G-structures and Flavouring non-Abelian T-duality

Alejandro Barranco; Carlos Nunez; Daniel C. Thompson; Jerome Gaillard; Niall Macpherson

REVIEW

G-structures and Flavouring non-Abelian T-duality

Alejandro Barranco, Jerome Gaillard, Niall Macpherson, Carlos Nunez, Daniel C. Thompson

High Energy Physics - Theory · Submitted 30 May 2013 · 36 references

Pith's verdict: A solid, honest paper that delivers on T^{1,1} and adds a genuinely new flavoured IIA solution, though the general SU(3)→SU(2) statement is asserted by analogy rather than proved.

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CitationAlejandro Barranco, Carlos Nunez, Daniel C. Thompson, Jerome Gaillard, Niall Macpherson. "G-structures and Flavouring non-Abelian T-duality." arXiv:1305.7229 (2013). https://pith.science/paper/5KHPOP6G

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The thing to know: this is the G-structure companion to Itsios-Nunez-Sfetsos-Thompson [7,8]. The unflavoured T-dual of KW was already in [8]; what's new here is (a) identifying that the T-dual carries SU(2)-structure, with the pure-spinor transformation rules Eq. 3.10, and (b) a flavoured construction — T-dualising the backreacted D7 solution of Benini et al. — with smearing forms, calibrations, and a careful discussion of how B-field gauge transformations redistribute D4/D6 source charge (Section 5.2). That last bit is the best physics in the paper and I'd point people to it first. The Page charge discussion around Eq. 5.25 is careful and correct in spirit, and the observation that the D4/D6 split is gauge-dependent while the total Maxwell charge is not is the kind of thing that saves a reader a week of confusion later. What's done well: the unflavoured example is worked cleanly. The rotated vielbeins (4.17) and the SU(2) forms (4.23) are explicit, and the check against the dilatino/gravitino equations is stated. The flavoured solution is checked against Bianchi identities and the full equations of motion in Appendix C, and the calibration conditions (5.38-5.40) are written down. Soft spots, in proportion: the crux rule Eq. 3.10 is asserted by analogy with Abelian T-duality, and the authors say so (Section 3, after Eq. 3.11: 'we have not verified the details'). Same for the RR potential rule Eq. 2.17. So the general claim rests on one geometry plus a heuristic about odd-dimensional isometry groups. That's a real gap but a modest one — the explicit construction is a genuine independent check, and the authors are upfront about it. Two smaller things: the flavoured SU(2) forms (5.28) are asserted by analogy with the unflavoured case rather than derived from the T-dual Killing spinors, and SUSY there is verified via EOM/calibration rather than the gravitino/dilatino equations. And the conclusion's claim that 'everything we have said holds true for the wide variety of N=1 backgrounds in [8]' points to forthcoming work — that's an unpublished forward reference and shouldn't be counted as support. Who it's for: people working on non-Abelian T-duality, G-structure compactifications, or flavoured AdS/CFT backgrounds. Narrow, but the right audience will use it. Recommendation: send to referees. The technical content is checkable, the limitations are stated, and the flavoured solution is a real addition. I'd cite it.

Referee report

Major objections (2)

[Section 3, Eq. (3.10) and paragraph after Eq. (3.11)] The general claim that SU(3)-structure maps to SU(2)-structure rests on the assertion that the Clifford multiplication in (3.10) commutes with the twisted differential d_H. The authors state explicitly that they 'have not verified the details' and that the situation 'appears to be exactly analogous' to the Abelian case. The explicit verification is on T^{1,1} (Section 4) and its flavoured deformation (Section 5). The abstract (abstract:s1) states the general map without qualification. Please either provide a proof sketch of the commuting of d_H with the Clifford action (even at the level of the Abelian argument in [16]), or qualify the abstract and Section 3 to make clear that the general statement is a conjecture supported by one explicit example. A second example, e.g. one of the [8] backgrounds, would also strengthen the claim.
[Section 5.3, Eqs. (5.28)-(5.29)] The SU(2)-structure forms for the flavoured T-dual are asserted by analogy with the unflavoured case, rather than derived from the T-dual Killing spinors as in Section 4 (Eqs. 4.13-4.19). The supersymmetry of the flavoured T-dual is checked via the equations of motion and calibration (Appendix C), not via the gravitino/dilatino conditions (B.9). Since the paper's claim is that this is a supersymmetric solution, please show the analogue of (4.18)-(4.19) for the flavoured case, or state explicitly that (B.9) is satisfied and give the check.

