Physical interpretation of the endpoint contribution

Determine the physical interpretation of the endpoint contribution associated with the singularity at zero field strength F=0 in the boundary-thimble analysis of the large-N epsilon-deformed and physical CP^{N-1} models.

Background

The paper resolves the unphysical-saddle problem in the large-N analysis of the epsilon-deformed CP{N-1} model by treating the singular point F=0 as a boundary of the integration contour. This introduces an endpoint thimble that can compete with the ordinary saddle thimble and, across a boundary Stokes transition, can leave the saddle inactive while the endpoint contribution saturates the integration cycle.

Although the endpoint contribution is shown to be mathematically essential for the proposed resolution and is estimated near F=0, its physical meaning is not identified. In particular, the paper does not establish what dynamical configuration, sector, or effective degree of freedom the endpoint contribution represents in the quantum field theory.

References

However, it is still an open problem to give the physical interpretation of the endpoint contribution due to the singularity at $F=0$.

— From Self-Dual to Physical $\mathbb{C}P^{N-1}$: Anomalies, Boundary Stokes Phenomenon, and Global Structure of $θ$-vacua  (2608.30160 - Hayashi et al., 31 Aug 2026) in Paragraph “Endpoint Stokes phenomenon,” Section 6 (Discussion)