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Gravitational higherform symmetries and the origin of hidden symmetries in KaluzaKlein compactifications
by Carmen GómezFayrén, Tomás Ortín, Matteo Zatti
Submission summary
Authors (as registered SciPost users):  Tomás Ortín 
Submission information  

Preprint Link:  https://arxiv.org/abs/2405.16706v1 (pdf) 
Date submitted:  20240806 19:42 
Submitted by:  Ortín, Tomás 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We show that, in presence of isometries and nontrivial topology, the EinsteinHilbert action is invariant under certain transformations of the metric which are not diffeomorphisms. These transformations are similar to the higherform symmetries of field theories with $p$form fields. In the context of toroidal KaluzaKlein compactifications, we show that these symmetries give rise to some of the ``hidden symmetries'' (dualities) of the dimensionallyreduced theories.
Current status:
Reports on this Submission
Strengths
1 The paper makes an attempt to clarify and render more exact some often sloppily used notion of the higherdimensional origin of certain global symmetries.
Weaknesses
1 According to the discussion in the introduction (after (1.11)): a main result is the new "global symmetry of the $\hat{d}$dimensional action" as stated in the introduction after (1.11) which is then "inherited by the $d$dimensional one”. However, the former one is only derived under the assumption of the existence of a Killing vector. Effectively this is already in the framework of a $d$dimensional action. Constructing a higherdimensional origin of a symmetry action using a Killing vector is essentially nothing but a rewriting of the lowerdimensional structure. It is not clear what can be learned from this.
2 It is not clear to what extent these results can improve the understanding of either, the higher or the lowerdimensional theory. Nor if they allow for any further applications.
Report
This short paper aims at setting up relations between certain hidden symmetries and a proposed gravitational analogue of higherform symmetries. However, the "new invariances" of the action only hold in presence of Killing vector fields. It should be justified why this structure is more than a mere rewriting of the wellknown lowerdimensional symmetries.
Requested changes
various imprecise statements and notation:
1 The use of ‘hidden symmetries’ in title and abstract as well as their discussion in the introduction is rather misleading. The split into two kinds, labelled 1. and 2. in the introduction, mixes up the notion of ‘hidden symmetry’ in the commonly used sense (no obvious higherdimensional origin) with the unrelated issue of invariance of an action vs equations of motion (which only arises in even dimensions). By this definition, the exceptional group E7 in fourdimensional maximal supergravity would fall into class 1 but the exceptional group E6 in fivedimensional maximal supergravity would fall into class 2. Yet, there is no obvious difference regarding the status of the higherdimensional origin of these groups. The paper then restricts to pure gravity theories whose symmetries are rarely described as ‘hidden’ (in compactifications above three dimensions).
2 footnote 3: is the higherdimensional origin of the former “unknown” as stated in the main text, or is there a “standard explanation” which is “not correct”, as stated in the footnote?
3 clarify the notion of $z$ vs ${\underline{z}}$
4 notation $\hat{\epsilon}$ is only introduced in footnote 8, but already used in equation (3.7)
5 equations (3.8a) (3.8b) are sufficient but not necessary for (3.7), why are these stronger equations introduced?
6 text before (3.9) “Thus, we expect the transformation of EH action to be proportional to $d\hat\epsilon$ and, perhaps, to vanish when $\hat\epsilon$ is closed.” Why ‘perhaps’? Would the transformation not necessarily vanish if it is proportional to $d\hat\epsilon=0$?
7 in (3.16), why do the terms have to vanish separately?
8 in (3.19) why are the statements not equivalent?
9 what is meant by “exactly invariant” before (3.22)?
10 what is $\alpha$ in (4.2)? is it defined by this equation? There is no $\alpha$ in (3.8) or the ansatz (4.1).
11 most importantly, the above weaknesses should be addressed: It should be justified why the new structure is more than a mere rewriting (in terms of a Killing vector) of the wellknown lowerdimensional symmetries.
Recommendation
Ask for major revision
Strengths
1) Good presentation
2) Correct results
3) Important observations
4) Wide scope of applications
Weaknesses
None.
Report
The paper shows that in addition to diffeomorphisms, the EinsteinHilbert action is invariant under a previously unknown global symmetry (3.21).
It is not a standard symmetry as it holds for a subset of configurations with isometries and nontrivial topology, but since operationaly no specific details on the backgrounds are required, for all practical purposes it works as an offshell symmetry prinicple at the level of the action. It is generated by a 1form whose contraction with the killing vector is constant, and is closed but no exact (3.8), and for this reason the authors refer to it as a higherform gravitational symmetry. Afetr a KaluzaKlein reduction, it descends to a known symmetry in the lower diemnsional action.
A new invariance of the EinsteinHilbert action is definitely an important result. The paper is clearly written and well organized, the computations are certainly correct, the results are intriguing and thought provoking and the scope of applications apears to be wide. I recommend the paper for publication as it stands.
As a small suggestion for future work, it could be interesting to explore the closure relations with diffeomorphisms, to understand the underlying algebraic structure, and see if they form a complete set of symmetries.
Requested changes
Is the second equality of Eq. (2.8b) a typo?
Recommendation
Publish (meets expectations and criteria for this Journal)
Author: Tomás Ortín on 20240912 [id 4776]
(in reply to Report 1 on 20240906)The second equality of Eq. (2.8b) is correct. It is the exterior product of a (p+1)form and a (dp2)form, which gives a (d1)form, the dual of a standard Noether current (we always use the dual forms, that can be directly integrated over (d1)dimensional hypersurfaces). On the other hand, this equation follows directly from (2.3b) and (2.6).