Department of Applied Mathematics and Theoretical Physics, University of
Cambridge,
Cambridge CB3 0WA, United Kingdom

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Spinning up the spool:
Massive spinning fields in 3d quantum gravity

Robert Bourne, Alejandra Castro, and Jackson R. Fliss

###### Contents

  1. 1 Introduction
  2. 2 Non-standard spinning representations of $\mathfrak{su}(2)$
  3. 3 Spinning spool on $S^{3}$
    1. 3.1 A group theory perspective on $S^{3}$ quasinormal modes
    2. 3.2 Constructing the spool
    3. 3.3 Testing the Wilson spool
  4. 4 Coupling quantum matter to dS3 quantum gravity
  5. 5 Spinning spool on AdS3
  6. 6 Discussion
  7. A Chern-Simons gravity dictionary
    1. A.1 Euclidean dS3
    2. A.2 Lorentzian AdS3
  8. B DHS for massive spinning fields on $S^{3}$
    1. B.1 Real massive scalar field
    2. B.2 Massive spin-1 field
    3. B.3 Massive spin-$\mathsf{s}$ fields

##  1 Introduction

Low-dimensional gravity is an exciting arena to explore and test the
gravitational path integral. In two and three spacetime dimensions, there is
no propagating graviton and all of the effective degrees of freedom are long-
range. A prime example of this phenomenon is the rewriting of pure Einstein
gravity with a cosmological constant (of either sign) as a Chern-Simons gauge
theory [3, 4] which is the quintessential example of a topological field
theory in three-dimensions. A full leveraging of this fact allows the exact
evaluation of the gravitational path integral either about a saddle point [5,
6, 7, 8, 9] or as a non-perturbative sum over saddles [10, 11, 12, 13]. While
Chern-Simons gravity is not a UV-complete model of quantum gravity [12, 14,
13], its all-loop exactness provides strong tests for potential microscopic
models in the spirit of e.g. [15, 16, 17, 18, 19, 8].

One feature that is expected of a UV-complete model of quantum gravity is that
it includes matter, in particular massive fields that couple to gravity. The
manifest topological invariance that makes Chern-Simons theory so powerful as
a description of the gravitational path integral also presents a challenge to
incorporating matter. On a practical level, this is simple to illustrate: the
action of a massive field theory minimally coupled to a geometry involves both
inverse metrics and metric determinants. The rewriting of these terms as
Chern-Simons connections is highly non-linear and indicates that integrating
out of the massive field will result in a non-local effective action. However,
we can take inspiration from the general philosophy that the low-energy avatar
of the worldline of a massive degree of freedom is a line-operator of the
effective gauge theory [20].

This philosophy was made precise in [1] for massive scalar fields minimally
coupled to gravity with a positive cosmological constant. The key result was
the expression of the one-loop determinant of a massive scalar field coupled
to a background metric, $g_{\mu\nu}$, as a gauge-invariant object of the
Chern-Simons connections, $A_{L/R}$. That is,

$Z_{\text{scalar}}[g_{\mu\nu}]=\exp\frac{1}{4}\mathbb{W}[A_{L},A_{R}]\leavevmode\nobreak\
.$ (1.1)

The object $\mathbb{W}[A_{L},A_{R}]$, coined the Wilson spool, is a collection
of Wilson loop operators wrapped many times around cycles of the base
geometry. The equality in (1.1) is expected to apply to three-dimensional
gravity of either sign of cosmological constant, and this was explicitly shown
for Euclidean black holes in Anti-de Sitter (i.e., Euclidean BTZ) and
Euclidean de Sitter (i.e., the three-sphere $S^{3}$) in [1, 2]. It has also
been upheld on $T\bar{T}$ deformations of AdS3 [21]. The importance of (1.1)
is not only conceptual, it is practical: it was additionally shown in [1] that
certain “exact methods” in Chern-Simons theory (such as Abelianisation [22,
23, 24] and supersymmetric localization [25, 26]) extend to three-dimensional
de Sitter (dS3) Chern-Simons gravity with the Wilson spool inserted into the
path integral. This allows a precise and efficient calculation of the quantum
gravitational corrections to $Z_{\text{scalar}}$ at any order of perturbation
theory of Newton’s constant, $G_{N}$.

An important element excluded from the dictionary established in [1] is the
description of massive fields with spin. The representation theory of dS3
spacetime admits a series of representations describing single-particle states
with both mass and integer spin [27, 28]. It is important to know how the
field theory of those particles fits into quantum gravity about the dS3
background. This consideration is more than pedestrian: we expect the massive
states of a UV description of quantum gravity to carry spin and it is sensible
to organize these states into the representation theory of the low energy
effective theory. In [2] a Wilson spool describing massive spinning fields on
Euclidean BTZ was conjectured and shown to reproduce the correct one-loop
determinants as $G_{N}\rightarrow 0$, however, no principled derivation was
provided there. More importantly, it was shown that a naïve generalization of
the Wilson spool does not correctly reproduce the physics of spinning fields
on $S^{3}$.

In this paper we address this mismatch by revisiting the derivation of
$\mathbb{W}[A_{L},A_{R}]$. The original derivation in [1] to obtain (1.1) used
some of the conditions behind the quasinormal mode method to evaluate one-loop
determinants developed in [29]. Here, we give a robust derivation where we
cast all aspects of the method in terms of conditions that rely solely on
group-theoretic concepts. These conditions apply to gravity with either sign
of cosmological constant, $\Lambda$. We show in explicit examples how from the
group-theoretic picture one can write the one-loop determinant in terms of
$(A_{L},A_{R})$ and representations of the algebra. This then gives the Wilson
spool, which describes the minimal coupling of massive spinning fields to
gravity.

To be specific, we state our main result, the generalization of (1.1) for
massive spinning fields, as the following. Consider the local path
integral,111Including any additional Stückelberg fields to fix its invariances
and associated ghosts. $Z_{\Delta,\mathsf{s}}$, of a spin-$\mathsf{s}$ field
$\Phi_{\mu_{1}\mu_{2}\ldots\mu_{s}}$ with mass

$\frac{m^{2}}{\Lambda}=(\Delta+\mathsf{s}-2)(\mathsf{s}-\Delta)\leavevmode\nobreak\
,$ (1.2)

minimally coupled to a metric geometry, $(M_{3},g_{M_{3}})$, where $M_{3}$ is
topologically either Euclidean BTZ or Euclidean dS3. Then, we propose

$\log
Z_{\Delta,\mathsf{s}}[g_{M_{3}}]=\frac{1}{4}\mathbb{W}_{j_{L},j_{R}}[A_{L},A_{R}]\leavevmode\nobreak\
,$ (1.3)

where

$\mathbb{W}_{j_{L},j_{R}}[A_{L},A_{R}]=\frac{i}{2}\int_{\mathcal{C}}\frac{\differential\alpha}{\alpha}\frac{\cos\left(\frac{\alpha}{2}\right)}{\sin\left(\frac{\alpha}{2}\right)}\left(1+\mathsf{s}^{2}\sin^{2}\left(\frac{\alpha}{2}\right)\right)\times\\\
\sum_{\mathsf{R}_{L}\otimes\mathsf{R}_{R}}\mathrm{Tr}_{\mathsf{R}_{L}}\left(\mathcal{P}e^{\frac{\alpha}{2\pi}\oint_{\gamma}A_{L}}\right)\mathrm{Tr}_{\mathsf{R}_{R}}\left(\mathcal{P}e^{-\frac{\alpha}{2\pi}\oint_{\gamma}A_{R}}\right)\leavevmode\nobreak\
.$ (1.4)

The details of $\mathbb{W}_{j_{L},j_{R}}$ will be made explicit as we
continue, however let us briefly summarize the parts appearing in (1.4). The
Chern-Simons connections, $A_{L/R}$, are related to the metric, $g_{M_{3}}$,
in (1.3) through the usual Chern-Simons gravity dictionary and they are
integrated over a non-trivial cycle, $\gamma$, of the base geometry. The
representations, $\mathsf{R}_{L/R}$, appearing in the Wilson loops are summed
over a set determined by the mass and spin, $(\Delta,\mathsf{s})$, of (1.3)
and labelled by weights $(j_{L},j_{R})$. Lastly the parameter $\alpha$ is
integrated along a contour, $\mathcal{C}$, determined by a regularization
scheme appropriate for the sign of cosmological constant. The ultimate effect
of the $\alpha$ integral is to implement a “winding” of the Wilson loop
operators around $\gamma$; this occurs through the summing the residues of the
poles of its measure (as well as any of representation traces themselves). We
coin the above object, (1.4), the spinning Wilson spool.

In the remainder of this paper we will establish the results upholding (1.3)
and (1.4) as well as the necessary details to utilize (1.4). Much of this
analysis will be done in the context of a positive cosmological constant,
where the difference between the scalar and spinning Wilson spools are most
apparent. In Sec. 2 we will establish the representation theory necessary for
describing massive spin-$\mathsf{s}$ particles in terms of the Euclidean
isometry group $\mathfrak{su}(2)_{L}\oplus\mathfrak{su}(2)_{R}$. In Sec. 3 we
will utilize this representation theory to build the spinning Wilson spool,
(1.4), when the background is classically $S^{3}$. This follows from a re-
organization of the quasinormal mode method for one-loop determinants [29] in
terms of representation theory following a set of broad guiding principles,
which we establish in Sec. 3.1 and apply to Chern-Simons gravity in Sec. 3.2.
In Sec. 4 we propose the spinning Wilson spool as an operator, allowing us to
insert it off-shell into the Chern-Simons path integral. The full power of
this is manifest in the context of a dS3 background where we show that the
Chern-Simons path integral with a spool insertion reduces to an ordinary
integral that can be evaluated to any order in $G_{N}$ perturbation theory. We
illustrate how this can be interpreted as either a renormalization of the
particle mass or Newton’s constant, depending on choice of renormalization
condition. Lastly in Sec. 5, to establish the scope of the results in Sec. 3,
we will also illustrate how (1.4) also holds when the background is Euclidean
BTZ mutatis mutandis. We conclude the paper in Sec. 6 with potential uses for
the spinning Wilson spool as well as a discussion of open problems. Details on
the dictionary relating Chern-Simons theory and gravity as well as details on
the spin-$\mathsf{s}$ quasinormal modes on $S^{3}$ are found in App. A and
App. B, respectively.

