###### Abstract

We propose a mechanism that couples matter fields to three-dimensional quantum
gravity, which can be used for theories with a positive or negative
cosmological constant. Our proposal is rooted in the Chern-Simons formulation
of three-dimensional gravity and makes use of the Wilson spool, a collection
of Wilson loops winding around closed paths of the background. We show that
the Wilson spool correctly reproduces the one-loop determinant of a free
massive scalar field on rotating black holes in AdS3 and Euclidean dS3 as
$G_{N}\to 0$. Moreover, we describe how to incorporate quantum metric
fluctuations into this formalism.

##  I Introduction

Chern-Simons theory is a compelling approach to quantum gravity that makes
manifest the topological nature of gravity in three-dimensions [1, 2]. For
negative cosmological constant, $\Lambda<0$, Chern-Simons theory nicely
characterizes the physics of black holes in Anti-de Sitter (AdS3) space and
their higher-spin generalizations [3, 4, 5], and dovetails the AdS3/CFT2
dictionary with the existence of edge modes [6, 7, 8]. Importantly, however,
Chern-Simons gravity provides a powerful computational foothold for three-
dimensional quantum gravity without recourse to holography. This is especially
key in the context of $\Lambda>0$, de Sitter (dS3) gravity, where the
corresponding dS/CFT dictionary is much less understood; see, though, [9, 10]
for recent developments. This context exhibits the true efficacy of Chern-
Simons gravity, allowing a characterization of loop corrections that are
relevant to quantum cosmology [11, 12, 13, 14, 15, 16, 17, 18].

It has been a long-standing problem to incorporate matter into Chern-Simons
gravity while retaining the topological features that make it natural as a
theory of quantum gravity. In this letter we address this problem. In short,
we introduce a new object we deem the Wilson spool which provides an effective
coupling of massive fields to quantum gravity directly as a gauge-invariant
operator. We define it precisely below, but intuitively, the spool represents
a Wilson loop winding arbitrarily many times around a closed path for which
the fields have non-trivial holonomy. This represents a pivotal entry into a
dictionary mapping geometric quantities to quantum operators in Chern-Simons
gravity [19, 20, 21, 22, 23, 24]. The Wilson spool will dictate how quantum
gravity alters the physics of quantum fields in a manner that is
quantitatively controlled by the gravitational coupling, $G_{N}$.

We will show that the Wilson spool is a natural object regardless of the sign
of cosmological constant. In the context of AdS3 we will exactly reproduce, at
tree-level $(G_{N}\to 0)$, the one-loop determinant of a scalar field on a
rotating BTZ black hole background. This computation is done directly “in the
bulk,” without reference to holography. In fact, it is in the context of dS3,
where holography is of limited utility, that we can make use of the full power
of this proposal. Certain exact results in Chern-Simons theory can be
meaningfully adapted to accommodate features necessary for de Sitter gravity
and its massive single-particle states. This provides a principled and
controlled method to computing $G_{N}$ corrections to one-loop determinants of
fields coupled to dynamical gravity. In this letter, we distill key results
that are presented in full detail in the companion paper [25], and also cast
them in a presentation that is unified for both signs of the cosmological
constant.

In the following we will present our proposal of the Wilson spool, and
illustrate its efficacy for AdS3 and dS3 gravity. We will evaluate it on AdS3
in Lorentzian signature, where the Chern-Simons gauge group is
$SL(2,\mathbb{R})\times SL(2,\mathbb{R})$, while for dS3 it will be treated in
Euclidean signature, where the gauge group is $SU(2)\times SU(2)$. This choice
facilitates making parallels between them, and hence makes evident the
robustness of our proposal, while also showing contrast when appropriate.

##  II The proposal

To illustrate our proposal it is instructive to cast some portions in the
metric formulation. Consider the path integral for a quantum scalar field
$\phi$, of mass $\mathsf{m}$, with no self-interactions, and minimally coupled
to the metric field, $g_{\mu\nu}$. This would be

$Z_{\scaleto{\text{scalar}}{3.5pt}}[g_{\mu\nu}]=\int[\mathcal{D}\phi]e^{iS_{\rm
matter}[\phi,g_{\mu\nu}]}~{}.$ (1)

Including the perturbative, quantum fluctuations of the metric leads to

$\big{\langle}Z_{\scaleto{\text{scalar}}{3.5pt}}[M]\big{\rangle}_{\scaleto{\text{grav}}{3.5pt}}:=\int[\mathcal{D}g_{\mu\nu}]_{\scaleto{M}{3.5pt}}e^{-I_{\rm
EH}[g_{\mu\nu}]}Z_{\scaleto{\text{scalar}}{3.5pt}}[g_{\mu\nu}]~{}.$ (2)

For concreteness we wrote here a Euclidean path integral, where $I_{\rm EH}$
is the Euclidean Einstein-Hilbert action plus a cosmological constant term.
The gravitational path integral is taken on a fixed topology $M$, and it
includes all perturbative corrections around that topology.

Our proposal for quantifying the coupling of matter to gravity is captured by
the following equality,111Since both theories, AdS3 and dS3 gravity, involve
the product of two gauge groups, we will introduce two Chern-Simons
connections, $(A_{L},A_{R}),$ corresponding to each group. These will encode
the metric field $g_{\mu\nu}$ as we will review in the subsequent sections.

