More Dual Fluxes and Moduli Fixing

A. Font; G. Aldazabal; L.E. Ibanez; P.G. Camara

REVIEW

More Dual Fluxes and Moduli Fixing

G. Aldazabal, P.G. Camara, A. Font, L.E. Ibanez

High Energy Physics - Theory · Submitted 9 Feb 2006 · 49 references

Pith's verdict: A careful, honest extension of non-geometric fluxes that adds an S-dual P sector and finds Kahler-fixing Minkowski vacua, with the main caveat flagged by the authors themselves.

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arXiv IDhep-th/0602089 (version 1)

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CitationA. Font, G. Aldazabal, L.E. Ibanez, P.G. Camara. "More Dual Fluxes and Moduli Fixing." arXiv:hep-th/0602089 (2006). https://pith.science/paper/QENKHOQO

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Editor's note

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The new content is the S-dual P flux sector in §4 and the Minkowski vacua in §5 where a Kahler modulus is fixed. The Q fluxes are Shelton-Taylor-Wecht; the paper says so up front. What's added is the observation that once Q is present, S-duality forces a partner P with the same tensor structure, giving W = ∫[(F3 - iSH3) + (Q - iSP)Jc]∧Ω. That's a clean idea and the superpotential terms in (4.9) are explicit enough to check by hand. The T-duality dictionary in tables 1-3 is careful, and the superpotentials (3.4), (3.12), (3.17) actually match under the stated exchanges. The Bianchi solutions in table 4 are explicit and the §5 examples give concrete numbers. The S-dual tadpole triplet in (4.11) lines up with the known Meessen-Ortin / Dall'Agata-Lechner-Tonin triplet of 8-forms, which is a real consistency check, not decoration. Citations are appropriate throughout; nothing looks padded or missing. On the stress-test concern: it's right that the algebra (3.22) doesn't include P, so the Jacobi identities generating (4.14), (4.15), (4.31) aren't computed, and if P enters [X,X] as Q+iP the Jacobi gives Q·Q = P·P rather than the authors' three separate constraints. But that cuts the other way: the authors' set is stronger than the naive algebra, so the risk is that they miss vacua, not that they include inconsistent ones. The §5 vacua solve a subset of what the weaker algebra would allow, so they aren't obviously wrong. The real soft spot is the one the authors name in §4.2: no independent 10D derivation of the P constraints. That's a completeness gap, not a soundness gap. Second soft spot: the isotropic truncation. The abstract says Kahler moduli are fixed; the paper itself notes that with independent Ti only one linear combination is fixed. That's a real weakening of the headline claim and worth saying plainly. Third: no vacua with full tadpole cancellation, acknowledged. Who it's for: anyone working on non-geometric fluxes or S-duality in flux compactifications. The 128-flux SL(2,Z)^7 conjecture in §6 is speculative but clearly flagged as such. Recommendation: send it to referees. The core proposal is worth the referee time even if the constraint derivation needs an independent check.

Referee report

Significance

If correct, the paper is a substantial contribution. The T-duality dictionary of tables 1–3 is a genuinely useful, checkable object, and the observation that the IIA and IIB superpotentials match only after Q fluxes are included is a clean and falsifiable statement. The S-dual P-flux proposal is more speculative but is presented with an explicit superpotential, explicit tadpole conditions, and explicit Bianchi constraints, so it can be tested rather than merely asserted. The Minkowski vacua of section 5, if they survive scrutiny, would be a real step beyond the Kachru–Schulz–Trivedi IIB vacua in that a Kähler modulus is fixed by fluxes alone. The paper is also candid about what it does not derive, which is a strength: the limitation in section 4.2 is stated in the authors' own words rather than hidden. The comparison with M-theory and heterotic compactifications in section 6 is a useful organising principle, and the identification f_i = ω_{K+1 K 11} of eq. (6.5) is a concrete, checkable statement that ties the new fluxes to ordinary metric fluxes in a dual frame.