##  2 Non-standard spinning representations of $\mathfrak{su}(2)$

In this section, we will describe single-particle states of massive
spin-$\mathsf{s}$ fields living on dS3, with de Sitter radius
$\ell_{\text{dS}}$, as representations of
$\mathfrak{su}(2)_{L}\oplus\mathfrak{su}(2)_{R}$. The guiding principle of the
construction is to mimic the unitary representations of the Lorentzian dS3
isometry group, $\mathfrak{so}(1,3)$. In [1] this was done for cases with
$\mathsf{s}=0$, and here, following the same arguments and conventions, we
extend that construction to include spin.

Unitary representations of $\mathfrak{so}(1,3)$ are labeled by a conformal
dimension $\Delta$ (the eigenvalue of the dilatation operator $D$) and a spin
$\mathsf{s}$ (the eigenvalue of the spin operator $iM$) and come in two
series:222See [1] for a basic introduction on $\mathfrak{so}(1,3)$
representation theory and for our notation here. See [27, 30, 31, 28, 32] for
a more thorough treatment.

$\displaystyle\Delta=$ $\displaystyle 1+\nu\leavevmode\nobreak\ ,$
$\displaystyle\nu\in(-1,1)\leavevmode\nobreak\ ,$
$\displaystyle\mathsf{s}=0\leavevmode\nobreak\ ,$ $\displaystyle\Delta=$
$\displaystyle 1-i\mu\leavevmode\nobreak\ ,$
$\displaystyle\mu\in\mathbb{R}\leavevmode\nobreak\ ,$
$\displaystyle\mathsf{s}\in\mathbb{Z}\leavevmode\nobreak\ .$ (2.1)

The first line is the complementary series describing light scalars with
masses $\ell_{\text{dS}}^{2}m^{2}=1-\nu^{2}$ while the second line is the
spinning principal series describing spinning particles with masses
$\ell_{\text{dS}}^{2}m^{2}=(\mathsf{s}-1)^{2}+\mu^{2}$.

To connect this to $\mathfrak{su}(2)_{L}\oplus\mathfrak{su}(2)_{R}$, let us
introduce some basic aspects of $\mathfrak{su}(2)$. The algebra is generated
by $L_{3}$ and $L_{\pm}$, where

$[L_{3},L_{\pm}]=\pm L_{\pm}\leavevmode\nobreak\
,\qquad[L_{+},L_{-}]=2L_{3}\leavevmode\nobreak\ .$ (2.2)

The Casimir of the representation is
$c_{2}^{\mathfrak{su}(2)}=\frac{1}{2}(L_{+}L_{-}+L_{-}L_{+})+L_{3}^{2}$.
Representations of $\mathfrak{su}(2)$ will be characterized by $j$, the
eigenvalue of $L_{3}$. The $\mathfrak{so}(1,3)$ algebra shares a
complexification with $\mathfrak{su}(2)_{L}\oplus\mathfrak{su}(2)_{R}$, such
that the dilatation and spin generators can be identified with the Cartan
elements of $\mathfrak{su}(2)$ as

$D=-L_{3}-\bar{L}_{3}\leavevmode\nobreak\ ,\qquad\qquad
M=iL_{3}-i\bar{L}_{3}\leavevmode\nobreak\ ,$ (2.3)

respectively. Similar relations follow for the remaining generators, which can
be found in [1]. In (2.3) we have distinguished the generators of
$\mathfrak{su}(2)_{R}$ from those of $\mathfrak{su}(2)_{L}$ by an overbar.
Within this complexification, the quadratic Casimir of $\mathfrak{so}(1,3)$ is
equal to the sum of the $\mathfrak{su}(2)$ Casimirs:

$c_{2}^{\mathfrak{so}(1,3)}=-2c_{2}^{\mathfrak{su}(2)_{L}}-2c_{2}^{\mathfrak{su}(2)_{R}}\leavevmode\nobreak\
.$ (2.4)

Equation (2.3) suggests that we need to look for
$\mathfrak{su}(2)_{L}\oplus\mathfrak{su}(2)_{R}$ representations with

$\Delta=-j_{R}-j_{L}\leavevmode\nobreak\
,\qquad\qquad\mathsf{s}=j_{R}-j_{L}\leavevmode\nobreak\ ,$ (2.5)

where $j_{L}$ is the eigenvalue of $L_{3}$ in $\mathfrak{su}(2)_{L}$, and
similarly for $j_{R}$.

The relations (2.5), together with (2), indicate to us that we will need to
construct representations with continuous and complex eigenvalues
$(j_{L},j_{R})$. These do not fall into the standard finite-dimensional
representation theory of $\mathfrak{su}(2)$. However in [33, 1] it was shown
how to construct an alternative inner product in $\mathfrak{su}(2)$ (or
equivalently, an alternative notion of Hermitian conjugation) allowing for
highest-weight representations with a continuous and complex weight while
preserving norm-positivity. Such representations were coined non-standard
representations in the latter reference. These representations are built from
a highest weight state, $|j,0\rangle$, satisfying

$L_{3}|j,0\rangle=j|j,0\rangle\leavevmode\nobreak\ ,\qquad\qquad
L_{+}|j,0\rangle=0\leavevmode\nobreak\ ,$ (2.6)

and $j\in\mathbb{C}$ is the weight of the representation. Acting with lowering
operators defines

$|j,p\rangle=N_{j,p}\left(L_{-}\right)^{p}|j,0\rangle\leavevmode\nobreak\ ,$
(2.7)

for some normalizations $N_{j,p}$ that we will determine shortly.

The key ingredient distinguishing the non-standard representations is a map,
$\mathcal{S}$, between highest weight representations

$\mathcal{S}|j,p\rangle=|j^{\ast},p\rangle\leavevmode\nobreak\ ,$ (2.8)

where $j^{\ast}$ is the complex conjugate of $j$. This map is utilized in
Hermitian conjugation in the following way:

$L_{3}^{\dagger}:=\mathcal{S}^{-1}L_{3}\mathcal{S}\leavevmode\nobreak\ ,\qquad
L_{\pm}^{\dagger}:=-\mathcal{S}^{-1}L_{\mp}\mathcal{S}\leavevmode\nobreak\ .$
(2.9)

Note that $j\overset{\mathcal{S}}{\rightarrow}j^{\ast}$ is consistent with the
action of $L_{3}^{\dagger}$. In [1], norm-positive inner products were
constructed for non-standard representations with $j=-\frac{1}{2}(1+\nu)$ or
$j=-\frac{1}{2}(1-i\mu)$ (and $\mu,\nu\in\mathbb{R}$) which relate to the
scalar complementary and principal series with $\mathsf{s}=0$. Here we will
relax this condition and show that a norm-positive inner product can be
constructed for any complex $j$.

We will first determine the normalizations $N_{j,p}$. We note that

$\langle j,p-1|L_{+}|j,p\rangle=\frac{N_{j,p}}{N_{j,p-1}}p(2j+1-p)\langle
j,p-1|j,p-1\rangle\leavevmode\nobreak\ ,$ (2.10)

which can be seen from replacing $L_{+}L_{-}$ with
$c_{2}^{\mathfrak{su}(2)}+L_{3}-L_{3}^{2}$ in the matrix element.
Alternatively, utilizing (2.9) and (2.8), this same matrix element is

$\langle j,p-1|L_{+}|j,p\rangle=\langle
j,p|L_{+}^{\dagger}|j,p-1\rangle^{\ast}=-\left(\frac{N_{j^{\ast},p-1}}{N_{j^{\ast},p}}\right)^{\ast}\langle
j,p|j,p\rangle\leavevmode\nobreak\ .$ (2.11)

If we assume that we can set $\langle j,p-1|j,p-1\rangle=1$ and also $\langle
j,p|j,p\rangle=1$, then (2.10) and (2.11) imply

$\left(\frac{N_{j^{\ast},p-1}}{N_{j^{\ast},p}}\right)^{\ast}=p(p-2j-1)\left(\frac{N_{j,p}}{N_{j,p-1}}\right)\leavevmode\nobreak\
.$ (2.12)

To solve this constraint, we write

$\alpha_{j,p}=\text{Arg}\left(N_{j,p}\right)\leavevmode\nobreak\
,\qquad\phi_{j,p}=\text{Arg}\left(p-2j-1\right)\leavevmode\nobreak\ ,$ (2.13)

leading to recurrence relations

$\displaystyle\frac{\absolutevalue{N_{j^{\ast},p-1}}}{\absolutevalue{N_{j^{\ast},p}}}=$
$\displaystyle
p\absolutevalue{p-2j-1}\frac{\absolutevalue{N_{j,p}}}{\absolutevalue{N_{j,p-1}}}\leavevmode\nobreak\
,$ $\displaystyle\alpha_{j^{\ast},p}-\alpha_{j^{\ast},p-1}=$
$\displaystyle\alpha_{j,p}-\alpha_{j,p-1}+\phi_{j,p}\;\;\;\text{mod}(2\pi)\leavevmode\nobreak\
.$ (2.14)

There is a lot of freedom in solving these recurrence relations. We will
choose a particular solution by fixing

$\absolutevalue{N_{j,p}}=\absolutevalue{N_{j^{\ast},p}}\leavevmode\nobreak\
,\qquad\qquad\alpha_{j,0}=\alpha_{j^{\ast},0}=0\leavevmode\nobreak\
,\qquad\qquad\alpha_{j,p}=-\alpha_{j^{\ast},p}\;\;\forall\;p\leavevmode\nobreak\
.$ (2.15)

This sets

$\frac{\absolutevalue{N_{j,p-1}}}{\absolutevalue{N_{j,p}}}=\frac{\absolutevalue{N_{j^{\ast},p-1}}}{\absolutevalue{N_{j^{\ast},p}}}=\sqrt{p\absolutevalue{p-2j-1}}$
(2.16)

and333Note that $\phi_{j^{\ast},p}=-\phi_{j,p}$.

