On Missing Bianchi Identities in Cohomology Formulation

TL;DR

This paper identifies missing Bianchi identities in the cohomology formulation of type IIA supergravity with fluxes, highlighting their importance for moduli stabilization and providing a conjecture for these identities.

Contribution

It reveals the non-equivalence of two flux formulations and proposes a model-independent set of missing identities in the cohomology approach.

Findings

Missing identities are necessary for equivalence between formulations.

Explicit examples support the conjecture of missing identities.

These identities are crucial for moduli stabilization in non-toroidal models.

Abstract

In this article, we perform a deep analysis of the Bianchi identities in the two known formulations developed for the four-dimensional effective type IIA supergravity theory with (non-)geometric fluxes. In what we call the `first formulation', fluxes are expressed in the real six-dimensional indices while in the `second formulation', fluxes are written in the cohomology form. We find that the set of flux constraints arising from these two known formulations are not equivalent, and there are missing identities in the cohomology version which need to be supplemented to match with the first formulation. By analyzing two explicit examples, we conjecture a model independent form for (the most of) the missing identities. These identities have been mostly overlooked in the previous attempts of studying moduli stabilization, particularly for the models developed in the beyond toroidal frameworks, where they could play some important role.