Fully stabilized Minkowski vacua in the $2^6$ Landau-Ginzburg model

We study moduli stabilization via fluxes in the $2^6$ Landau-Ginzburg model. Fluxes not only give masses to scalar fields but can also induce higher order couplings that stabilize massless fields. We investigate this for several different flux choices in the $2^6$ model and find two examples that are inconsistent with the Refined Tadpole Conjecture. We also present, to our knowledge, the first 4d $\mathcal{N}=1$ Minkowski solution in string theory without any flat direction.