Higher-form symmetry and chiral transport in real-time Abelian lattice gauge theory

1. The first classical lattice simulations of scalar QED were presented to test the validity of MHD and chiral MHD, which are formulated using the action principle and the EFT method.

2 Clarifying the conceptual issues related to the resistivity and conductivity of plasmas with dynamical gauge fields.

3. The EFT formulation is well elaborated.

1. The simulation based on scalar QED is illustrative. The extracted resistivity might not be relevant for the applications in cosmology and heavy-ion collisions.

I would recommend publishing this paper in SciPost.

1. I suggest that the authors address the following question. In usual fluctuating hydrodynamics, the fluctuations lead to the long-time hydrodynamic tail (see, e.g., https://arxiv.org/abs/hep-th/0303010v1). What is the analog of the hydrodynamic tail in MHD (Chiral MHD)? Can one observe it from numerical simulations?

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1- Explicit and independent checks regarding the lack of meaning of conductivity in strong coupling regimes.
2- Application of the developed formalism to chiral transport.
3- Clarity in the explanations.

1- In presenting the numerical results, it is not specified the units of the displayed dimensionful quantities nor are they recast into dimensionless quantities. This hinders the physical interpretation of the results and the direct comparison with results from the literature.

The authors employ the magnetohydrodynamics (MHD) formalism to study the effective evolution of electrodynamics coupled to charged matter at finite temperature, further employing lattice techniques to assess the microscopic properties of the system and compute the transport coefficients. Very interestingly, it is argued and checked that the notion of electric conductivity loses its meaning in the strong coupling regime, or more generically when a certain separation of scales is not present in the system. Furthermore, they apply their formalism to a chiral system and compute the relaxation rate for the axial charge arising from the ABJ anomaly. Overall, the paper provides interesting and innovative insight and deserves publication provided the authors address the concerns presented below.

1- In page 8 it is stated that regulating the classical field theory in thermal equilibrium via discretising space or via QFT are "obviously inequivalent" while "They do however belong to the same universality class". Both statements are in tension since it is not clear what is meant in this context by belonging to the same universality class. Indeed the authors themselves point out a non-trivial difference between both approaches in Appendix D by computing the thermal mass of the scalar field and obtain different scalings with temperature. In addition, they also discuss that the separation of scales required to have a well defined conductivity is "not present in the lattice simulations presented in this work" (page 25), while it should be present in the weak coupling QFT approach. I believe that the authors should review and explicitly state the limitations in comparing both theories.

2- The authors compare their results to the conductivity computed in [48] and report an O(10) discrepancy. However, in [48] employs QFT techniques as opposed to the lattice regularisation. Once again, both approaches do not seem to be directly comparable, as the lattice regularisation lacks of the separation of scales required to defined the conductivity as explained in the submitted paper. This issue should be clarified in accordance with the first proposed change above.

3- In page 8, it is stated that "the universal dynamics that we will discuss does not depend on the precise form of the potential". I would suggest that the authors are more explicit with what the mean by "universal dynamics" in this context and provide some form of justification as to why the form of the potential is apparently not relevant.

4-In page 4 it is stated that the chiral "decay rate is controlled by the resistivity". Indeed this is shown in the paper in the context of abelian gauge theories. It is a genuine question whether a similar statement could be made for the topologically induced axial charge dissipation stemming from the axial anomaly in non-abelian theories.

5 - A major difficulty in interpreting the results of the paper comes from the fact that dimensionful quantities are presented without units. This happens in the figures and more importantly in the estimation of resistivity in page 12 and the computation of the chiral decay rate in page 22 (fig. 6). This complicates the direct comparison with the results of [48] as well as with the chiral decay rate as computed for non-abelian theories existing in the literature. I kindly ask the authors to remedy this situation.

6- The boundary conditions used in the lattice are relevant for the existence of absence of non-zero fluxes. In page 30 the authors imply that they use twisted boundary conditions in certain cases while in page 10 (bottom of Sec 2) they suggest the use of periodic boundary conditions. It would be useful to have a clear statement as to when the authors choose either of the boundary conditions.

7-Typos: In page 11, Sec. III A, there is a reference to the upper left hand side of Fig.5 which seems to be wrong (there is a single figure in Fig 5, no notion of upper lower left or right). The description does not match what is shown in Fig. 5 either. The same happens in page 12. I would kindly ask the authors to check that all the hyperlinks included refer to the proper object.
Page 5 line 1 "an conventional"
Page 23 "showed the the response"

Ask for minor revision