Major objections (4)

[Abstract and §5] The abstract states that the new vacua fix 'not only dilaton and complex moduli but also Kähler moduli' (abstract:s2), but the explicit construction in §5 is carried out only in the isotropic truncation T_i = T, U_i = U with the flux ansätze (3.5) and (4.20), and the authors themselves state in the paragraph following eq. (5.25) that 'if we analyze the more general case with independent T_i, U_i fields, generically only one linear combination of the Kähler moduli T_i is fixed.' These two statements are in tension at the level of the headline claim. Either the abstract should be qualified to say that a Kähler modulus (or one combination) is fixed in the isotropic slice, or the paper should exhibit at least one non-isotropic solution in which more than one combination of the T_i is fixed. As written, a reader who takes the abstract at face value will overestimate what has been demonstrated.
[§4.2, eqs. (4.14), (4.15), (4.31)] The authors acknowledge in §4.2 that 'we do not have at the moment a prescription, e.g. by doing generalized dimensional reduction of a 10-dimensional theory, to obtain the S-dual equivalent of the Bianchi identities,' and that the constraints on P are obtained by taking SL(2,Z)_S orbits of the known Q constraints. That acknowledgment is appropriate and I do not treat it as a hidden defect. What remains unresolved is a structural point that the manuscript does not address: the extended algebra (3.22) contains H and Q but not F3 and P, so it is not manifestly SL(2,Z)_S covariant, and the Jacobi identities that would generate (4.14), (4.15) and (4.31) are therefore not exhibited anywhere. Since the vacua of §5 are obtained by solving precisely these extrapolated constraints (table 4), it would materially strengthen the paper to display the S-dual completion of (3.22) — with (F3, H3) and (Q, P) as doublets — and to verify that its Jacobi identities reproduce (4.14), (4.15) and (4.31) and do not produce additional conditions. If such a completion is not available, the §5 results should be explicitly labelled as conditional on the conjectured constraint set, both in §5 and in the abstract.
[§5, eq. (5.18)] In the first worked example the tadpole conditions of eq. (5.18) read N_D3 − 288 = 16, −N_D7 + 80 = 0, −N_NS7 + 40 = 0 and N_I7 + 32 = 0. The last of these requires N_I7 = −32, i.e. a negative multiplicity of I7-branes, which is not a physical source count unless there is an orientifold-like object of that type in the spectrum. The text discusses the signs of the C4, C8 and C̃8 flux contributions at some length but does not comment on this entry. Please state how N_I7 = −32 is to be interpreted, or replace the example with one for which all four source multiplicities are non-negative. The same question should be checked for the second example of eq. (5.19) and for eq. (5.26), where N_I7 + 112 = 0 likewise gives a negative multiplicity.
[§5, discussion of tadpole signs] The paper reports the 'surprising result' that with S-dual fluxes the C4 tadpole can receive a contribution of either sign, in contrast to the standard ISD statement that the H3–F3 contribution is always positive. This claim is used to argue for model-building relevance, so it deserves a direct cross-check. Since the P fluxes have no known ten-dimensional origin, the supersymmetry of these configurations is established only at the level of the four-dimensional superpotential; it would be valuable to show, for at least one example, that the corresponding ten-dimensional flux configuration satisfies the expected supersymmetry conditions (or to explain why such a check is not currently possible). Absent that, the sign statement should be phrased as a property of the four-dimensional effective theory rather than as a statement about the ten-dimensional tadpole.

Minor objections (8)

[§2.1, eq. (2.3)] The definition of the complex structure parameter τ_j in eq. (2.3) is garbled in the current text ('τj = 1e2jx (Aj + i ejx· ejy )'); please rewrite it so that the normalisation by the lattice vector lengths is unambiguous.
[§3.1, eq. (3.5)] The isotropic flux ansatz contains the self-referential line 'h̄_i = h̄_i'; presumably this should read h̄_i = h̄. Please correct.
[§2.2] The statement that 'to avoid subtleties with exotic orientifold planes we take the flux integers to be even' is used throughout but not justified; a sentence of explanation or a citation would help the reader judge how restrictive this is for the vacua of §5.
[Table 2] The caption describes the table as 'Non-geometric IIB/O3 fluxes and their T-duals', but the table also contains the metric fluxes ω and the parameters b_ij, b̄_ij. A caption that names all three classes would make the dictionary easier to use.
[§7] The section heading reads 'Final comments and conclussions'; please correct the spelling.
[§1] In the second paragraph of the introduction the text reads 'were rather neglected in the past b ut are are starting to receive more attention'; please remove the duplicated 'are'.
[Appendix, table 6] The spinorial embedding table is difficult to read in its current form, with the brace constructs obscuring which entries carry which signs. A cleaner rendering, or a worked example mapping one flux multiplet onto its weights, would make the appendix usable.
[§5, general] It would help to state explicitly, near the start of §5, that the examples presented are not tadpole-free and require the addition of localised sources, and to say which of the four tadpole conditions each example leaves to be cancelled by branes.