Minor objections (5)

[Section 4, Eq. (4.9)] The coefficient of σ̂3² is typeset as λ1²λ2 Δ x1², whereas the T-dual construction and the B-field in the same equation suggest it should be λ1²λ2 x1²/Δ. Please check.
[Section 6, first paragraph] The parenthetical definition of 'static SU(2)-structure' as 'the pure spinors are of type (2,1) everywhere' does not obviously match the explicit forms in (3.8) and (4.23), where Ψ+ has (2,0) and (2,2) pieces. Please clarify the terminology or correct the parenthetical.
[Section 5.2] The interpretation of B-field gauge transformations as an 'inverse' of the Myers effect is suggestive but not derived; please soften the language or provide a concrete argument. Similarly, the statement 'We do not believe it is possible to find a gauge in which we only have explicit D8 sources' is a belief rather than a proof; please flag it as such.
[Abstract (abstract:s3) and Section 5.1] The flavoured T-dual has F0 = Nf/(√2π x2), which is neither constant nor quantised, and the paper acknowledges that it should be regarded as a solution of IIA with D8 sources rather than of massive IIA. Please state the source content explicitly in the abstract or introduction.
[Footnote 2] The symbol Ω is used both for the spinor transformation matrix and for the SU(3)-structure three-form. The authors acknowledge this, but a different symbol for one of them would improve readability.

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The paper

The paper claims

Non-Abelian T-duality along an SU(2) isometry converts a supersymmetric background with SU(3)-structure into one with SU(2)-structure, and the paper gives the explicit rule for how the underlying pure spinors transform.

The reading

The paper asks what happens to the geometry of a supersymmetric string background when you perform non-Abelian T-duality along an SU(2) isometry — a solution-generating transformation that has recently produced a family of new type IIA backgrounds whose field-theory meaning is unclear. The authors argue that the answer is uniform: a background whose internal six dimensions carry SU(3)-structure (a single conserved internal spinor, encoded by a two-form $J$ and a three-form $Omega$) is mapped to one carrying SU(2)-structure (two internal spinors that are nowhere parallel, encoded by a vector $v+iw$, a two-form $j$ and a two-form $omega$). They give transformation rules for the pure spinors, the differential-form objects that package this data, and then check the claim concretely on the T-dual of the Klebanov–Witten geometry and on its flavoured deformation with backreacted D7 branes. The payoff, if the map holds, is that a whole class of newly generated type IIA solutions can be analysed with the standard machinery of G-structures and generalised calibrations rather than by guesswork: one can read off which branes can wrap which cycles and which sources the dual geometry supports.

What carries the argument

The load-bearing device is the pair of pure-spinor transformation rules inherited from the spinor rotation $Omega$ that relates the left- and right-moving frames of the dual $\sigma$ model. Because $Omega$ is built from the same data that defines the original SU(3)-structure, the T-dual pure spinors can be written directly in terms of the original ones, and the resulting objects are of SU(2) type rather than SU(3) type. The second device is a compensating vielbein rotation, with parameters $zeta_1, zeta_2, zeta_3$ built from the T-dual coordinates, that brings the dual Killing spinor into a form as simple as the Abelian case; in that basis the projections on the original spinor are untouched while the second spinor obeys flipped projections, which is exactly the statement that the two spinors are nowhere parallel. The third device is the calibration polyform $Psi_{cal} = -8 e^{3A-Phi} mathrm{Im}Psi_- wedge e^B$, which converts the SU(2)-structure into a practical test of which D-branes can wrap which cycles.

