Compatibility of the integrated Gibbons-Maldacena-Nunez constraint with the four-dimensional trace constraint in brane-uplifted de Sitter vacua of type IIB string theory

In the KKLT construction, metastable de Sitter (dS) vacua of type IIB string flux compactifications are produced by uplifting a supersymmetric anti-de Sitter vacuum with an anti-D3-brane (or a magnetized-D7-brane D-term). Cabo Bizet and Hirano (arXiv:1607.01139) showed that two constraints following from the ten-dimensional type IIB supergravity equations, integrated over the internal Calabi-Yau space, then pull in opposite directions: the Gibbons-Maldacena-Nunez (GMN) no-go inequality demands net negative localized-source contributions, while the inequality obtained from the trace of the four-dimensional Einstein equations demands net positive contributions; positive-tension uplift sources help the latter and aggravate the former. The problem asks whether the two integrated inequalities can be satisfied simultaneously in a controlled compactification with brane-based uplifts (or whether a general impossibility argument exists), whether the proposed 'reversed-KKLT' construction - fixing complex structure and dilaton at a positive-potential no-scale point before adding non-perturbative and alpha-prime/string-loop corrections that fix the Kaehler moduli and render the GMN inequality consistent - yields controlled metastable dS vacua, and whether a full ten-dimensional treatment of localized-source singularities and anti-D3 backreaction resolves or modifies the obstruction. The question remains unresolved in the literature.

Background

Superstring theory is consistent in ten spacetime dimensions, so any application to our world requires six dimensions to be compactified on a microscopic internal manifold. In type IIB string theory the most studied route is the flux compactification of Giddings, Kachru and Polchinski (GKP): the six extra dimensions form a Calabi-Yau orientifold - a Calabi-Yau manifold (a compact six-dimensional Ricci-flat space with special holonomy) modded out by an orientation reversal, which forces the presence of orientifold planes (O-planes, extended defects carrying negative tension and negative D3 charge) - together with D-branes (extended objects on which open strings end) and quantized three-form fluxes $G_3=F_3-\tau H_3$ threading internal cycles, where $\tau$ is the axio-dilaton (the string coupling combined with an axion). The fluxes generate a potential that fixes the complex-structure (shape) moduli and $\tau$, while the Kaehler moduli (volumes of internal even-dimensional cycles) remain unfixed at leading order; the low-energy four-dimensional (4d) description is $\mathcal{N}=1$ supergravity with scalar potential $V$.

Kachru, Kallosh, Linde and Trivedi (KKLT) completed the stabilization and produced de Sitter (dS) space - the maximally symmetric spacetime of positive cosmological constant approximating our universe's far future - in two steps. First, non-perturbative effects (gaugino condensation on D7-branes wrapping an internal four-cycle, or Euclidean D3-brane instantons) fix the Kaehler moduli at a supersymmetric anti-de Sitter (AdS, negative cosmological constant) vacuum. Second, an anti-D3-brane, a brane with negative D3 charge and positive tension, is placed at the tip of a strongly warped throat region (the Klebanov-Strassler solution is the prototypical example); its positive energy density uplifts the AdS minimum to a metastable dS vacuum. A variant due to Burgess, Kallosh and Quevedo replaces the anti-D3-brane by magnetic flux on D7-branes, whose D-term (the auxiliary scalar potential of a gauged symmetry) supplies the uplifting energy. If such vacua are abundant they underlie the string landscape and anthropic reasoning about the small observed cosmological constant; whether they actually exist in the full ten-dimensional (10d) theory is among the most contested questions in string theory.

Two integrated constraints on any dS vacuum follow from the 10d type IIB supergravity equations. The first, due to Gibbons and independently to Maldacena and Nunez (GMN), comes from integrating the trace of the four-dimensional components of the 10d Einstein equations over the internal manifold: for a compactification with positive 4d Ricci scalar $R_4$, the positive contributions - the potential $V$, the flux and warp-factor terms, and the positive tension of D-branes - must be outweighed by negative localized contributions, normally supplied by O-planes; this is the classical no-go theorem that GKP/KKLT-type compactifications evade precisely because they contain O-planes. The second constraint, emphasized by Cabo Bizet and Hirano, comes from the trace of the 4d Einstein equations, $R_4=4\kappa_4^2 V+\cdots$ with $\kappa_4$ the 4d gravitational coupling, which is itself an integral of a specific linear combination of the same 10d equations over the internal space; for a dS vacuum it requires positive total contributions.

Cabo Bizet and Hirano observe that these two requirements pull in opposite directions under brane-based uplifts. Writing $T^{\rm loc}_4$ for the trace of the localized sources' energy-momentum tensor over the 4d spacetime (entering the GMN combination) and $T^{\rm loc}$ for the corresponding internal combination (entering the 4d-trace combination), a dS vacuum with corrections retained must satisfy simultaneously the two sign conditions of their eqs. (4.1)-(4.2), which follow from the genuinely integrated constraints (their eqs. (3.1) and (3.3), weighted internal integrals with $R_4$ inside the integrand) once a positive 4d Ricci scalar $R_4>0$ is imposed:

$2\kappa_{10}^2 e^{2A}\left(V+V'_{\rm np}+V'_{\alpha'+g_s}\right)+2\kappa_{10}^2 e^{2A}\,T^{\rm loc}_4+e^{-4A}(\tilde\nabla\alpha)^2\;<\;0,$

