Holographic Anomalous and Magnetohydrodynamic Transport

Holographic anomalous transport follows from bulk Chern–Simons inflow together with a declared boundary current convention. Magnetic response also requires deciding whether the magnetic field is an external source or a dynamical hydrodynamic variable. Without those choices, chiral conductivities can differ by local polynomials and magnetization currents can be mistaken for transport.

Required background. Charged and Anomalous Hydrodynamics supplies anomaly-induced constitutive terms. Kubo Formulae and Horizon Response supplies the causal response limits.

Helpful background. Magnetohydrodynamics and Higher-Form Symmetries supplies dynamical magnetic flux. Anomaly Polynomials and Inflow supplies the cohomological normalization of anomalies.

Inflow and current conventions

Consider a five-dimensional gauge sector

$$S=-\frac{1}{4g_5^2}\int\sqrt{\lvert g\rvert}\,F_{ab}F^{ab} +\frac{\kappa}{3}\int A\wedge F\wedge F+S_{\mathrm{ct}}.$$

The gauge variation of the Chern–Simons term is a boundary term proportional to $\kappa\int\lambda F\wedge F$. It fixes the boundary anomaly once orientation, charge normalization, and finite counterterms are fixed. We define $\mathcal C$ in the chosen renormalization scheme by

$$\nabla_\mu J_{\mathrm{cons}}^\mu =\frac{\mathcal C}{24}\epsilon^{\mu\nu\rho\sigma} F_{\mu\nu}F_{\rho\sigma}.$$

$J_{\mathrm{cons}}$ is the functional derivative of the generating functional. Adding the Bardeen–Zumino polynomial produces a covariant current $J_{\mathrm{cov}}$. The two have different local magnetic terms but encode the same anomaly after the convention is stated. Switching between them halfway through a Kubo calculation changes the apparent chiral conductivity.

A radial chiral-magnetic flux

Apply a weak, homogeneous magnetic field $B^i=\tfrac12\epsilon^{ijk}F_{jk}$ to a charged black brane. In the stationary, zero-momentum sector, the Maxwell equation plus Chern–Simons term gives a conserved combination. Choose $\mathcal C$ so that this combination reads

$$\mathcal J^i(r) =-\frac{1}{g_5^2}\sqrt{\lvert g\rvert}\,F^{ri} +\mathcal C A_t(r)B^i, \qquad \partial_r\mathcal J^i=0.$$

Use the regular horizon gauge $A_t(r_h)=0$ and $A_t(\partial)=\mu$. Matching the horizon and boundary values yields a term in the covariant transport current of the form

$$J_{\mathrm{cov}}^i=\mathcal C\mu B^i+J_{\mathrm{diss}}^i,$$

with the displayed coefficient tied to the definitions above. In systems with separate vector and axial currents, vector-current conservation fixes an additional Bardeen counterterm and redistributes anomaly coefficients. Holographic charged-fluid calculations recover the corresponding magnetic and vortical terms from regular black-brane solutions Erdmenger et al. 2009.

If the anomaly includes a mixed gauge–gravitational term, the vortical coefficient can contain a $T^2$ contribution. Its holographic origin is a mixed gauge–gravitational Chern–Simons interaction Landsteiner, Megías, and Peña-Benitez 2011. This term is fixed by anomaly data in the appropriate nondissipative equilibrium observable; it is not a generic temperature correction to every current definition.

External fields versus magnetohydrodynamics

With a nondynamical boundary gauge source, $B$ is imposed and ordinary charged hydrodynamics responds to it. In magnetohydrodynamics the photon is dynamical, magnetic flux is a slow conserved quantity, and the constitutive variables include the magnetic field or an equivalent conserved two-form current. The mode spectrum then contains Alfvén and magnetosonic branches rather than merely current diffusion in a fixed background.

At nonzero background $B$, the boundary current also contains equilibrium magnetization circulation. A transport current subtracts it:

$$J_{\mathrm{tr}}^i=J_{\mathrm{tot}}^i-\epsilon^{ijk}\partial_jM_k,$$

with an analogous energy-magnetization subtraction for heat transport. Although a homogeneous infinite system can hide the curl locally, thermoelectric Kubo formulae and finite samples depend on this distinction.

Current-switching adversary

Recompute the chiral magnetic response first with $J_{\mathrm{cons}}$, then with $J_{\mathrm{cov}}=J_{\mathrm{cons}}+J_{\mathrm{BZ}}$, carrying the same counterterm through the Ward identity and Kubo formula. The numerical coefficients should differ by exactly the Bardeen–Zumino contribution. If they do not, the bulk normalization or boundary variation is inconsistent.

As a second control, compare total and transport currents in a background with spatially varying magnetization. A nonzero equilibrium curl that vanishes after subtraction is not dissipative transport. Finally, promote the boundary gauge field to a dynamical field only after specifying its coupling and boundary condition; doing so changes the mode content and cannot be inferred from a Dirichlet-source calculation.

Exercises

Why does choosing $A_t(r_h)=0$ not set the chemical potential to zero?

Solution

The chemical potential is the gauge-invariant potential difference between boundary and horizon, $\mu=A_t(\partial)-A_t(r_h)$. Horizon regularity sets the second term to zero in a convenient gauge, leaving $\mu=A_t(\partial)$.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Erdmenger, Johanna, Michael Haack, Matthias Kaminski, and Amos Yarom. “Fluid Dynamics of R-Charged Black Holes.” Journal of High Energy Physics 2009, 055 (2009). DOI.
• Landsteiner, Karl, Eugenio Megías, and Francisco Peña-Benitez. “Gravitational Anomaly and Transport.” Physical Review Letters 107, 021601 (2011). DOI.