Type II Theories Compactified on Calabi-Yau Threefolds in the Presence of Background Fluxes

TL;DR

The paper analyzes Type II string compactifications on Calabi–Yau threefolds in the presence of background fluxes, deriving the bosonic 4D action and showing it forms a non-canonical $N=2$ supergravity with a massive two-form $B_2$. RR fluxes (and the dualized 4D constant $e_0$) organize into a symplectic vector, preserving a generalized symplectic invariance of the equations of motion, and enabling a consistent gauged supergravity framework. Type IIA RR fluxes map under mirror symmetry to Type IIB RR fluxes at the level of the effective action, with the symplectic data exchanged; NS fluxes lead to a distinct gauging pattern where hypermultiplet scalars are primarily affected and mirror relations are less transparent. The work illuminates the role of a massive $B_2$ in maintaining symplectic invariance and clarifies the conditions under which flux backgrounds yield consistent gauged $N=2$ theories, while highlighting open questions for NS-flux mirror duals and potential extensions to include holomorphic superpotentials or $N=2 o N=1$ reductions.

Abstract

Compactifications of type II theories on Calabi-Yau threefolds including electric and magnetic background fluxes are discussed. We derive the bosonic part of the four-dimensional low energy effective action and show that it is a non-canonical N=2 supergravity which includes a massive two-form. The symplectic invariance of the theory is maintained as long as the flux parameters transform as a symplectic vector and a massive two-form which couples to both electric and magnetic field strengths is present. The mirror symmetry between type IIA and type IIB compactified on mirror manifolds is shown to hold for R-R fluxes at the level of the effective action. We also compactify type IIA in the presence of NS three-form flux but the mirror symmetry in this case remains unclear.