Compactifications on twisted tori with fluxes and free differential algebras

TL;DR

The work analyzes how fluxes and geometric twisting in M-theory compactifications give rise to gauge algebras that are more general than ordinary Lie algebras. It formulates a Free Differential Algebra (FDA) with non-abelian 1-forms and 2-forms, derives the zero-curvature (Maurer–Cartan) equations ${\cal F}^{\Lambda}=dA^{\Lambda} + \frac{1}{2} f^{\Lambda}{}_{\Sigma\Gamma} A^{\Sigma} \wedge A^{\Gamma} + m^{\Lambda i} B_i = 0$ and ${\cal H}_i = dB_i + (T_{\Lambda})_{i}{}^{j} A^{\Lambda} \wedge B_j + k_{i\Lambda\Sigma\Gamma} A^{\Lambda} \wedge A^{\Sigma} \wedge A^{\Gamma} = 0$, and identifies the integrability constraints $f^{\Lambda}{}_{\Sigma[\Gamma} f^{\Sigma}{}_{\Pi\Delta]} + 2 m^{\Lambda i} k_{i\Gamma\Pi\Delta} = 0$, etc. In M-theory on twisted tori with fluxes, the FDA is realized with Λ = {I, IJ}, i = I, yielding 28 vectors and 7 two-forms; depending on whether form flux $g_{IJKL}$ and/or Scherk–Schwarz flux $\tau_{IJ}^{K}$ are present, the gauge algebra can be a central extension, a Scherk–Schwarz algebra, or a genuine FDA, with integrability conditions $\tau_{[IJ}^{M}\tau_{K]M}^{L}=0$ and $\tau_{[IJ}^{N} g_{KLM]N}=0$ (and the stronger $\tau_{IJ}^{N} g_{KLMN}=0$ recovering a Lie algebra). The work also discusses nonzero curvature, Green–Schwarz-type couplings, a contractible generator in $\mathcal H_I$, and a gauge-invariant 3-form curvature tied to the 4-form flux, as well as implications for moduli stabilization and potential differences between FDAs and Lie algebras in four-dimensional effective theories.

Abstract

We describe free differential algebras for non-abelian one and two form gauge potentials in four dimensions deriving the integrability conditions for the corresponding curvatures. We show that a realization of these algebras occurs in M-theory compactifications on twisted tori with constant four-form flux, due to the presence of antisymmetric tensor fields in the reduced theory.