$Z_{\scaleto{\text{scalar}}{3.5pt}}[g_{\mu\nu}]=\exp\left(\frac{1}{4}\mathbb{W}_{j}[A_{L},A_{R}]\right)~{}.$
(3)

On the right hand side we have the Wilson spool

$\displaystyle\mathbb{W}_{j}[A_{L},A_{R}]:=i\int_{\mathcal{C}}\frac{\differential\alpha}{\alpha}\frac{\cos\alpha/2}{\sin\alpha/2}$
$\displaystyle\qquad\qquad\times\text{Tr}_{R_{j}}\left(\mathcal{P}e^{\frac{\alpha}{2\pi}\oint
A_{L}}\right)\text{Tr}_{R_{j}}\left(\mathcal{P}e^{-\frac{\alpha}{2\pi}\oint
A_{R}}\right)~{},$ (4)

which captures $Z_{\scaleto{\text{scalar}}{3.5pt}}$ by using solely objects in
Chern-Simons theory. Starting from the right, we have path ordered
exponentials of $A_{L}$ and $A_{R}$ which encode $g_{\mu\nu}$ above: this is
the portion that captures the information of the geometry. Next, we have
traces over a representation $R_{j}$: here is where we encode the single-
particle representations of the field. The mass of the field is related to the
Casimir of the representation

$c_{2}=-\frac{\mathsf{m}^{2}}{4\Lambda}~{},$ (5)

with $\Lambda$ the cosmological constant. Finally, in (II) we have an integral
over $\alpha$. The measure and contour ${\mathcal{C}}$ of this integral serve
two purposes [25]: First, they make $\mathbb{W}$ free of UV divergences.
Second, as will become clear in detail, evaluating this contour integral as a
sum over poles implements a sum over Wilson loops with arbitrary winding,
which is what makes it a “spool.” The specific details of ${\mathcal{C}}$, and
allowed deformations, depend on the holonomies of $A_{L,R}$; these will be
specified for the backgrounds considered in the following sections.

In this letter we want to uphold (II) on two fronts. First, we believe (II)
applies to any smooth three-dimensional background, including but also
extending beyond dS3 quantum gravity. Second, it is a useful expression to
quantify quantum gravity effects: the Chern-Simons formulation allows us to
integrate out the scalar field, and obtain an explicit functional in terms of
the connections $A_{L,R}$. In this context, the main appeal of our proposal is
the ability to quantify

$\displaystyle\big{\langle}\log Z_{\rm
scalar}\big{\rangle}_{\scaleto{\text{grav}}{3.5pt}}=\frac{1}{4}\big{\langle}\mathbb{W}_{j}\big{\rangle}_{\scaleto{\text{grav}}{3.5pt}}$
(6) $\displaystyle=\frac{1}{4}\int\mathcal{D}A_{L/R}\,e^{ik_{L}S_{\rm
CS}[A_{L}]+ik_{R}S_{\rm CS}[A_{R}]}\mathbb{W}_{j}[A_{L},A_{R}]~{},$

where $S_{\rm CS}[A]$ is the Chern-Simons action, $k_{L/R}$ are the levels.
For brevity, here $\mathcal{D}A_{L/R}$ accounts for the measure for the two
copies of the gauge group. The brackets mean that one is accounting for
gravitational fluctuations around a fixed topology. In the Chern-Simons
language, this means that we fix a background holonomy for $A_{L,R}$ in
addition to the topology. It is important to stress that (6) is a non-trivial
function of $G_{N}$ and the mass of scalar field.

##  III Wilson spools in AdS3 gravity

As a first example of the utility of our proposal we will focus on Chern-
Simons gravity with negative cosmological constant,
$\Lambda=-\ell_{\text{AdS}}^{-2}$. This is quantum gravity on spaces that are
locally asymptotically Anti-de Sitter (AdS). In Lorentzian signature, the
relevant isometry group is $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ and
massive particles in this space can be organized into
$\mathfrak{sl}(2,\mathbb{R})$ representation theory. The Lorentzian Einstein-
Hilbert action is given by the difference in $SL(2,\mathbb{R})$ Chern-Simons
theories

$S_{\text{EH}}=k\left(S_{\text{CS}}[A_{L}]-S_{\text{CS}}[A_{R}]\right)~{},$
(7)

where $k=\frac{\ell_{\text{AdS}}}{4G_{N}}$. The connections are related to the
co-frame and the spin connection via

$A_{L}=(\omega^{a}+e^{a}/\ell_{\text{AdS}})L_{a}~{},~{}~{}A_{R}=(\omega^{a}-e^{a}/\ell_{\text{AdS}})\bar{L}_{a}~{}.$
(8)

Above $L_{a}$ and $\bar{L}_{a}$ generate the independent
$\mathfrak{sl}(2,\mathbb{R})$ algebras. Our conventions follow those in [3,
23]. At the classical level, the background geometries of interest are
generated by flat background connections

$\displaystyle a_{L}=$ $\displaystyle
L_{0}\,\differential\rho+\left(e^{\rho}L_{+}-e^{-\rho}\frac{2\pi\mathcal{L}}{k}L_{-}\right)\differential
x^{+}~{},$ $\displaystyle a_{R}=$
$\displaystyle-\bar{L}_{0}\,\differential\rho-\left(e^{\rho}\,\bar{L}_{-}-e^{-\rho}\frac{2\pi\bar{\mathcal{L}}}{k}\,\bar{L}_{+}\right)\differential
x^{-}~{},$ (9)

where $x^{\pm}=t\pm\varphi$ and $\varphi\sim\varphi+2\pi$. Upon using (8),
these are rotating BTZ black hole geometries [26, 27] with mass, $M$, and
angular momentum, $J$, given by

$\mathcal{L}=\frac{M\ell_{\text{AdS}}+J}{4\pi}~{},\qquad\bar{\mathcal{L}}=\frac{M\ell_{\text{AdS}}-J}{4\pi}~{}.$
(10)

In Euclidean signature, we can rotate $(x^{+},x^{-})$ to complex coordinates
$(z,-\bar{z})$. Periodicity in $\varphi$ plus smoothness of the horizon
implies these parameterize a complex torus, $(z,\bar{z})\sim(z+2\pi m+2\pi
n\tau,\bar{z}+2\pi m+2\pi n\bar{\tau})$, $m,n\in\mathbb{Z}$, with modular
parameter

