REVIEW
Dualisation of Dualities, I
High Energy Physics - Theory · Submitted 15 Oct 1997 · 38 references
Pith's verdict: A careful, explicit account of how the choice of dualisation changes the global symmetry of toroidally reduced maximal supergravity; the general root-deletion rule is stated more broadly than it is proved, but the named cases are done properly.
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Referee report
The paper
The paper claims
Which fields you dualise decides which rigid symmetry group a toroidally compactified supergravity has, and the answer can differ even at the level of the equations of motion.
The reading
What carries the argument
The load-bearing construction is a Borel-type (triangular) parameterisation of the coset, $V=V_1V_2V_3cdots$, with the dilatons exponentiating the Cartan generators and the axions exponentiating the positive-root generators in a fixed anti-lexical order. Computing $dV,V^{-1}$ then produces, term by term, exactly the Kaluza-Klein-modified field strengths of the dimensionally reduced Lagrangian, with the dilaton vectors of the axions playing the role of the positive roots. This makes the global symmetry manifest as the invariance of $mathrm{tr}(partial M^{-1}partial M)$ with $M=V^{#}V$ under constant right multiplication, and it makes the dualisation dictionary exact: a scalar that is inversely dualised to a $(D-2)$-form has its root generator contracted and then truncated, which forces the generators at the next level down to commute, turning the algebra into a semidirect product. A second piece of machinery is the 'doubled' Lagrangian for even $D$, in which an extra set of $(D/2)$-form potentials is introduced so that the full duality group acts on a Lagrangian, with a twisted self-duality constraint $H=Omega M{*}H$ imposed afterwards to recover the original theory.
Extended reading notes
Core claim
The central claim is that the global (rigid) symmetry realised on the bosonic sector of a toroidally compactified maximal supergravity is a function of the dualisation choices, and that this remains true for the symmetry of the field equations and Bianchi identities, not merely of the Lagrangian. When all $(D-2)$-forms are dualised to scalars so that the scalar count is maximal (the 'E series'), the scalar Lagrangian is a $\sigma$ model on the symmetric space $K(E_{11-D})backslash E_{11-D}$, with $K$ the maximal compact subgroup; without dualisation one instead gets $GL(11-D,mathbb{R})ltimes mathbb{R}^{q}$ with $q=tfrac{1}{6}(11-D)(10-D)(9-D)$; dualising only the Ramond-Ramond fields gives $O(10-D,10-D)ltimes mathbb{R}^{s}$ for $Dle 6$; and the direct reductions of type IIB give $SL(2,mathbb{R})times GL(9-D,mathbb{R})ltimes mathbb{R}^{r}$. The unifying statement is that the non-E series are obtained from the E series by contracting and truncating the positive-root generators whose axions were inversely dualised, so that the residual symmetry is a homogeneous space $K(G)backslash Gltimes mathbb{R}^{s}$ rather than a symmetric space. The paper also shows that the dimensions of the maximal abelian shift symmetries can be read off as the dimensions of maximal abelian subalgebras of the positive-root system, and tabulates them for each series.
Load-bearing premise
The load-bearing premise is that every choice of dualisation can be read as deleting a commuting set of positive roots from $E_{11-D}$ and contracting the rest, a prescription the paper verifies for its four named series and a handful of examples but does not prove for arbitrary subsets.
Editorial extensions
If this is right
- There is no single 'the' maximal supergravity in a given dimension: inequivalent theories, related by non-local field redefinitions, carry different rigid symmetry groups, and the group must be quoted together with the field content.
- The fully dualised E series provides a generating theory from which the others follow by root deletion, so the symmetry of any partial-dualisation variant can be read off a Dynkin diagram rather than recomputed.
- In $D=6$ the $O(5,5)$ symmetry of the scalar sector cannot be realised on the full theory, even on shell, unless the 3-form potential is dualised to a vector, because bare Kaluza-Klein potentials otherwise appear unavoidably in the Bianchi identities or in the trilinear terms.
- The maximal number of commuting axion shift symmetries in each variant is fixed by the dimension of a maximal abelian subalgebra of the positive-root system, giving the table $1,2,3,6,10,16,27,36$ for full dualisation and $1,2,3,4,10,20,35,56$ for no dualisation.
- For even dimensions the doubled-Lagrangian formulation supplies a Lagrangian, not just equations of motion, on which the full duality group acts, with the original theory recovered by a consistent truncation.
