REVIEW
SU(3) x SU(3) compactification and mirror duals of magnetic fluxes
High Energy Physics - Theory · Submitted 21 Dec 2006 · 72 references
Pith's verdict: A careful, honest completion of the authors' SU(3)xSU(3) program; the new p^A mirror flux is real, but the non-geometric extrapolation is checked only at superpotential level.
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Referee report
The paper
The paper claims
Type II string compactifications on backgrounds with a pair of SU(3) structures give a manifestly mirror-symmetric low-energy theory, and supply the mirror partner of Calabi–Yau compactifications with magnetic three-form flux.
The reading
What carries the argument
The load-bearing object is the pair of complex pure spinors $Phi_+ = e^{-B}eta_{1+}bar{eta}_{2+}$ and $Phi_- = e^{-B}eta_{1+}bar{eta}_{2-}$ of $O(6,6)$, which encode the $SU(3)times SU(3)$ structure on the generalized tangent bundle $F oplus F^*$. Two ingredients make them work. First, the Hitchin function $H(Phi_pm) = i langle Phi_pm, bar{Phi}_pm rangle$ supplies the Kähler potentials $e^{-K_pm}$ of the vector and hypermultiplet moduli spaces, so the kinetic terms come for free once the spinors are chosen. Second, the Mukai pairing turns the supersymmetry variations of the gravitinos into integrals like $int langle Phi_-, dPhi_+ rangle$ and $int langle Phi_-, G_- rangle$, which is how the flux and torsion charges enter the prepotentials. The truncation is organized by a charge matrix $Q$ mapping the odd basis to the even basis, with nilpotency of the generalized derivative imposing the quadratic constraints $Q S^+ Q^T = 0$. In the twisted-torus examples $Q$ is filled in by the conventional $H$-flux and geometric $f$-flux, and the remaining entries require the non-geometric $Q$- and $R$-fluxes of T-duality chains.
Extended reading notes
Core claim
The central claim is that the mirror dual of a type IIB compactification on a Calabi–Yau threefold with magnetic NS three-form flux is a type IIA compactification on a six-manifold with $SU(3)times SU(3)$ structure, and that this duality is visible directly in the low-energy data. Concretely, the authors compute the gravitino mass matrix $S_{AB}$ and hence the Killing prepotentials $P^x$ in terms of the two pure spinors $Phi_pm$ and the Ramond–Ramond field strengths $G_pm$, finding the same functional form as in the single-$SU(3)$ case but now with the extra one- and five-form components that $Phi_-$ can carry. With a truncation to finitely many modes, expanded in a symplectic basis of even and odd forms, the prepotentials of type IIA and type IIB become identical under $X^A leftrightarrow Z^I$, $F_A leftrightarrow F_I$, $e^I leftrightarrow e^A$ and $m^I leftrightarrow -p^A$. The new entry $p^A$ plays the role of a dual magnetic flux: it gives mass to an antisymmetric tensor via a Stueckelberg mechanism in type IIA and to a vector in type IIB, matching what magnetic flux does on the Calabi–Yau side. The authors are explicit that such backgrounds are generically non-geometric, so the supergravity derivation is not strictly valid, but they argue the naive geometric expressions still give the correct effective theory.
Load-bearing premise
The argument assumes that a consistent truncation to a finite set of modes exists satisfying the stated differential conditions, and that formulas derived under the assumption of an ordinary geometric compactification remain valid for backgrounds the authors themselves describe as non-geometric.
Editorial extensions
If this is right
- Mirror symmetry extends to compactifications with magnetic NS flux: the mirror of a Calabi–Yau with $H_3$ flux is an $SU(3)times SU(3)$ background whose odd basis contains one- and five-forms, with the new charges $p^A$ playing the role of dual magnetic flux.
- The type IIA and type IIB Killing prepotentials computed from the general charge matrix $Q$ are related by a symplectic transpose, so all previously known flux compactifications drop out as special cases of one formula.
- A charge matrix with generic entries cannot come from an honest exterior derivative acting on forms of definite degree, which forces the conclusion that the corresponding backgrounds are non-geometric, patched by T-duality or worse.
- Comparing the general prepotentials with the conjectured non-geometric superpotentials of the twisted-torus literature reproduces those superpotentials term by term, including the terms attributed to $Q$- and $R$-flux.
- The number of independent charges in a generic $Q$ is fixed by the nilpotency constraints to $(2b^+ + 3)(b^- + 1)$ when $b^- geq b^+$, giving a concrete count against which candidate backgrounds can be checked.
Reading between the lines
- If the geometric expressions really do survive the loss of a global geometric description, the practical consequence is that one can compute flux effective actions without ever constructing the mirror manifold — a shortcut that would be worth testing on a case where both sides are independently known.