$\tau=\frac{i}{2}\sqrt{\frac{k}{2\pi\mathcal{L}}}~{},\qquad\bar{\tau}=-\frac{i}{2}\sqrt{\frac{k}{2\pi\bar{\mathcal{L}}}}~{}.$
(11)

What separates this complex torus from the torus defining the boundary of
thermal AdS3 is how we “fill it in” in the bulk: in particular, the black hole
geometry is filled in so that the thermal-cycle is bulk contractible while the
spatial-cycle is not. The holonomies of the background connections $a_{L/R}$
around this cycle, $\gamma_{\varphi}$, are easily computed to be

$\displaystyle\mathcal{P}\exp\left(\oint_{\gamma_{\varphi}}a_{L/R}\right)=$
$\displaystyle u_{L/R}^{-1}\,e^{i2\pi\mathsf{h}_{L/R}L_{0}}\,u_{L/R}~{},$ (12)

where $u_{L/R}$ are periodic group elements and

$\mathsf{h}_{L}=-\frac{1}{\tau}~{},\qquad\mathsf{h}_{R}=-\frac{1}{\bar{\tau}}~{}.$
(13)

Next we will show that the Wilson spool of these background connections,
$a_{L/R}$, reproduces the one-loop determinant of a massive scalar field on
the BTZ background. That is, we will test the relation

$\displaystyle\log Z_{\scaleto{\text{scalar}}{3.5pt}}[\text{BTZ}]$
$\displaystyle=\log\det(-\nabla_{\scaleto{\rm
BTZ}{3.5pt}}^{2}+\mathsf{m}^{2}\ell_{\text{AdS}}^{2})^{-1/2}$ (14)
$\displaystyle=\frac{1}{4}\mathbb{W}_{j}[a_{L},a_{R}]~{}.$

Here $j$ labels a lowest-weight representation of
$\mathfrak{sl}(2,\mathbb{R})$ related to the mass of the bulk field via

$j=\frac{1}{2}\left(1+\sqrt{\mathsf{m}^{2}\ell_{\text{AdS}}^{2}+1}\right)\equiv\frac{1}{2}\Delta~{}.$
(15)

The expression for the Wilson spool, (II), can be derived for this background.
Given (12), we have

$\displaystyle\mathbb{W}_{j}[a_{L},a_{R}]=i\int_{\mathcal{C}}\frac{\differential\alpha}{\alpha}\frac{\cos\alpha/2}{\sin\alpha/2}\chi_{j}\left(\frac{\alpha}{2\pi}\mathsf{h}_{L}\right)\chi_{j}\left(-\frac{\alpha}{2\pi}\mathsf{h}_{R}\right)~{},$
(16)

where $\mathsf{h}_{L/R}$ are given by (13) and

$\chi_{j}(z)=\text{Tr}_{R_{j}}\left(e^{i2\pi zL_{0}}\right)=\frac{e^{i\pi
z(2j-1)}}{2\sinh(-i\pi z)}$ (17)

is the character of the lowest-weight representation, $R_{j}$. The contour,
$\mathcal{C}$, is given by twice the contour running up the imaginary $\alpha$
axis to the right of zero: $\mathcal{C}=2\mathcal{C}_{+}$. This follows from
the procedure in [25] but assigning an $i\epsilon$ prescription appropriate
for representations and holonomies relevant to AdS3. Further detail of this is
also described in the supplementary material.

Because $\tau,\bar{\tau}\in i\mathbb{R}$ (equivalently, $\mathcal{L}$ and
$\bar{\mathcal{L}}$ are real and positive), all poles in the $\alpha$
integrand in (16) to the right of zero are simple poles at
$2\pi\mathbb{Z}_{>0}$ and arise from the measure,
$\frac{\cos\alpha/2}{\sin\alpha/2}$. We then deform the contour $\mathcal{C}$
to the right where the $\alpha$ integrand is damped, picking up the residues
of the simple poles. We can then write

$\displaystyle\mathbb{W}_{j}[a_{L},a_{R}]=$
$\displaystyle\sum_{n=1}^{\infty}\frac{e^{-i\pi
n(\tau^{-1}-\bar{\tau}^{-1})(2j-1)}}{n\,\sin\left(\frac{\pi
n}{\tau}\right)\sin\left(\frac{\pi n}{\bar{\tau}}\right)}~{}.$ (18)

This can be easily rewritten into

$\frac{1}{4}\mathbb{W}_{j}[a_{L},a_{R}]=\log\prod_{l,\bar{l}=0}^{\infty}\left(1-q^{\frac{\Delta}{2}+l}\bar{q}^{\frac{\Delta}{2}+\bar{l}}\right)^{-1}~{},$
(19)

where $q=e^{-i\frac{2\pi}{\tau}}$ and $\bar{q}=e^{i\frac{2\pi}{\bar{\tau}}}$.
This matches exactly $\log Z_{\scaleto{\text{scalar}}{3.5pt}}[\text{BTZ}]$ for
a real massive field propagating on the rotating BTZ background [28].

This is the tree-level ($G_{N}\to 0$) contribution to $\big{\langle}\log
Z_{\scaleto{\text{scalar}}{3.5pt}}\big{\rangle}_{\scaleto{\text{grav}}{3.5pt}}$.
Promoting the background connections to dynamical fields, $a_{L/R}\rightarrow
A_{L/R}$, the Chern-Simons path-integral provides a way forward, in principle,
for calculating perturbative $G_{N}$ corrections to (19). The non-compact
gauge group and the non-compact background topology make this program still
difficult. However, one may still make progress using large-$k$ Chern-Simons
perturbation theory similar to [22, 29, 30]. Below we will see that in the
context of positive cosmological constant, our ability to calculate $G_{N}$
corrections is under even better control.

