Construct gauge-invariant nonzero BTZ fluctuation modes

Construct the explicit gauge-invariant nonzero modes of the extremal BTZ kinetic operator and use them, together with degenerate perturbation theory and an admissible compensating gauge transformation, to determine the validity of the perturbative treatment of rotational zero modes in the near-extremal BTZ geometry.

Background

The near-extremal BTZ calculation involves rotational zero modes whose first-order eigenvalue correction vanishes. The paper explains that this vanishing does not by itself establish a breakdown of perturbation theory, because the degeneracy may remain unlifted at first order and corrections to the fluctuations can receive contributions from nonzero modes of the extremal background.

The authors state that the relevant gauge-invariant nonzero modes have not been explicitly constructed and that a complete analysis must also determine whether a compensating transformation can preserve the chosen gauge slice without generating non-normalizable fluctuations.

References

To the best of our knowledge, the explicit forms of the gauge invariant non zero modes are not yet constructed. We therefore believe that a more detailed computation is required before drawing any conclusion regarding the validity of the perturbative approach.

— Logarithmic corrections to the entropy for near-extremal rotating black holes  (2609.01039 - Banerjee et al., 1 Sep 2026) in Section 2, discussion of tensor and rotational zero modes