Analyze the Euclidean positive-curvature Iwasawa construction

Develop a systematic analysis of the Euclidean positive-cosmological-constant case for the split BF to Regge construction, including the complexified algebra and the reality conditions required to recover real geometry.

Background

The split BF to Regge construction uses an Iwasawa decomposition of the BF gauge group. For Euclidean positive cosmological constant, the relevant gauge algebra is the compact algebra so(3)\mathfrak{so}(3), which does not contain the required non-compact real subalgebra. The construction therefore passes to the complexification and must impose suitable reality conditions to recover the intended real geometry.

The paper does not provide this analysis and explicitly defers it, leaving the positive-curvature Euclidean case unresolved within the presented framework.

References

The treatment of the corresponding case $g_{1,1}$ therefore requires a more detailed analysis of the complexified algebra, together with the associated reality conditions. Such technical developments go beyond the scope of the present paper, and in particular beyond the argument we wish to develop here. We therefore leave a systematic analysis of this case for future work.

— Regge Dilaton Gravity from BF Theory  (2610.21058 - Girelli et al., 15 Oct 2026) in Section 4, subsection “The Iwasawa decomposition”