Anti-D3 branes and moduli in non-linear supergravity

Published for SISSA by Springer Received: August 14, 2017 Accepted: September 25, 2017 Published: October 26, 2017 Anti-D3 branes and moduli in non-linear supergravity a Departamento de Fisica, Universidad de Antofagasta, Antonio Toro 851, Casilla 170, Antofagasta, Chile b Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool, L69 7ZL U.K. c Departamento de Ciencias de la Naturaleza, CUSUR, Universidad de Guadalajara, Enrique Arreola Silva 883, C.P. 49000, Cd. Guzmán, Jalisco, México d Department of Physics, Swansea University, Swansea, Singleton Park, SA2 8PP U.K. E-mail: , , , Abstract: Anti-D3 branes and non-perturbative effects in flux compactifications spontaneously break supersymmetry and stabilise moduli in a metastable de Sitter vacua. The low energy 4D effective field theory description for such models would be a supergravity theory with non-linearly realised supersymmetry. Guided by string theory modular symmetry, we compute this non-linear supergravity theory, including dependence on all bulk moduli. Using either a constrained chiral superfield or a constrained vector field, the uplifting contribution to the scalar potential from the anti-D3 brane can be parameterised either as an F-term or Fayet-Iliopoulos D-term. Using again the modular symmetry, we show that 4D non-linear supergravities that descend from string theory have an enhanced protection from quantum corrections by non-renormalisation theorems. The superpotential giving rise to metastable de Sitter vacua is robust against perturbative string-loop and α′ corrections. Keywords: D-branes, Flux compactifications, Supergravity Models, Supersymmetry Breaking ArXiv ePrint: 1707.07059 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP10(2017)185 JHEP10(2017)185 Maria P. Garcia del Moral,a Susha Parameswaran,b Norma Quirozc and Ivonne Zavalad Contents 1 2 Spontaneous supersymmetry breaking by D3-branes 2.1 Setup 2.2 D3-brane in orientifolded flat space 2.3 D3-brane in flux compactifications 4 5 6 7 3 Non-linear supergravity from D3-branes 3.1 Constrained superfields and their couplings to supergravity and matter 3.2 Non-linear supergravity for KKLT 3.2.1 Modular invariance 3.2.2 Nilpotent chiral superfield and modular invariance 3.2.3 Modular invariance with constrained superfields X or V 3.2.4 Equivalence between the F-term and FI D-term uplift 10 10 13 14 16 19 22 4 Non-renormalisation theorem 4.1 R-symmetry and Peccei-Quinn symmetry 4.2 Spontaneously broken SL(2, R) and spurions 4.3 Proof of the non-renormalisation theorem 22 23 23 24 5 Discussion 25 A Notation and conventions 27 B Modular transformation of the worldvolume fermion 28 1 Introduction One of the main challenges in connecting String Theory to our observed Universe is to provide a string theoretic description of the early and late time accelerated expansions. This requires us to identify well-controlled string theory vacua whose 4D geometry corresponds to de Sitter (dS) or quasi-dS, with all moduli stabilised. Moduli stabilisation into a dS vacuum has been notoriously difficult to achieve, and the no-go theorems [1, 2] made clear what ingredients would be necessary. In particular, taking the classical two-derivative 10D string supergravities, including localised Dp-brane, Dp-branes and Op-plane sources, the Einstein’s and dilaton equations imply that one needs negative tensions and negative internal curvature to source dS. A way to evade these restrictions is to include higher derivative corrections. –1– JHEP10(2017)185 1 Introduction 1 Alternative methods include D-term uplifting via gauge fluxes on wrapped D7-branes [3, 4], F-term uplifting via complex structure [5], α′ corrections to the Kähler potential in no-scale flux compactifications [6] and F-term uplifting from dilaton dependent non-perturbative terms [7]. 2 At energies above the visible sector superpartner masses, the latter can still be parameterised by soft susy breaking terms. –2– JHEP10(2017)185 Although this makes the explicit construction of dS vacua — and moreover metastable dS vacua — difficult, several mechanisms have been proposed. Arguably the most used construction is to uplift1 a Minkowski or adS vacuum to dS with the addition of a positive energy density from an D3-brane. For a small number of probe D3-branes at the tip of a highly warped throat, an effective field theory analysis shows that such a configuration is metastable [8]. There is a non-perturbative instability to antibrane-flux annihilation, but the timescale of this stability can be far longer than the age of the Universe. Moreover, if we place the D3-brane on top of an O3-plane, then any concerns about tachyonic instabilities that might appear when going beyond the probe approximation (see [9, 10] and references therein) are simply projected out. The original D3-brane uplift scenario, by Kachru, Kallosh, Linde and Trivedi (KKLT) [11], was presented in three steps. Firstly, a Giddings, Kachru, Polchinski (GKP) [12] type IIB flux compactification stabilises the dilaton and complex structure moduli in a non-supersymmetric vacuum. Next, the resulting runaway in the Kähler modulus is stabilised into a supersymmetry restoring vacuum by non-perturbative effects, such as gaugino condensation on wrapped D7-branes and/or Euclidean D3-branes. Finally, the supersymmetric adS vacuum is uplifted to a supersymmetry breaking dS vacuum by the D3-brane. Note that the dS vacuum is achieved with a combination of the D3-brane and non-perturbative effects — without the non-perturbative effects, the anti-brane would just give a runaway towards decompactification — so, as expected, quantum corrections are essential to evade the dS no-go theorems. The D3-brane, as well as uplifting the classical vacuum energy to dS, spontaneously breaks supersymmetry. Any string compactification with spontaneously broken supersymmetry would have a non-linearly realised local supersymmetry (“non-linear supergravity”) as its effective field theory description at energies below2 the mass of the goldstino’s superpartner (usually the sgoldstino). That is, the action is invariant under non-linear supersymmetry transformations, and the non-linear supersymmetry transformation for the goldstino implies that all solutions spontaneously break supersymmetry. The goldstino is eaten by the gravitino in the super-Higgs mechanism. Non-linear supergravity can be written in terms of non-linear or constrained supermultiplets, which contain a single elementary field (either bosonic or fermionic) and the goldstino. This superfield description makes it easy to couple to supergravity and matter, starting with [13, 14]. Recently, [15–18] computed the component form for supergravity coupled to a nilpotent chiral superfield, S 2 = 0, which carries the goldstino, and general matter. Although the original KKLT construction parameterised the D3-brane contribution to the (...truncated)