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The paper

The paper claims

Type IIB S-duality forces a second family of non-geometric fluxes, and with it the Kähler moduli can be fixed.

The reading

The paper asks what happens to the flux superpotential of a type IIB toroidal orientifold once the previously proposed non-geometric $Q$ fluxes are included. Those fluxes restore T-duality with the type IIA side, but they break the S-duality that type IIB string theory has built in. The authors argue that S-duality can only be recovered by adding a partner set of fluxes, called $P$, transforming together with $Q$ as a doublet under $SL(2, Z)$, and they write down the resulting superpotential. With these extra terms the superpotential becomes linear in the Kähler moduli $T_i$, so a search for $N=1$ supersymmetric Minkowski vacua can fix not only the dilaton and complex structure moduli but also, in the isotropic case, the Kähler modulus. The paper closes by arguing that matching against M-theory on twisted seven-tori and heterotic compactifications points to a much larger set of 128 fluxes carrying an $SL(2, Z)^7$ structure.

What carries the argument

The machinery is a T-duality chain relating $H_{MNP}$, the metric flux $omega_{MN}{}^P$, the non-geometric $Q_M{}^{NP}$ and $R_{MNP}$ across the type IIA/O6, type IIB/O3 and type IIB/O9 orientifolds, promoted to an $SL(2, Z)_S$ doublet structure by pairing $Q$ with a new $P$. Contraction rules such as $(QX)_{L M_1 ... M_{p-2}} = (1/2) Q^{AB}{}_{[L} X_{M_1 ... M_{p-2}]AB}$ let the fluxes be wedged with the complexified forms $Omega$, $J_c$, $J_c^{(2)}$ and $J_c^{(3)}$ to produce superpotential terms and tadpole contributions as integrals over $T^6$. The same contraction, applied to the extended algebra of isometry and B-field generators $[Z_M, Z_N]$, $[Z_M, X^P]$ and $[X^M, X^N]$, generates the Bianchi identities that constrain the fluxes, and $SL(2, Z)_S$ covariance then dictates the constraints obeyed by $P$ and by mixed $Q$-$P$ combinations.

Extended reading notes

Core claim

The central claim is that the non-geometric flux programme is incomplete without a second tensor, $P^{MNP}$, with exactly the same index structure as the non-geometric $Q^{MNP}$ but opposite behaviour under S-duality. The proposed superpotential in eq. (4.5) pairs $(F_3, H_3)$ and $(Q, P)$ as $SL(2, Z)$ doublets, so that $S$ to $1/S$ exchanges $F_3$ with $H_3$ and $Q$ with $-P$. The $P$ terms contribute monomials of the form $S T_i$ times a cubic polynomial in the $U_j$, which is precisely the ingredient missing from earlier superpotentials that were at most quadratic in the $T_i$. Using an isotropic ansatz with $T_i = T$ and $U_i = U$, the authors exhibit explicit Minkowski vacua with all moduli fixed, for instance taking $a = -8$, $b = -4$, $h = -4$, $gamma = -4$, $bar a = 12$, $h_0 = 8$ and $e = -16$ gives $U = S = sqrt{2}$ and $T = (14 sqrt{2} + 16 i)/9$. They also report that the flux contribution to the $C_4$ tadpole can take either sign, unlike the purely imaginary-self-dual case where it is always positive.

Load-bearing premise

The constraints obeyed by the new $P$ fluxes, and hence the vacua built on them, are obtained by applying S-duality to the known $Q$-flux Bianchi identities rather than derived independently from a ten-dimensional reduction, a gap the authors state explicitly in section 4.2.

Editorial extensions

If this is right

Every flux-induced superpotential term of the form $S T_i$ times a polynomial in the $U_j$ becomes available, so Kähler moduli need no longer be flat directions in type IIB toroidal orientifolds.
The $C_8$ and $tilde C_8$ tadpoles become an $SL(2, Z)_S$ triplet with $C_8$, sourcing D7-, NS7- and I7-branes, and the flux contributions to all three can have either sign, vanish, or cancel.
The Bianchi identities close under S-duality into a triplet of conditions $Q cdot Q = 0$, $P cdot P = 0$ and $Q cdot P + P cdot Q = 0$, plus the singlet $Q cdot H_3 - P cdot F_3 = 0$.
Comparison with M-theory on $T^7/(Z_2 ^{-3})$ identifies the $f_i$ fluxes with ordinary metric fluxes $omega_{K+1,K,11}$ of the seven-torus, and heterotic self-T-duality requires still further fluxes $F'_3$, $Q'$, $H'_3$ and $P'$.
The full duality-invariant superpotential is conjectured to contain $2^7 = 128$ flux parameters, one for each monomial in the seven moduli that is at most linear in each field, filling a spinorial representation of $SO(7,7;Z)$.