##  IV Wilson spools in dS3 gravity

Our next example will be Chern-Simons gravity with positive cosmological
constant, $\Lambda=\ell_{\rm dS}^{-2}$, which describes quantum gravity on de
Sitter space. We will be working with a Euclidean action given by two copies
of the $SU(2)$ Chern-Simons action; its relation to dS3 gravity is via

$I_{\text{EH}}-i\delta I_{\text{GCS}}=-ik_{L}S_{\rm CS}[A_{L}]-ik_{R}S_{\rm
CS}[A_{R}]~{},$ (20)

where $I_{\text{EH}}$ is the Euclidean Einstein-Hilbert action and
$I_{\text{GCS}}$ is the gravitational Chern-Simons action. The Chern-Simons
couplings $k_{L,R}$ are now in general complex,

$k_{L}=\delta+is~{},\qquad k_{R}=\delta-is~{},$ (21)

with an imaginary part that is related to Newton’s constant,
$s=\frac{\ell_{\text{dS}}}{4G_{N}}$, and a real part, $\delta$, giving the
coefficient of a gravitational Chern-Simons action. Quantum effects lead to a
renormalization in the coupling constants [31]

$k_{L}\rightarrow r_{L}=k_{L}+2~{},\qquad k_{R}\rightarrow r_{R}=k_{R}+2~{},$
(22)

which amounts to a renormalization of
$\delta\rightarrow\hat{\delta}=\delta+2$.

The above matching is facilitated by relating the gauge fields to the vielbein
and spin connection via

$\displaystyle
A_{L}=i(\omega^{a}+\frac{e^{a}}{\ell_{\text{dS}}})L_{a}~{},~{}~{}A_{R}=i(\omega^{a}-\frac{e^{a}}{\ell_{\text{dS}}})\bar{L}_{a}~{},$
(23)

where now $\\{L_{a}\\}$ and $\\{\bar{L}_{a}\\}$ generate the two independent
$\mathfrak{su}(2)$’s. We will focus presently on the round $S^{3}$ saddle
whose classical background geometry is generated by background connections

$\displaystyle a_{L}=$ $\displaystyle
iL_{1}\,\differential\rho+i\left(\sin\rho L_{2}-\cos\rho
L_{3}\right)(\differential\varphi-\differential\tau)~{},$ $\displaystyle
a_{R}=$
$\displaystyle-i\bar{L}_{1}\,\differential\rho-i\left(\sin\rho\bar{L}_{2}+\cos\rho\bar{L}_{3}\right)(\differential\varphi+\differential\tau)~{}.$
(24)

These background connections have a singularity at the causal horizon
($\rho=\pi/2$) and the holonomy around that singularity is given by222The
$S^{3}$ geometry is regular everywhere; the holonomies add up to give a smooth
background.

$\displaystyle\begin{aligned}
\mathcal{P}\exp\oint_{\gamma}a_{L/R}&=u_{L/R}^{-1}e^{i2\pi
L_{3}\mathsf{h}_{L/R}}u_{L/R}~{},\end{aligned}$ (25)

with $u_{L/R}$ periodic group elements and

$\mathsf{h}_{L}=1~{},\qquad\mathsf{h}_{R}=-1~{}.$ (26)

Now let us add matter and test our proposal. We will begin with the tree-level
check which evaluates the spool on the background connections:

$\displaystyle\log Z_{\rm scalar}[S^{3}]$
$\displaystyle=\frac{1}{4}\mathbb{W}_{j}[a_{L},a_{R}]~{}.$ (27)

The representation, $R_{j}$, that appears in $\mathbb{W}_{j}$ is a highest-
weight representation of $\mathfrak{su}(2)$ related to the mass of the scalar
field as

$j=-\frac{1}{2}\left(1+\sqrt{1-\mathsf{m}^{2}\ell_{\text{dS}}^{2}}\right)~{}.$
(28)

Note that $j$ is a continuous parameter and can even become complex for large
enough masses. Thus these representations do not correspond to the standard
finite dimensional representations of $SU(2)$. Instead they correspond to
infinite dimensional “non-standard” representations whose weight spaces line
up with the dS3 quasi-normal mode spectrum [24, 25]. Despite not lying in the
standard $SU(2)$ representation theory, the representations obeying (28) can
be equipped with an inner product such that all states have positive norm [24,
25]. Additionally they admit well-defined characters:

$\chi_{j}(z)=\mbox{Tr}_{R_{j}}\left(e^{2\pi izL_{3}}\right)=\frac{e^{i\pi
z(2j+1)}}{2i\sin{(\pi z)}}~{}.$ (29)

We can then express (IV) in terms of the holonomies (26) as

$\displaystyle\mathbb{W}_{j}[a_{L},a_{R}]=$ $\displaystyle
i\int_{\mathcal{C}}\frac{\differential\alpha}{\alpha}\frac{\cos\alpha/2}{\sin\alpha/2}\chi_{j}\left(\frac{\alpha}{2\pi}\mathsf{h}_{L}\right)\chi_{j}\left(-\frac{\alpha}{2\pi}\mathsf{h}_{R}\right)$
$\displaystyle=$
$\displaystyle-\frac{i}{4}\int_{\mathcal{C}}\frac{\differential\alpha}{\alpha}\frac{\cos\alpha/2}{\sin^{3}\alpha/2}e^{i(2j+1)\alpha}~{}.$
(30)

