Generalized structures of N=1 vacua

Alessandro Tomasiello; Mariana Gra\~na; Michela Petrini; Ruben Minasian

REVIEW

Generalized structures of N=1 vacua

Mariana Graña, Ruben Minasian, Michela Petrini, Alessandro Tomasiello

High Energy Physics - Theory · Submitted 24 May 2005 · 37 references

Pith's verdict: The two pure spinor equations and the hybrid classification are the real result and hold up; the 'necessary and sufficient' phrasing leans on a representation count in §3.2 that the paper never displays.

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arXiv IDhep-th/0505212 (version 3)

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CitationAlessandro Tomasiello, Mariana Gra\~na, Michela Petrini, Ruben Minasian. "Generalized structures of N=1 vacua." arXiv:hep-th/0505212 (2005). https://pith.science/paper/4MVQ4V5N

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Editor's note

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Worth reading and worth citing, but don't take the 'necessary and sufficient' phrasing at face value. The real contribution is the pair of equations (3.1)-(3.4) plus the norm relations (3.10): N=1 in type II with RR flux becomes two pure spinor equations, one twisted-closed, one with the RR field acting as a source. That is a genuine extension of the authors' earlier SU(3) work (hep-th/0406137) to generic SU(3)xSU(3), and the classification that falls out — IIA generically symplectic, IIB generically hybrid complex-symplectic, degenerating to pure complex or static SU(2) at special loci — is the part people actually use. The observation that every Minkowski vacuum carries a topological model, and the appendix B treatment of the NS-only (2,1) case via Courant brackets, are both clean and useful. Where it is soft: §3.2. Necessity is derived properly, via the bispinor map (3.11) plus the gravitino and dilatino variations. Sufficiency is a representation count — four equations in the (1,1), eight with one index, four with two, and 'of course they agree.' Equal counts do not prove the two linear systems are the same system unless the map between them is invertible, and the authors withhold the component expressions as 'not particularly enlightening.' So the 'iff' is asserted rather than shown. It is a modest gap, since the equations have been used and cross-checked downstream for two decades, but if you plan to solve (3.1)-(3.4) in place of the Killing spinor equations, you are leaning on a step the paper does not fully display. One correction to the record: the classifier reports 'limitations (none found).' That is wrong. The paper states in §3 that (3.1)-(3.4) do not cover F=0 N=1, which that limit sends to N=2, and in the Discussion that Bianchi identities and flux equations of motion are not imposed, so compact Minkowski solutions hit the usual no-go. The authors are honest about scope; the record simply missed it. Who it is for: anyone working on flux compactifications or generalized geometry. Send it out. The referee should press on §3.2 and let the rest through.

Referee report

Major objections (2)

[§3.2] The claim that (3.1)-(3.4) plus (3.10) are necessary and sufficient rests on counting independent equations per SU(3)×SU(3) representation. The authors state 'we confirm that the number of independent equations is the same... (and of course they agree with the latter)' but explicitly withhold the component expressions ('not particularly enlightening'). Equal counts per representation do not by themselves establish that the two linear systems are equivalent; they establish equal solution-space dimension only if the map between them is invertible. A concrete test: exhibit the component map for at least one nontrivial representation, e.g., the (1,1) or (3,3) sector, showing that the pure spinor equations are literally the same linear combinations as δλ=0 and δψ_m=0, or state the invertibility of the bispinor map (2.5). The counting itself ('four for (1,1), eight for one-index, four for two-index') is also not derived in the text, so the reader cannot check it.
[Abstract and Introduction] The characterization excludes the F=0, N=1 case, as acknowledged after eq. (3.10) ('equations (3.1-3.4) do not cover the case F=0, N=1 vacua'). The body argues that F=0 with both a,b nonzero gives N=2, and F=0 with one of them zero makes both pure spinors vanish. The abstract [abstract:s0] and [abstract:s3] state the characterization without the RR≠0 qualification, so the headline claim is broader than what is proven. The abstract should be qualified to N=1 vacua with RR flux, or the F=0 case should be connected to the standard SU(3) results.