- The argument leaves open whether the truncation to a finite mode basis can always be found; a counterexample would be a background whose light spectrum does not close into the symplectic bases assumed here, and searching the twisted-torus family for such a case is a concrete next step.
- The identification of $p^A$ with a dual magnetic flux suggests that the quantization of $H_3$ on one side should translate into integrality of $p^A$ on the other; checking that against known T-fold monodromy data would be a sharp test.
- Since the same formalism covers $R$-flux, where even a local geometric picture is claimed to fail, the success or failure of the naive formulas there is the most informative stress test of the whole approach.
Original abstract
This paper analyses type II string theories in backgrounds which admit an SU(3) x SU(3) structure. Such backgrounds are designed to linearly realize eight out of the original 32 supercharges and as a consequence the low-energy effective action can be written in terms of couplings which are closely related to the couplings of four-dimensional N=2 theories. This generalizes the previously studied case of SU(3) backgrounds in that the left- and right-moving sector each have a different globally defined spinor. Given a truncation to a finite number of modes, these backgrounds lead to a conventional four-dimensional low-energy effective theory. The results are manifestly mirror symmetric and give terms corresponding to the mirror dual couplings of Calabi-Yau compactifications with magnetic fluxes. It is argued, however, that generically such backgrounds are non-geometric and hence the supergravity analysis is not strictly valid. Remarkably, the naive generalization of the geometrical expressions nonetheless appears to give the correct low-energy effective theory.
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
This paper analyses type II string theories in backgrounds which admit an SU(3) x SU(3) structure. Such backgrounds are designed to linearly realize eight out of the original 32 supercharges and as a consequence the low-energy effective action can be written in terms of couplings which are closely related to the couplings of four-dimensional N=2 theories. This generalizes the previously studied case of SU(3) backgrounds in that the left- and right-moving sector each have a different globally defined spinor. Given a truncation to a finite number of modes, these backgrounds lead to a conventional four-dimensional low-energy effective theory. The results are manifestly mirror symmetric and give terms corresponding to the mirror dual couplings of Calabi-Yau compactifications with magnetic fluxes. It is argued, however, that generically such backgrounds are non-geometric and hence the supergravity analysis is not strictly valid. Remarkably, the naive generalization of the geometrical expressions nonetheless appears to give the correct low-energy effective theory.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper analyses type II string theories in backgrounds which admit an SU(3) x SU(3) structure. | contribution 0.95 | 0.87 | 0.04 | asserted 1.00 |
| s1 | Such backgrounds are designed to linearly realize eight out of the original 32 supercharges and as a consequence the low-energy effective action can be written in terms of couplings which are closely related to the couplings of four-dimensional N=2 theories. | result 0.41 | 0.28 | 0.08 | asserted 0.99 |
| s2 | This generalizes the previously studied case of SU(3) backgrounds in that the left- and right-moving sector each have a different globally defined spinor. | contribution 0.71 | 0.31 | 0.05 | asserted 1.00 |
| s3 | Given a truncation to a finite number of modes, these backgrounds lead to a conventional four-dimensional low-energy effective theory. | result 0.82 | 0.69 | 0.09 | asserted 0.99 |
| s4 | The results are manifestly mirror symmetric and give terms corresponding to the mirror dual couplings of Calabi-Yau compactifications with magnetic fluxes. | result 0.91 | 0.88 | 0.47 | asserted 0.99 |
| s5 | It is argued, however, that generically such backgrounds are non-geometric and hence the supergravity analysis is not strictly valid. | limitation 1.00 | 0.50 | 0.18 | hedged possibility 0.64 |
| s6 | Remarkably, the naive generalization of the geometrical expressions nonetheless appears to give the correct low-energy effective theory. | result 0.75 | 0.72 | 0.33 | hedged possibility 0.65 |
What was examined, and what was not
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References
How later papers used it
72 later papers cite it
1 later paper by other authors is listed below: 1 used its method or tool.
Uses its method or tool 1 of 3 papers with this label
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Obtained its flux Bianchi identities by imposing that the twisted differential in this paper satisfies the idempotency constraint when applied on the p-form basis.
Systematics of Type IIA moduli stabilisation (2020) · 0.82 · paraphrased: the licence does not allow quoting
How the literature uses it
68 arXiv papers cite this in 231 sentences
Papers citing it · 160 sentences without a confident label
Builds on it 4 papers
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Generalizing a previous analysis [ref] done for $SU(3)$ structures, the authors of [ref] reduced the type II gravitino susy variations on $SU(3)×SU(3)$ structure backgrounds, and determined in this way an expression for the susy transformation of the 4d $N=2$ gravitini in terms of the higher dimensional fields.