Deviating slightly from the procedure for the BTZ computation, the $i\epsilon$
prescription appropriate for de Sitter results in a contour,
$\mathcal{C}=\mathcal{C}_{-}\cup\mathcal{C}_{+}$, that is, the union of
contours running up the imaginary $\alpha$ axis both to the left and right of
zero [25]. We pull both $\mathcal{C}_{\pm}$ towards the positive real axis to
pick up residue of the quadruple pole at $\alpha=0$ as well as twice the
residues of the poles at $\alpha\in 2\pi\mathbb{Z}_{>0}$ along the positive
real line; these latter poles are now third-order. Doing so we find

$\displaystyle\frac{1}{4}\mathbb{W}_{j}[a_{L},a_{R}]=$ $\displaystyle
i\frac{\pi(2j+1)^{3}}{6}-\frac{1}{4\pi^{2}}\text{Li}_{3}\left(e^{i2\pi(2j+1)}\right)$
$\displaystyle+i\frac{(2j+1)}{2\pi}\text{Li}_{2}\left(e^{i2\pi(2j+1)}\right)$
$\displaystyle-\frac{(2j+1)^{2}}{2}\text{Li}_{1}\left(e^{i2\pi(2j+1)}\right)~{},$
(31)

where $\text{Li}_{q}(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{n^{q}}$ are
polylogarithm functions. Upon using (28), this answer matches precisely the
finite contribution to the scalar one-loop determinant on $S^{3}$; see for
instance [17]. This provides a second non-trivial check of the physical
relevance of the Wilson spool.

In the context of de Sitter gravity we can actually make substantial progress
in discussing the Wilson spool beyond its classical expectation value. This is
because there exists a library of “exact techniques” for $SU(2)$ Chern-Simons
theories on compact topologies. These techniques reduce the Chern-Simons path-
integral, as well as the expectation values of certain operators, to ordinary
integrals. In the present context, care is needed to alter these methods to
accommodate the features necessary for Chern-Simons gravity (i.e. complex
levels, non-zero background connections, and non-standard representations).
However several exact methods remain amenable to these alterations [25]. For
instance, a suitable alteration of Abelianisation [32, 33, 34] reduces the
expectation value of $\mathbb{W}_{j}[A_{L},A_{R}]$ with dynamical connections
to a pair of simple integrals [25]:

$\displaystyle\frac{1}{4}\big{\langle}\mathbb{W}_{j}\big{\rangle}_{\scaleto{\text{grav}}{3.5pt}}=\frac{i}{4}\,e^{ir_{L}S_{\rm
CS}[a_{L}]+ir_{R}S_{\rm CS}[a_{R}]}$
$\displaystyle\times\int\differential\sigma_{L}\differential\sigma_{R}\,\Big{\\{}e^{i\frac{\pi}{2}r_{L}\sigma_{L}^{2}+i\frac{\pi}{2}r_{R}\sigma_{R}^{2}}\sin^{2}(\pi\sigma_{L})\sin^{2}(\pi\sigma_{R})$
$\displaystyle\times\int_{\mathcal{C}}\frac{\differential\alpha}{\alpha}\frac{\cos\alpha/2}{\sin\alpha/2}\chi_{j}\left(\frac{\alpha(\sigma_{L}+\mathsf{h}_{L})}{2\pi}\right)\chi_{j}\left(-\frac{\alpha(\sigma_{R}+\mathsf{h}_{R})}{2\pi}\right)\Big{\\}}~{},$
(32)

which we can equate to the quantum gravity corrected one-loop determinant
about the $S^{3}$ saddle, $\big{\langle}\log
Z_{\scaleto{\text{scalar}}{3.5pt}}[S^{3}]\big{\rangle}_{\scaleto{\text{grav}}{3.5pt}}$.
The integral appearing in (IV) is difficult to evaluate analytically, however
it can systematically be evaluated in a $s^{-1}$ Taylor expansion. This
provides a controlled procedure to computing $G_{N}$ corrections to $\log
Z_{\scaleto{\text{scalar}}{3.5pt}}[S^{3}]$. These corrections are naturally
identified with a mass renormalization. To extract them one simply computes
$\big{\langle}\log
Z_{\scaleto{\text{scalar}}{3.5pt}}[S^{3}]\big{\rangle}_{\scaleto{\text{grav}}{3.5pt}}$,
normalized by the gravitational path integral, using (IV); the resulting
renormalized mass to $O(G_{N}^{2})$ is given by333In (33) we have assumed that
the (renormalized) gravitational Chern-Simons coupling, $\hat{\delta}$, does
not scale with $G_{N}$.

$\mathsf{m}^{2}_{R}\ell^{2}_{\text{dS}}=\mathsf{m}^{2}\ell^{2}_{\text{dS}}+\frac{96}{5}\mathsf{m}^{4}\ell^{4}_{\text{dS}}e^{-2\pi\,\absolutevalue{\mathsf{m}\ell_{\text{dS}}}}\,\left(\frac{G_{N}}{\ell_{\text{dS}}}\right)^{2}+\ldots~{}.$
(33)

Above we have kept the leading term in a large mass expansion,
$\mathsf{m}^{2}\ell_{\text{dS}}^{2}\gg 1$, however (IV) provides an expression
for the $G_{N}^{2}$ renormalization of the mass that can be calculated
analytically [25]. We emphasize that this is a concrete predictive statement
about how dynamical quantum gravity renormalizes quantum field theory.

##  V Discussion

We have introduced a new object, the Wilson spool, which allows one to have
matter fields in the Chern-Simons formulation of three-dimensional quantum
gravity while keeping manifest the key topological aspects of gravity. To test
this object, we have shown that, for $G_{N}\rightarrow 0$, it reproduces
correctly one-loop determinants of massive scalar fields on a curved
background. The proposal works for spacetimes that are widely different, such
as the spinning BTZ black hole and Euclidean de Sitter spacetime.

