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Gravitational higher-form symmetries and the origin of hidden symmetries in Kaluza-Klein compactifications
by Carmen Gómez-Fayrén, Tomás Ortín, Matteo Zatti
Submission summary
| Authors (as registered SciPost users): | Tomás Ortín |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202411_00011v1 (pdf) |
| Date submitted: | 2024-11-06 00:22 |
| 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 non-trivial topology, the Einstein--Hilbert action is invariant under certain transformations of the metric which are not diffeomorphisms. These transformations are similar to the higher-form symmetries of field theories with $p$-form fields. In the context of toroidal Kaluza--Klein compactifications, we show that these symmetries give rise to some of the ``hidden symmetries'' (dualities) of the dimensionally-reduced theories.
Author comments upon resubmission
Dear referee,
These are our answers to your questions and requests, following the same
numeration.
1.- In this paper we show that the higher-dimensional origin of some
well-known symmetries was, actually, unknown. In most of the literature, it
was assumed that those symmetries were diffeomorphisms in higher dimensions,
but, as we show, this is not correct. Then, the higher-dimensional origin of
those symmetries was unknown and they truly deserve the name of hidden
symmetries''. In this sense on can call most of the duality symmetries of
lower-dimensional supergravities hidden symmetries''. Some of them involve
electric-magnetic transformations and are only symmetries of the equations
of motion and some of them don't and are symmetries of the action. This
distinction is very clear and well known. We do not apply this distinction
to whole groups. In general, duality groups contain transformations of both
kinds. The SL(2,R) duality group of the t3 model contains symmetries of
both kinds, for instance and, as we have said, the higher-dimensional origin
of both kinds was unkown, strictly speaking.
Nevertheless, we have rephrased the first paragraphs of the introduction in
order to make this more clear.
Concerning pure gravity theories, the common understanding is that they do
not have any global symmetries. They are only invariant under
diffeomorphisms. If, after compactification, the theory exhibits some global
symmetries, it is fair to say that their higher-dimensional origin was
unknown. We focus on those theories for the sake of simplicity but we are
working on the inclusion of matter.
2.- If the standard explanation is wrong, we would say that there is no
valid explanation or that the explanation is unknown. We have moved this
footnot to the text, to the end of the paragraph, and we have rewritten it
tom make our point more clear.
3.- In footnote 4 (in the revised version) we state that we use the notation
and conventions of Ref.[11]. That notation is explained there, but we have
added a sentence explaning it.
4.-We have moved the footnote to the sentence above Eq.~(3.7).
5.- They are introduced because they provide a simple solution of Eq.~(3.7)
which proves to be enough for our purposes.
6.- We have removed the perhaps''. It was just a rhetorical recourse.
7.- We do not know any other way to make (3.16) vanish. Actually, we do not
think there is any other way to make it vanish, but proving it can be very
difficult and time consuming and, in the end, useless, since the solution
that we find is quite satisfactory.
8.- We are not saying anywhere that they are not equvalent. We just say that
one implies the other, which is completely correct. The other implication
may be true or not: it implies that the function is locally constant, but it
could just be picewise constant, unless we state that the derivative is zero
everywhere. We feel that it is totally unnecessary to discuss this point.
9.- Symmetries can leave the action exactly invariant or invariant up to
total derivatives. This one lease it exactly invariant.
10.- α is just another constant. We have explained this in the
paragraph that follows Eq.~(4.2), which has also been modified to explain
its meaning.
11.- First of all, we would like to explain that mere rewritings'' are very
important in Physics because, often, they lead to deeper understandings and
further developments. One could argue that Special Relativity in terms of
the Minkowski metric and tensors is a mere rewriting'' of the original
equations published by Einstein but one cannot argue that this rewriting led
to many fruitful insights and developments. The same can be said about
Lagrangian and Hamiltonian mechanics, for instance.
Our results give a new and correct explanation (we believe the referee
agrees with us in this point) of the higher-dimensional origin of some
well-known symmetries. We think that the wrong identification of the origin
of those symmetries prevalent in the literature is due, precisely, to a
mere'' (but unprecise and careless) rewriting'' of those symmetries in
higher dimensions. We do not simply rewrite the action of the symmetries on
the higher-dimensional fields. We show how these symmetries arise in the
higher-dimensional theory when the topology is non-trivial. We believe that
further developments may follow from this approach.
