Supersymmetry and F-theory realization of the deformed conifold with three-form flux

TL;DR

This work demonstrates that the Klebanov-Strassler deformed conifold solution with three-form flux preserves ${\cal N}=1$ supersymmetry in four dimensions and admits a streamlined F-theory description as a product of a non-compact Calabi–Yau threefold with a torus, specifically with $\tau = i$. By performing a detailed analysis of the type IIB supersymmetry variations, it shows that the flux must satisfy $\tilde{*} G_{(3)} = i G_{(3)}$ and that $G_{(3)}$ is a closed $(2,1)$ form, leading to an internal $SU(3)$ holonomy and a warped geometry where the warp factor is tied to the flux via the Bianchi identities. The connection to F-theory is established through the standard $G_{(4)}$-form construction on a CY$_4$ that reduces to the product $T^2 \times$ CY$_3$ for constant $\tau$, and the usual tadpole constraint becomes $\chi/24 = n_{D3} + \int \mathrm{Im} G_{(3)} \wedge G_{(3)}^{*}$, which vanishes for the product geometry, motivating non-constant $\tau$ to realize a fully compact warped F-theory background. Overall, the paper provides a concrete local model where warped RR backgrounds, supersymmetry, and F-theory duality intersect, highlighting potential pathways to moduli stabilization and gauge/string dualities in warped geometries.

Abstract

It is shown that the deformed conifold solution with three-form flux, found by Klebanov and Strassler, is supersymmetric, and that it admits a simple F-theory description in terms of a direct product of the deformed conifold and a torus. Some general remarks on Ramond-Ramond backgrounds and warped compactifications are included.