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SO(4) homotopy fixed points for modular categories and orientation of fully extended TQFTs

Determine whether every modular category admits a homotopy fixed point structure for the SO(4) action that upgrades the invertible fully extended framed topological field theory furnished by the cobordism hypothesis to an oriented fully extended topological field theory.

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Background

The paper proves that modular categories yield invertible skein-module maps and discusses their relationship to fully extended framed topological field theories obtained via the cobordism hypothesis. Passing from a framed to an oriented theory requires a homotopy fixed point structure under the SO(4) action, but the existence of such a structure is not established.

Clarifying this would connect algebraic data from modular categories with the geometric requirement for orientation in fully extended TQFTs, and would align the invertibility of certain skein maps with the expected properties of oriented theories.

References

It is not known whether a modular category also comes with a homotopy fixed point structure under SO(4) to make this an oriented theory.

— Reflection Equivariance and the Heisenberg Picture for Spaces of Conformal Blocks (2507.22820 - Woike, 30 Jul 2025) in Section 6.2 (Invertibility of handlebody skein modules)