+q^2-3\right) \left( z \left( z \left( z \left( h^2+q^2\right) -1\right) -1\right) -1\right) }\\&+ \frac{i \gamma _y q \left( h^2 (3-4 z)+q^2 (3-4 z)+3\right) }{4 h (z-1) \left( z \left( z \left( z \left( h^2+q^2\right) -1\right) -1\right) -1\right) (h W(z)+q Z_0)} \end{aligned} \end{aligned}$$

(A.2)

and similarly for $\mathcal {A}^0_y(u)$.

Solution for Fluctuations for the Anisotropic Model

$$\begin{aligned} \begin{aligned} \gamma _y = \frac{\left( h^2+q^2-3\right) }{3 h k_x \left( h^2+q^2+1\right) }&\left( -i h k_x \int _0^1 \frac{4 i k_y W(z) \left( h^2+k_x k_y q^2\right) (\gamma _x q z-\alpha _x)}{q (k_x k_y)^{3/2}} \, dz \right. \\&\quad \quad \left. -4 \alpha _x Z_0 \left( h^2+k_x k_y q^2\right) \right. \\ \quad&\left. +\gamma _x q \left( k_x k_y \left( h^2+2 q^2 Z_0+q^2+3\right) +2 h^2 Z_0\right) \right) \end{aligned} \end{aligned}$$

(B.1)

and

$$\begin{aligned} \begin{aligned} \gamma _x = \frac{\left( h^2+q^2-3\right) }{3 h k_y \left( h^2+q^2+1\right) }&\left( i h k_y \int _0^1 \frac{4 i k_x W(z) \left( h^2+k_x k_y q^2\right) (\gamma _y q z-\alpha _y)}{q (k_x k_y)^{3/2}} \, dz\right. \\&\quad \quad \left. -4 \alpha _y Z_0 \left( h^2+k_x k_y q^2\right) \right. \\&\left. \quad +\gamma _y q \left( k_x k_y \left( h^2+2 q^2 Z_0+q^2+3\right) +2 h^2 Z_0\right) \right) \end{aligned} \end{aligned}$$

(B.2)