REVIEW
The O_{D,D} Geometry of String Theory
High Energy Physics - Theory · Submitted 27 Mar 2013 · 65 references
Pith's verdict: A real teleparallel derivation of the DFT action — the Scherk-Schwarz extra term drops out for free — but the uniqueness claim in §4.4 is asserted, not shown.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
The double field theory action is rebuilt from the torsion of a flat connection, with curvature playing no role.
The reading
What carries the argument
The load-bearing object is the Weitzenbock connection, a flat metric-compatible connection familiar from teleparallel gravity, here lifted to the doubled space. Its curvature vanishes identically, so no generalised Riemann tensor can source the action; instead the generalised torsion $tau^{a}{}_{bc}$ carries all the information. Two further pieces make the argument work. The first is the identity $Y^{ad}{}_{ce} Gamma^{e}{}_{bd} = - Gamma^{a}{}_{bc}$, which holds for any connection preserving $eta_{ab}$ and reduces the space of independent torsion-squared contractions to four. The second is the covariant derivative of the doubled dilaton, $nabla_a d = partial_a d + (1/2) Gamma^{c}{}_{ca}$, which is weakly a covector even though $d$ itself is not a scalar. Local $H$-invariance is the filter: the connection and its torsion are not $H$-invariant, so the authors look for $H$-invariant combinations of the torsion-squared and dilaton-derivative scalars and report that only one exists.
Extended reading notes
Core claim
The claim is that the dynamics of double field theory live in torsion, not curvature. The authors adopt the Weitzenbock connection $Gamma^{a}{}_{bc} = e^{mu a} partial_b e_{mu c}$, built from the generalised vielbein, and verify three properties: it is metric compatible, $nabla_a M_{bc} = 0$; it preserves the $O_{D,D}$ structure, $nabla_a eta_{bc} = 0$; and its Riemann tensor vanishes, so the usual curvature route to an action is closed off. What survives is the generalised torsion $tau^{a}{}_{bc} = T^{a}{}_{bc} + Y^{ad}{}_{ce} Gamma^{e}{}_{db}$, which is only weakly a generalised tensor, and the covariant derivative of the doubled dilaton. The authors then state that requiring invariance under the local $H = O_D imes O_D$ symmetry singles out one combination of the four independent torsion-squared scalars and the dilaton terms, giving a Lagrangian that reproduces the standard double field theory action. The same construction survives a generalised Scherk-Schwarz reduction, where the reduced torsion and the Scherk-Schwarz gaugings $f_{BC}{}^{A}$ enter on the same footing, and the gauged supergravity potential follows.
Load-bearing premise
The paper asserts, without showing an enumeration or the transformation algebra, that only one combination of the four torsion-squared scalars and the dilaton-derivative scalars is invariant under local $H$ transformations; if a second independent invariant exists, the action is no longer fixed by the construction and the overall normalisation is left to matching the literature.
Editorial extensions
If this is right
- The double field theory action acquires a derivation rather than a verification: it is the unique locally $H$-invariant scalar built from the torsion of a flat, metric-compatible connection.
- The term $(1/2) eta^{ab} eta^{cd} partial_a e^{mu}{}_{c} partial_b e^{mu}{}_{d}$, previously inserted by hand to make Scherk-Schwarz reductions reproduce gauged supergravity, appears automatically in the new Lagrangian.
- Under a generalised Scherk-Schwarz reduction the reduced torsion and the gaugings $f_{BC}{}^{A}$ appear on the same footing, giving a structural reason why the gauged action takes the form it does.
- Because the construction uses only the vielbein, derivatives, and the $Y$-tensor, the authors argue it should carry over to the exceptional $E_d$ geometries relevant to M-theory.
- The framework suggests a route to higher-derivative corrections, since terms quartic in the torsion are the natural next objects to write down.
Reading between the lines
- If the uniqueness of the $H$-invariant combination holds, the normalisation of the action is fixed by the same argument rather than by matching to the literature, which would make the construction predictive rather than reconstructive.
- The teleparallel reading suggests a sharper statement than the paper makes: ordinary Riemannian geometry would be what the doubled geometry looks like after a section is chosen, so curvature is an emergent, frame-dependent notion rather than a fundamental one.
- The appearance of the gaugings alongside the reduced torsion hints that non-geometric fluxes and Scherk-Schwarz twists are the same kind of object viewed from different frames, which is testable by checking whether the torsion-squared invariants reproduce the known flux potentials of gauged supergravity without extra input.
- A natural check the paper does not perform: whether the torsion-squared invariants, evaluated on a generic doubled background, reproduce the known $alpha'$ corrections to the string effective action, which would test the framework beyond two derivatives.
Original abstract
We construct an action for double field theory using a metric connection that is compatible with both the generalised metric and the O_{D,D} structure. The connection is simultaneously torsionful and flat. Using this connection one may construct a proper covariant derivative for double field theory. We then write the doubled action in terms of the generalised torsion of this connection. This action then exactly reproduces that required for doubled field theory and gauged supergravity.