$\alpha_{j,p}=-\frac{1}{2}\sum_{p^{\prime}=1}^{p}\phi_{j,p^{\prime}}\leavevmode\nobreak\
.$ (2.17)

For concreteness we summarize the generator actions on this representation as

$\displaystyle L_{3}|j,p\rangle$
$\displaystyle=(j-p)|j,p\rangle\leavevmode\nobreak\ ,$ $\displaystyle
L_{-}|j,p\rangle$
$\displaystyle=e^{i\phi_{j,p}/2}\sqrt{(p+1)\absolutevalue{p-2j}}|j,p+1\rangle\leavevmode\nobreak\
,$ $\displaystyle L_{+}|j,p\rangle$
$\displaystyle=-e^{i\phi_{j,p}/2}\sqrt{p\absolutevalue{p-2j-1}}|j,p-1\rangle\leavevmode\nobreak\
,$ (2.18)

with the action on $|j^{\ast},p\rangle$ obtained by simply replacing
$j\rightarrow j^{\ast}$ in the above formulas. We can now show norm-positivity
by induction starting with $\langle j,0|j,0\rangle=1$. Let us investigate the
first descendant state:

$\displaystyle\absolutevalue{L_{-}|j,0\rangle}^{2}=$ $\displaystyle-\langle
j,0|\mathcal{S}^{-1}L_{+}\mathcal{S}L_{-}|j,0\rangle$ $\displaystyle=$
$\displaystyle
e^{i\phi_{j,1}/2+i\phi_{j^{\ast},1}/2}\sqrt{\absolutevalue{2j}}\sqrt{\absolutevalue{2j}}\langle
j,0|j,0\rangle$ $\displaystyle=$
$\displaystyle\absolutevalue{2j}>0\leavevmode\nobreak\ .$ (2.19)

Similarly norm-positivity of $|j,p\rangle$ induces norm-positivity of
$|j,p+1\rangle$444We are assuming $j$ is a generic complex number and ignoring
the potential for possible null states when $j\in\frac{1}{2}\mathbb{N}_{0}$.
Of course, in these cases the representation simply terminates and we recover
the standard, finite-dimensional, representations of $\mathfrak{su}(2)$.:

$\displaystyle\absolutevalue{L_{-}|j,p\rangle}^{2}=$ $\displaystyle-\langle
j,p|\mathcal{S}^{-1}L_{+}\mathcal{S}L_{-}|j,p\rangle$ $\displaystyle=$
$\displaystyle
e^{i\phi_{j,p+1}/2+i\phi_{j^{\ast},p+1}/2}\sqrt{(p+1)\absolutevalue{p-2j}}\sqrt{(p+1)\absolutevalue{p-2j}}\langle
j,p|j,p\rangle$ $\displaystyle=$
$\displaystyle(p+1)\absolutevalue{p-2j}\langle
j,p|j,p\rangle>0\leavevmode\nobreak\ .$ (2.20)

This establishes norm-positivity for all states in the representation. We
additionally note that these representations have well-defined characters

$\chi_{j}(z)=\mathrm{Tr}_{j}\left(e^{i2\pi
zL_{3}}\right)=\sum_{p=0}^{\infty}e^{i2\pi
z(j-p)}=\frac{e^{i\pi(2j+1)z}}{2i\sin(\pi z)}\leavevmode\nobreak\ .$ (2.21)

In [1] the restrictions of the highest-weights, $j=-\frac{1}{2}(1+\nu)$ or
$j=-\frac{1}{2}(1-i\mu)$, followed from Hermiticity of the $\mathfrak{su}(2)$
Casimirs, separately. In this paper we will take a more Lorentzian
perspective, and impose reality of the $\mathfrak{so}(1,3)$ Casimir at the
level of (2.4); doing so ties the $\mathfrak{su}(2)_{L/R}$ representations
together and the highest weights take the generic form

$j_{L}=-\frac{1}{2}(1+\mathsf{s}-i\mu)\leavevmode\nobreak\ ,\qquad
j_{R}=-\frac{1}{2}(1-\mathsf{s}-i\mu)\leavevmode\nobreak\ .$ (2.22)

Let us briefly show this now. Assuming that $j_{L}$ and $j_{R}$ are
generically complex the condition for reality of the $\mathfrak{so}(1,3)$
Casimir is that

$\text{Im}\left(j_{L}(j_{L}+1)\right)=-\text{Im}\left(j_{R}(j_{R}+1)\right)\leavevmode\nobreak\
,$ (2.23)

or equivalently

$\text{Im}(j_{L})\left(1+2\text{Re}(j_{L})\right)=-\text{Im}(j_{R})\left(1+2\text{Re}(j_{R})\right)\leavevmode\nobreak\
.$ (2.24)

If both sides are simultaneously zero then we must have either
$j\in\mathbb{R}$ or $j\in-\frac{1}{2}+i\mathbb{R}$ (for either $j_{L}$ or
$j_{R}$) which lead to the complimentary and principal-type representations
discussed in [1]. More generally there will be a family of highest weight
solutions satisfying (2.24). However, if we further insist555At this point we
will impose this by hand; we will see later that in the quantum theory only
representations with $j_{L}-j_{R}\in\mathbb{Z}$ contribute to the matter one-
loop determinant. that $j_{L}-j_{R}\in\mathbb{R}$ then it must be the case
that

$\text{Im}(j_{L})=\text{Im}(j_{R})\leavevmode\nobreak\
,\qquad\qquad\text{Re}(j_{L})+\text{Re}(j_{R})=-1\leavevmode\nobreak\ ,$
(2.25)

which lead to the highest-weights appropriate for the spinning principal
series, (2.22). We will refer to such representations as the spinning
principal-type representations.

As noted already in [1], our non-standard characters can be massaged into the
suggestive Lorentzian form via

$\chi_{j_{L}}(z_{L})\chi_{j_{R}}(z_{R})+\chi_{j^{\ast}_{L}}(z_{L})\chi_{j^{\ast}_{R}}(z_{R})=\frac{w^{\mathsf{s}}q^{\Delta}+w^{-\mathsf{s}}q^{\bar{\Delta}}}{(1-w^{-1}q)(1-wq)}\leavevmode\nobreak\
,$ (2.26)

where $q=e^{-i\pi(z_{L}+z_{R})}$ and $w=e^{i\pi(z_{L}-z_{R})}$ and we have
identified (2.5) as well as $\bar{\Delta}=2-\Delta$. This matches the Harish-
Chandra character, $\mathrm{Tr}\left(q^{D}w^{iM}\right),$ for the
$\mathfrak{so}(1,3)$ spinning principal series [31].

##  3 Spinning spool on $S^{3}$

In this section, we present the construction of the Wilson spool for massive
spin-$\mathsf{s}$ fields on $S^{3}$. That is, we will derive an expression for
the one-loop determinant of these fields on $S^{3}$ in terms of the
representations of $\mathfrak{su}(2)_{L}\otimes\mathfrak{su}(2)_{R}$
constructed in the previous section.

To start, let us describe the path integral of a single massive
spin-$\mathsf{s}$ field, with no self-interactions. The local partition
function for this theory contains a symmetric spin-$\mathsf{s}$ tensor,
$\Phi_{\mu_{1}\mu_{2}\ldots\mu_{\mathsf{s}}}$, as well as a tower of
Stückelberg fields which enforce that
$\Phi_{\mu_{1}\mu_{2}\ldots\mu_{\mathsf{s}}}$ is transverse and traceless
[34]:

$\nabla^{\nu}\Phi_{\nu\mu_{2}\ldots\mu_{\mathsf{s}}}={\Phi^{\nu}}_{\nu\mu_{3}\ldots\mu_{\mathsf{s}}}=0\leavevmode\nobreak\
.$ (3.1)

As emphasized in [8], on a compact manifold the path integral over symmetric,
transverse, traceless (STT) tensors leads to non-local divergences which
cannot be canceled by local counterterms. This path integral must be
compensated by the path integral over the Stückelberg fields and ghosts which
leave behind a finite product from integrating over normalizable zero modes.
To that end, we write

$Z_{\Delta,\mathsf{s}}=Z_{\text{zero}}Z_{\text{STT}}\leavevmode\nobreak\ ,$
(3.2)

where

$Z_{\text{STT}}=\int[\mathcal{D}\Phi_{\mu_{1}\mu_{2}\ldots\mu_{\mathsf{s}}}]_{\text{STT}}e^{-\frac{1}{2}\int\Phi\left(-\nabla^{2}_{(\mathsf{s})}+\ell_{\text{dS}}^{2}\bar{m}^{2}_{\mathsf{s}}\right)\Phi}\leavevmode\nobreak\
.$ (3.3)

Above $\nabla_{(\mathsf{s})}^{2}$ is the Laplace-Beltrami operator

$\left[\nabla_{(\mathsf{s})}^{2}\Phi\right]_{\mu_{1}\mu_{2}\ldots\mu_{\mathsf{s}}}=\nabla_{\nu}\nabla^{\nu}\Phi_{\mu_{1}\mu_{2}\ldots\mu_{\mathsf{s}}}\leavevmode\nobreak\
,$ (3.4)

and $\bar{m}^{2}_{\mathsf{s}}$ is an “effective mass”

$\ell_{\text{dS}}^{2}\bar{m}_{\mathsf{s}}^{2}=\ell_{\text{dS}}^{2}m^{2}+3\mathsf{s}-\mathsf{s}^{2}\leavevmode\nobreak\
,$ (3.5)

where we recall that $m^{2}$ is the standard mass parameter in dS3 [8], and
related to the representation theory of Sec. 2 via

$\ell_{\text{dS}}^{2}m^{2}=(\Delta+\mathsf{s}-2)(\mathsf{s}-\Delta)\leavevmode\nobreak\
.$ (3.6)

The zero mode contribution in (3.2) follows from counting conformal Killing
tensor modes on $S^{3}$ and is given by [8]666This is true for $\mathsf{s}\geq
2$ while for $\mathsf{s}=0,1$ the product over $n$ is replaced by 1.