Reading between the lines

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

If the $P$ fluxes are genuine, the same doublet structure should appear in the four-dimensional gauged supergravity description, where $Q$ and $P$ would be electric and magnetic counterparts of one another; checking the embedding tensor against the paper's Bianchi identities would be an independent test.
The observation that flux tadpoles can mimic orientifold planes rather than branes suggests a model-building route: flux contributions of the wrong sign could cancel the charge of an orientifold plane locally, allowing brane configurations that would otherwise be forbidden by tadpole cancellation.
Because the superpotential is only linear in the $T_i$, at most one linear combination of the three Kähler moduli is fixed in the general non-isotropic case; the remaining flat directions may be exactly what is needed for the Green-Schwarz mechanism that gives mass to anomalous $U(1)$ gauge bosons on D-branes.
The paper's duality-extrapolated constraints are a natural target for a direct worldsheet or ten-dimensional derivation; if such a derivation produces a different set of conditions on $P$, the numerical vacua in section 5 would have to be revisited.

Original abstract

We generalize the recent proposal that invariance under T-duality leads to additional non-geometric fluxes required so that superpotentials in type IIA and type IIB orientifolds match. We show that invariance under type IIB S-duality requires the introduction of a new set of fluxes leading to further superpotential terms. We find new classes of N=1 supersymmetric Minkowski vacua based on type IIB toroidal orientifolds in which not only dilaton and complex moduli but also Kahler moduli are fixed. The chains of dualities relating type II orientifolds to heterotic and M-theory compactifications suggests the existence of yet further flux degrees of freedom. Restricting to a particular type IIA/IIB or heterotic compactification only some of these degrees of freedom have a simple perturbative and/or geometric interpretation.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We generalize the recent proposal that invariance under T-duality leads to additional non-geometric fluxes required so that superpotentials in type IIA and type IIB orientifolds match. We show that invariance under type IIB S-duality requires the introduction of a new set of fluxes leading to further superpotential terms. We find new classes of N=1 supersymmetric Minkowski vacua based on type IIB toroidal orientifolds in which not only dilaton and complex moduli but also Kahler moduli are fixed. The chains of dualities relating type II orientifolds to heterotic and M-theory compactifications suggests the existence of yet further flux degrees of freedom. Restricting to a particular type IIA/IIB or heterotic compactification only some of these degrees of freedom have a simple perturbative and/or geometric interpretation.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We generalize the recent proposal that invariance under T-duality leads to additional non-geometric fluxes required so that superpotentials in type IIA and type IIB orientifolds match.	contribution 0.84	0.60	0.15	asserted 0.92
s1	We show that invariance under type IIB S-duality requires the introduction of a new set of fluxes leading to further superpotential terms.	contribution 0.59	0.93	0.20	asserted 1.00
s2	We find new classes of N=1 supersymmetric Minkowski vacua based on type IIB toroidal orientifolds in which not only dilaton and complex moduli but also Kahler moduli are fixed.	result 0.97	0.95	0.47	asserted 1.00
s3	The chains of dualities relating type II orientifolds to heterotic and M-theory compactifications suggests the existence of yet further flux degrees of freedom.	interpretation 0.67	0.16	0.08	hedged possibility 1.00
s4	Restricting to a particular type IIA/IIB or heterotic compactification only some of these degrees of freedom have a simple perturbative and/or geometric interpretation.	interpretation 0.49	0.28	0.06	asserted 0.95

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.95.

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sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
relations to other papers

levels run

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K. Becker, M. Becker, K. Dasgupta and S. Prokushkin, Properties of heterotic vacua from superpotentials, Nucl. Phys. B666 (2003) 144, hep-th/0304001

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S. Gurrieri, A. Lukas and A. Micu, Heterotic on half-ﬂat , Phys. Rev. D70 (2004) 126009, hep-th/0408121

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How the literature uses it

86 arXiv papers cite this in 319 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 · 233 sentences without a confident label

Contests it 1 paper

However, the Minkowski vacua given in [ref] are not under good perturbative control.