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 construct an action for double field theory using a metric connection that is compatible with both the generalised metric and the O_{D,D} structure. The connection is simultaneously torsionful and flat. Using this connection one may construct a proper covariant derivative for double field theory. We then write the doubled action in terms of the generalised torsion of this connection. This action then exactly reproduces that required for doubled field theory and gauged supergravity.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We construct an action for double field theory using a metric connection that is compatible with both the generalised metric and the O_{D,D} structure. | contribution 0.98 | 0.95 | 0.19 | asserted 1.00 |
| s1 | The connection is simultaneously torsionful and flat. | result 0.79 | 0.78 | 0.08 | asserted 1.00 |
| s2 | Using this connection one may construct a proper covariant derivative for double field theory. | contribution 0.39 | 0.74 | 0.14 | hedged possibility 0.99 |
| s3 | We then write the doubled action in terms of the generalised torsion of this connection. | method 0.58 | 0.93 | 0.12 | asserted 0.99 |
| s4 | This action then exactly reproduces that required for doubled field theory and gauged supergravity. | result 0.99 | 0.87 | 0.65 | asserted 1.00 |
What was examined, and what was not
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- references and citing sentences
- relations to other papers
- P abstract
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- R cross-paper relations
References
How the literature uses it
50 arXiv papers cite this in 88 sentences
Papers citing it · 50 sentences without a confident label
Builds on it 1 paper
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One can easily deduce that the double-vielbein formalism in strongly constrained DFT [ref] would be easily generalized to the weakly constrained DFT case through the $∘$-product.
Towards Weakly Constrained Double Field Theory · Conclusion · 0.95
Uses its method or tool 2 papers
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While the motivation of our work comes from the Dynkin diagram prediction [ref] and the $E_11$ proposal [ref] , for the actual construction of the $$ U-gravity, we heavily employ the differential geometry tools from [ref] which were developed to provide an underlying `stringy' differential geometry for DFT (see Table [ref] for its characteristic).[Alternative approaches include [ref] .] The methods have been successfully applied to construct Yang-Mills DFT [ref] , coupling to fermions [ref] , coupling to RR-sector [ref] , $=1, D=10$ (full order) super DFT [ref] , $=2, D=10$ (full order) super DFT [ref] , and the completely covariant string action on doubled-yet-gauged spacetime [ref] .
U-gravity : ${\mathbf{SL}(N)}$ · Introduction · 1.00
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We will use a Weitzenböck-like connection [equation] and derive $V_1$ in terms of the generalised torsion of this connection, mirroring the construction in [ref] .[This is related to the flux formulation of DFT and EFT but is different to the construction used in [ref] .] The generalised torsion is defined as [equation] where $_ab$ is an irrep of the generalised torsion.
A geometric formulation of exceptional field theory · Scalar potential · 0.80
Compares with it 5 papers
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We should also note that, the results of [ref] was also obtained by [ref] , by considering the duality twisted reductions of the DFT action they constructed in terms of a torsionful, flat generalized connection, called the Weitzenböck connection[This formulation of the DFT action has the added advantage that it already includes an extra term, which has to be added by hand in the original formulation.
Duality Twisted Reductions of Double Field Theory of Type II Strings · Introduction · 0.84
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This is exactly as expected from the double field theory case, where the Lagrangian resulting from requiring invariance under generalised Lorentz transformations led exactly to the analogous term needed for gauged double field theory [ref] .
Geometry and fluxes of SL(5) exceptional field theory · Constructing the action · 0.96
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This is analogous to [ref] where the scalar potential was rewritten in terms of the Weitzenböck torsion by requiring invariance under local $d ×d$ or $5$ transformations, in the case of DFT and $5$ EFT, respectively.
Half-maximal supersymmetry from exceptional field theory · Reformulating double field theory · 0.83
Cites it as background 24 papers
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The role of the Weitzenböck connection and the generalised torsion in exceptional field theories was first discussed in [ref] and later for DFT in [ref] .
Exotic branes in Exceptional Field Theory: the SL(5) duality group · Introduction · 0.95
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It provides a unified geometric framework for the entire closed string massless NS-NS sector encoded in DFT field variables [ref] : the generalized metric $ℋ_MN$ and the DFT dilaton $d$.
Kerr-Schild Double Field Theory and Classical Double Copy · Introduction · 0.88
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Double field theory (DFT) [ref] is a low energy effective field theory of strings with manifest $𝑂(d,d)$ T-duality [ref] .
Heterotic Kerr-Schild Double Field Theory and Classical Double Copy · Introduction · 1.00
What it cites · 56 sentences without a confident label
Uses the method or tool of 5 papers
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We do this by an analogue of the Scherk-Schwarz mechanism applicable to generalised geometry [ref] .
The effective action of Double Field Theory · Discussion · 0.91
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We do this by an analogue of the Scherk-Schwarz mechanism applicable to generalised geometry [ref] .
Gauged Double Field Theory · Discussion · 0.91
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We do this by an analogue of the Scherk-Schwarz mechanism applicable to generalised geometry [ref] .
Duality Invariant M-theory: Gauged supergravities and Scherk-Schwarz reductions · Discussion · 0.91
Compares with 2 papers
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It is straightforward to check that this action exactly reproduces the gauged DFT action of [ref] .
Gauged Double Field Theory · The action · 0.96
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Inserting the expressions for the torsion and covariant derivative of $d$ we find that we can write [ref] as [equation] This agrees with the original formulation [ref] up to the final term which vanishes by the section condition.
Generalized metric formulation of double field theory · Constructing the action · 0.96
Cites as support 1 paper
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The action can therefore be used for a Scherk-Schwarz ansatz, as has been checked in [ref] by an explicit computation.
Gauged Double Field Theory · Discussion of weak tensorial properties · 0.99
Cites as background 36 papers
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It has recently gone through a renaissance [ref] .
Double Field Theory · Introduction · 0.94
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It has recently gone through a renaissance [ref] .
Generalized metric formulation of double field theory · Introduction · 0.94
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It has recently gone through a renaissance [ref] .
The gauge algebra of double field theory and Courant brackets · Introduction · 0.94
Discussion
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