Minor objections (6)

[Eq. (2.11) and Table 1] The two pure spinors are interchanged. Eq. (2.11) gives Φ+ = e^{-ij}∧(v+iw), Φ− = ω∧e^{-iv∧w}, while Table 1 gives Φ+ = ω∧e^{-iv∧w}, Φ− = e^{-ij}∧(v+iw). The general formula (2.13) with c=0 and the type assignment in §3.4 ('types ... are 2 and 1 for Φ+ and Φ−') both agree with Table 1, so (2.11) appears to be the typo.
[Abstract] 'A twisted generalized Calabi-Yau manifold of a hybrid complex-symplectic type' is not universal. §3.4 states that IIA is generically symplectic (type 0) and IIB generically hybrid (type 1), with the type jumping at points where c=0 or v+iw=0. The abstract should reflect this theory-dependence.
[Reference [32]] The arXiv identifier 'math.DG/04090993' appears to have an extra digit; the standard format is YYMM.NNNN.
[Eqs. (3.2)-(3.3)] Formatting: in (3.2) the coefficient '116' should be '1/16', and in (3.3) '3 ie^{-A}' should be clarified as a coefficient multiplying the Im term.
[Eq. (3.10)] The norm relations (3.10) are stated without derivation and delegated to [7]. Since [7] is published this is supported, but a brief indication of how they follow from the supersymmetry variations would help the reader.
[§4] The claim that 'all N=1 Minkowski vacua have an underlying topological model' is argued for RR≠0 via (3.1)/(3.4). For F=0, N=1 the standard A/B models apply, but the paper does not explicitly connect this case; a sentence would close the gap.

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The paper

The paper claims

Unbroken N=1 type II vacua with RR flux are claimed to be equivalent to two pure-spinor equations on the internal six-manifold.

The reading

The paper asks what internal geometry is equivalent to an unbroken N=1 vacuum of type II string theory when RR flux is present. Its answer is that the tangent-plus-cotangent bundle of the six-dimensional internal space carries an SU(3)×SU(3) structure, encoded by two pure spinors $Φ_1$ and $Φ_2$, and that supersymmetry is captured by two differential equations: $(d+H∧)Φ_1=0$ and $(d+H∧)Φ_2=F_{RR}$, with warp-factor, dilaton and norm corrections. The first equation says the internal space is a twisted generalized Calabi-Yau of hybrid complex-symplectic type; the second says the RR fields measure the failure of the second spinor to be integrable. This matters because it unifies the older SU(3) and SU(2) pictures, links every such Minkowski vacuum to a topological model, and suggests mirror symmetry as an exchange of the two spinors. The authors are explicit that the equations do not cover the RR-free N=1 case: setting $F=0$ forces constant spinor norms and, if both survive, gives N=2 rather than N=1.

What carries the argument

The argument is carried by pure spinors of Clifford(6,6), which are formal sums of even or odd differential forms on the internal six-manifold. Each pure spinor is the Clifford map of a bispinor $η_{1+}⊗η_{2±}^†$, so the two ten-dimensional supersymmetry parameters reduce to two six-dimensional spinors; their common annihilators reduce the structure group of $T(M)⊕T^*(M)$ from O(6,6) to SU(3)×SU(3). The differential equations are then statements about the twisted Courant bracket: $(d+H∧)Φ=0$ says the annihilator distribution of $Φ$ is closed under that bracket, i.e. the associated generalized complex structure is integrable. The paper classifies the possible $Φ$'s by type, the number of holomorphic coordinates in a local $e^A∧ω_k$ decomposition, which is how it distinguishes symplectic, complex and hybrid complex-symplectic cases. To argue sufficiency, the authors decompose the supersymmetry variations and the pure-spinor equations into the same SU(3)×SU(3) Hodge diamond and compare the number of independent equations in each representation; they state that the counts match, but they do not display the component expressions.