Searching for Inflation in Simple String Theory Models: An Astrophysical Perspective · The potential energy challenge · contest 0.75

Uses its method or tool 7 papers

-1.0cm V_IIB^tad = 1/2 e^K^(Q) ∫_X_3 ⟨ [Im Φ_c^even ] , D F_RR ⟩, where the multi-form $D F_RR$ can be expanded using the flux actions in the generalized twisted differential operator given as under [ref] ,

Dictionary for the type II nongeometric flux compactifications · Tadpoles cancellation conditions and Bianchi identities : · 0.85
One way to derive these identities is via the Jacobi identities of the following Lie brackets for the NS-NS fluxes [ref] , [Z_i, Z_j ] = ω_ij^k Z_k - H_ijk X^k, [ Z_i, X^j ] = Q_i^jk Z_k - ω_ik^j X^k , [ X^i, X^j] = Q_k^ij X^k - R^ijk Z_k , where $Z_i$ and $X^i$'s are generators of the gauge transformations corresponding to the two gauge groups consisting of two sets of $d$-dimensional vectors obtained, from the metric and the $B$-field respectively, via the reduction of type II superstring theory on a $d$-dimensional torus.

On Missing Bianchi Identities in Cohomology Formulation · First formulation · 0.83
We will find that the general NS-NS fluxes are most naturally parametrized by their action on the untwisted cohomology, along the lines described in [ref] .

Toroidal Orientifolds in IIA with General NS-NS Fluxes · Introduction · 0.87

Uses its data or values 1 paper

D3-QP which are taken from [ref] .

IIB String Theory and Sequestered Inflation · Effective 4D theory and tadpole cancellation conditions · 0.91

Compares with it 7 papers

Comparing this potential, using the decomposition ([ref]) and the parametrization ([ref]), with those obtained from Scherk-Schwarz reductions with fluxes [ref] as well as with their non-geometric generalizations [ref] , leads us to identify $H_abc$ with the standard NS-NS flux, $τ_ab^abc$ with the geometric flux, $Q_c^cab$ with the locally geometric flux and $R^abc$ with the so-called non-geometric flux.

Conformal chiral boson models on twisted doubled tori and non-geometric string vacua · Twisted doubled tori and $N=4$ gauged supergravities · 1.00
In order to explicitly determine the structures of the $P^', Q^', H^'$ and $F^'$ flux components, we compare our results with the non-symplectic formulation presented in [ref] .

Symplectic formulation of the type IIB scalar potential with U-dual fluxes · More insights of the flux components in the cohomology basis · 0.97
In particular, one may check that ([ref]) reproduces the results derived by similar arguments in [ref] for the case of $Γ=ℤ_2×ℤ_2$ and $G=SL(2,ℝ)^7$.[For completeness we present here the embedding of $SL(2,ℝ)^7⊂E_7$ selected by the $ℤ_2×ℤ_2$ orbifold.

U-dual fluxes and Generalized Geometry · Type IIB orientifold compactifications · 0.97

Cites it as support 1 paper

It turns out that adding these extra ingredients really does alleviate some of the problems mentioned above [ref] .

D-Terms from Generalized NS-NS Fluxes in Type II · Introduction · 1.00

Cites it as background 41 papers

They arise from type IIB generalised flux compactifications on $𝕋^6/ (ℤ_2 ×ℤ_2)$ isotropic orientifolds with O3/O7-planes [ref] .

Complete classification of Minkowski vacua in generalised flux models · The $𝒩=1$ SUGRA models · 0.86
We are working with a type IIB string compactification on the $ℤ_2 ×ℤ_2$ orbifold considered in [ref] , where the space-time topology is $ℳ_10=ℳ_4 ×ℳ_6$, $ℳ_6$ our compact internal space, initially a $6$-torus, $𝕋^6$ and $ℳ_4$ the noncompact space-time.

Non-geometric flux vacua, S-duality and algebraic geometry · Conventions and 3-form fluxes. · 0.91
Implementing the successive chain of T- and S-dualities, leads to a U-dual completion of the flux superpotential which has been studied in [ref] .

Symplectic formulation of the type IIB scalar potential with U-dual fluxes · Introduction · 0.81

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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