$Z_{\text{zero}}=\left[(\Delta-1)(\bar{\Delta}-1)\right]^{\frac{\mathsf{s}^{2}}{2}}\prod_{n=0}^{\mathsf{s}-2}\left[(\Delta+n)(\bar{\Delta}+n)\right]^{-(n+1+\mathsf{s})(n+1-\mathsf{s})}\leavevmode\nobreak\
,$ (3.7)

where we recall that $\bar{\Delta}=2-\Delta$.

In the following, we will show that

$\log
Z_{\Delta,\mathsf{s}}=\frac{1}{4}\mathbb{W}_{j_{L},j_{R}}\leavevmode\nobreak\
,$ (3.8)

for fields on $S^{3}$. That is, we will express $Z_{\Delta,\mathsf{s}}$ as a
function of the Chern-Simons connections which we will explicitly construct
utilizing the non-standard representation theory of Sec. 2. While our
construction takes place on a fixed classical background, we will see that
$\mathbb{W}_{j_{L},j_{R}}$ is an integral over gauge-invariant Wilson loop
operators and naturally generalizes into an off-shell operator that can be
inserted into the Chern-Simons path integral, which we will discuss in Sec 4.

The roadmap to derive (3.8) is as follows. We will first focus on
$Z_{\text{STT}}$. This determinant can be evaluated via the method of
quasinormal modes pioneered by Denef, Hartnoll, and Sachdev (DHS) [29]. We
will adapt this method such that each component has a group theoretic
interpretation: we will show how defining properties of quasinormal modes can
be translated to conditions on the representation of the fields. This follows
[1], however, for spinning fields, we will take particular care with the role
of global conditions (i.e., Euclidean solutions are regular and single-valued)
in isolating physical contributions to the quasinormal mode product. The
additional contribution of $Z_{\text{zero}}$, which is not part of the
quasinormal mode product but crucial for maintaining locality of
$Z_{\Delta,\mathsf{s}}$, will permit a Schwinger parameterization of $\log
Z_{\Delta,\mathsf{s}}$, regularized by an $i\varepsilon$ prescription. This
will organize the quasinormal mode sum into an integral over representation
traces of the background holonomies. From this follows our main result.

###  3.1 A group theory perspective on $S^{3}$ quasinormal modes

As the first step in our construction, we will recast the functional
determinant

$Z_{\text{STT}}=\det\left(-\nabla_{(\mathsf{s})}^{2}+\ell_{\text{dS}}^{2}\bar{m}_{\mathsf{s}}^{2}\right)^{-\frac{1}{2}}$
(3.9)

in $\mathfrak{su}(2)$ representation theoretic language. We recall that the
DHS method instructs us to treat $Z_{\text{STT}}^{2}$ as meromorphic function
of $\Delta$. Then, up to a holomorphic function, $Z_{\text{STT}}^{2}$ is equal
to the product containing the same zeros and poles. Here $Z_{\text{STT}}$ only
has poles on states satisfying
$(-\nabla^{2}_{(\mathsf{s})}+\bar{m}_{\mathsf{s}}^{2})\Phi_{\mu_{1}\mu_{2}\ldots\mu_{\mathsf{s}}}=0$.
These are precisely the spin-$\mathsf{s}$ quasinormal modes. In App. B, we
explicitly compute these modes and their product to obtain $Z_{\text{STT}}$
directly. Here we will reinterpret these modes in terms of $\mathfrak{su}(2)$
representation theory to obtain an expression natural to the Chern-Simons
theory formulation of gravity.

We note that the isometry algebra of the three-sphere is generated by two
mutually commuting sets of $\mathfrak{su}(2)$ vector fields $\\{\zeta_{a}\\}$
and $\\{\bar{\zeta}_{b}\\}$ which are the infinitesimal left and right group
actions acting on $S^{3}\simeq SU(2)$. On spin-$\mathsf{s}$ STT tensors the
Casimirs of their Lie derivatives, $\\{\mathcal{L}_{\zeta_{a}}\\}$ and
$\\{\mathcal{L}_{\bar{\zeta}_{b}}\\}$, act as the Laplace-Beltrami operator
[35]:

$-2\delta^{ab}\left(\mathcal{L}_{\zeta_{a}}\mathcal{L}_{\zeta_{b}}+\mathcal{L}_{\bar{\zeta}_{a}}\mathcal{L}_{\bar{\zeta}_{b}}\right)\Phi_{\mu_{1}\mu_{2}\ldots\mu_{\mathsf{s}}}=\left[\nabla^{2}_{(\mathsf{s})}-\mathsf{s}(\mathsf{s}+1)\right]\Phi_{\mu_{1}\mu_{2}\ldots\mu_{\mathsf{s}}}\leavevmode\nobreak\
.$ (3.10)

Hence we can write (3.9) suggestively as

$Z_{\text{STT}}=\det\left(2c_{2}^{\mathfrak{su}(2)_{L}}+2c_{2}^{\mathfrak{su}(2)_{R}}+\Delta(2-\Delta)-\mathsf{s}^{2}\right)^{-\frac{1}{2}}\leavevmode\nobreak\
.$ (3.11)

Following the DHS methodology, we then expect $Z_{\text{STT}}^{2}$ to have
pole contributions from states in
$\mathfrak{su}(2)_{L}\oplus\mathfrak{su}(2)_{R}$ representations satisfying

$\left(-2c_{2}^{\mathfrak{su}(2)_{L}}-2c_{2}^{\mathfrak{su}(2)_{R}}\right)|\psi\rangle=\left[\Delta(2-\Delta)-\mathsf{s}^{2}\right]|\psi\rangle\leavevmode\nobreak\
.$ (3.12)

This is precisely the condition satisfied by the non-standard representations
constructed in Sec. 2 with highest weights
$(j_{L},j_{R})=(-\frac{\Delta+\mathsf{s}}{2},-\frac{\Delta-\mathsf{s}}{2})$.
We are interested in the poles in $Z_{\text{STT}}^{2}$ arising from weights of
the representations $\mathsf{R}_{j_{L}}\otimes\mathsf{R}_{j_{R}}$ as we
continue $\Delta$ in the complex plane.777It will be important as we progress
to take special care of the cases when $\Delta$ is such that
$j_{L/R}\in\frac{1}{2}\mathbb{N}$; in these cases the weight spaces terminate
discontinuously to finite-dimensional representations. In principle we should
consider all representations that satisfy (3.12), so for a given
$(\Delta,\mathsf{s})$, we also encounter poles associated to highest weight
representations arrived at by sending $\Delta\rightarrow\bar{\Delta}=2-\Delta$
as well as $\mathsf{s}\rightarrow-\mathsf{s}$.888This is consistent with the
Lorentzian picture: a spin-$\mathsf{s}$ field is built out of
$\mathfrak{so}(1,3)$ representations labelled by both $(\Delta,\pm\mathsf{s})$
while the $(\Delta,\mathsf{s})$ representation is isomorphic to that labelled
by $(\bar{\Delta},-\mathsf{s})$ through the $\mathfrak{so}(1,3)$ shadow map
[32]. If we define
$(j_{L},j_{R})=(-\frac{\Delta+\mathsf{s}}{2},-\frac{\Delta-\mathsf{s}}{2})$
then we will denote

$(\bar{j}_{L},\bar{j}_{R})=\left(-\frac{\bar{\Delta}+\mathsf{s}}{2},-\frac{\bar{\Delta}-\mathsf{s}}{2}\right)\leavevmode\nobreak\
,$ (3.13)

while $\mathsf{s}\rightarrow-\mathsf{s}$ is equivalent to
$j_{L}\leftrightarrow j_{R}$. We will then have pole contributions from any of
the representations appearing in

$\mathcal{R}_{\Delta,\mathsf{s}}=\left\\{\mathsf{R}_{j_{L}}\otimes\mathsf{R}_{j_{R}},\mathsf{R}_{\bar{j}_{L}}\otimes\mathsf{R}_{\bar{j}_{R}},\mathsf{R}_{j_{R}}\otimes\mathsf{R}_{j_{L}},\mathsf{R}_{\bar{j}_{R}}\otimes\mathsf{R}_{\bar{j}_{L}}\right\\}\leavevmode\nobreak\
.$ (3.14)

We make a special note that for scalar representations ($j_{L}=j_{R}=j$) it is
sufficient to consider the smaller set

$\mathcal{R}_{\Delta,\text{scalar}}=\left\\{\mathsf{R}_{j}\otimes\mathsf{R}_{j},\mathsf{R}_{\bar{j}}\otimes\mathsf{R}_{\bar{j}}\right\\}$
(3.15)

as in [1].

The “mass-shell condition” (3.12) is only a necessary condition to contribute
a physical pole to $Z_{\text{STT}}^{2}$. Functional determinants come with
boundary and regularity conditions on their functional domain and we must
impose these on weight spaces satisfying (3.12) to obtain a physical answer.
We will state these up-front in an $\mathfrak{su}(2)$ natural language as the
following:

Condition I. Single-valued solutions:

A configuration constructed from a representation
$\mathsf{R}_{L}\otimes\mathsf{R}_{R}\in\mathcal{R}$ must return to itself
under parallel transport around any closed cycle in the Euclidean manifold.

