A spool for every quotient: One-loop partition functions in AdS$_3$ gravity

1. Problem Statement

The paper addresses the calculation of one-loop determinants for massive spinning fields (scalar and higher-spin $s$) in three-dimensional Euclidean gravity with a negative cosmological constant ($\Lambda < 0$).

• Context: In 3D gravity, the Einstein-Hilbert action can be reformulated as a Chern-Simons theory. While the classical equivalence is well-known, the quantum treatment of matter coupling remains challenging, particularly for manifolds with complex topology.
• Limitation of Previous Work: Previous constructions of the "Wilson spool" (a topological line operator representing the one-loop determinant) were largely restricted to simple topologies like the BTZ black hole or thermal AdS$_3$, where the fundamental group is generated by a single element (cyclic).
• The Gap: There was no general prescription for smooth, cusp-free hyperbolic three-manifolds where the fundamental group $\pi_1(M)$ is generated by multiple non-commuting elements (e.g., multi-boundary wormholes, compact hyperbolic manifolds like the Weeks manifold). The challenge lies in correctly summing over all non-contractible loops and handling the intricate relations within the fundamental group.

2. Methodology

The authors propose a generalized prescription called the Wilson Spool, which expresses the one-loop partition function as a sum over topological line operators (Wilson loops) wrapping the non-contractible cycles of the manifold.

Key Frameworks

1. Chern-Simons Formulation: The authors utilize the equivalence between 3D gravity and $SL(2, \mathbb{R})_L \times SL(2, \mathbb{R})_R$ Chern-Simons theory. The metric $g_{\mu\nu}$ is encoded in two gauge connections, $A_L$ and $A_R$, which are linear combinations of the coframe and spin connection.
2. Hyperbolic Quotients: The target manifolds $M$ are treated as quotients of hyperbolic 3-space: $M = \mathbb{H}^3 / \Gamma$, where $\Gamma \subset PSL(2, \mathbb{C})$ is a discrete, torsion-free subgroup (Kleinian group). The fundamental group $\pi_1(M)$ is identified with $\Gamma$.
3. Holonomy-Geometry Correspondence: Closed geodesics in $M$ correspond to conjugacy classes $[\gamma]$ in $\Gamma$. The holonomy of the Chern-Simons connection around a geodesic $\gamma$ is directly related to the complex length $\hat{l}_\gamma = l_\gamma + i\theta_\gamma$ (length + torsion) of the geodesic.

The Prescription

The one-loop determinant $\log Z_{\Delta, s}$ for a field of mass $\Delta$ and spin $s$ is given by:
$$\log Z_{\Delta, s}[g_{\mu\nu}] = W_\Gamma[A_L, A_R]$$
Where $W_\Gamma$ is the generalized Wilson spool:
$$W_\Gamma = \sum_{[\gamma] \in [\Gamma]_+} \frac{1}{n_\gamma} \sum_{R_{LW}} \left[ \text{Tr}_{R_L} \mathcal{P} \exp \oint_\gamma A_L \right] \left[ \text{Tr}_{R_R} \mathcal{P} \exp \oint_\gamma (-A_R) \right]$$

• $[\Gamma]_+$: The set of unoriented conjugacy classes of $\Gamma$ with positive geodesic length.
• $n_\gamma$: The multiplicity of the conjugacy class (the power $n$ such that $\gamma = \gamma_0^n$ for a primitive element $\gamma_0$). This acts as a symmetry factor.
• $R_{LW}$: The sum runs over lowest-weight representations of $sl(2, \mathbb{R})$ determined by the mass and spin of the field.

3. Key Contributions

A. Generalization to Arbitrary Smooth Hyperbolic Quotients

The primary contribution is extending the Wilson spool from single-cycle topologies (BTZ) to any smooth, cusp-free hyperbolic 3-manifold. This includes:

• Multi-boundary wormholes (asymptotically AdS$_3$).
• Compact hyperbolic manifolds (e.g., the Weeks manifold) which are not holographic in the conventional sense.

B. Three Independent Derivations

The authors validate their prescription through three distinct physical and mathematical perspectives:

1. Selberg Trace Formula:

   • They relate the spectrum of the Laplacian on the manifold to the spectrum of closed geodesics.
   • By choosing a specific test function, they reconstruct the one-loop determinant as a sum over conjugacy classes, matching the Wilson spool structure exactly. This confirms the result for on-shell metrics.

2. Worldline Quantum Mechanics:

   • They express the one-loop determinant as a path integral over closed loops (worldlines) in the Euclidean geometry.
   • By lifting the path integral from the quotient $M$ to the covering space $\mathbb{H}^3$, they demonstrate that the sum over loops decomposes into sectors labeled by conjugacy classes of $\Gamma$.
   • Crucial Insight: The factor $1/n_\gamma$ arises naturally as a symmetry factor when lifting the path integral, providing a topological (off-shell) justification for the prescription. They show the calculation is "two-loop exact," reproducing the correct characters.

3. Quasinormal Mode (QNM) Method:

   • They extend the DHS (Denef-Hartnoll-Sachdev) formalism used for thermal AdS to general quotients.
   • They impose conditions on eigenfunctions (single-valuedness and regularity) which translate to constraints on the weights of $sl(2, \mathbb{R})$ representations.
   • This approach reproduces the product structure of the determinant and links the Wilson spool to the Selberg Zeta function.

C. Off-Shell Interpretation

While the derivations rely on on-shell (hyperbolic) metrics, the final operator $W_\Gamma$ is expressed entirely in terms of gauge-invariant holonomies and topological data (conjugacy classes). The authors argue that this allows the Wilson spool to be promoted to an off-shell operator within the gravitational path integral, valid for metric fluctuations around a saddle point.

4. Results

• Explicit Formula: The paper provides the explicit formula (Eq. 1.3 and 3.2) for the one-loop determinant of massive spinning fields ($s \geq 0$) on any smooth hyperbolic quotient.
• Reproduction of Known Results: For $s=0$ (scalar) and $s=1$ (vector), the result reproduces the known one-loop determinants calculated by Giombi, Maloney, and Yin (GMY).
• New Physics: For massive fields with spin $s \geq 2$, the paper provides the first known expression for their one-loop determinant on general hyperbolic quotients.
• Integral Representation: The sum over conjugacy classes can be rewritten as a contour integral (Eq. 1.4), which unifies the result with previous spool literature and highlights the pole structure associated with the representation theory.

5. Significance

• Topological Nature of 3D Gravity: The work reinforces the idea that in 3D, quantum gravity observables (like one-loop determinants) can be captured by topological operators (Wilson loops) that are insensitive to local geometric details but sensitive to global topology.
• Bridge to Number Theory: The connection between the Wilson spool and the Selberg Zeta function deepens the link between 3D quantum gravity and analytic number theory (spectral geometry).
• Foundation for Future Work:
  • It sets the stage for studying orbifolds and cusped spaces (though these require handling elliptic/parabolic elements and divergent characters, which the paper discusses as future work).
  • It suggests a pathway to embedding the Wilson spool into the Virasoro TQFT for a fully quantum treatment of 3D gravity coupled to matter, moving beyond the classical Chern-Simons equivalence.
  • It provides tools to analyze "multi-boundary wormholes" and compact hyperbolic manifolds, which are relevant for understanding the ensemble nature of holography and the statistical mechanics of 3D gravity.

In summary, the paper successfully generalizes a powerful computational tool (the Wilson spool) to the full class of smooth hyperbolic 3-manifolds, unifying geometric, topological, and representation-theoretic approaches to one-loop quantum gravity.