Define reflection positivity for extended noninvertible topological field theories

Develop a precise, mathematically rigorous definition of reflection positivity for extended topological field theories that are not necessarily invertible, formulated within the modern higher-categorical frameworks for bordism and field theories, so that it interfaces appropriately with existing classification results for invertible theories and enables the systematic study of noninvertible symmetries.

Background

In the paper, reflection positivity is used as a structural requirement in the classification of invertible topological field theories via maps from bordism spectra into Anderson duals. For invertible theories, a definition of reflection positivity using Z/2-actions on the bordism category and target is available and underpins key classification results of Freed–Hopkins and others.

However, for extended topological field theories that are not necessarily invertible, a general, accepted definition of reflection positivity has not yet been established. The authors note ongoing work in this direction and highlight the importance of such a definition for extending anomaly and symmetry-breaking analyses beyond the invertible setting.