Condition II. Globally regular solutions:

A configuration constructed from a representation
$\mathsf{R}_{L}\otimes\mathsf{R}_{R}\in\mathcal{R}$ must be globally regular
on the Euclidean manifold. When the base space is homogeneous this means
$\mathsf{R}_{L}\otimes\mathsf{R}_{R}$ lifts from a representation of the
isometry algebra to a representation of the isometry group.

Let us first expand upon Condition I for spin-$\mathsf{s}$ fields on $S^{3}$.
A field $\Phi$ living in a representation
$\mathsf{R}_{L}\otimes\mathsf{R}_{R}$ is parallel transported around a cycle,
$\gamma$, through the background connections:

$\Phi_{f}=\mathsf{R}_{L}\left[\mathcal{P}\exp\left(\oint_{\gamma}a_{L}\right)\right]\,\Phi_{i}\,\mathsf{R}_{R}\left[\mathcal{P}\exp\left(-\oint_{\gamma}a_{R}\right)\right]\leavevmode\nobreak\
.$ (3.16)

When $a_{L/R}$ are flat this conjugation is trivial. However for the
backgrounds appropriate for describing the $S^{3}$ metric geometry, the
background connections take non-trivial holonomies

$\mathcal{P}\exp\left(\oint_{\gamma}a_{L}\right)=u_{L}^{-1}\,e^{i2\pi
L_{3}\mathsf{h}_{L}^{(\gamma)}}\,u_{L}^{-1}\leavevmode\nobreak\
,\qquad\quad\mathcal{P}\exp\left(\oint_{\gamma}a_{R}\right)=u_{R}^{-1}\,e^{i2\pi\bar{L}_{3}\mathsf{h}_{R}^{(\gamma)}}\,u_{R}^{-1}\leavevmode\nobreak\
,$ (3.17)

when $\gamma$ wraps one of two lines on the base $S^{3}$ [1]. These lines are
Hopf-linked and Wick-rotate to the coordinate positions of the static patch
origin and horizon. They yield respective holonomies

$\gamma_{\text{orig.}}\colon\;\;\left(\mathsf{h}_{L},\mathsf{h}_{R}\right)=(1,1)\leavevmode\nobreak\
,\qquad\gamma_{\text{hor.}}\colon\;\;\left(\mathsf{h}_{L},\mathsf{h}_{R}\right)=(1,-1)\leavevmode\nobreak\
.$ (3.18)

More details can be found in App. A.1. The salient point is that a single-
valued field will satisfy

$\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R}\in\mathbb{Z}\leavevmode\nobreak\
,$ (3.19)

for each of the two sets of holonomies in (3.18) and for all weights
$(\lambda_{L},\lambda_{R})$ in the representation
$\mathsf{R}_{L}\otimes\mathsf{R}_{R}$.

From cycles wrapping the origin weights must satisfy

$\lambda_{L}-\lambda_{R}\in\mathbb{Z}$ (3.20)

to contribute a pole to $Z_{\text{STT}}^{2}$. Weights of a highest-weight
representation $\mathsf{R}_{j_{L/R}}$ necessarily take the form

$\lambda_{L/R}=j_{L/R}-p_{L/R}\leavevmode\nobreak\ ,\qquad
p_{L/R}\in\mathbb{N}_{0}\leavevmode\nobreak\ ,$ (3.21)

simply via the structure of the $\mathfrak{su}(2)$ algebra. Single-valuedness
around the static patch origin then requires

$j_{L}-j_{R}\in\mathbb{Z}\qquad\Leftrightarrow\qquad\mathsf{s}\in\mathbb{Z}\leavevmode\nobreak\
.$ (3.22)

This condition is the same for all other representations in
$\mathcal{R}_{\Delta,\mathsf{s}}$. We pause here to note that while the
representation theory in Sec. 2 only relies on $j_{L}-j_{R}\in\mathbb{R}$, we
now see that only fields with quantized spin can contribute physical poles to
$Z_{\text{STT}}^{2}$. We will thus fix $\mathsf{s}\in\mathbb{Z}$ and consider
the analytic structure of $Z^{2}_{\text{STT}}$ as a function of $\Delta$. This
analytic structure is constrained by single-valuedness around the static patch
horizon, which requires

$\lambda_{L}+\lambda_{R}\in\mathbb{Z}\leavevmode\nobreak\ .$ (3.23)

We will return to this condition shortly.

We now address Condition II, that configurations contributing to
$Z^{2}_{\text{STT}}$ are globally regular. Without loss of generality we will
state this for
$\mathsf{R}_{j_{L}}\otimes\mathsf{R}_{j_{R}}\in\mathcal{R}_{\Delta,\mathsf{s}}$.
For the $S^{3}$ background in question the isometry group acts transitively.
Thus regularity at a point guarantees global regularity as long as the
isometry group, $SU(2)_{L}\times SU(2)_{R}$, acts faithfully on the field in
question: that is, $\mathsf{R}_{j_{L}}\otimes\mathsf{R}_{j_{R}}$ lifts to a
representation of the isometry group. The Peter-Weyl theorem states that these
must be finite-dimensional representations of
$\mathfrak{su}(2)_{L}\otimes\mathfrak{su}(2)_{R}$ (see, for example, [36]),
where such representations have weights (3.21) satisfying

$\lambda_{L/R}=j_{L/R}-p_{L/R}\leavevmode\nobreak\ ,\qquad 0\leq p_{L/R}\leq
2j_{L/R}\leavevmode\nobreak\ ,\qquad
j_{L/R}\in\frac{1}{2}\mathbb{N}_{0}\leavevmode\nobreak\ .$ (3.24)

To be clear about interpretation: the DHS method instructs us to consider the
structure of $Z_{\text{STT}}^{2}$ as $\Delta$ continues to the complex plane.
The mass-shell condition, (3.12), then instructs us to consider
representations with generically complex highest weights,
$j_{L/R}\in\mathbb{C}$. Such representations are non-standard and infinite-
dimensional. However Condition II simply tells us that the poles of
$Z^{2}_{\text{STT}}$ are located at $\Delta\in\mathbb{Z}_{\leq-\mathsf{s}}$
and the orders of these poles are correctly counted not by weights of an
infinite dimensional representation but instead by (3.24). In this counting we
notice that weights of finite dimensional representations of $SU(2)$ are
centered about zero and so for any weight satisfying
$\lambda_{L}+\lambda_{R}=N>0$ there is a corresponding weight with
$\lambda_{L}+\lambda_{R}=-N$. Thus for the purposes of counting the number of
weights contributing to a particular pole, we can restate (3.23) as

$\lambda_{L}+\lambda_{R}=\absolutevalue{N}\leavevmode\nobreak\ ,\qquad
N\in\mathbb{Z}\leavevmode\nobreak\ $ (3.25)

as a necessary condition for incorporating Condition II.

We pause to note that for minimally coupled scalar fields (3.25) is also
sufficient to imply Condition II. Thus the scalar one-loop determinant can be
written as

$Z_{\text{scalar}}=\prod_{\overset{(\lambda_{L},\lambda_{R})}{\;\;\;\;\in\mathsf{R}_{j}\otimes\mathsf{R}_{j}}}\prod_{N\in\mathbb{Z}}\left(\absolutevalue{N}-\lambda_{L}-\lambda_{R}\right)^{-1/2}\times\prod_{\overset{(\bar{\lambda}_{L},\bar{\lambda}_{R})}{\;\;\;\;\in\mathsf{R}_{\bar{j}}\otimes\mathsf{R}_{\bar{j}}}}\prod_{\bar{N}\in\mathbb{Z}}\left(\absolutevalue{\bar{N}}-\bar{\lambda}_{L}-\bar{\lambda}_{R}\right)^{-1/2}\leavevmode\nobreak\
,$ (3.26)

where we have written explicitly the product over the two representations
appearing in $\mathcal{R}_{\Delta,\text{scalar}}$. From here the expression of
$Z_{\text{scalar}}$ as a Wilson spool follows the procedure in [1].

For massive spin-$\mathsf{s}$ fields, (3.25) is no longer sufficient and we
must impose additional constraints to reproduce $Z_{\text{STT}}^{2}$. For a
given $\absolutevalue{N}$ in (3.25), Condition II additionally implies the
weights $\lambda_{L/R}=j_{L/R}-p_{L/R}$ must satisfy

$p_{L}\geq-\absolutevalue{N}-\mathsf{s}\leavevmode\nobreak\ ,\qquad\qquad
p_{R}\geq-\absolutevalue{N}+\mathsf{s}\leavevmode\nobreak\ .$ (3.27)

While the first of these is always satisfied (for positive $\mathsf{s}$) the
second is an additional constraint on counting the order of the poles
appearing in $Z^{2}_{\text{STT}}$ and is only non-trivial when
$\absolutevalue{N}\leq\mathsf{s}$. For these $2\mathsf{s}+1$ cases we observe
that $\tilde{p}_{R}=p_{R}+\absolutevalue{N}-\mathsf{s}\geq 0$ and (3.25) is
equivalently written

$j_{L}+j_{R}-p_{L}-\tilde{p}_{R}=\mathsf{s}\leavevmode\nobreak\ ,\qquad
p_{L},\tilde{p}_{R}\geq 0\leavevmode\nobreak\ .$ (3.28)

We can thus treat this as a condition on the weights
$(\lambda_{L},\tilde{\lambda}_{R})=(j_{L}-p_{L},j_{R}-\tilde{p}_{R})$ of
highest-weight representations $\mathsf{R}_{j_{L}}\otimes\mathsf{R}_{j_{R}}$
and for each pole arising from this condition being satisfied, it arises
$2\mathsf{s}+1$ times.

