AdS3 Boundary Gravitons and Vacuum Characters

AdS3 Einstein gravity has no local propagating gravitons, but it does have boundary gravitons: finite-energy excitations generated by asymptotic diffeomorphisms whose charges do not vanish. Around thermal AdS3, their perturbative spectrum forms the Virasoro vacuum module, and the one-loop determinant reproduces its character. This is a statement about one saddle and one representation, not a complete CFT partition function.

Required background. AdS3/CFT2 and the Brown–Henneaux Central Charge supplies the physical asymptotic generators, and Characters and Conformal Multiplet Counting supplies the representation-theoretic count.

Helpful background. Chiral Blocks, Sewing, and Modular Invariance explains how a character enters a torus amplitude, while Finite-N Smoothing, Tunneling, and Metastability gives the general warning against promoting a large-parameter saddle to an exact finite-system result.

Boundary gravitons as vacuum descendants

Let $|0\rangle$ be the global AdS3 vacuum. The modes $L_{-1},L_0,L_1$ generate its exact $SL(2,\mathbb R)$ isometries, so $L_{-1}|0\rangle=0$ rather than creating a physical excitation. Modes $L_{-n}$ with $n\ge2$ change the boundary stress tensor and carry nonzero surface charge. Acting with independent left- and right-moving modes gives states

$$L_{-n_1}\cdots L_{-n_r} \bar L_{-\bar n_1}\cdots\bar L_{-\bar n_s}|0\rangle, \qquad n_i,\bar n_j\ge2.$$

Their levels are integer partitions using parts at least two. For generic $c>1$, with no additional vacuum null vectors, the holomorphic vacuum character is

$$\chi_{\rm vac}(q) =q^{-c/24}\prod_{n=2}^{\infty}\frac{1}{1-q^n}.$$

The omitted $n=1$ factor is not a convention: it removes the global translation descendant that annihilates the vacuum.

The one-loop determinant

Gauge fixing metric fluctuations around Euclidean thermal AdS3 leaves a ratio of transverse fluctuation and ghost determinants. After zero modes are handled consistently, the perturbative saddle gives

$$Z_{\text{thermal AdS}}(\tau,\bar\tau) =|q|^{-c/12} \prod_{n=2}^{\infty}\frac{1}{|1-q^n|^2} \left[1+O\!\left(\frac1c\right)\right], \qquad q=e^{2\pi i\tau}.$$

The classical action supplies $|q|^{-c/12}$ and the one-loop fluctuations supply the product. In pure three-dimensional Einstein gravity, higher-loop corrections around this saddle are strongly constrained; nevertheless the expression remains a perturbative contribution associated with thermal AdS, not a sum over all fillings or a nonperturbative definition.

First application. Quantize linearized boundary gravitons around thermal AdS and reproduce the expected vacuum-character product. Enumerating level two gives $L_{-2}|0\rangle$; level three gives $L_{-3}|0\rangle$; level four gives $L_{-4}|0\rangle$ and $L_{-2}^2|0\rangle$. These coefficients agree with the expansion of $\prod_{n\ge2}(1-q^n)^{-1}$ and with the determinant calculation.

Null states and extra sectors

The generic product overcounts whenever the vacuum Verma module has additional null relations. Minimal-model central charges are the canonical examples: null descendants must be quotiented out, changing the exact character. Conversely, a complete CFT normally contains nonvacuum primary modules, so the vacuum character undercounts the full torus partition function. Modular completion also mixes low- and high-energy data; it cannot be inferred by declaring the vacuum sector complete.

Adversarial control. Evaluate the generic product at a central charge with a known vacuum null vector, or modularly transform it as though it were the entire partition function. The first test yields spurious states; the second does not generally produce a discrete spectrum with nonnegative integer multiplicities. Both failures preserve the valid conclusion—boundary gravitons realize the perturbative vacuum module—while rejecting the stronger claim of an exact quantum spectrum.

Evidence ceiling

The agreement between canonical boundary modes, the Virasoro vacuum character, and the one-loop determinant is a three-way perturbative check. It determines neither the nonvacuum primary content nor the allowed topology, and it does not decide whether “pure” AdS3 gravity has a unitary modular-invariant completion at finite $c$.

The one-loop determinant’s vacuum-character organization is computed in Giombi, Maloney, and Yin 2008; it counts perturbative boundary gravitons and does not construct a complete nonperturbative spectrum.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Brown, J. David, and Marc Henneaux. “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity.” Communications in Mathematical Physics 104 (1986): 207–226. DOI.
• Giombi, Simone, Alexander Maloney, and Xi Yin. “One-Loop Partition Functions of 3D Gravity.” Journal of High Energy Physics 2008, no. 8 (2008): 007. DOI; Open PDF.
• Maloney, Alexander, and Edward Witten. “Quantum Gravity Partition Functions in Three Dimensions.” Journal of High Energy Physics 2010, no. 2 (2010): 029. DOI; Open PDF.