Extended reading notes

Core claim

The central claim is that non-Abelian T-duality along an SU(2) group is type-changing for the internal geometry: SU(3)-structure goes in, SU(2)-structure comes out. The authors motivate this by an Abelian analogy — dualising a pair of directions that the complex structure pairs leaves the structure alone, while dualising directions it does not pair breaks it — and then make it precise with the transformation rules $/Psi^{SU(2)}_+ = i /Psi^{SU(3)}_- Omega^{-1}$ and $/Psi^{SU(2)}_- = /Psi^{SU(3)}_+ Omega^{-1}$, where $Omega$ is the spinor rotation induced by the duality. For the T-dual of $AdS_5 imes T^{1,1}$ they exhibit the two internal spinors explicitly, show that after a suitable rotation of the vielbein the second spinor takes the simple form $ildeeta_{2-} = -i ildetilde{gamma}_r eta_+$, and read off the SU(2)-structure forms $v+iw = -ilde{e}_3 + i ilde{e}_r$, $j = ilde{e}_{hetaphi} + ilde{e}_{21}$, $omega = (ilde{e}_2 + i ilde{e}_1) wedge (ilde{e}_heta + i ilde{e}_phi)$. They extend the same analysis to the flavoured Klebanov–Witten background with smeared D7 sources, obtaining a type IIA solution supported by D6 and D8 sources whose potentials and calibrations are expressed entirely in the SU(2)-structure data, and they check that the DBI and Wess–Zumino actions cancel for calibrated cycles.

Load-bearing premise

The general claim rests on the assumption that the pure-spinor transformation rule commutes with the twisted differential $d_H$, which the paper states it has not verified in detail, and on the analogous unproved generality of the Ramond–Ramond potential transformation rule.

Editorial extensions

If this is right

Any N=1 background with an SU(2) isometry can be dualised and then analysed with SU(2)-structure technology, rather than requiring a case-by-case Killing-spinor computation.
The T-dual of a flavoured background carries explicit D6 and D8 sources, and the paper shows how the smearing forms transform in parallel with the Ramond–Ramond fields, giving a general recipe for dualising brane sources.
Because the source content depends on the choice of $B$-field gauge, the split between explicit D6 and D4 charge is gauge-dependent, while the total Maxwell charge and the equations of motion are invariant.
The calibrated cycles found in the dual geometry — spacetime-filling D4, D6 and D8 branes wrapping specific submanifolds — give candidate domain walls and defects in whatever field theory the background duals.
The same procedure should apply to the wider family of N=1 backgrounds previously generated by non-Abelian T-duality, and to more general toric Calabi–Yau geometries.

Reading between the lines

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

If the SU(3)-to-SU(2) map is generic, it suggests a rough counting statement: each SU(2) isometry dualised costs one complex structure pairing, so backgrounds with larger dualised isometry groups should land further down the G-structure hierarchy.
The gauge-dependence of the D6/D4 split is reminiscent of the Myers dielectric effect run backwards; a reader could ask whether the same redistribution of charge appears when the duality is applied to a single localised (unsmeared) flavour brane rather than a smeared stack.
The paper leaves the periodicity of the new coordinates $x_1, x_2$ open; until that is fixed, the flux quantisation and hence the field-theory rank assignments of the dual background remain ambiguous.
Since the transformation rule is asserted by analogy with the Abelian case rather than proved, an independent check would be to verify that the Clifford multiplication commutes with $d_H$ on a background with non-trivial $H$, which the paper does not do.

Original abstract

We examine non-Abelian T-duality on backgrounds preserving N=1 supersymmetry. We show how backgrounds with SU(3)-structure are mapped to backgrounds with SU(2)-structure. We provide the transformation rules for the corresponding pure spinors. We use these techniques to construct new flavoured solutions of type IIA supergravity.

