Calabi-Yau Fourfolds with Flux and Supersymmetry Breaking

TL;DR

The paper investigates partial supersymmetry breaking in M-theory on Calabi-Yau fourfolds with 4-form flux by recasting the problem in three-dimensional $N=2$ gauged supergravity with translational (Peccei-Quinn) gaugings. It develops a redualized, flux-coupled Lagrangian with a no-scale structure and derives explicit conditions for preserving $N=2$ or breaking to $N=1$, expressed through flux data via the holomorphic superpotential $W= frac{1}{16} ext{∫} olimits \,ar{oldsymbol{ abla}} \,igl( ext{Ω} igr) \, F_4$ and the real function $T= frac{1}{16} ext{∫} olimits J ext{∧}J ext{∧}F_4$. An explicit toric Calabi-Yau fourfold example demonstrates a consistent $N=1$ vacuum with flux satisfying tadpole and integrality constraints, highlighting how flux and geometry control SUSY breaking and the cosmological constant in M-theory/F-theory contexts.

Abstract

In Calabi-Yau fourfold compactifications of M-theory with flux, we investigate the possibility of partial supersymmetry breaking in the three-dimensional effective theory. To this end, we place the effective theory in the framework of general N=2 gauged supergravities, in the special case where only translational symmetries are gauged. This allows us to extract supersymmetry-breaking conditions, and interpret them as conditions on the 4-form flux and Calabi-Yau geometry. For N=2 unbroken supersymmetry in three dimensions we recover previously known results, and we find a new condition for breaking supersymmetry from N=2 to N=1, i.e. from four to two supercharges. An example of a Calabi-Yau hypersurface in a toric variety that satisfies this condition is provided.