If our proposal merely provided a way to match onto known one-loop
determinants, it would have limited utility. However, it is also possible to
use this techniques to make a _prediction_ for quantum corrections to $\log
Z_{\rm grav}$, having allowed for quantum fluctuations of the metric around a
classical background. For certain AdS geometries (see, e.g., [22]), the $1/c$
corrections to Wilson lines have been computed in a holographic setting. Our
techniques give us a way to extend this beyond the holographic setting to
quantum gravity in spacetimes like de Sitter, which are more realistic
approximations of our observed universe but where techniques of holography are
largely out of reach.

There are two future directions in this area that we would like to highlight.
A more in depth discussion is presented in [25].

### Massive spinning fields.

We have focused on massive scalar fields in this letter, for which we have a
direct construction of the spool. But both AdS3 and dS3 admit massive fields
of arbitrary integer spin. We can ask if the Wilson spool is of utility in
coupling these excitations to dynamical gravity. We do not presently have a
first-principles construction for $\mathbb{W}$ applied to spinning fields (as
we do for scalars). However, in the context of AdS3, the definition of the
Wilson spool, (II), can be intuitively extended to reproduce the one-loop
determinant for a massive spinning field on the rotating BTZ background. The
details of this can be found in the supplementary material. This is a strong
indication that $\mathbb{W}$ is useful for the physics of spinning fields. We
can also make a first pass application of $\mathbb{W}$ to spinning fields in
de Sitter; this relies on further guess work since the appropriate non-
standard $\mathfrak{su}(2)$ representations have not yet been identified. In
this case, while we can reliably reproduce certain contributions to the one-
loop determinant, $\mathbb{W}$ currently misses an important “edge
subtraction” arising from properly subtracting normalizable transverse-
traceless zero modes on $S^{3}$ [17]. A proper derivation of the spool to
capture the physics of spinning fields will be addressed in future work.

### Sum over topologies.

Here we have kept the topology of the background, and holonomies of the
connections, fixed. It is of great interest to allow for these non-
perturbative contributions in (2), albeit it comes with difficulties: if only
metric degrees of freedom are incorporated, both the AdS3 thermal partition
function [35, 36] and the Euclidean dS3 path integral [16] suffer from
pathologies. But it is also expected that adding matter can fix some of these
problems [37]. It would interesting to investigate the fate of the Wilson
spool under these pathologies, i.e., loop in matter in the sum over manifolds
and quantify its imprint. This is specially of interest in dS3, where a
holographic dictionary is still nascent, but the Wilson spool gives a concise
path to quantify the effect of fields at all orders in $G_{N}$.

###### Acknowledgements.

## Acknowledgments

We would like to thank Kurt Hinterbichler and Sergei Cherkis for useful
comments and discussions. JRF thanks the University of Amsterdam for
hospitality. The work of AC and JRF has been partially supported by STFC
consolidated grant ST/T000694/1. The work JRF has been also partially
supported by Simons Foundation Award number 620869. IC has been partially
supported by the ERC starting grant H2020 ERC StG No.640159. CZ has been
supported by a UM Duluth Higholt Professorship.

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Supplemental Material

##  Appendix A A. Details on constructing the Wilson spool in AdS3

The construction of $\mathbb{W}_{j}$ as described in [25] makes use of the
physical reasoning established by Denef, Hartnoll, and Sachdev (DHS) [38].
Namely, the one-loop determinant

$\left(Z_{\text{scalar}}\right)^{2}=\left(\text{det}\left(-\nabla^{2}+\mathsf{m}^{2}\right)\right)^{-1}~{},$
(34)

interpreted as a meromorphic function of the mass, is equal to the rational
function containing the same zeros and poles (up to multiplication by an
entire holomorphic function). $\left(Z_{\text{scalar}}\right)^{2}$ will have
poles in the complex $\mathsf{m}^{2}$ plane on solutions to

$\left(-\nabla^{2}+\mathsf{m}^{2}\right)\phi=0~{},$ (35)

subject to appropriate boundary conditions.

For asymptotically AdS backgrounds, we interpret this statement in the
language of $\mathfrak{sl}(2,\mathbb{R})$ representation theory by noting that
there exist vector fields, $\\{\zeta_{0},\zeta_{+},\zeta_{-}\\}$ and
$\\{\bar{\zeta}_{0},\bar{\zeta}_{+},\bar{\zeta}_{-}\\}$, spanning
$\mathfrak{sl}(2,\mathbb{R})_{L}$ and $\mathfrak{sl}(2,\mathbb{R})_{R}$,
respectively, whose summed Casimirs act as the Laplacian [23]:

$\ell_{\text{AdS}}^{2}\nabla^{2}=\zeta_{+}\zeta_{-}+\zeta_{-}\zeta_{+}-2\zeta_{0}^{2}+\bar{\zeta}_{+}\bar{\zeta}_{-}+\bar{\zeta}_{-}\bar{\zeta}_{+}-2\bar{\zeta}_{0}^{2}=2c_{2,L}+2c_{2,R}~{}.$
(36)

The DHS procedure instructs us to write

$\left(Z_{\text{scalar}}\right)^{2}=\left(\text{det}(-2c_{2,L}-2c_{2,R}+\mathsf{m}^{2}\ell^{2}_{\text{AdS}})\right)^{-1}$
(37)

as the simple product function containing the same poles. These poles lie
along weights of scalar representations of
$\mathfrak{sl}(2,\mathbb{R})_{L}\oplus\mathfrak{sl}(2,\mathbb{R})_{R}$
satisfying $c_{2,L}=c_{2,R}=\frac{1}{4}\mathsf{m}^{2}\ell^{2}_{\text{AdS}}$.
These are highest/lowest-weight representations with $j_{L}=j_{R}\equiv j$ and

