Asymptotic Symmetries and the Kerr/CFT Central Charge

Kerr/CFT begins with a choice of boundary conditions on the NHEK throat. The diffeomorphisms preserving those falloffs include one Virasoro algebra, and their covariant phase-space charges have central charge $c_L=12J$ in units $\hbar=1$. Neither the algebra nor its central term is a property of the local NHEK metric alone: finiteness, conservation, and integrability of the charges are part of the construction. This page keeps that dependence explicit.

Required background. Gauge Theory and String-Theory Preparation provides symmetry-algebra and normalization conventions.

Helpful background. Quantum Gravity supplies covariant phase-space interpretation; Quantum Information supplies later state-counting distinctions; Black-Hole Geometry and Mechanics provides extremal Kerr charges; and Gravitational Radiation and Waveforms supplies Kerr mode conventions.

Boundary conditions define the phase space

Write $g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu}$ with $\bar g$ the global NHEK metric. Use dimensionless global coordinates $(\tau,\phi,\theta,r)$, with $\phi\sim\phi+2\pi$, and work at the $r\to+\infty$ end. The original componentwise falloffs are

$$h_{\mu\nu}=\begin{pmatrix} O(r^2)&O(1)&O(r^{-1})&O(r^{-2})\\ O(1)&O(1)&O(r^{-1})&O(r^{-1})\\ O(r^{-1})&O(r^{-1})&O(r^{-1})&O(r^{-2})\\ O(r^{-2})&O(r^{-1})&O(r^{-2})&O(r^{-3}) \end{pmatrix}.$$

They are supplemented by $Q_{\partial_\tau}=0$, which fixes the departure-from-extremality charge. The boundary conditions determine which asymptotic symmetries are nontrivial and which variations are tangent to the phase space Guica et al. 2009, § 5, Eq. (5.1), and § 6, Eq. (6.7).

The leading preserving vector fields are

$$\zeta_\epsilon= \epsilon(\phi)\,\partial_\phi -r\epsilon'(\phi)\,\partial_r+\cdots .$$

For $\epsilon_n(\phi)=-e^{-in\phi}$, their Lie brackets are

$$i[\zeta_m,\zeta_n]=(m-n)\zeta_{m+n}.$$

This is a Witt algebra before central extension. Exact NHEK isometries form only $\mathrm{SL}(2,\mathbb R)\times\mathrm{U}(1)$; the Virasoro transformations are asymptotic symmetries of the chosen phase space, not additional exact Killing fields. Guica, Hartman, Song, and Strominger 2009, §§4–6, pp. 6–10 specify the falloffs, preserving vectors, and charges.

Covariant charges and their central term

For Einstein gravity, the charge variation associated with $\zeta$ is a surface integral

$$\delta Q_\zeta[h;\bar g] =\frac{1}{8\pi G}\int_{\partial\Sigma} k_\zeta[h;\bar g].$$

The two-form $k_\zeta$ is fixed by the covariant phase-space prescription up to standard boundary ambiguities. One must verify that the integral is finite, conserved between allowed cuts, and integrable in field space. Integrability means there exists a function $Q_\zeta[g]$ whose variation is the displayed one; otherwise the putative generator is not a well-defined charge.

Evaluating the charge bracket gives

$$\{Q_m,Q_n\}=Q_{[m,n]}+K(\zeta_m,\zeta_n),$$

where the direct surface integral gives a central term proportional to $J(m^3+2m)$. Shift the charge zero mode by defining $L_n=Q_n+(3J/2)\delta_{n,0}$ in units $\hbar=1$. This converts the linear term to the conventional Virasoro form,

$$i\{L_m,L_n\}=(m-n)L_{m+n} +\frac{c_L}{12}m(m^2-1)\delta_{m+n,0}, \qquad c_L=12J .$$

With $\hbar$ restored, $\hbar L_n=Q_n+(3J/2)\delta_{n,0}$ and $c_L=12J/\hbar$; the quantum commutator replaces $i$ times the classical bracket. $G$ has already entered through the physical angular momentum and the charge integral. The cubic coefficient fixes $c_L$, while a constant zero-mode shift changes only the term linear in $m$ Guica et al. 2009, § 7, Eqs. (7.6)–(7.9).

Finiteness, integrability, and altered falloffs

The leading falloffs permit some variations that can produce nonintegrable flux unless a supplementary condition fixes the appropriate zero mode. The original analysis discusses this issue explicitly in Guica et al. 2009, Appendix B, pp. 15–16. Later refinements can use different boundary counterterms or phase spaces; they must be compared as complete packages rather than by quoting only $c_L$.

An independent check evaluates the central term both from the surface-charge bracket and from the transformation of the boundary stress tensor when such a description is available. A dimensional check confirms that $J/\hbar$ is dimensionless. A zero-mode check ensures that $Q_0$ measures the declared angular sector rather than a shifted generator.

Changing the falloffs can make a previously allowed $\zeta_n$ pure gauge, cause its charge to diverge, admit flux through the boundary, or introduce a different algebra. Thus $c_L=12J$ is conditional on the boundary conditions, angular period, and charge prescription. It is not a coordinate-invariant number attached to every treatment of NHEK.

There is a simple normalization trap here. Rescaling the angular coordinate changes the Fourier-mode period and the normalization of both generators and temperatures. Quoting the same numerical central charge after such a rescaling is inconsistent unless the charge basis is transformed too. The product entering the Cardy entropy can remain invariant, but its factors must be translated together.

What the central charge does and does not show

The calculation constructs a classical asymptotic symmetry algebra with a central extension. It is strong evidence for a chiral conformal organization of the extremal throat. It does not construct a unitary modular-invariant microscopic CFT, prove that the full asymptotically flat Kerr Hilbert space is represented, or establish Cardy asymptotics at finite $J$.

The result is also sector-specific. Excitations that leave the near-superradiant scaling region, change extremality, or violate the falloffs are outside this phase space. Matching them to the full Kerr exterior requires additional boundary data. These qualifications are why the subsequent entropy match is a conditional consistency calculation rather than a completed duality proof.

Exercises

Verify the Witt bracket of the preserving vector fields and explain why a finite but nonintegrable surface-charge variation is not yet a valid symmetry generator.

Solution

Substitution of $\zeta_\epsilon=\epsilon\partial_\phi-r\epsilon'\partial_r$ into the Lie bracket gives $\zeta_{\epsilon_1\epsilon_2'-\epsilon_2\epsilon_1'}$. For Fourier modes this becomes $i[\zeta_m,\zeta_n]=(m-n)\zeta_{m+n}$. Finiteness only says the surface integral for $\delta Q$ has a finite value. Integrability additionally requires path independence in the space of allowed metrics, so that $\delta Q$ is the variation of one charge function. Without it, repeated transformations do not act through well-defined Hamiltonian generators even if each infinitesimal integral is finite.

References

• Guica, Monica, Thomas Hartman, Wei Song, and Andrew Strominger. “The Kerr/CFT Correspondence.” Physical Review D 80 (2009) 124008. arXiv:0809.4266.