Effective actions and N=1 vacuum conditions from SU(3) x SU(3) compactifications · $N=2$ Killing prepotentials from the dimensional reduction · 0.90
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This can be extended [ref] to include the T duality images of the NS-NS field strength, making the twisted Calabi-Yau into a generalised non-geometric one with covariant exterior derivative$$.
The Generalised Geometry of Type II Non-Geometric Fluxes Under T and S Dualities · Type IIA Fluxes · 0.92
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The extension of the setup in [ref] to more general type II string compactifications leading to 4d $=2$ gauged supergravities has been performed in [ref] .
Torsion in cohomology and dimensional reduction · More general $=2$ compactifications · 0.81
Uses its method or tool 3 papers
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Following the approach of [ref] , let us introduce a twisted differential acting on $p$-forms.
A Flux-Scaling Scenario for High-Scale Moduli Stabilization in String Theory · Twisted differential and Bianchi identities · 1.00
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In our setup, these can be obtained by imposing that the twisted differential $D$ in [ref] satisfies the idempotency constraint $D^2 = 0$ when applied on the $p$-form basis of table [ref] [ref] .
Systematics of Type IIA moduli stabilisation · Constraints from Bianchi identities · 0.82
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Using an approach similar to the ones of [ref] we clarify in this paper how these $N=1$ supersymmetric flux compactifications to both AdS or flat $ℝ^1,3$ space-time can be understood from a four-dimensional point of view.
From ten to four and back again: how to generalize the geometry · Introduction · 0.86
Compares with it 6 papers
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Indeed, in [ref] , the type IIA `pure spinor equations' were derived assigning positive chirality to $ϵ^1$ and negative chirality to $ϵ^2$, while in ([ref]) we have done the opposite choice (following the conventions of [ref] ).
Effective actions and N=1 vacuum conditions from SU(3) x SU(3) compactifications · $N=1$ equations from the ten dimensional analysis · 0.90
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By comparing their result with [ref] we can read off the corresponding embedding tensor as [equation] $Θ^λ̃_Λ$ precisely coincides with the `doubly symplectic' charge matrix $Q$ discussed in [ref] .
Spontaneous N=2 to N=1 Supersymmetry Breaking in Supergravity and Type II String Theory · Realisation in String Theory · 0.98
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We propose an $7$ form for $_a$ which coincides with the known expressions given in [ref] when decomposed under the $$ subgroup of $7$.
E7(7) formulation of N=2 backgrounds · Introduction · 0.96
Cites it as support 3 papers
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Motivated by the above considerations, we are then led to take this topological fact as a necessary condition for compactifications of type II supergravity to yield an $N=2$ effective action in 4d [ref] .
Reducing democratic type II supergravity on SU(3) x SU(3) structures · Introduction · 0.99
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The assumption relies on that fact that it seems there is no reason why turning them on should be forbidden: The effective four-dimensional theories can be derived from generalised compactifications [ref] .
Low Energy Supersymmetry from Non-Geometry · Non-geometry and Calabi-Yaus · 0.99
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Furthermore, it supports the prescription given in ref.$$ [ref] of projecting out all the fields transforming in the vector representation of $O(6,6)$ when decomposing the 10d supergravity fields on the basis ([ref]), due to the fact that they would assemble to define spin $3/2$ multiplets in 4d, which correspond to non-standard couplings of $N=2$ supergravity.
Effective actions and N=1 vacuum conditions from SU(3) x SU(3) compactifications · Deformations · 0.96
Cites it as background 39 papers
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One formulation involves fluxes denoted by the real six dimensional indices (e.g. $H_ijk$ etc.) [ref] while in the latter one, all flux components are written out using cohomology indices of the complex threefold $X_3$; e.g. $H_K, ω_a K$ etc., where $K ∈{0, 1, ..,h^2,1(X_3)}$ and $a ∈{1, 2, .., h^1,1_-(X_3) }$ [ref] .
On Missing Bianchi Identities in Cohomology Formulation · Introduction · 0.92
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The superpotential and Kähler potential of the effective $𝒩=1$ supergravity have been derived in various ways in [ref] (based on earlier work of [ref] ).
On the effective theory of type IIA AdS_4 compactifications · Effective supergravity · 0.88
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In subsect.$$[ref] we briefly reviewed how refs.$$ [ref] got the Killing prepotentials of the $N=2$ theory which is defined starting from a 10d background preserving eight supercharges.
Effective actions and N=1 vacuum conditions from SU(3) x SU(3) compactifications · $N=1$ superpotential · 0.99
Discussion
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