Characterize representation-theoretic state-sum structures

Determine whether the state sum associated with the two-dimensional split BF and homogeneous Regge models can be organized in terms of representations of deformed symmetry structures that reduce to ordinary Poincaré representations in the flat limit.

Background

The construction integrates out the dilaton variables and is formulated primarily in terms of group holonomies and Regge geometric variables. The paper does not develop a representation-theoretic formulation of the resulting model.

Because the underlying BF theory has a matched-pair structure and the curved cases involve ANAN-type factors, the authors identify a possible connection with κ-Poincaré- or Euclidean-type deformations, but leave the existence and organization of such a state sum unresolved.

References

We have also restricted our attention to the simplest dilaton-gravity model. More general two-dimensional dilaton gravities are naturally described by non-linear Poisson sigma models , rather than by an ordinary BF theory, and include models such as Liouville-type gravity. In such cases, one expects the underlying algebraic structure to be correspondingly deformed, potentially involving further quantum-group-like structures. It would be interesting to determine to what extent the present discretization can be generalized to these non-linear theories and whether the split triangulation/dual-complex picture survives in an appropriate form.

— Regge Dilaton Gravity from BF Theory  (2610.21058 - Girelli et al., 15 Oct 2026) in Conclusion