String moduli stabilization at the conifold

Published for SISSA by Springer Received: July 6, 2016 Accepted: August 11, 2016 Published: August 18, 2016 Ralph Blumenhagen, Daniela Herschmann and Florian Wolf Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München, Germany E-mail: , , Abstract: We study moduli stabilization for type IIB orientifolds compactified on CalabiYau threefolds in the region close to conifold singularities in the complex structure moduli space. The form of the periods implies new phenomena like exponential mass hierarchies even in the regime of negligible warping. Integrating out the heavy conic complex structure modulus leads to an effective flux induced potential for the axio-dilaton and the remaining complex structure moduli containing exponentially suppressed terms that imitate nonperturbative effects. It is shown that this scenario can be naturally combined with the large volume scenario so that all moduli are dynamically stabilized in the dilute flux regime. As an application of this moduli stabilization scheme, a string inspired model of aligned inflation is designed that features a parametrically controlled hierarchy of mass scales. Keywords: Flux compactifications, Cosmology of Theories beyond the SM, Superstring Vacua ArXiv ePrint: 1605.06299 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP08(2016)110 JHEP08(2016)110 String moduli stabilization at the conifold Contents 1 2 Type IIB fluxes on CY threefolds 2.1 Massless fields 2.2 Three-form flux 2.3 Large volume scenario 3 3 4 6 3 Moduli stabilization close to the conifold 3.1 Periods of the quintic 3.2 Flux induced exponential mass hierarchies 3.3 Conic LVS scenario 7 7 9 13 4 Towards axion alignment 4.1 Periods for P11226 [12] 4.2 Freezing axio-dilaton and complex structures 4.3 Axion alignment 15 15 17 19 5 Conclusions 24 1 Introduction Moduli stabilization is one of the most important challenges to relate compactifications of string theory to our four-dimensional world. Despite its importance, we think it is fair so say that comparably few concrete scenarios have been discussed in the literature. The ingredients used are tree-level fluxes as well as perturbative and non-perturbative corrections to the leading order quantities. The work of Giddings-Kachru-Polchinski (GKP) [1] revealed that Type IIB compactifications on warped Calabi-Yau threefolds equipped with localized sources and NS-NS and R-R three-form fluxes are leading order solutions to the string equations of motion. There, a scalar potential for the complex structure and the axio-dilaton moduli is generated, while due to its no-scale structure, the Kähler moduli remain as massless moduli. It is from this scenario that the landscape idea of string vacua arose. In a second step, by turning on non-perturbative corrections to the superpotential, the no-scale structure was broken and the Kähler moduli could be stabilized. This led to the KKLT [2] and the large volume scenario (LVS) [3]. In [1] it was pointed out that one can dynamically freeze the complex structure moduli in the vicinity of a conifold singularity. For that purpose, as for the Klebanov-Strassler throat [4], one turns on a three-form flux on the three-cycle that vanishes at the conifold –1– JHEP08(2016)110 1 Introduction –2– JHEP08(2016)110 locus and an additional flux on its symplectic dual three-cycle. In this case, at the tip of the throat the warp factor becomes large, which leads to red-shifted masses of the modes localized there. From the statistical analysis [5] it even followed that the number of vacua enhances close to a conifold locus. This was explicitly verified for a concrete example in [6]. In this respect it is important to keep in mind that this analysis was employing the usual effective supergravity action described by the leading order Kähler potential and the Gukov-Vafa-Witten (GVW) superpotential [7, 8]. In the case of strong warping, this action is not any longer expected to be trustable as, due to red-shifting, certain KaluzaKlein modes can become light and one can have non-trivial mixings among the modes. The effective action for warped compactifications was studied in [9–12]. If one wants to use the standard supergravity action one has to ensure that one is working consistently in a dilute flux limit, where the backreaction is suppressed. In this paper we systematically study such compactifications close to a conifold locus in the complex structure moduli space in the dilute flux limit. Our main concern is to analyze the appearing moduli mass scales and the implications for string cosmology model building. It is a known and celebrated result that, including a non-trivial warp factor, exponential hierarchies among the masses of the moduli are generated. This is true in particular for modes localized in a strongly warped throat. As a matter of fact, this is the idea of the Randall-Sundrum scenario [13]. Inflationary models in warped throats with red-shifted inflaton masses have recently been discussed in [14–16]. In this paper we find that even in the dilute flux limit, where the usual effective supergravity theory is applicable, exponential mass hierarchies between bulk modes are generated. This observation is very interesting, as it allows to generate exponential hierarchies in a controllable supergravity theory, a problem that was recently of crucial relevance for potential realizations of large-field inflation in concrete string theory set-ups. To build models with large tensor-to-scalar ratios r > 0.01 one needs a rolling of the inflaton over trans-Planckian field ranges, which are hard to control in a perturbation expansion (see e.g. [17] for a string realization). Here the perturbative shift symmetry of axions does help and various models of axion inflation were proposed (see e.g. [18–23] and [24] for review). However, it turned out to be quite a difficult task to dynamically stabilize all the moduli at a higher mass scale than the inflaton in a controllable way [25–27]. Moreover, such models of axion inflation came under pressure also via the weak-gravity conjecture (WGC) [28–31]. Having this motivation in mind, in this paper we will proceed as follows: in section 2 we briefly provide the main ingredients for Type IIB moduli stabilization, including geometric fluxes and instanton effects. For the quintic we recall the form of the periods close to the conifold singularity. The distinguished new issue is the appearance of a logarithmic term for a certain period. Computing the Kähler potential close to the conifold point, one realizes the appearance of axion-like shift-symmetries [32], making this regime interesting for realizing large field inflation. The latter has been subject of intense studies during the last two years, where it has become clear that the weak-gravity conjecture and concrete model building attempts severely constrain the viability of such models. In section 3, we fi (...truncated)