Wilson Spool in Quantum Gravity

• Wilson spool is a gauge-invariant operator that encodes the complete one-loop determinant for massive, spinning fields using contour integrals over infinite-winding Wilson loops.
• It organizes spectral data from gauge-theoretic traces by summing residues at poles corresponding to quasinormal or normal modes in various geometries such as AdS₃, dS₃, and flat spacetime.
• The operator unifies quantum corrections in matter-coupled gravity by linking group characters, Selberg trace formulas, and worldline quantum mechanics across multiple dimensions.

A Wilson spool is a topological, gauge-invariant operator in the Chern–Simons or BF gauge-theory formulation of gravity whose expectation value reproduces the complete one-loop determinant for massive, spinning fields on a fixed background, and can be systematically generalized to capture quantum corrections from metric fluctuations. The Wilson spool encodes the functional determinant for a matter field as a contour integral over products of infinite-winding Wilson loops in non-standard one-particle representations, with a universal integration measure. It applies naturally across three-dimensional AdS, dS, and flat spacetime, as well as in two-dimensional JT gravity, providing a unifying framework for coupling matter to quantum gravity in a manifestly gauge-theoretic manner.

1. Formal Definition and Construction

The Wilson spool operator arises in three-dimensional quantum gravity with cosmological constant $\Lambda$ as a functional of the Chern–Simons connections $A_L, A_R$. For a scalar field of mass $m$ (with quadratic Casimir $C_2(R_j) = -m^2/\Lambda$), the core definition is

$$W_j[A_L, A_R] = i \int_\mathcal{C} d\alpha\, \frac{\cos(\alpha/2)}{\sin(\alpha/2)} \sum_{m \in \mathrm{weights}(R_j)} \langle m | \mathcal{P}\exp(\oint_\gamma A_L) | m \rangle \langle m | \mathcal{P}\exp(\oint_\gamma A_R) | m \rangle.$$

Here, $\gamma$ is a chosen closed path (e.g., the contractible thermal circle in BTZ or horizon circle in dS$_3$), and $R_j$ is a highest- or lowest-weight representation of the gauge algebra (e.g., $\mathfrak{sl}(2)$ or $\mathfrak{su}(2)$), generally infinite-dimensional for $\Lambda > 0$. The integration measure and contour $\mathcal{C}$ organize the sum over all winding numbers by expanding in residues at poles in the integrand. The matter partition function is then $\exp(W_j[A_L, A_R])$, and its insertion in the gravitational path integral produces the scalar-plus-gravity partition function including quantum metric fluctuations (Castro et al., 2023, Castro et al., 2023).

This generic construction is directly generalized to spinning fields by replacing $j \to (j_L, j_R)$, with the spinning Wilson spool defined for AdS$_3$/dS$_3$ as

$$W_{j_L, j_R}[A_L, A_R] = \frac{i}{2} \int_{\mathcal{C}} \frac{d\alpha}{\alpha} \frac{\cos(\alpha/2)}{\sin(\alpha/2)} [1 + 2 s^2 \sin^2(\alpha/2)] \sum_{R_L \otimes R_R} \mathrm{Tr}_{R_L}\bigl(\mathcal P e^{(\alpha/2\pi)\oint_\gamma A_L}\bigr)\mathrm{Tr}_{R_R}\bigl(\mathcal P e^{-(\alpha/2\pi)\oint_\gamma A_R}\bigr),$$

where $s$ is the spin and the sum is over non-standard representations determined by the mass and spin via Casimir constraints; the additional factor encodes edge/polarization corrections for nonzero spin (Bourne et al., 12 Jul 2024).

2. Representation-Theoretic and Group-Theoretic Origin

The Wilson spool directly packages the spectral data of bulk fluctuations: traces over Wilson loops in $\mathsf{R}_j$ compute group characters whose spectrum matches quasinormal modes or Euclidean normal modes. In AdS$_3$, the relevant representations are lowest-weight modules of $\mathfrak{sl}(2)_L \oplus \mathfrak{sl}(2)_R$, labeled by $(j_+, j_-) = ((\Delta+s)/2, (\Delta-s)/2)$ for conformal dimension $\Delta$ and spin $s$, satisfying the mass-shell condition $$c_2^{(L)} + c_2^{(R)} = \frac12(\Delta(\Delta-2) - s)$$ (Bourne et al., 7 Jul 2025).

For dS$_3$ ($S^3$ background), the construction requires non-standard infinite-dimensional highest-weight representations of $\mathfrak{su}(2)$, classified into complementary or principal series depending on $m^2\ell^2$ (“complementary type” for $m^2\ell^2 < 1$, “principal type” for $m^2\ell^2 > 1$), both having positive-definite norm structures and explicit character formulas $\chi_j(z)$ (Castro et al., 2023).

The group-theoretic derivation leverages the fact that functional determinants in these geometries are meromorphic functions whose poles correspond to solutions of the representation-theoretic Casimir eigenvalue equations, with physical single-valuedness and global regularity conditions picking out allowed representations (see e.g. Section 2 of (Bourne et al., 12 Jul 2024, Fliss, 11 Mar 2025, Haupfear et al., 7 Jul 2025)).

3. Relation to One-Loop Determinants and Quantum Corrections

Insertion of the Wilson spool correctly reproduces the standard one-loop determinant for massive (possibly spinning) fields in both tree-level and quantum-corrected backgrounds:

• For BTZ black holes in AdS$_3$, evaluation of the spool yields the product formula matching the determinant $\det(-\nabla^2 + m^2)$ on the thermal background: summing residues at $\alpha=2\pi n$ reconstructs the heat-kernel and Selberg-trace expansion of the determinant (Castro et al., 2023, Haupfear et al., 7 Jul 2025).
• For $S^3$ in Euclidean dS$_3$, the contour manipulation picks up multiple poles, and the result is a finite sum of polylogarithms precisely matching the known zeta-regularized one-loop determinant (Castro et al., 2023).