Reading between the lines
- If the root-deletion picture is right, the classification of allowed symmetry groups should be readable purely from gradings of the $E_{11-D}$ Dynkin diagram, which suggests a finite combinatorial classification of all dualisation variants rather than a case-by-case analysis.
- The paper notes that the Ramond-Ramond-only groups $O(6,6)ltimesmathbb{R}^{32}$ and $O(7,7)ltimesmathbb{R}^{64}$ are not subgroups of $E_7$ and $E_8$, so the E series is not an upper bound in any naive sense; a useful follow-up would be a partial order on the variants by subgroup inclusion rather than by root counting.
- Because the commuting-shift count equals the dimension of a maximal abelian subalgebra, the same numbers should govern how many axions can simultaneously be given a Scherk-Schwarz-type linear dependence on a compact coordinate, which makes the table directly checkable against the spectrum of any resulting massive deformation.
- The observation that bare potentials, not field strengths, can be forced into the equations of motion suggests a sharper criterion for when a duality survives dualisation: one should ask whether the relevant multiplet can be assembled from potentials of a single degree, not whether the field strengths fill a representation.
Original abstract
We analyse the global (rigid) symmetries that are realised on the bosonic fields of the various supergravity actions obtained from eleven-dimensional supergravity by toroidal compactification followed by the dualisation of some subset of fields. In particular, we show how the global symmetries of the action can be affected by the choice of this subset. This phenomenon occurs even with the global symmetries of the equations of motion. A striking regularity is exhibited by the series of theories obtained respectively without any dualisation, with the dualisation of only the Ramond-Ramond fields of the type IIA theory, with full dualisation to lowest degree forms, and finally for certain inverse dualisations (increasing the degrees of some forms) to give the type IIB series. These theories may be called the GL_A, D, E and GL_B series respectively. It turns out that the scalar Lagrangians of the E series are sigma models on the symmetric spaces K(E_{11-D})\backslash E_{11-D} (where K(G) is the maximal compact subgroup of G) and the other three series lead to models on homogeneous spaces K(G) \backslash G\semi \R^s. These can be understood from the E series in terms of the deletion of positive roots associated with the dualised scalars, which implies a group contraction. We also propose a constrained Lagrangian version of the even dimensional theories exhibiting the full duality symmetry and begin a systematic analysis of abelian duality subalgebras.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We analyse the global (rigid) symmetries that are realised on the bosonic fields of the various supergravity actions obtained from eleven-dimensional supergravity by toroidal compactification followed by the dualisation of some subset of fields. In particular, we show how the global symmetries of the action can be affected by the choice of this subset. This phenomenon occurs even with the global symmetries of the equations of motion. A striking regularity is exhibited by the series of theories obtained respectively without any dualisation, with the dualisation of only the Ramond-Ramond fields of the type IIA theory, with full dualisation to lowest degree forms, and finally for certain inverse dualisations (increasing the degrees of some forms) to give the type IIB series. These theories may be called the GL_A, D, E and GL_B series respectively. It turns out that the scalar Lagrangians of the E series are sigma models on the symmetric spaces K(E_{11-D})\backslash E_{11-D} (where K(G) is the maximal compact subgroup of G) and the other three series lead to models on homogeneous spaces K(G) \backslash G\semi \R^s. These can be understood from the E series in terms of the deletion of positive roots associated with the dualised scalars, which implies a group contraction. We also propose a constrained Lagrangian version of the even dimensional theories exhibiting the full duality symmetry and begin a systematic analysis of abelian duality subalgebras.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We analyse the global (rigid) symmetries that are realised on the bosonic fields of the various supergravity actions obtained from eleven-dimensional supergravity by toroidal compactification followed by the dualisation of some subset of fields. | contribution 1.00 | 0.93 | 0.08 | asserted 0.94 |
| s1 | In particular, we show how the global symmetries of the action can be affected by the choice of this subset. | result 0.88 | 0.93 | 0.30 | asserted 1.00 |
| s2 | This phenomenon occurs even with the global symmetries of the equations of motion. | result 0.99 | 0.71 | 0.11 | asserted 1.00 |
| s3 | A striking regularity is exhibited by the series of theories obtained respectively without any dualisation, with the dualisation of only the Ramond-Ramond fields of the type IIA theory, with full dualisation to lowest degree forms, and finally for certain inverse dualisations (increasing the degrees of some forms) to give the type IIB series. | result 0.97 | 0.84 | 0.15 | asserted 1.00 |
| s4 | These theories may be called the GL_A, D, E and GL_B series respectively. | result 0.44 | 0.72 | 0.07 | hedged possibility 0.56 |
| s5 | It turns out that the scalar Lagrangians of the E series are sigma models on the symmetric spaces K(E_{11-D})\backslash E_{11-D} (where K(G) is the maximal compact subgroup of G) and the other three series lead to models on homogeneous spaces K(G) \backslash G\semi \R^s. | result 0.99 | 0.87 | 0.30 | asserted 1.00 |
| s6 | These can be understood from the E series in terms of the deletion of positive roots associated with the dualised scalars, which implies a group contraction. | interpretation 0.98 | 0.77 | 0.13 | asserted 0.85 |
| s7 | We also propose a constrained Lagrangian version of the even dimensional theories exhibiting the full duality symmetry and begin a systematic analysis of abelian duality subalgebras. | contribution 0.90 | 0.90 | 0.12 | asserted 0.70 |
What was examined, and what was not
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References
How later papers used it
97 later papers cite it
3 later papers by other authors are listed below: 3 used its method or tool.