Extended reading notes

Core claim

The central claim is an equivalence: for type II compactifications to warped Minkowski or AdS with nonzero RR flux, unbroken N=1 supersymmetry is equivalent to the pair of pure-spinor equations plus the two norm relations (3.10). In schematic form the equations are $(d+H∧)Φ_1=0$ and $(d+H∧)Φ_2=F_{RR}$, where $Φ_1$ has the parity of the RR fields and $Φ_2$ the opposite parity; the full versions in (3.1)-(3.4) include the warp factor $A$, dilaton $φ$, cosmological constant $μ$, and the weighted RR sums. The first equation makes the internal manifold a twisted generalized Calabi-Yau, generically symplectic in IIA and hybrid complex-symplectic in IIB, with jumps at loci where the two internal spinors become orthogonal or parallel. The second equation makes the RR flux an integrability defect for $Φ_2$, not a closure condition. The authors also stress the boundary of the claim: when $F=0$ the equations lose information, and the RR-free N=1 vacua with only NS flux must be described by the older SU(3) torsion equations instead.

Load-bearing premise

The load-bearing premise is that the two pure-spinor equations plus the norm relations are sufficient as well as necessary for N=1 supersymmetry, a claim supported only by matching representation counts while the explicit component expressions are omitted.

Editorial extensions

If this is right

The SU(3)-structure and static SU(2)-structure flux vacua become special cases of one SU(3)×SU(3) framework rather than separate case-by-case analyses.
Every N=1 Minkowski vacuum in this class carries an underlying topological model, because one pure spinor is closed under $d+H∧$.
IIA and IIB vacua are governed by the same pair of equations with even and odd spinors exchanged, making a mirror-symmetry interpretation concrete.
The local type of the integrable spinor predicts that IIA generically prefers symplectic geometry and IIB generically prefers hybrid complex-symplectic geometry, with pure complex or static SU(2) loci as special points.
The second equation suggests that RR flux can be treated as a deformation of a generalized Calabi-Yau structure, opening a route to counting moduli by coupling deformations of $Φ_2$ to deformations of the RR potential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the RR term really is an integrability defect, one can try to define a generalized complex structure whose Nijenhuis tensor is exactly $F_{RR}$; that would turn the second equation into a cohomological classification of allowed RR fluxes by the failure of closure.
The claimed topological model for every N=1 Minkowski vacuum suggests a protected sector whose BRST cohomology depends only on the closed pure spinor $Φ_1$ and not on the RR deformation; checking this on a nilmanifold or a static SU(2) background would test whether the topological model is more than formal.
Because the equations are essentially identical in IIA and IIB with even and odd spinors exchanged, an explicit mirror pair of hybrid complex-symplectic backgrounds would be a sharp test of the proposed mirror symmetry.
The sufficiency argument could be made checkable by publishing the component map between the intrinsic torsion $W^{ij}$ and the fluxes; without it, the equivalence remains a representation-counting statement rather than an explicit dictionary.