As a matter of fact, in this direction, we have to mention that we have
shown in recent and yet unpublished work that the Killing condition is not
necessary and that there are weaker conditions which still ensure the
existence of a global symmetry. We think that this shows quite convincingly
that we are not merely rewriting'' known results.
Yours, sincerely,
Tomas Ortin
These are our answers to your questions and requests, following the same
numeration.
1.- In this paper we show that the higher-dimensional origin of some
well-known symmetries was, actually, unknown. In most of the literature, it
was assumed that those symmetries were diffeomorphisms in higher dimensions,
but, as we show, this is not correct. Then, the higher-dimensional origin of
those symmetries was unknown and they truly deserve the name of hidden
symmetries''. In this sense on can call most of the duality symmetries of
lower-dimensional supergravities hidden symmetries''. Some of them involve
electric-magnetic transformations and are only symmetries of the equations
of motion and some of them don't and are symmetries of the action. This
distinction is very clear and well known. We do not apply this distinction
to whole groups. In general, duality groups contain transformations of both
kinds. The SL(2,R) duality group of the t3 model contains symmetries of
both kinds, for instance and, as we have said, the higher-dimensional origin
of both kinds was unkown, strictly speaking.
Nevertheless, we have rephrased the first paragraphs of the introduction in
order to make this more clear.
Concerning pure gravity theories, the common understanding is that they do
not have any global symmetries. They are only invariant under
diffeomorphisms. If, after compactification, the theory exhibits some global
symmetries, it is fair to say that their higher-dimensional origin was
unknown. We focus on those theories for the sake of simplicity but we are
working on the inclusion of matter.
2.- If the standard explanation is wrong, we would say that there is no
valid explanation or that the explanation is unknown. We have moved this
footnot to the text, to the end of the paragraph, and we have rewritten it
tom make our point more clear.
3.- In footnote 4 (in the revised version) we state that we use the notation
and conventions of Ref.[11]. That notation is explained there, but we have
added a sentence explaning it.
4.-We have moved the footnote to the sentence above Eq.~(3.7).
5.- They are introduced because they provide a simple solution of Eq.~(3.7)
which proves to be enough for our purposes.
6.- We have removed the perhaps''. It was just a rhetorical recourse.
7.- We do not know any other way to make (3.16) vanish. Actually, we do not
think there is any other way to make it vanish, but proving it can be very
difficult and time consuming and, in the end, useless, since the solution
that we find is quite satisfactory.
8.- We are not saying anywhere that they are not equvalent. We just say that
one implies the other, which is completely correct. The other implication
may be true or not: it implies that the function is locally constant, but it
could just be picewise constant, unless we state that the derivative is zero
everywhere. We feel that it is totally unnecessary to discuss this point.
9.- Symmetries can leave the action exactly invariant or invariant up to
total derivatives. This one lease it exactly invariant.
10.- α is just another constant. We have explained this in the
paragraph that follows Eq.~(4.2), which has also been modified to explain
its meaning.
11.- First of all, we would like to explain that mere rewritings'' are very
important in Physics because, often, they lead to deeper understandings and
further developments. One could argue that Special Relativity in terms of
the Minkowski metric and tensors is a mere rewriting'' of the original
equations published by Einstein but one cannot argue that this rewriting led
to many fruitful insights and developments. The same can be said about
Lagrangian and Hamiltonian mechanics, for instance.
Our results give a new and correct explanation (we believe the referee
agrees with us in this point) of the higher-dimensional origin of some
well-known symmetries. We think that the wrong identification of the origin
of those symmetries prevalent in the literature is due, precisely, to a
mere'' (but unprecise and careless) rewriting'' of those symmetries in
higher dimensions. We do not simply rewrite the action of the symmetries on
the higher-dimensional fields. We show how these symmetries arise in the
higher-dimensional theory when the topology is non-trivial. We believe that
further developments may follow from this approach.
As a matter of fact, in this direction, we have to mention that we have
shown in recent and yet unpublished work that the Killing condition is not
necessary and that there are weaker conditions which still ensure the
existence of a global symmetry. We think that this shows quite convincingly
that we are not merely rewriting'' known results.
Yours, sincerely,
Tomas Ortin
List of changes
The changes are listed in the comments.
Current status:
In refereeing