Dynamical Phases of Higher Dimensional Floquet CFTs

The mathematical analysis presented in the paper is mathematically sound, and it extends known results on driven (1+1)d CFTs, which is interesting both theoretically and experimentally.

The manuscript currently reads more like a collection of notes and results than a cohesive work with clear physical motivation and a well-defined goal. I worry that the results may not reach the community effectively in this form.

The paper analyses higher-dimensional CFTs out of equilibrium, in particular systems evolved under exactly solvable Floquet drives. Similar drives have already been studied in detail for (1+1)d CFTs, and the authors extend these results to higher-dimensional CFTs using quaternionic representations. They then connect various dynamical phases of such drives with the presence or absence of Killing horizons in AdS$_d$, as a putative dual to a CFT$_d$.

The mathematical analysis presented in the paper is mathematically sound, and it extends known results on driven (1+1)d CFTs, which is interesting both theoretically and experimentally. However, I have several comments regarding the conceptual structure and presentation of the manuscript.

1) The introduction is hard to follow: it is quite long and presents several technical details on both the broader field of driven CFTs and the manuscript itself, leaving little space for the underlying physics. Likewise, Section 2 delves immediately into the technical development. A softer start would benefit accessibility.

2) In Section 3.5, the Authors state that $\beta_0 = 1$ "leads to linear growth or decay of correlators." Perhaps they meant 'polynomial' behavior of correlators? I find it counterintuitive for correlators to behave linearly.

3) In Section 4.2.2, the Authors find a “phase with damped oscillatory response”. Could they clarify the physics behind this phase? In these systems one typically finds heating (energy/entropy growth) or non-heating (oscillations), with a critical phase in between. Damped oscillations are usually a sign of non-unitary evolution; however, the evolution here appears unitary. What, then, is the origin of the damping?

4) Related to the previous point, what is the main difference between the higher-dimensional case and (1+1)-d CFTs? While the group structure differs (which I agree with), the physical differences were unclear to me. It would help to highlight more explicitly what changes in higher dimensions.

5) Section 6 on holography and AdS space feels somewhat detached. The earlier CFT methods rely only on conformal symmetry and thus apply more generally than to holographic CFTs alone. Please clarify what the holographic perspective adds, particularly to the central goal of understanding higher-dimensional driven CFTs.

6) In the Discussion the Authors write that the AdS interpretation “reveals heating as a generalised Unruh effect.” Given that the methods are not restricted to holographic CFTs, wouldn’t this then hold for any higher-dimensional CFT, not just holographic ones? Why is the geometric picture needed to interpret it as a generalised Unruh effect?

Furthermore, the manuscript presents several typos.

i) In eq. (1) there is an additional "dt".

ii) In eq. (4.28), perhaps the third Hamiltonian should be H_2(\beta_2)? And the time cycles should be $nT < t < nT + T_0$ , $ nT + T_0 < t < nT + T_0 + T_1$ , $ nT + T_0 + T_1 < t < (n+1)T$ ? In the present form they do not make sense to me.

iii) Above eq. (4.39) there are two colons.

iv) Some "Eq. ..." are without parenthesis, while some are with parenthesis but without space between "Eq." and the reference. Please standardise.

v) Above Section 6.3, "plug-in" should be written "plug in".

Overall, the manuscript currently reads more like a collection of notes and results than a cohesive work with clear physical motivation and a well-defined goal. I worry that the results may not reach the community effectively in this form.

Ask for major revision

This work investigates non-equilibrium dynamical phases of driven conformal field theories (CFTs) in higher dimensions, extending the driving protocols proposed in the authors’ earlier work [1] where the underlying algebra is SU(1,1). Analytically solvable setups in non-equilibrium quantum systems are very rare, and the framework developed here provides new insights into possible dynamical phases of driven higher-dimensional CFTs. The topic is timely, the analysis is solid and clearly presented, and I expect this work will generate broad interest within both the conformal field theory and non-equilibrium physics communities.

I recommend publication after the following issues are addressed:

-- Choice of parameter space. Throughout the paper, the dynamical phase diagrams are presented in terms of the parameters T1 and T2. This choice may not be the most natural. In (1+1)d driven CFTs, an additional parameter "L" characterizes the length scale of the deformed Hamiltonian, and the phase diagram is best expressed in terms of the dimensionless ratios T1/L and T2/L. For higher-dimensional driven CFTs, is there an analogous way to define a dimensionless parameter space? Clarifying this point would strengthen the physical interpretation of the results.

-- Physical intuition and initial states. The current analysis considers the ground state as the initial state, for which the time evolution is relatively trivial. If one instead starts from other states, such as primary excited states or thermal states, the dynamics would become richer. What qualitative features of physical observables (e.g., entanglement entropy evolution) might be expected in different phases under these alternative initial conditions? Adding some intuitive discussion here would broaden the scope of the work.

-- Geometric or group-theoretic perspective. The identification of different dynamical phases relies on analyzing the eigenvalues of matrices at each step of the driving protocol. Is there a possible geometric understanding of this classification? In (1+1)d, for instance, different types of Möbius transformations correspond to distinct features in the time evolution of operators. A comment on whether a similar geometric or group-theoretic picture exists in higher dimensions would be very valuable.

-- Some refereces should be cited at the appropriate positions. For example, in the discussion section, it is mentioned that the dynamics starting from a thermal state has been done in 1+1 dimensions. Relevant references added here will be helpful for curious readers. There are also some other places with similar issues.

-- There are some typos throughout the work. The authors need to go through the writing carefully.

Ask for minor revision