Meromorphic flux compactification

Received: February Published for SISSA by Springer Open Access c The Authors. 0 Departamento de F sica, Universidad de Guanajuato We present exact solutions of four-dimensional Einstein's equations related to Minkoswki vacuum constructed from Type IIB string theory with non-trivial uxes. Flux compacti cations; String Duality - Meromorphic ux compacti cation ering only 3-form uxes and the dilaton, as functions on the internal sphere coordinates, we by the use of G-theory. Meromorphicity on functions constructed in terms of uxes and warping factors guarantees that ux and 5-brane contributions to the scalar curvature vanish while ful lling stringent constraints as tadpole cancelation and Bianchi identities. Di erent Einstein's solutions are shown to be related by U-dualities. We present three soft terms. We also construct a non-supersymmetric solution and study its stability. 1 Introduction 2 Flux compacti cation 2.1 2.2 2.3 The anzatz Einstein equations Flux contribution Local source's contribution 2.5 Soft terms 3 SUSY solutions 3.1 Solution 1: H 6= 0 and F = 0 3.2 Solution 2: H = 0 and F 6= 0 Soft terms for H 6= 0 and F = 0 3.3 SUSY solution with H 6= 0 and F 6= 0 Soft terms for H = 0 and F 6= 0 Soft terms for H 6= 0 and F 6= 0 4 Non-SUSY case Soft terms for the non-SUSY case 4.2 Stability analysis 4.2.1 An example 5 Conclusions and nal remarks A Notation B Useful gamma identities C Non-zero components of spin connection D Global residue theorem D.1 Bianchi identity E E ective DBI theory String ux compacti cation has been extensively studied in the last decade opening up a constructed in the absence of uxes corresponding to Calabi-Yau compacti cations. Even more, ux compacti cation solves the so called moduli stabilization problem and gives us At the present stage it is commonly accepted that dS vacua can be gathered from a compacti cation in the presence of orientifold planes and anti D3 branes1 [5, 6] or by the inclusion of non-geometric uxes [7{9]. Also, it is well known that for supersymmetric ux backgrounds, Einstein's equation is satis ed if we demand Bianchi identity and su ux compacti cation brings a consistent scenario for dS (see for instance [10, 11] for some interesting discussions), the presence of localized sources introduce singular points at which the uxes, for most of the cases [5], have not an analytical expression and are not exact solutions of the equations of motion. This is a consequence of taking trivial uxes (not depending on internal coordinates or moduli), an assertion valid only in a dilute ux limit. Another problem faced by ux string compacti cation involves reproducing a minimal extension of the Standard Model of particles while preserving chirality and solving the hierarchy problem for the Higgs boson. However, as result of the last experiments run in the LHC, the possible presence of supersymmetry at low scales as TEV's is close to be discarded and therefore, supersymmetry appears to be non essential for solving the hierarchy problem. Although such a problem remains unsolved it opens up the possibility to scale is close to the string compacti cation scale [12]: Mp > Ms > MKK > Mcomp mSUSY > min aton: Hence, it is desirable to consider more generic ux scenarios which allow us to face these kind of problems. One possibility concerns turning on non-constant uxes. Compacti cation in the presence of uxes depending on the internal coordinates or moduli have been considered previously [13, 14], while examples of U-folds with ux in string theory and M-theory were studied in [15]. Further studies on non-trivial ux compacti cations were considered in [16{18]. More recently, it was constructed a family of exact solutions of compacti cations threaded by uxes depending on internal coordinates by [1, 2] and sourced by branes of diverse dimensionalities. Speci cally the authors show that for a compacti cation on a bered internal space given by a warped product of a four-dimensional torus of the sphere. In the same way as F-theory, these ux compacti cations with meromor1See for instance [3, 4] for recent discussions on possible classical extra constraints. (see [19, 20]), by replacing the tori by an auxiliary K3. In this work we study generic conditions upon which a ux con guration depending on the same internal coordinates of the sphere, with a similar compacti cation on T 4 satisfy Einstein's equations. By turning on 3-form uxes and the dilaton, sourced by 5and 7-branes respectively, we nd that a family of solutions of Einstein's equations are given precisely by meromorphic functions on S2. For that, we have make an extensive use of the Global Residual Theorem [21] in complex analysis, which states that on a compact space with singular points the total sum of residues related to meromorphic functions vanishes. This allows us to prove that by the simple use of \meromorphic uxes" on the no-go theorem [5] as having a constant warping factor in the absence of a or (...truncated)