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We examine non-Abelian T-duality on backgrounds preserving N=1 supersymmetry. We show how backgrounds with SU(3)-structure are mapped to backgrounds with SU(2)-structure. We provide the transformation rules for the corresponding pure spinors. We use these techniques to construct new flavoured solutions of type IIA supergravity.

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s0	We examine non-Abelian T-duality on backgrounds preserving N=1 supersymmetry.	contribution 0.74	0.90	0.07	asserted 0.87
s1	We show how backgrounds with SU(3)-structure are mapped to backgrounds with SU(2)-structure.	result 0.72	0.93	0.29	asserted 1.00
s2	We provide the transformation rules for the corresponding pure spinors.	contribution 0.71	0.94	0.10	asserted 1.00
s3	We use these techniques to construct new flavoured solutions of type IIA supergravity.	contribution 0.77	0.95	0.48	asserted 1.00

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References

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33 arXiv papers cite this in 63 sentences

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Papers citing it · 39 sentences without a confident label

Uses its method or tool 2 papers

Fortunately we can exploit a map for the $SU(2)$ transformation of the pure spinors that was proposed in [ref] Ψ̂_±= Ψ_∓ Ω^-1_SU(2), where in general, in the frame of eq ([ref]) Ω_SU(2)= 1/√(1+ζ_a^2)Γ^(10)(-Γ_456 +∑_a=1^3ζ_aΓ^a+3), for $ζ_a$ defined as in [ref] , which for our parametrisation of ABJM specifically is ζ_1= 4/L^2 cosζsin^2ζy_1 cosξ, ζ_2= 4/L^2 cosζsin^2ζy_1 sinξ, ζ_3= 4 /L^2 sin^2ζy_2.

A $\mathcal{N}=2$ Supersymmetric $AdS_4$ Solution in M-theory with Purely Magnetic Flux · A $U(1)$ of $SU(2)$-structures in the Non-Abelian T-dual · 0.99
[ref] via the map given in [ref] Ψ_±= Ψ_∓Ω^-1.

Holographic Flows in non-Abelian T-dual Geometries · Type IIA/M-theory non-Abelian T-dual backgrounds of the flow ansatz · 0.93

Compares with it 5 papers

* The non-conformality of the geometry is enabled by a non-zero Romans' mass. * Whilst the UV geometries proposed in [ref] are characterized by static $SU(2)$ structure [ref] the full IR complete geometry of this paper has dynamic $SU(2)$ structure. * The transition to dynamic $SU(2)$ structure gives a geometric realization of confinement and permits supersymmetric D2 branes that act as domain walls in the IR.

Dualising the Baryonic Branch: Dynamic SU(2) and confining backgrounds in IIA · Conclusions and Future Directions · 0.82
In the UV, this structure flows to the orthogonal $SU(2)$ structure of the KW solution presented in [ref] .

Non-Abelian T-Dualizing the Resolved Conifold with Regular and Fractional D3-Branes · Comments on the G-structure of the solution · 0.89
The latter could then be interpreted as the field of$$̱-supergravity, providing a new understanding of a dynamical SU(2) structure.[The non-abelian T-duality transformation described in [ref] looks analogous to [ref]; it would be interesting to compare the two, given our discussion and the result of [ref] .] Let us reformulate this idea more precisely. ]

Supersymmetry with non-geometric fluxes, or a $\beta$-twist in Generalized Geometry and Dirac operator · $$̱-twist and intermediate or dynamical SU(2) structure solutions$ · 0.80

Cites it as support 1 paper

Although we have not provided a proof that one implies the other there is mounting evidence that this is indeed the case [ref] , and a proof for the situation for $SU(2)⊂SO(4)$ [ref] .