$j=\frac{1}{2}\left(1\pm\sqrt{\mathsf{m}^{2}\ell^{2}_{\text{AdS}}+1}\right)~{},$
(38)

where $j$ is a lowest (highest)-weight if it is positive (negative). However,
unlike in de Sitter, it is not the case that both roots of (38) can
contribution poles to $\left(Z_{\text{scalar}}\right)^{2}$. This is because we
must impose boundary conditions: in particular, at asymptotic infinity we fix
a Dirichlet boundary condition on the scalar field, $\Phi$, with normalizable
fall-off. This then fixes $\Phi$ to correspond to a unique lowest-weight
representation. For the rest of this letter we will fix this to the lowest-
weight representation with the positive root,
$j=\frac{1}{2}\left(1+\sqrt{\mathsf{m}^{2}\ell^{2}_{\text{AdS}}+1}\right)$. We
should note that for $-1<\mathsf{m}^{2}\ell^{2}_{\text{AdS}}<0$ the other
solution is also positive and so is also compatible with a normalizable
boundary condition; the results for this “alternative quantization” follow
exactly the same procedure laid out below.

Additionally we must impose periodicity along the non-contractible spatial
cycle, $\gamma_{\varphi}$. For a field living in the $R_{j}\otimes R_{j}$
representation of
$\mathfrak{sl}(2,\mathbb{R})_{L}\oplus\mathfrak{sl}(2,\mathbb{R})_{R}$, its
parallel transport along this cycle is

$\phi\rightarrow
R_{j}\left(\mathcal{P}e^{\oint_{\gamma_{\varphi}}a_{L}}\right)\cdot\phi\cdot
R_{j}\left(\mathcal{P}e^{-\oint_{\gamma_{\varphi}}a_{R}}\right)~{},$ (39)

and so a weight $(\lambda_{L},\lambda_{R})\in R_{j}\otimes R_{j}$ can
contribute a pole to $\left(Z_{\text{scalar}}\right)^{2}$ if

$\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R}=\pm n~{},\qquad
n\in\mathbb{Z}~{},$ (40)

where $\mathsf{h}_{L/R}$ are the holonomies of the background connections,
(13). The $\pm$ arises from imposing periodicity on both the left and right-
propagating modes. Alternatively, it arises from choosing the sign of the
square-root in arriving at (13). In the metric language this is the same as
imposing periodicity on both ingoing and outgoing quasi-normal modes [38, 39].
Note that in de Sitter this contribution is accounted for by including the
shadow representation [38]. Thus we arrive at the following expression for
$\left(Z_{\text{scalar}}\right)^{2}$:

$\displaystyle\left(Z_{\text{scalar}}\right)^{2}=\prod_{\overset{(\lambda_{L},\lambda_{R})}{\in
R_{j}\otimes
R_{j}}}\prod_{n\in\mathbb{Z}}\left(|n|-\lambda_{L}\mathsf{h}_{L}+\lambda_{R}\mathsf{h}_{R}\right)^{-1}\left(|n|+\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R}\right)^{-1}~{}.$
(41)

We now take the log. We will implement this with a Schwinger parameter $\log
M=-\int_{\times}^{\infty}\frac{\differential\alpha}{\alpha}e^{-\alpha M}$,
where in the lower limit we indicate a need to regulate the UV divergence at
$\alpha\sim 0$. Additionally we will need to regulate the sum over
representation weights $(\lambda_{L},\lambda_{R})$. Both of these can be taken
care of simultaneously through an $i\epsilon$ prescription which we implement
now. Noting
$\sum\limits_{n\in\mathbb{Z}}e^{-|n|\alpha}=\frac{\cosh\alpha/2}{\sinh\alpha/2}$,
we write

$\log
Z_{\scaleto{\text{scalar}}{3.5pt}}=\frac{1}{2}\int_{\times}^{\infty}\frac{\differential\alpha}{\alpha}\frac{\cosh\alpha/2}{\sinh\alpha/2}\sum_{\lambda_{L},\lambda_{R}}\left(e^{\alpha\left(\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R}\right)}+e^{-\alpha\left(\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R}\right)}\right)~{}.$
(42)

Note that $\mathsf{h}_{L}$ is equal to $i$ times a positive quantity, while
$\mathsf{h}_{R}$ is $i$ times a negative quantity. Also, since $R_{j}$ is a
lowest-weight representation of $\mathfrak{sl}(2,\mathbb{R})$, all
$\lambda_{L/R}$ appearing in the sum are real and positive. It then follows
that the first (second) sum converges by giving $\alpha$ a small positive
(negative) imaginary part. Under the replacement of integration variable,
$\alpha\rightarrow-\alpha$, in the second sum, we can write the regulated
$\log Z_{\scaleto{\text{scalar}}{3.5pt}}$ succinctly as

$\log{Z_{\scaleto{\text{scalar}}{3.5pt}}}=\frac{1}{2}\int_{-\infty+i\epsilon}^{\infty+i\epsilon}\frac{\differential\alpha}{\alpha}\frac{\cosh\alpha/2}{\sinh\alpha/2}\sum_{\lambda_{L},\lambda_{R}}e^{\alpha\left(\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R}\right)}~{}.$
(43)

We recognize the sum over representation weights times holonomies as trace
path-ordered exponentials of the background connections. Finally, we make the
integration variable replacement $\alpha\rightarrow i\alpha$ to write

