Determine the near-extremal rotational zero-mode contribution for rotating Kerr black holes

Determine the rotational zero modes of the four-dimensional near-horizon extremal Kerr geometry and establish their contribution to the near-extremal logarithmic temperature correction to the entropy.

Background

The paper identifies three tensor zero modes associated with the SL(2,R) isometries of NHEK and one expected rotational zero mode associated with axial U(1) symmetry. Unlike the tensor modes, the four-dimensional rotational zero modes had not been successfully constructed in the authors’ treatment.

This issue matters because the rotational mode may affect the logarithmic temperature correction, but its four-dimensional harmonic-gauge representative must satisfy regularity and normalizability conditions on the polar sphere.

References

While the 4-dimensional tensor zero modes have been constructed, the rotational zero modes are still not found.

— Logarithmic corrections to the entropy for near-extremal rotating black holes  (2609.01039 - Banerjee et al., 1 Sep 2026) in Section 3, paragraph preceding Section 3.1, “Issues with rotational zero modes”

It suggests one can never find a constant $c_2$ such that the integrand of the squared norm $\langle\hat\xi_{-n}{\rm rot}\,|\,\hat\xi_n{\rm rot}\rangle$ is regular over the interval $\theta\in[0,\pi]$. Hence, the apparent compensating part of the gauge transformation, even though restores harmonic gauge, is non-normalizable.

— Logarithmic corrections to the entropy for near-extremal rotating black holes  (2609.01039 - Banerjee et al., 1 Sep 2026) in Section 3.1, “Issues with rotational zero modes”

While both of these approaches look promising, a complete construction is still under construction. We will report on this construction in a future study.

— Logarithmic corrections to the entropy for near-extremal rotating black holes  (2609.01039 - Banerjee et al., 1 Sep 2026) in Section 3.1, concluding paragraph of “Issues with rotational zero modes”