A new AdS_4/CFT_3 Dual with Extended SUSY and a Spectral Flow · Conclusions · 1.00

Cites it as background 12 papers

The T-dual solution that we consider here has been studied in detail in [ref] .

The non-Abelian T-dual of Klebanov-Witten Background and its Penrose Limits · The Non-Abelian T-dual of the Klebanov-Witten Background · 0.86
Despite these open problems, recently we have witnessed a small resurgence in interest in non-Abelian T-duality, spurred on by the extension of non-Abelian T-duality to incorporate RR fields [ref] .[See also [ref] for the role the Fourier-Mukai transform plays in the transformation of the RR fields.] Provided one is careful about the R symmetry, it is possible to generate supersymmetry preserving solutions of relevance in the context of gauge/gravity duality [ref] as well as to study the implications for G-structures [ref] .

Hints of 5d Fixed Point Theories from Non-Abelian T-duality · Introduction · 0.94
This problem is addressed in various papers, notably in [ref] .

Pin(d,d) covariance of pure spinor equations for supersymmetric vacua and Non-Abelian T-duality · Introduction · 0.92

What it cites · 52 sentences without a confident label

Builds on 3 papers

In section 4 we look at examples of the T-dual of the un-flavoured Klebanov-Witten model studied in [ref] and explicitly construct its $SU(2)$-structure.

On Non-Abelian T-Duality and new N=1 backgrounds · Introduction · 0.80
In section 4 we look at examples of the T-dual of the un-flavoured Klebanov-Witten model studied in [ref] and explicitly construct its $SU(2)$-structure.

Non-Abelian T-duality and the AdS/CFT correspondence:new N=1 backgrounds · Introduction · 0.80
We will then consider the background obtained by adding fundamental fields (quarks) to the Klebanov-Witten field theory [ref] (conversely, we will consider the addition of source-branes to the Klebanov-Witten background).

Unquenched Flavors in the Klebanov-Witten Model · G-structures and their transformations · 0.86

Uses the method or tool of 1 paper

To compactly display the results it is convenient to perform a supplementary rotation as detailed in equation (3.21) of [ref] .

Non-Abelian T-duality and the AdS/CFT correspondence:new N=1 backgrounds · The T-dual · 0.99

Compares with 1 paper

For instance applying an $SU(2)$ non-Abelian T-duality to the internal space of the Klebanov–Witten background ($AdS^5 ×T^1,1$) results in a solution of type IIA which retains the $AdS_5$ factor and has a lift to M-theory which corresponds to the geometries obtained in [ref] from wrapping M5 branes on an $S^2$.

Four-Dimensional SCFTs from M5-Branes · Introduction · 0.87

Cites as support 1 paper

If on the other hand this vanishes only for some projected subset of Killing spinors then we expect only a corresponding projected amount of supersymmetry to be preserved in the T-dual.[In [ref] this was confirmed to be true in general for a large class of backgrounds.] In this paper we consider the case of $N= 1$ supersymmetry which is invariant under the above action of $G$ so that the non-Abelian duality should preserve supersymmetry.

Non-Abelian T-duality and consistent truncations in type-II supergravity · Non-Abelian T-duality · 0.97

Cites as background 14 papers

The recent work of Itsios et al. [ref] considered the application of this duality transformation in IIB supergravity backgrounds preserving $N=1$ supersymmetry.

Non-Abelian T-duality and the AdS/CFT correspondence:new N=1 backgrounds · Introduction · 0.91
The recent work of Itsios et al. [ref] considered the application of this duality transformation in IIB supergravity backgrounds preserving $N=1$ supersymmetry.

On Non-Abelian T-Duality and new N=1 backgrounds · Introduction · 0.91
Although the idea of generalising T-duality to non-Abelian isometry groups has rather old roots [ref] , it is only recently that it has been studied as full solution-generating symmetry of supergravity [ref] .

Non-abelian T-duality, Ramond Fields and Coset Geometries · Introduction · 1.00

Discussion

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