Original abstract

We characterize N=1 vacua of type II theories in terms of generalized complex structure on the internal manifold M. The structure group of T(M) + T*(M) being SU(3) x SU(3) implies the existence of two pure spinors Phi_1 and Phi_2. The conditions for preserving N=1 supersymmetry turn out to be simple generalizations of equations that have appeared in the context of N=2 and topological strings. They are (d + H wedge) Phi_1=0 and (d + H wedge) Phi_2 = F_RR. The equation for the first pure spinor implies that the internal space is a twisted generalized Calabi-Yau manifold of a hybrid complex-symplectic type, while the RR-fields serve as an integrability defect for the second.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,327 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.64 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We characterize N=1 vacua of type II theories in terms of generalized complex structure on the internal manifold M. The structure group of T(M) + T*(M) being SU(3) x SU(3) implies the existence of two pure spinors Phi_1 and Phi_2. The conditions for preserving N=1 supersymmetry turn out to be simple generalizations of equations that have appeared in the context of N=2 and topological strings. They are (d + H wedge) Phi_1=0 and (d + H wedge) Phi_2 = F_RR. The equation for the first pure spinor implies that the internal space is a twisted generalized Calabi-Yau manifold of a hybrid complex-symplectic type, while the RR-fields serve as an integrability defect for the second.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We characterize N=1 vacua of type II theories in terms of generalized complex structure on the internal manifold M.	contribution 1.00	0.89	0.44	asserted 1.00
s1	The structure group of T(M) + T*(M) being SU(3) x SU(3) implies the existence of two pure spinors Phi_1 and Phi_2.	result 0.48	0.37	0.11	asserted 1.00
s2	The conditions for preserving N=1 supersymmetry turn out to be simple generalizations of equations that have appeared in the context of N=2 and topological strings.	result 0.73	0.82	0.14	asserted 0.99
s3	They are (d + H wedge) Phi_1=0 and (d + H wedge) Phi_2 = F_RR.	result 0.96	0.81	0.28	asserted 1.00
s4	The equation for the first pure spinor implies that the internal space is a twisted generalized Calabi-Yau manifold of a hybrid complex-symplectic type, while the RR-fields serve as an integrability defect for the second.	interpretation 0.92	0.83	0.44	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Localized O6-plane solutions with Romans mass
hep-th 2012-01 conditional novelty 6.0 of 10

Localized O6 orientifold planes in massive type IIA supergravity appear to exist, with the Romans mass removing the unphysical region that surrounds the source in the massless theory.
T-duality and closed string non-commutative (doubled) geometry
hep-th 2010-10 conditional novelty 6.0 of 10

Closed string coordinates in twisted-torus and H-flux backgrounds fail to commute, with a commutator proportional to the string's momentum (or winding) in the third direction.
Generalized non-supersymmetric flux vacua
hep-th 2008-07 conditional novelty 6.0 of 10

A classification of non-supersymmetric type II flux vacua by which D-branes remain BPS, yielding no-scale Minkowski examples and new AdS4 solutions.
T-duality, Generalized Geometry and Non-Geometric Backgrounds
hep-th 2008-07 conditional novelty 6.0 of 10

T-duality is realized as an O(d,d) rotation of generalized vielbeins and pure spinors, and the four-dimensional fluxes f, H, Q, R are identified with Courant-bracket structure constants of those vielbeins.
M-theory, exceptional generalised geometry and superpotentials
hep-th 2008-04 conditional novelty 6.0 of 10

N=1 M-theory backgrounds on seven-dimensional spaces are characterised by an element of the 912 representation of E_7(7), and the four-dimensional superpotential is an E_7(7)-invariant functional of this element.
From ten to four and back again: how to generalize the geometry
hep-th 2007-07 conditional novelty 6.0 of 10

The ten-dimensional supersymmetry equations of warped SU(3)xSU(3) flux vacua are exactly the F- and D-flatness conditions of a Weyl-invariant four-dimensional N=1 supergravity, and non-perturbative corrections deform ...
Supersymmetric sources, integrability and generalized-structure compactifications
hep-th 2007-06 conditional novelty 6.0 of 10

Orientifolds are compatible with dynamic SU(3)xSU(3)-structure, and source-corrected Einstein and dilaton equations follow from supersymmetry plus form Bianchi identities.
A scan for new N=1 vacua on twisted tori
hep-th 2006-09 conditional novelty 6.0 of 10

The paper classifies N=1 Minkowski vacua of type II strings on compact nilmanifolds and solvmanifolds, finding a few new non-Calabi-Yau backgrounds that so far exist only as smeared large-volume solutions.
Topological strings in generalized complex space
hep-th 2006-03 conditional novelty 6.0 of 10

A BV/AKSZ construction defines a topological sigma-model on generalized Calabi-Yau targets, recovering the A- and B-models and deriving the Kontsevich star product in the open-string sector.
D-branes on general N=1 backgrounds: superpotentials and D-terms
hep-th 2006-02 conditional novelty 6.0 of 10