Applying this same procedure to all representations appearing in
$\mathcal{R}_{\Delta,\mathsf{s}}$ we arrive at

$Z_{\text{STT}}=\prod_{\mathcal{R}_{\Delta,\mathsf{s}}}\prod_{(\lambda_{L},\lambda_{R})}\left((\mathsf{s}-\lambda_{L}-\lambda_{R})^{-\frac{2\mathsf{s}+1}{2}}\prod_{\begin{subarray}{c}N\in\mathbb{Z},\\\
\absolutevalue{N}>\mathsf{s}\end{subarray}}(|N|-\lambda_{L}-\lambda_{R})^{-\frac{1}{2}}\right)\leavevmode\nobreak\
,$ (3.29)

where the first product is understood to take the product over all pairs
$\mathsf{R}_{L}\otimes\mathsf{R}_{R}\in\mathcal{R}_{\Delta,\mathsf{s}}$ and
the second product is taken over all weights
$(\lambda_{L},\lambda_{R})\in\mathsf{R}_{L}\otimes\mathsf{R}_{R}$ of a
particular pair in $\mathcal{R}_{\Delta,\mathsf{s}}$. Where it does not cause
confusion we will maintain this shorthand (in both products and sums) for
compactness of notation.

As mentioned at the beginning to this section, the local spin-$\mathsf{s}$
partition function, $Z_{\Delta,\mathsf{s}}$, includes, in addition to this
quasinormal product, the product from integrating over normalizable zero
modes, (3.7):

$Z_{\Delta,\mathsf{s}}=Z_{\text{zero}}Z_{\text{STT}}\leavevmode\nobreak\ .$
(3.30)

In the next section we will show how this combination, with the expression of
$Z_{\text{STT}}$ as a product over representation weights, (3.29), will lead
to the Wilson spool.

###  3.2 Constructing the spool

The procedure to cast $\log Z_{\Delta,\mathsf{s}}$ as an integral over Wilson
loop operators starts by rearranging (3.7) and (3.29). We first make use of
the Schwinger parameterization of the logarithm

$\log M=-\int_{\times}^{\infty}\frac{\text{d}\alpha}{\alpha}e^{-\alpha
M}\leavevmode\nobreak\ ,$ (3.31)

with a regularization of the divergence at $\alpha\rightarrow 0$ that we will
leave unspecified for now. We will address this regularization through a
suitable $i\varepsilon$ prescription below. Applying (3.31) to (3.29), we
first see that the sum over weights in $\log Z_{\text{STT}}$ can then be
organized into representation traces

$\sum_{(\lambda_{L},\lambda_{R})}e^{\alpha(\lambda_{L}+\lambda_{R})}=\Tr_{\mathsf{R}_{L}}\big{(}e^{\alpha
L_{3}}\big{)}\Tr_{\mathsf{R}_{R}}\big{(}e^{\alpha\bar{L}_{3}}\big{)}\leavevmode\nobreak\
,$ (3.32)

which are the characters of the non-standard representation in (2.21). This
then leads to

$\displaystyle\log(Z_{\text{STT}})=\frac{1}{2}\int_{\times}^{\infty}\frac{\text{d}\alpha}{\alpha}\left(\sum_{\begin{subarray}{c}N\in\mathbb{Z}\\\
\absolutevalue{N}>\mathsf{s}\end{subarray}}e^{-|N|\alpha}\right.$
$\displaystyle+(2\mathsf{s}+1)e^{-\mathsf{s}\alpha}\left.\vphantom{\sum_{\begin{subarray}{c}N\in\mathbb{Z}\\\
\absolutevalue{N}>\mathsf{s}\end{subarray}}}\right)\times$
$\displaystyle\sum_{\mathcal{R}_{\Delta,\mathsf{s}}}\Tr_{\mathsf{R}_{L}}\big{(}e^{\alpha
L_{3}}\big{)}\Tr_{\mathsf{R}_{R}}\big{(}e^{\alpha\bar{L}_{3}}\big{)}\leavevmode\nobreak\
.$ (3.33)

Similarly, we can introduce a Schwinger parameter to $\log Z_{\text{zero}}$,
where now (3.7) reads

$\displaystyle\log Z_{\text{zero}}$
$\displaystyle=\int_{\times}^{\infty}\frac{\text{d}\alpha}{\alpha}\left(\sum_{n=0}^{\mathsf{s}-2}((n+1)^{2}-\mathsf{s}^{2})e^{(j_{L}+j_{R}-n)\alpha}-\frac{\mathsf{s}^{2}}{2}e^{(j_{L}+j_{R}+1)\alpha}\right)+(j_{L/R}\to\bar{j}_{L/R})$
$\displaystyle=\frac{1}{2}\int_{\times}^{\infty}\frac{\text{d}\alpha}{\alpha}(e^{\alpha}-1)^{2}e^{-2\alpha}\left(\sum_{n=0}^{\mathsf{s}-2}((n+1)^{2}-\mathsf{s}^{2})e^{-n\alpha}-\frac{\mathsf{s}^{2}}{2}e^{\alpha}\right)\times$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\sum_{\mathcal{R}_{\Delta,\mathsf{s}}}\Tr_{\mathsf{R}_{L}}\big{(}e^{\alpha
L_{3}}\big{)}\Tr_{\mathsf{R}_{R}}\big{(}e^{\alpha\bar{L}_{3}}\big{)}\leavevmode\nobreak\
.$ (3.34)

In the first line, we used $\Delta=-j_{L}-j_{R}$, and in the second, the
characters (2.21) to cast this as a trace. The zero mode contribution (3.2)
combines nicely with $\log\left(Z_{\text{STT}}\right)$ to give

$\log
Z_{\Delta,\mathsf{s}}=\frac{1}{2}\int_{\times}^{\infty}\frac{\text{d}\alpha}{\alpha}\left(\frac{\cosh\left(\frac{\alpha}{2}\right)}{\sinh\left(\frac{\alpha}{2}\right)}-\mathsf{s}^{2}\sinh(\alpha)\right)\sum_{\mathcal{R}_{\Delta,\mathsf{s}}}\Tr_{\mathsf{R}_{L}}\big{(}e^{\alpha
L_{3}}\big{)}\Tr_{\mathsf{R}_{R}}\big{(}e^{\alpha\bar{L}_{3}}\big{)}\leavevmode\nobreak\
,$ (3.35)

where we used

$\sum_{n\in\mathbb{Z}}e^{-\absolutevalue{n}\alpha}=\frac{\cosh\left(\frac{\alpha}{2}\right)}{\sinh\left(\frac{\alpha}{2}\right)}\leavevmode\nobreak\
.$ (3.36)

At this point we use the even parity of the integrand to regulate the
divergence through the following $i\varepsilon$ prescription:

$\int_{\times}^{\infty}\frac{\differential\alpha}{\alpha}f(\alpha):=\lim_{\varepsilon\rightarrow
0}\frac{1}{4}\sum_{\pm}\int_{-\infty}^{\infty}\frac{\differential\alpha}{\alpha\pm
i\varepsilon}f(\alpha\pm i\varepsilon)\leavevmode\nobreak\ .$ (3.37)

This is a choice of regularization scheme for the one-loop determinant.
Finally, under a change of integration variables $\alpha\rightarrow-i\alpha$
we can write the partition function as

$\displaystyle\\!\log
Z_{\Delta,\mathsf{s}}=\frac{i}{8}\\!\int_{\mathcal{C}}\frac{\text{d}\alpha}{\alpha}\\!\left(\frac{\cos(\frac{\alpha}{2})}{\sin(\frac{\alpha}{2})}+2\mathsf{s}^{2}\cos(\frac{\alpha}{2})\sin(\frac{\alpha}{2})\right)\\!\sum_{\mathcal{R}_{\Delta,\mathsf{s}}}\Tr_{\mathsf{R}_{L}}\big{(}e^{i\alpha
L_{3}}\big{)}\Tr_{\mathsf{R}_{R}}\big{(}e^{i\alpha\bar{L}_{3}}\big{)}\leavevmode\nobreak\
,$ (3.38)

where the contour $\mathcal{C}$ runs upwards along the imaginary $\alpha$ axis
to the left and right of the divergence at the origin, as depicted in Fig. 1.

$\text{Re}(\alpha)$$\text{Im}(\alpha)$$\mathcal{C}$ Figure 1: The integration
contour regulating the $\alpha\rightarrow 0$ divergence.

As a final step we now rewrite the holonomies inside the traces to restore the
background connections, arriving at (3.8) with $\mathbb{W}_{j_{L},j_{R}}$ the
spinning Wilson spool:

$\mathbb{W}_{j_{L},j_{R}}[a_{L},a_{R}]:=\frac{i}{2}\int_{\mathcal{C}}\frac{\differential\alpha}{\alpha}\frac{\cos\left(\frac{\alpha}{2}\right)}{\sin\left(\frac{\alpha}{2}\right)}\left(1+2\mathsf{s}^{2}\sin^{2}\left(\frac{\alpha}{2}\right)\right)\times\\\
\sum_{\mathcal{R}_{\Delta,\mathsf{s}}}\Tr_{\mathsf{R}_{L}}\left(\mathcal{P}e^{\frac{\alpha}{2\pi}\oint_{\gamma}a_{L}}\right)\Tr_{\mathsf{R}_{R}}\left(\mathcal{P}e^{-\frac{\alpha}{2\pi}\oint_{\gamma}a_{R}}\right)\leavevmode\nobreak\
,$ (3.39)

where above $\gamma=\gamma_{\text{hor.}}$ is a cycle wrapping the singular
point corresponding to the horizon.

At this point let us make several comments:

  * •

The spinning Wilson spool takes a form similar to that of the scalar spool
found in [1, 2]; importantly the “operator pieces” of the expression (3.39)
have been organized into gauge-invariant Wilson loop operators. The only
modification the spinning spool brings is in the integration measure.

  * •

The modification to the integration measure, proportional to $\mathsf{s}^{2}$,
will have the effect of lowering the degree of each pole at $\alpha\in
2\pi\mathbb{Z}$ by two. As we will shortly see, this effect reproduces the
“edge partition function” of [8].

