Remark on Calabi-Yau vacua of the string theory and the cosmological constant problem

Abstract

We propose, based on the viewpoint that our three-dimensional space is a stack of BPS D3-branes located at the conifold singularity of the Calabi-Yau three-fold, a new mechanism to address the cosmological constant problem in the framework of the conventional compactifications where the $n$-form fluxes including NS-NS three-form are all turned off. In this mechanism the four-dimensional cosmological constant $\lambda$ appears as two types, NS-NS type and R-R type, of vacuum energies on the brane plus supersymmetry breaking term, which constitute a brane action density ${\hat I}_{\rm brane}$, and sum of these three terms of ${\hat I}_{\rm brane}$ are forced to vanish by field equations so that $\lambda$ adjusts itself to zero as a result. Also in this mechanism the $d=4$ supersymmetry is broken in the brane region, while still maintaining $\lambda =0$. The supersymmetry breaking occurs as a result of the gauge symmetry breaking of the R-R four-form arising at the quantum level. We generalize the above mechanism to the case of the flux compactifications where the fluxes are all turned on to stabilize the moduli. In the generalized theory $\lambda$ appears as ${\hat I}_{\rm brane}$ plus the scalar potential ${\mathcal V}_{\rm scalar}$ for the moduli, in contrast to the case of the ordinary flux compactifications where $\lambda$ is simply given by ${\mathcal V}_{\rm scalar}$. Also in this theory any nonzero ${\mathcal V}_{\rm scalar}$ arising from perturbative or nonperturbative corrections is gauged away by the gauge arbitrariness of ${\hat I}_{\rm brane}$ and the condition $\lambda=0$. So $\lambda$ is again expressed only as a brane action density as before, or it simply vanishes by the cancelation between ${\hat I}_{\rm brane}$ and ${\mathcal V}_{\rm scalar}$.