A single universal superpotential, built from the background's integrable pure spinor, generates the F-flatness conditions for spacetime-filling D-branes on general N=1 flux backgrounds, while the D-flatness condition...
Supersymmetric D-branes and calibrations on general N=1 backgrounds
hep-th 2005-07 conditional novelty 6.0 of 10

Supersymmetric D-branes in general N=1 flux backgrounds wrap generalized complex submanifolds and are calibrated by a form built from the non-integrable pure spinor plus the Ramond-Ramond potentials.
Hitchin Functionals in N=2 Supergravity
hep-th 2005-05 accept novelty 6.0 of 10

Type II supergravity on SU(3) x SU(3)-structure backgrounds admits a special Kähler geometry on its metric moduli space, with Kähler potentials given by Hitchin functionals, and its gravitino transformations yield N=2...
On the Cosmology of Type IIA Compactifications on SU(3)-structure Manifolds
hep-th 2008-12 accept novelty 5.5 of 10

A refined no-go theorem rules out de Sitter vacua and slow-roll inflation for six of seven SU(3)-structure coset compactifications of type IIA string theory, and the surviving SU(2)xSU(2) case yields only saddle point...
Stability of non-supersymmetric vacua from calibrations
hep-th 2025-07 unverdicted novelty 5.0 of 10

Uses calibrations to protect examined non-supersymmetric AdS vacua in type II string theory from D-brane mediated decays, including abelian bound states.
Type IIB Supergravity Solutions with AdS${}_5$ From Abelian and Non-Abelian T Dualities
hep-th 2014-10 conditional novelty 5.0 of 10

Chaining non-Abelian and Abelian T-dualities on AdS5 x S5, T^{1,1} and Y^{p,q} produces new type IIB backgrounds with an AdS5 factor, non-trivial axion-dilaton and three-form flux, and no five-form flux.
NS-branes, source corrected Bianchi identities, and more on backgrounds with non-geometric fluxes
hep-th 2014-02 conditional novelty 5.0 of 10

The authors propose source-corrected Bianchi identities for the NS5-brane, KK-monopole and Q-brane, recover the sourceless identities from a nilpotent Dirac operator, and show that a class of geometric beta-supergravi...
Non-commutative/non-associative IIA (IIB) Q- and R-branes and their intersections
hep-th 2013-03 conditional novelty 5.0 of 10

Intersecting Q- and R-branes, obtained by T-duality from NS5-branes and KK-monopoles, yield supersymmetric AdS4 x M6 backgrounds whose internal tori carry non-geometric Q- and R-fluxes and exhibit non-commutative or n...
Double Field Theory and N=4 Gauged Supergravity
hep-th 2011-09 conditional novelty 5.0 of 10

Compactifying Double Field Theory on a twisted doubled torus yields N=4 electric gauged supergravity, and the embedding-tensor closure conditions follow from a weaker differential constraint than the full strong constraint.
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets
hep-th 2008-06 conditional novelty 5.0 of 10

Explicit four-dimensional effective theories for type IIA AdS4 compactifications on nilmanifolds and cosets show all moduli stabilized in all but one coset model, with a Kaluza-Klein cross-check confirming the mass spectra.
Flux compactifications in string theory: a comprehensive review
hep-th 2005-09 accept novelty 3.0 of 10

A structured survey of type II flux compactifications: which fluxes preserve N=1, why compact spaces need orientifold planes, how fluxes generate superpotentials, and how far they go toward fixing moduli.
Moduli Stabilization in String Theory
hep-th 2023-10 accept novelty 2.0 of 10

This is a review of moduli stabilization in string compactifications, organizing the KKLT and Large Volume Scenario constructions, their extensions to other string theories, and their cosmological and particle-physics...
The Geometry, Branes and Applications of Exceptional Field Theory
hep-th 2020-06 accept novelty 2.0 of 10

Exceptional field theory recasts the maximal supergravities as a single extended geometry whose local symmetry is the exceptional U-duality group Ed(d), with extra coordinates conjugate to brane winding modes.
Non-geometric backgrounds in string theory
hep-th 2018-11 accept

The review maps how T-duality, generalised geometry, and flux compactifications describe string backgrounds that cannot be given an ordinary Riemannian geometry.