  * •

Mathematically the holonomies corresponding to $\gamma_{\text{hor.}}$ appear
in the one-loop determinant because they are sensitive to $\Delta$ on which
$Z_{\text{STT}}$ is treated as an meromorphic function. The physics behind
this is clear: we are reproducing a one-loop determinant of massive fields. In
the worldline quantum mechanics framework this corresponds to averaging over
worldlines of a massive particle in the static patch. Such wordlines are
timelike and rotate to a contour gauge equivalent to $\gamma_{\text{hor.}}$.

###  3.3 Testing the Wilson spool

We now uphold equations (3.8) and (3.39) by verifying that it indeed
reproduces the correct path integral of a massive spinning field on $S^{3}$.
There are several ways how to evaluate this path integral, and here we will
focus on two approaches to use as a comparison. The first is to implement the
DHS method traditionally: in App. B we evaluate this path integral by
explicitly listing the quasinormal modes and applying DHS. The second approach
we can compare to are the expressions found in [8], which cast the results in
terms of $\mathfrak{so}(1,3)$ characters.

Since we are not turning on gravity ($G_{N}\to 0$), we evaluate the Wilson
loop operators in (3.39) as characters with the appropriate holonomies in
(3.18). Using the form of our non-standard representation character, (2.21),
we then write

$\log
Z_{\Delta,\mathsf{s}}=-\frac{i}{8}\int_{\mathcal{C}}\frac{\text{d}\alpha}{\alpha}\left(\frac{\cos(\frac{\alpha}{2})}{\sin^{3}(\frac{\alpha}{2})}+2\mathsf{s}^{2}\frac{\cos(\frac{\alpha}{2})}{\sin(\frac{\alpha}{2})}\right)e^{i\alpha(1-\Delta)}\leavevmode\nobreak\
,$ (3.40)

where we note that the sum over representations in
$\mathcal{R}_{\Delta,\mathsf{s}}$ is already neatly packaged into our two
contours. We recognize the first term in the parentheses of (3.40) as twice
the on-shell scalar Wilson spool found in [1]. We evaluate both terms by
deforming the $\alpha$ contours to run above and below the real $\alpha$ axis
to pick up the residues at the poles lying at $2\pi\mathbb{Z}_{\neq 0}$. This
deformation is depicted in Fig. 2.

$\text{Re}(\alpha)$$\text{Im}(\alpha)$$\mathcal{C}$ Figure 2: We deform the
$\alpha$ integration contour to wrap the poles lying along the real axis.

Summing the towers of poles and expressing the $\mathfrak{su}(2)_{L/R}$
highest-weight labels in terms of $\mu$ and $\mathsf{s}$ as in (2.22), we
write this as

$\displaystyle\begin{split}\log
Z_{\Delta,\mathsf{s}}&=\sum_{\pm}\left(-\frac{1}{4\pi^{2}}\text{Li}_{3}\left(e^{\mp
2\pi\mu}\right)\mp\frac{\mu}{2\pi}\text{Li}_{2}\left(e^{\mp
2\pi\mu}\right)-\frac{\mu^{2}+\mathsf{s}^{2}}{2}\text{Li}_{1}\left(e^{\mp
2\pi\mu}\right)\right)\\\ &=2\log
Z_{\text{scalar}}-\frac{\mathsf{s}^{2}}{2}\sum_{\pm}\text{Li}_{1}\left(e^{\mp
2\pi\mu}\right)\leavevmode\nobreak\ ,\end{split}$ (3.41)

where

$\text{Li}_{p}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{p}}$ (3.42)

is the polylogarithm. As mentioned above, $\log Z_{\text{scalar}}$ is the one-
loop determinant of a massive scalar field on $S^{3}$ with a mass set by
$\ell_{\text{dS}}^{2}m^{2}=\Delta(2-\Delta)=1+\mu^{2}$. Up to an overall
phase, unfixed by the quasinormal mode method, (3.41) matches the path
integral of a massive spin-$\mathsf{s}$ field on $S^{3}$ via DHS in App. B and
the results reported in [8].

##  4 Coupling quantum matter to dS3 quantum gravity

Having constructed the spinning Wilson spool on $S^{3}$ in Sec. 3, in this
section we will promote it to an off-shell object with the aim of
incorporating quantum effects from gravity around $S^{3}$. More concretely, we
posit the following: let $A_{L}$ and $A_{R}$ be $\mathfrak{su}(2)_{L}$ and
$\mathfrak{su}(2)_{R}$ connections respectively, yielding a non-degenerate
dreibein $e^{a}=-i\left(A_{L}^{a}-A_{R}^{a}\right)$ with an associated metric
geometry $(M_{3},g_{M_{3}})$ that is topologically equivalent to the round
$S^{3}$. Then the partition function of massive spin-$\mathsf{s}$ fields
minimally coupled to $g_{M_{3}}$ is determined by the spinning Wilson spool
given by (3.39) with $a_{L/R}$ replaced with $A_{L/R}$:

$\log
Z_{\Delta,\mathsf{s}}[g_{M_{3}}]=\frac{1}{4}\mathbb{W}_{j_{L},j_{R}}[A_{L},A_{R}]\leavevmode\nobreak\
.$ (4.1)

For scalar fields it was argued in [1] that the manipulations leading to the
Wilson spool remain valid for off-shell geometries. This argumentation relied
on the expression of the Laplacian $-\nabla^{2}_{g_{M_{3}}}$ as a Casimir of
local $\mathfrak{su}(2)_{L/R}$ action (see App. D of [1]). While this remains
true when acting on spin-$\mathsf{s}$ fields (via the transverse-traceless
condition), various manipulations leading to (3.39) make the leap to (4.1)
less rigorous than for scalar fields. This includes both manipulations relying
on the integer nature of the holonomies, $\mathsf{h}_{L/R}$, as well as the
explicit form of $Z_{\text{zero}}$. Barring a detailed analysis of
normalizable zero modes for the ghost and Stückelberg fields appearing in
$Z_{\Delta,\mathsf{s}}$ on generic three-geometries, at this point we take
(4.1) as a proposal. This proposal is upheld on the grounds that it utilizes
and generalizes naturally the gauge-invariant observables in Chern-Simons
theory (namely its Wilson loops); these observables appear in a form that
reduces to twice999This stems simply from the fact that a STT field contains
roughly two scalars corresponding to the polarizations with $\mathsf{s}$ and
$-\mathsf{s}$ and has nothing to do with the ambiguities of a compact space
and possible zero modes. This counting is obviously not continuous as
$\mathsf{s}\rightarrow 0$. the scalar path integral when
$\mathsf{s}\rightarrow 0$ and, as we will see in the following section,
continues to work for negative cosmological constant.

With this proposal in hand, the Wilson spool gives us a concrete route for
calculating finite $G_{N}$ effects101010To be clear on scope: here we mean to
all orders in $G_{N}$ perturbation theory about the $S^{3}$ saddle. We do not
consider topology change or other non-perturbative effects here. to $\log
Z_{\Delta,\mathsf{s}}$. That is we can consider

$\displaystyle\big{\langle}\log Z_{\Delta,\mathsf{s}}\big{\rangle}_{\rm
grav}:=$
$\displaystyle\int\left[\mathcal{D}g_{M_{3}}\right]_{S^{3}}\,e^{-I_{\text{EH}}[g_{M_{3}}]}\log
Z_{\Delta,\mathsf{s}}[g_{M_{3}}]$ $\displaystyle\equiv$
$\displaystyle\int[\mathcal{D}A_{L}\mathcal{D}A_{R}]_{S^{3}}\,e^{iS[A_{L},A_{R}]}\left(\frac{1}{4}\mathbb{W}_{j_{L},j_{R}}[A_{L},A_{R}]\right)\leavevmode\nobreak\
,$ (4.2)

where the second line follows from the rewriting of the gravitational
variables and action (in a first-order formalism) as two Chern-Simons path
integrals.111111In App. A.1 we review dictionary between Chern-Simons theory
and dS3 gravity. Additionally, in the first line,
$[\mathcal{D}g_{M_{3}}]_{S^{3}}$ indicates that we integrate over metric
geometries topologically equivalent to $S^{3}$, and in the second line,
$[\mathcal{D}A_{L}\mathcal{D}A_{R}]_{S^{3}}$ indicates we perform the Chern-
Simons path integral on the base $S^{3}$ topology.

The power of the second line of (4) lies in the breadth of techniques for
evaluating Wilson loop observables for $SU(2)$ Chern-Simons theories on the
three-sphere [22, 23, 24, 25, 26, 37, 38]. In [1] it was shown how to adapt
two such techniques, Abelianisation [22, 23, 24] and localization through a
$\mathcal{N}=2$ supersymmetric extension [25, 26], for the evaluation of
Wilson loop expectation values in light of the features unique to Chern-Simons
gravity: non-trivial background connections, complex levels, and non-standard
representations appearing in Wilson loop operators. While prima facie these
are two very different techniques, they both lead to the expression of a
Wilson loop expectation value as a deformation of its on-shell (i.e. flat)
background value integrated over a single modulus:

$\int[\mathcal{D}A]e^{ik\,S_{\text{CS}}[A]}\Tr_{\mathsf{R}}\mathcal{P}e^{\oint_{\gamma}A}=e^{irS_{\text{CS}}[a]}\int_{\mathbb{R}}\differential\sigma\,\sin^{2}(\pi\sigma)e^{ir\,\frac{\pi}{2}\sigma^{2}}\chi_{\mathsf{R}}\left(\sigma+\mathsf{h}\right)\leavevmode\nobreak\
,$ (4.3)

where $r=k+2$ is the renormalized level, $a$ is the on-shell background
connection, $\mathsf{h}$ is its holonomy about the contour $\gamma$, and
$\chi_{\mathsf{R}}(z)=\Tr_{\mathsf{R}}e^{i2\pi z\,L_{3}}$ is the
$\mathfrak{su}(2)$ representation character. Equation (4.3) holds for complex
levels and non-standard $\mathfrak{su}(2)$ representations, including those
found in Sec. 2. Applying this to (4), we can write

