Long-string interpretation of the normalizable hair

Determine whether the normalizable non-Riemannian hair mode W_1 admits an interpretation as a long-string condensate.

Background

The non-Riemannian branch contains an independent normalizable mode W_1 that couples directly to the Gomis–Ooguri long-string worldsheet through a hair vertex. Unlike ordinary charged excitations, this mode contributes neither to the asymptotic surface charges nor to the renormalized on-shell action under the stated interior prescription.

The physical interpretation of W_1 is therefore unresolved. In particular, the paper leaves open whether this charge-neutral soft hair represents a condensate of long strings.

References

Whether the normalizable hair W_{1} admits an interpretation as a long-string condensate is also open.

— Non-Riemannian Holography: Long Strings and Soft Hair  (2610.25669 - Hampton et al., 19 Oct 2026) in Conclusion

It is a bosonic matter Virasoro condition; a type-II super-BRST and GSO analysis is not supplied here. Exact marginality at finite W_1 is not established here.

— Non-Riemannian Holography: Long Strings and Soft Hair  (2610.25669 - Hampton et al., 19 Oct 2026) in Supplemental Material, Section SM 4.2, “Fradkin–Tseytlin Improvement and the Radial Hair”