How later papers used it

118 later papers cite it

Evidence: substantial. 81 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

Pith read 25 of the 302 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool. 2 more papers by the same authors are listed: 2 used its method or tool.

Uses its method or tool 4 of 17 papers with this label · 2 by the same authors

Adopted the language of generalized complex geometry as in this paper to describe the background geometry.

D-branes on AdS flux compactifications (2007) · 0.97 · paraphrased: the licence does not allow quoting
Used the supersymmetry conditions of massive IIA as written down in this paper.

Missing Mirrors: Type IIA Supergravity on the Resolved Conifold (2010) · 0.88 · paraphrased: the licence does not allow quoting

By the same authors

Used the pure spinor formalism of this paper, with its precise conversions.

Minimal flux Minkowski classification (2016) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
Used its Ansatz in the generalized complex geometry formalism of this paper, where AdS4 x M6 solutions of type II supergravity were reformulated in terms of pure spinor equations.

AdS_4 compactifications of AdS_7 solutions in type II supergravity (2015) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

113 arXiv papers cite this in 294 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 194 sentences without a confident label

Contests it 1 paper

While the spinor components of such equations reproduce those of [ref] and are therefore true conditions for susy vacua, other components failed to reproduce supersymmetry conditions.].

N=1 vacua in Exceptional Generalized Geometry · Introduction · contest 0.85

Uses its method or tool 18 papers

In order to find sourceless vacua, we use the pure spinors formalism developed in [ref] .

A method to find $\mathcal{N}=1$ AdS$_4$ vacua in type IIB · Introduction · 1.00
To describe the backreaction of the gaugino condensate on the bulk fields, we must relax the Calabi-Yau condition and employ generalized complex geometry, as in [ref] .

de Sitter Vacua from Ten Dimensions · D7-brane gaugino couplings · 0.93
Preserved supersymmetry is equivalent to the following “pure spinor equations” [ref] on $M_6$:

Minimal flux Minkowski classification · Pure spinors · 0.88

Compares with it 6 papers

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
Equation ([ref]) reduces to a condition about the norm of the spinors, while ([ref]) reproduces all of the pure spinor equations in [ref] .

Generalized structures of ten-dimensional supersymmetric solutions · Reproducing the pure spinor equations for four-dimensional vacua · 0.94
In this sense, they play a role similar to that of the RR field strengths in the physical string theories, which are obstructions to the integrability of one of the pure spinors in the generalized Kähler structure [ref] .

Topological strings live on attractive manifolds · The action and equations of motion · 0.83

Cites it as support 1 paper

GMPT describes all $𝒩=1$ solutions for which the two internal spinors are nowhere vanishing – this is an open condition and hence will not be changed by small deformations [ref] .

Generalising G$_\text{2}$ geometry: involutivity, moment maps and moduli · Example 2: GMPT geometries · 0.96

Cites it as background 45 papers

$=1$ supersymmetry has been reformulated as a geometrical condition in [ref] .

Reformulating Supersymmetry with a Generalized Dolbeault Operator · Introduction · 0.83
In this section we review the results of [ref] (in the language of GCG, as in [ref] ) and [ref] where the conditions for respectively $𝒩=2$ supersymmetry with NS flux only, and $𝒩=1$ with NS and RR fluxes were found.

N=1 vacua in Exceptional Generalized Geometry · Introduction · 0.93
Supersymmetric vacua obey a system of pure spinor equations [ref] (Sec. [ref]).

Stability of non-supersymmetric vacua from calibrations · Introduction · 0.90

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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