Uses its method or tool 3 of 8 papers with this label
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Followed this paper to analyse the toroidal reduction down to three dimensions.
Aspects of electric-magnetic dualities in maximal supergravity (2018) · 0.93 · paraphrased: the licence does not allow quoting
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It is convenient here to use the “doubled Lagrangian” formalism in [ref] , which is valid for even dimensions D=2k.
The role of Sp(2n) duality in quantum theory (2024) · GZ duality in even and odd D/2=k : Sp(2n) and SO(n,n) · 1.00
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Followed this paper's dualisation prescription.
Polyvector deformations in eleven-dimensional supergravity (2020) · 0.96 · paraphrased: the licence does not allow quoting
How the literature uses it
88 arXiv papers cite this in 164 sentences
Papers citing it · 99 sentences without a confident label
Uses its method or tool 7 papers
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Using the information given in [ref] as well as the explicit formulae for dimensional reduction given in [ref] , we can get the explicit embedding of the STU model in $=8$ supergravity.
$E_{7(7)}$ invariant non-extremal entropy · Embedding of the STU model · 0.83
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We start by analysing the toroidal reduction of [ref] down to three dimensions, following [ref] .
Aspects of electric-magnetic dualities in maximal supergravity · Part [ref]: Six-dimensional exotic fields · 0.93
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It is convenient here to use the “doubled Lagrangian” formalism in [ref] , which is valid for even dimensions D=2k.
The role of Sp(2n) duality in quantum theory · GZ duality in even and odd D/2=k : Sp(2n) and SO(n,n) · 1.00
Compares with it 1 paper
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These correspond exactly to the root vectors $b⃗_i,i+1$ and $a⃗_123$ as they appear in the analysis of [ref] , except for the additional factor of $1√(2)$ needed to compensate for the fact that the aforementioned reference has an additional factor of $2$ in the Killing form.
Spacelike Singularities and Hidden Symmetries of Gravity · Starting at the bottom – Overextensions of finite-dimensional Lie algebras · 0.96
Cites it as support 1 paper
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Thus, given the evidence for the relation between dualisation of fields before and after reduction [ref] , from an eleven-dimensional perspective this potential is indeed to be interpreted as a dual gravity field.
E8 duality and dual gravity · Non-linear realisation of $E_8$ · 1.00
Cites it as background 46 papers
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The doubled formalism was described by Hull in [ref] and drew from previous duality covariant models [ref] .
Aspects of M-Theory Brane Interactions and String Theory Symmetries · Introducing the Formalism · 0.93
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(This is nicely reviewed in [ref] ; see also [ref] and appendix [ref].) As we saw in the previous chapter, the group that appears in maximal supergravity is not [ref] but the much larger $E_n(n)$, of which [ref] is a parabolic subgroup.
Aspects of electric-magnetic dualities in maximal supergravity · Infinite-dimensional symmetries · 0.96
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On top of that, even dimensional supergravities have a Gaillard-Zumino (GZ) electro-magnetic duality symmetry proposed in [ref] in 4D and in [ref] in 6D and studied in general in [ref] and in [ref] .
The role of Sp(2n) duality in quantum theory · Introduction · 0.83
Discussion
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