W=0 Complex Structure Moduli Stabilization on CM-type K3 x K3 Orbifolds:---Arithmetic, Geometry and Particle Physics---

Abstract

It is an important question in string compactification whether complex structure moduli stabilization inevitably ends up with a vacuum expectation value of the superpotential W of the order of the Planck scale cubed. Any thoughts on volume stabilization and inflation in string theory, as well as on phenomenology of supersymmetric Standard Models, will be affected by the answer to this question. In this work, we follow an idea for making W ≃ 0 where the internal manifold has a vacuum complex structure with arithmetic characterization, and address Calabi-Yau fourfold compactification of F-theory. The moduli space of K3 × K3 orbifolds contain infinitely many such vacua. Arithmetic conditions for a W = 0 flux are worked out, and then all the K3 moduli have supersymmetric mass. Possible gauge groups, matter representations and discrete symmetries are studied for the case of Z 2 -orbifolds.

Algorithmically solving the Tadpole Problem

“…Finally, it does not seem to be completely unrealistic to obtain Q min by purely analytic arguments, at least for some special lattices. In particular there exist connections between our problem and number theory, 20 see for example [75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90], as also suggested by the exposition in 19 This probability density could roughly be estimated in the following way: The number of flux configurations with a certain charge Q is given by the number of lattice points in a shell of radius Q and thickness 1. In the continuum limit, for Q 1, the number of points is proportional to the volume of the shell.…”

A novel counterexample to the Nelson-Seiberg theorem

“…Just like any SUSY vacuum in R-symmetric models, the SUSY vacua in the new counterexample give W = 0 at the SUSY vacuum [10,11,16], and the supergravity version of the model also gives SUSY vacua with zero vacuum energy. One may hope use the supergravity model as a low energy effective description for flux compactification of type IIB string theory [17,18,19,20], and such string constructions of W = 0 SUSY vacua [21,22,23,24,25,26,27] serve as the first step toward vacua with small superpotentials [28]. But the R-symmetry breaking feature of the vacua means that some complex structure moduli obtain nonzero VEVs, which send the Calabi-Yau manifold away from the R-symmetric point in its moduli space.…”

A formal notion of genericity and term-by-term vanishing superpotentials at supersymmetric vacua from R-symmetric Wess-Zumino models

“…This result constrains the form of an R-symmetric Wess-Zumino model which leads to a SUSY vacuum: each term of the superpotential must contain at least one field with a zero expectation value. Such a constraint may lead to new extensions of the Nelson-Seiberg theorem, contribute to a refined classification of R-symmetric Wess-Zumino models, and find applications in string constructions of W = 0 SUSY vacua [26,27,28,29,30] as the first step toward vacua with small superpotentials [31].…”