$\log
Z_{\scaleto{\text{scalar}}{3.5pt}}=\frac{i}{4}\int_{2\mathcal{C}_{+}}\frac{\differential\alpha}{\alpha}\frac{\cos\alpha/2}{\sin\alpha/2}\text{Tr}_{R_{j}}\left(\mathcal{P}e^{\frac{\alpha}{2\pi}\oint_{\gamma_{\varphi}}a_{L}}\right)\text{Tr}_{R_{j}}\left(\mathcal{P}e^{-\frac{\alpha}{2\pi}\oint_{\gamma_{\varphi}}a_{R}}\right)\equiv\frac{1}{4}\mathbb{W}_{j}[a_{L},a_{R}]~{}.$
(44)

Thus appears the Wilson spool. The contour $\mathcal{C}_{+}$ runs upwards
along the imaginary $\alpha$ axis to the right of zero. We write it
conventionally with the factor of two to uniformize the notation with the de
Sitter spool (where the ingoing vs. outgoing periodicity constraints give rise
to distinct contours [25]). We derived $\mathcal{C}_{+}$ by ensuring
representation sums converge, however, in the body of the letter we verify a
posteriori that it also regulates the UV divergence in $\log
Z_{\scaleto{\text{scalar}}{3.5pt}}$.

##  Appendix B B. Massive spinning fields

We provide additional details on a “first-pass” attempt to incorporate
spinning fields into the Wilson spool. This is easiest to address in the
context of AdS3, where massive particles of spin $\mathsf{s}$ are already
included in
$\mathfrak{sl}(2,\mathbb{R})_{L}\oplus\mathfrak{sl}(2,\mathbb{R})_{R}$
representation theory through lowest-weights

$j^{+}=\frac{\Delta+\mathsf{s}}{2}~{},\qquad
j^{-}=\frac{\Delta-\mathsf{s}}{2}~{}.$ (45)

Noting that in [25] that the construction of $\mathbb{W}$ arises from a sum of
Casimirs, $c_{2,L}+c_{2,R}$, it is natural to conjecture that both
contributions, $(j_{L},j_{R})=(j^{+},j^{-})$ and
$(j_{L},j_{R})=(j^{-},j^{+})$, should be included the spool since both choices
have the same sum of Casimirs:

$\displaystyle\mathbb{W}_{j^{+},j^{-}}[A_{L},A_{R}]=i\int_{\mathcal{C}}\frac{\differential\alpha}{\alpha}\frac{\cos\alpha/2}{\sin\alpha/2}\sum_{\pm}\text{Tr}_{R_{j^{\pm}}}\left(\mathcal{P}e^{\frac{\alpha}{2\pi}\oint
A_{L}}\right)\text{Tr}_{R_{j^{\mp}}}\left(\mathcal{P}e^{-\frac{\alpha}{2\pi}\oint
A_{R}}\right)~{}.$ (46)

Indeed doing so, we can calculate for the BTZ background, (III), that the
tree-level contribution to the spinning spool is

$\displaystyle\frac{1}{4}\mathbb{W}_{j^{+},j^{-}}[a_{L},a_{R}]=$
$\displaystyle\sum_{\pm}\sum_{n=1}^{\infty}\frac{1}{n}\frac{q^{\frac{n}{2}(\Delta\pm\mathsf{s})}\bar{q}^{\frac{n}{2}(\Delta\mp\mathsf{s})}}{(1-q^{n})(1-\bar{q}^{n})}=-\sum_{\pm}\sum_{l,\bar{l}=0}^{\infty}\log\left(1-q^{\frac{1}{2}(\Delta\pm\mathsf{s})+l}\bar{q}^{\frac{1}{2}(\Delta\mp\mathsf{s})+\bar{l}}\right)~{},$
(47)

which is the correct one-loop determinant for a massive spinning field on the
rotating BTZ background [40, 41, 39]. We stress that, as of present, we do not
have a first principles construction for $\mathbb{W}_{j^{+},j^{-}}$ to justify
the above calculation. However in the context of AdS3 we have reason to trust
the above: massive spinning fields in AdS3 have only two real physical
polarizations [40] which we can package into $\mathfrak{sl}(2,\mathbb{R})$
lowest-weights $(j^{+},j^{-})$. If the one-loop determinant continues to
depend only on the summed Casimirs, as it does for scalar fields, then is hard
to imagine any other appropriately group theoretic object to construct besides
(46).

Trying to extend this reasoning to de Sitter gravity is more subtle. Part of
the issue is that we already need novel $\mathfrak{su}(2)$ representations to
capture the physics of scalar excitations; as of yet we have not constructed
the non-standard representations necessary to accommodate $\mathsf{s}\neq 0$.
We view this as technical hurdle as opposed to a conceptual no-go. As a first
pass, we can analytically continue the $\mathfrak{su}(2)$ weights to
accommodate spin via

$j^{+}=-\frac{\Delta+\mathsf{s}}{2}~{},\qquad
j^{-}=-\frac{\Delta-\mathsf{s}}{2}~{}.$ (48)

Investigating (46) for the classical $S^{3}$ background, (IV), we can
implement this analytic continuation at the level of the characters. We find
that $\mathbb{W}_{j^{+},j^{-}}[a_{L},a_{R}]$ is, in fact, $\mathsf{s}$
independent and essentially doubles the scalar answer, (IV). This almost
reproduces one piece of the one-loop determinant of massive spinning fields on
$S^{3}$,

$\log
Z_{\Delta,\mathsf{s}}[S^{3}]=\frac{1}{4}\mathbb{W}_{j^{+},j^{-}}[a_{L},a_{R}]-\log
Z_{\text{edge}}~{},$ (49)

however it misses an additional “edge” subtraction

$\log
Z_{\text{edge}}=\mathsf{s}^{2}\log\left(\frac{e^{i\pi(\Delta-1)}}{1-e^{i2\pi(\Delta-1)}}\right)~{},$
(50)

which arises from properly subtracting normalizable transverse-traceless zero
modes [17]. We hope that a suitable modification of $\mathbb{W}$ can capture
this edge term, which we leave for future work.