Beyond the tree level, inserting the spool into the full quantum-gravity path integral produces an expansion in $G_N$ (Newton’s constant), systematically accounting for graviton loops. E.g., for $S^3$, evaluation via abelianization/localization yields mass renormalization at leading order,

$$m^2_{\rm eff}\ell^2 = m^2\ell^2 + \frac{G_N}{96\ell}\bigl|m\ell\bigr| + \cdots$$

(Castro et al., 2023), and higher orders can be computed by Taylor-expanding the integrand and integrating over saddle parameters (Bourne et al., 12 Jul 2024).

4. Extensions: General Topologies, Spin, and Dimensional Reduction

The Wilson spool construction extends beyond maximally symmetric backgrounds:

• AdS$_3$ quotients: For any smooth, cusp-free hyperbolic manifold $M = \mathbb{H}^3 /\Gamma$, the spool is defined as a sum over conjugacy classes $[\gamma]_+$ with multiplicities $n_\gamma$, and characters evaluated at the holonomy $\hat\ell_\gamma$ around each primitive loop (Bourne et al., 7 Jul 2025).
• Higher genus and compact examples: In handlebodies and closed manifolds, the summation encompasses all primitive nontrivial geodesics with proper symmetry factors.
• Lens spaces and higher-dimensional spheres: Analogous spool operators exist in lens spaces $L(p,q)$ (with sums over $\mathbb{Z}_p$ holonomies) and odd-dimensional spheres $S^d$, each preserving the universal integration measure and group-character structure (Haupfear et al., 7 Jul 2025).
• JT gravity (2D): The Wilson spool maps under dimensional reduction to a line operator in BF theory, encoding the functional determinant for massive scalars on hyperbolic and spherical backgrounds, and is directly related to its higher-dimensional ancestor (Fliss, 11 Mar 2025).

A plausible implication is the emergence of a unified language for one-loop determinants across dimensions, where the Wilson spool is the gauge-theoretic avatar for spectral invariants of worldline quantum mechanics.

5. Gauge Invariance, Geometry Independence, and Path Integral Role

A key property of the Wilson spool is its manifest gauge invariance: it depends only on the holonomy class of the Chern–Simons (or BF) connections around non-contractible cycles, not on the explicit metric. This feature extends to the flat-space case, where holonomy determines the spectrum, and the spool is geometry-independent except for its dependence on cycle topology and gauge field monodromy (Pannier, 12 Nov 2025). For off-shell or non-classical backgrounds, the operator can be straightforwardly generalized by integrating over connection moduli in the gravitational path integral.

$$\mathbb{W}[A] = \text{(universal contour integral)} \times \text{(representation-theoretic trace)}.$$

This suggests the Wilson spool serves both as a physical observable (summed over worldline windings) and as a master generating functional for matter-coupled quantum gravity.

6. Interpretative Perspectives and Physical Significance

Three complementary approaches illuminate the Wilson spool construction:

• Selberg trace formula: The spool matches the spectral sum over closed geodesics, with group characters encoding classical action and quantum fluctuations (Bourne et al., 7 Jul 2025, Haupfear et al., 7 Jul 2025).
• Worldline quantum mechanics: The contour integral representation arises naturally via first-quantized path integrals over winding particle worldlines; winding sums are encoded via residue expansion in the $\alpha$ parameter (Bourne et al., 7 Jul 2025).
• Quasinormal-mode method (DHS): The poles of the integrand coincide with the eigenvalues from the quasinormal-mode spectrum; representation theory correctly organizes the spectrum and physical boundary conditions (Bourne et al., 12 Jul 2024, Fliss, 11 Mar 2025, Haupfear et al., 7 Jul 2025).

In all cases, the Wilson spool unifies spectral, algebraic, and topological perspectives on quantum corrections to gravity-matter systems. As an operator, it efficiently packages all IR-finite, scheme-independent, and gauge-invariant content of one-loop determinants in a single line integral. Its natural extension to off-shell situations facilitates controlled analysis of quantum gravity corrections and ties into the broad program of constructing quantum-corrected partition functions in holography and beyond.

7. Open Problems and Further Developments

Recent work has extended the Wilson spool to flat spacetime (Pannier, 12 Nov 2025), JT gravity (Fliss, 11 Mar 2025), and arbitrary AdS$_3$ quotients (Bourne et al., 7 Jul 2025). Outstanding directions include:

• Full classification of non-standard representations for general values of $\Lambda$ and for higher-spin fields.
• Complete mapping of dimensional reduction between 3D and 2D spools, including Kaluza-Klein sectors and dilaton couplings.
• Systematic computation of spool expectation values in higher-genus or nontrivial topology, with possible improvements to convergence of gravitational path integrals in the presence of matter (Castro et al., 2023).
• Applications to holographic renormalization, genus expansions, and dual boundary descriptions, e.g., genuine edge/boundary partition functions in putative dS/CFT scenarios (Castro et al., 2023).
• Extension to higher dimensions and incorporation of supersymmetric localization.

The spool formalism promises a robust framework for organizing matter-gravity coupling, computing quantum corrections, and elucidating the interplay between gauge theory, spectral geometry, and quantum topology in modern gravitational physics.