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Chiral anomalies give rise to dissipationless transport phenomena such as the chiral magnetic and vortical effects. In these notes I review the theory from a quantum field theoretic, hydrodynamic and holographic perspective. A physical interpretation of the otherwise somewhat obscure concepts of consistent and covariant anomalies will be given. Vanishing of the CME in strict equilibrium will be connected to the boundary conditions in momentum space imposed by the regularization. The role of the gravitational anomaly will be explained. That it contributes to transport in an unexpectedly low order in the derivative expansion can be easiest understood via holography. Anomalous transport is supposed to play also a key role in understanding the electronics of advanced materials, the Dirac- and Weyl (semi)metals. Anomaly related phenomena such as negative magnetoresistivity, anomalous Hall effect, thermal anomalous Hall effect and Fermi arcs can be understood via anomalous transport. Finally I briefly review a holographic model of Weyl semimetal which allows to infer a new phenomenon related to the gravitational anomaly: the presence of odd viscosity.
A derivation of the anomaly-induced transport phenomena---the chiral magnetic/vortical effect---is revisited based on the imaginary-time formalism of quantum field theory. Considering the simplest anomalous system composed of a single Weyl fermion, w
In this paper, we discuss relativistic hydrodynamics for a massless Dirac fermion in $(2+1)$ dimensions, which has the parity anomaly -- a global t Hooft anomaly between $mathrm{U}(1)$ and parity symmetries. We investigate how hydrodynamics implement
We study some of the transport processes which are specific to an ideal gas of relativistic Weyl fermions and relate the corresponding transport coefficients to various anomaly coefficients of the system. We propose that these transport processes can
Higher-spin diffeomorphisms are to higher-order differential operators what diffeomorphisms are to vector fields. Their rigorous definition is a challenging mathematical problem which might predate a better understanding of higher-spin symmetries and
We describe numerous properties of the Sachdev-Ye-Kitaev model for complex fermions with $Ngg 1$ flavors and a global U(1) charge. We provide a general definition of the charge in the $(G,Sigma)$ formalism, and compute its universal relation to the i