Symmetry Operators and Gravity

1-Clear bottom up understanding of violations to topologicalness of QFT symmetry operators upon coupling to gravity for large classes of symmetries.
2-Inclusion of many explicit examples.
3-Concise presentation in logic and formatting.

1-Arguments apply only in the setting of continuous invertible n-form (n≠-1) symmetries acting on bosonic fields (with action containing the standard two-derivative kinetic terms)

This paper begins with discussion of topological symmetry operators in quantum field theory. The authors point out that the textbook definition of continuous invertible n-form symmetry operators via Noether currents requires regularization due to contact terms when acting on bosonic fields (with action containing the standard two-derivative kinetic terms). The regularization is discussed within a finite width approximation to the symmetry operator and derived requiring a well-defined zero-width limit - reproducing the original textbook setting. In the finite width approximation the symmetry operator picks up dynamics with kinetic term given by a Nambu-Goto action for the collective coordinates. It is argued that the Nambu-Goto action reproduces the topological symmetry operator in the zero-width limit which freezes out the collective coordinates.

The paper goes on to discuss the zero-width limit in gravitational settings. The metric field now enters the Nambu-Goto action for the collective coordinates and contributes a tadpole term to leading order in a linear approximation. The putative symmetry operator is seen to act as source for a the gravitation field and the infinite zero-width limit then results in a black object. In particular, it does not produce a topological operator.

Arguments are well presented, clear and the referee finds no fault with them. The paper meets the acceptance criteria of Scipost Physics.

Going beyond the scope of the paper, arguments seemingly do not apply to symmetries acting on fermionic fields (with action containing the standard one-derivative kinetic terms) discrete symmetries, -1-form symmetries (either continuous or discrete) or invertible symmetries.

For non-invertible symmetries this is expected, as the linearized approach in this work is not sensitive to spacetime topology change. The referee agrees that in the fermionic setting no regularization of symmetry operators is required, why the arguments starting from equation (18) do not hold in this case is less clear. In the setting of discrete symmetries it would be interesting to extend the authors approach to approximate global symmetries, whenever the discrete group can be thought of as a subgroup of a continuous group, and combine the zero-width limit with a limit localizing back onto the subgroup. -1-form symmetries seem out of reach with the presented methodology.

• Typo, “Goldstone” -> “Goldstone modes” below (21)
• The referee cannot verify footnote [26], instead they find lambda f’=f(1-f). Also, Sqrt(1+x)=1+x/2+… and similarly for Sqrt(det()) as relevant in expansions of the Nambu-Goto action. Consequently the referee can not reproduce numerical factors in equations (19), (20), (21), (22), unnumbered equation after (22), (28), (31), (37), (38), (41), (42), (43), (44).
• Please expand on the technical issues encountered in the fermionic case, as mentioned in the "Summary and Outlook" section

We ask the authors to check their computations. The numerical factors are of no consequence for the conclusions of this paper and the referee sees this paper as fit for publication once the above comments are addressed.

Ask for minor revision

This is a very interesting paper. The fact that there should be no symmetries in a theory of gravity is well established, but the arguments are often somewhat indirect, and involve consistency with black hole physics.

In this paper the authors provide an argument that is more direct: they define a family of operators that regularize the standard symmetry operator (in the sense that the standard operator is obtained as a limiting member of this family), and show that in the presence of gravity the limiting object is not well defined any more.

The paper is clearly preliminary in some respects (a fact that the authors acknowledge), but it makes interesting progress in understanding exactly why symmetries are not allowed in gravity.

The paper is written quite clearly, although I have a couple of suggestions for presentation detailed below.

1- In principle one needs to worry about Jacobian factors introduced by (10). In practice I believe the Jacobian is harmless, but I would ask the authors to state this explicitly (perhaps with a short justification), for the benefit of the reader.

2- I find the discussion around the last (unnumbered) equation in pg. 3 a bit quick. As an illustration, the authors state: "In order to get the expected result...", but they never state what the expected result is. It's not hard to understand what it's meant here, but I would like the authors to rework/expand the arguments around this equation a bit, since it's a key point in the paper and it's explained rather briefly, leaving some important details for the reader to work out.

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