$4\kappa_{10}^2 e^{2A}\left(V+V_{\rm np}+V_{\alpha'+g_s}\right)+4\kappa_{10}^2 e^{2A}\,T^{\rm loc}+e^{-4A}(\tilde\nabla\alpha)^2+\frac{e^{4A}|\tilde\nabla\tau|^2}{({\rm Im}\,\tau)^2}-2e^{4A}\tilde R_6\;>\;0,$

where $A(y)$ is the warp factor, $\tilde\nabla$ denotes the covariant derivative of the internal metric, $\alpha$ is the internal scalar sourcing the self-dual Ramond-Ramond five-form flux (the axionic real part of the complexified Kaehler modulus $\rho$), $\tilde R_6$ the possibly non-zero Ricci scalar of the internal manifold in the presence of 7-branes, and $V_{\rm np}$, $V_{\alpha'+g_s}$ the non-perturbative and $\alpha'$ (higher-derivative, inverse string tension) plus string-loop ($g_s$) corrections, which enter the two combinations differently. In these displays $V$ denotes the source's ten-dimensional potential density $\mathcal{V}$, whose internal integral $V=({\rm Im}\,\rho)^{-3}\int d^6y\sqrt{\tilde g}\,\mathcal{V}$ gives the 4d scalar potential defined above (source, eq. (2.4)); the primed terms $V'_{\rm np}$, $V'_{\alpha'+g_s}$ entering the first (GMN) inequality are the corresponding corrected potential-density contributions in the source's eq. (3.5), and they differ from the unprimed terms entering the second inequality through the implicit Weyl rescaling of $R_4$ from the 10d to the 4d Einstein frame and through the three-form-flux contributions to $\mathcal{V}$ (source, footnote 8); the source gives no closed-form expressions for the primed terms. Crucially, anti-D3-branes and the magnetized-D7 D-term shift both $T^{\rm loc}_4$ and $T^{\rm loc}$ positively: they help the second inequality while aggravating the first, and the negative O-plane contributions needed for the first are already tied up in the D3 tadpole-cancellation condition (the Gauss-law constraint that the total D3 charge, including flux-induced and curvature-induced charges, sums to zero). Cabo Bizet and Hirano conclude that in brane-based uplift scenarios the two constraints are 'very hard, if not impossible' to satisfy simultaneously, while explicitly weakening the claim: they did not prove the two constraints incompatible. They also point to a possible escape route: reversing the KKLT order of operations. One starts from no-scale compactifications in the manner of Saltman and Silverstein, in which the complex structure and dilaton are stabilized at a positive-potential local minimum while the Kaehler moduli are still unfixed, and only afterwards adds non-perturbative and/or $\alpha'/g_s$ corrections so that the Kaehler moduli settle at fixed values, hoping the GMN inequality becomes consistent while the vacuum energy stays positive. Finally, they stress that their manipulations treat localized sources and warp-factor singularities schematically: a rigorous treatment of brane-source singularities - including the backreaction of anti-D3-branes and magnetized D7-branes, possibly through an M-theory lift of the type IIB background - is required before the obstruction can be considered definitive.

Problem Statement

Determine whether brane-based uplifts to metastable de Sitter vacua in type IIB flux compactifications are consistent with the full ten-dimensional equations of motion, in the precise sense of Cabo Bizet and Hirano (arXiv:1607.01139): for a GKP/KKLT-type compactification whose 4d effective theory admits an uplifted dS minimum (so the 4d-trace constraint of the source holds - eq. (4.2) in sign-condition form, following from the integrated constraint (3.3)), decide whether the two integrated constraints of the source (eqs. (4.1) and (4.2) in sign-condition form, following from the integrated constraints (3.1) and (3.3)) can hold simultaneously in the presence of brane-based uplift sources, or whether a general impossibility argument covering such settings holds. The sub-questions of this single objective are as follows:

(1) Compatibility: exhibit a concrete compactification - with specified internal topology, quantized fluxes, brane and orientifold content, and a stated treatment of non-perturbative, $\alpha'$ and string-loop corrections - in which the GMN constraint (eq. (4.1) of the source in sign-condition form, following from the integrated constraint (3.1)) and the 4d-trace constraint (eq. (4.2), following from (3.3)) are satisfied simultaneously, together with the D3 tadpole-cancellation condition, the flux Bianchi identities, and control hierarchies (weak coupling, large volume, validity of the effective theory); and show that the resulting vacuum is metastable against all scalar directions.

(2) Impossibility: prove, within a stated class of compactifications and retained corrections that does not exclude the localized-source and $\alpha'/g_s$-corrected settings in which the obstruction is contested, that the two constraints cannot hold simultaneously in the presence of brane-based uplift sources (anti-D3-branes or magnetized-D7 D-terms), i.e. that the apparent obstruction is generic and not an artifact of approximations such as smearing the sources or truncating $\alpha'/g_s$ corrections.

(3) Reversed-order settings: whether the two integrated constraints can hold simultaneously in the class of no-scale compactifications in which the complex-structure moduli and axio-dilaton are stabilized at a positive-potential local minimum (in the sense of Saltman and Silverstein) before the Kaehler moduli are fixed by non-perturbative and/or $\alpha'/g_s$ corrections, with the resulting vacuum metastable, controlled, and of positive vacuum energy.

(4) Ten-dimensional treatment: settle the correct ten-dimensional treatment of the localized sources - including the singularities and backreaction of anti-D3-branes and magnetized D7-branes - and determine whether it removes, modifies, or confirms the integrated constraints in a way that permits or definitively forbids brane-based uplifts.

The sub-questions above are parts of one objective: deciding whether the two integrated constraints derived from the ten-dimensional type IIB supergravity equations are compatible with the existence of brane-uplifted de Sitter vacua. That objective is decided by the joint status of the two integrated constraints of the source, eqs. (4.1)-(4.2) in sign-condition form, and by its consequences for the remaining sub-questions; the ten-dimensional treatment of the sources (4) bears on it only insofar as it settles whether brane-based uplifts are permitted or forbidden.

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