$\displaystyle\\!\big{\langle}\log Z_{\Delta,\mathsf{s}}\big{\rangle}_{\rm
grav}=\frac{i}{8}e^{ir_{L}S_{\text{CS}}[a_{L}]+ir_{R}S_{\text{CS}}[a_{R}]}$
$\displaystyle\int\differential\sigma_{L}\differential\sigma_{R}\,e^{ir_{L}\frac{\pi}{2}\sigma_{L}^{2}+ir_{R}\frac{\pi}{2}\sigma_{R}^{2}}\sin^{2}(\pi\sigma_{L})\sin^{2}(\pi\sigma_{R})\times$
$\displaystyle\sum_{\mathcal{R}_{\Delta,\mathsf{s}}}\int_{\mathcal{C}}\frac{\differential\alpha}{\alpha}\frac{\cos\left(\frac{\alpha}{2}\right)}{\sin\left(\frac{\alpha}{2}\right)}\left(1+2\mathsf{s}^{2}\sin^{2}\left(\frac{\alpha}{2}\right)\right)\times$
$\displaystyle\qquad\;\;\chi_{\mathsf{R}_{L}}\left(\frac{\alpha}{2\pi}(1+\sigma_{L})\right)\chi_{\mathsf{R}_{R}}\left(\frac{\alpha}{2\pi}(1-\sigma_{R})\right)\leavevmode\nobreak\
,$ (4.4)

where we used the off-shell version of the spool in (3.39). Upon writing

$r_{L}=\hat{\delta}+is\leavevmode\nobreak\ ,\qquad
r_{R}=\hat{\delta}-is\leavevmode\nobreak\ ,\qquad
s=\frac{\ell_{\text{dS}}}{4G_{N}}\leavevmode\nobreak\
,\qquad\hat{\delta}\in\mathbb{Z}\leavevmode\nobreak\ ,$ (4.5)

per the Chern-Simons gravity dictionary outlined in App. A.1, we posit that
(4) is exact (that is, holding to all orders) in $G_{N}$ perturbation theory
about the $S^{3}$ saddle.

While this claim has power in principle, in practice (4) is complicated as an
integral. By rescaling $\sigma_{L/R}\rightarrow r_{L/R}^{-1/2}\sigma_{L/R}$,
we can proceed systematically in $\ell_{\text{dS}}^{-1}G_{N}$ perturbation
theory which amounts to a Taylor expansion in $\sigma_{L/R}$ of non-Gaussian
pieces of the integrand in $\eqref{eq:AbelianizedlogZEV}$. At any order in
this expansion the Gaussian integrals over $\sigma_{L/R}$ can be performed.
This leaves the contour integral over $\alpha$ which can be deformed to pick
up its poles (which remain at $2\pi\mathbb{Z}_{\neq 0}$ at each order of
perturbation theory) in a similar spirit as our computation in Sec. 3.3. This
procedure completely mirrors that outlined for the scalar Wilson spool in [1]
and can be efficiently implemented on a computer algebra system.

To illustrate this concretely, we evaluate $\big{\langle}\log
Z_{\Delta,\mathsf{s}}\big{\rangle}_{\rm grav}$ normalized by the gravitational
path integral to the first non-zero order of $\ell_{\text{dS}}^{-1}G_{N}$
perturbation theory. The gravitational path integral on $S^{3}$ has a close
form given by

$\displaystyle\mathcal{Z}_{\text{grav}}=$ $\displaystyle
e^{ir_{L}S_{\text{CS}}[a_{L}]+ir_{R}S_{\text{CS}}[a_{R}]}\int\differential\sigma_{L}\differential\sigma_{R}\,e^{ir_{L}\frac{\pi}{2}\sigma_{L}^{2}+ir_{R}\frac{\pi}{2}\sigma_{R}^{2}}\sin^{2}(\pi\sigma_{L})\sin^{2}(\pi\sigma_{R})$
$\displaystyle=$ $\displaystyle e^{2\pi
s}\left(ie^{-i2\pi\frac{\hat{\delta}}{\hat{\delta}^{2}+s^{2}}}\right)\frac{2}{\sqrt{\hat{\delta}^{2}+s^{2}}}\,\absolutevalue{\sin\left(\frac{\pi}{\hat{\delta}+is}\right)}^{2}\leavevmode\nobreak\
,$ (4.6)

as outlined in [1] (see also [13, 8, 39]). By implementing the rescaling
$\sigma_{L/R}\rightarrow r_{L/R}^{-1/2}\sigma_{L/R}$ in (4), to leading order
in the coupling we find

$\frac{\big{\langle}\log Z_{\Delta,\mathsf{s}}\big{\rangle}_{\rm
grav}}{\mathcal{Z}_{\text{grav}}}=\log
Z_{\Delta,\mathsf{s}}[S^{3}]+\left(\frac{G_{N}}{\ell_{\text{dS}}}\right)^{2}\left[\log
Z_{\Delta,\mathsf{s}}\right]_{(2)}+\ldots\leavevmode\nobreak\ ,$ (4.7)

where the dots correspond to subleading corrections in
$\ell_{\text{dS}}^{-1}G_{N}$, and the first non-trivial correction is

$\left[\log
Z_{\Delta,\mathsf{s}}\right]_{(2)}=\sum_{\pm}\sum_{i=0}^{3}z_{i}^{(2)}[\pm\mu,\mathsf{s}]\text{Li}_{-i}\left(e^{\mp
2\pi\mu}\right)\leavevmode\nobreak\ ,$ (4.8)

with

$\displaystyle z_{0}^{(2)}[\mu,\mathsf{s}]=$
$\displaystyle-\left(\frac{16\pi}{3}-\frac{8}{\pi}-8i\hat{\delta}\right)\mu^{3}-\left(16\pi-\frac{144}{\pi}-24i\hat{\delta}+16\pi\mathsf{s}^{2}-\frac{216}{\pi}\mathsf{s}^{2}-24i\hat{\delta}\mathsf{s}^{2}\right)\mu\,,$
$\displaystyle z_{1}^{(2)}[\mu,\mathsf{s}]=$
$\displaystyle\left(\frac{16\pi^{2}}{3}+12-i8\pi\hat{\delta}\right)\mu^{4}+\left(16\pi^{2}-276-i24\pi\hat{\delta}-432\mathsf{s}^{2}\right)\mu^{2}$
$\displaystyle-(16\pi^{2}-372-i24\pi\hat{\delta})\mathsf{s}^{4}+(16\pi^{2}-588-i24\pi\hat{\delta})\mathsf{s}^{2}-48\,,$
$\displaystyle z_{2}^{(2)}[\mu,\mathsf{s}]=$
$\displaystyle-\frac{24\pi}{5}\mu^{5}+\left(104\pi+96\pi\mathsf{s}^{2}\right)\mu^{3}+\left(96\pi-168\pi\mathsf{s}^{4}+792\pi\mathsf{s}^{2}\right)\mu\,,$
$\displaystyle z_{3}^{(2)}[\mu,\mathsf{s}]=$
$\displaystyle-\left(8\pi^{2}-16\pi^{2}\mathsf{s}^{2}\right)\mu^{4}-\left(32\pi^{2}+40\pi^{2}\mathsf{s}^{4}+160\pi^{2}\mathsf{s}^{2}\right)\mu^{2}$
$\displaystyle+8\pi^{2}\mathsf{s}^{6}+24\pi^{2}\mathsf{s}^{4}-32\pi^{2}\mathsf{s}^{2}\leavevmode\nobreak\
.$ (4.9)

We note that taking $\mathsf{s}\rightarrow 0$ doubles the corrections to the
scalar one-loop determinant [1] as expected. At leading order in a large $\mu$
expansion (which amounts to a large mass expansion while holding $\mathsf{s}$
fixed)

$[\log
Z_{\Delta,\mathsf{s}}]_{(2)}=-\frac{48\pi\,\mu^{5}}{5}e^{-2\pi\mu}+\left(24-\frac{16\pi^{2}}{3}+32\pi^{2}\mathsf{s}^{2}-i16\pi\hat{\delta}\right)\mu^{4}\,e^{-2\pi\mu}+\ldots\leavevmode\nobreak\
,$ (4.10)

where we have kept to next-to-leading order in the large mass expansion where
the first contribution from spin appears.

It is natural at this stage to interpret this as a renormalization of the
mass121212Importantly the spin of the field does not get renormalized. of the
spin-$\mathsf{s}$ field. Writing an expansion for the renormalized mass

$\mu_{R}=\mu+\left(\frac{G_{N}}{\ell_{\text{dS}}}\right)^{2}\delta_{\mu}^{(2)}+\ldots\leavevmode\nobreak\
,$ (4.11)

we note to $O(G_{N}^{2}\ell_{\text{dS}}^{-2})$

$\log Z_{\Delta,\mathsf{s}}=\log
Z_{\Delta_{R},\mathsf{s}}-\pi\frac{\cosh(\pi\mu_{R})}{\sinh(\pi\mu_{R})}(\mu_{R}^{2}+\mathsf{s}^{2})\left(\frac{G_{N}^{2}}{\ell_{\text{dS}}^{2}}\right)\delta_{\mu}^{(2)}+\ldots\leavevmode\nobreak\
,$ (4.12)

where $\Delta_{R}=1-i\mu_{R}$. Interpreting the corrections due to quantum
gravity as a mass renormalization then sets as a renormalization condition

$\frac{\big{\langle}\log Z_{\Delta,\mathsf{s}}\big{\rangle}_{\rm
grav}}{\mathcal{Z}_{\text{grav}}}\overset{!}{=}\log
Z_{\Delta_{R},\mathsf{s}}\leavevmode\nobreak\ ,$ (4.13)

with $G_{N}$ held fixed. This then determines the renormalized mass as

$\displaystyle\delta_{\mu}^{(2)}=$
$\displaystyle\frac{1}{\pi}\frac{\tanh(\pi\mu_{R})}{\mu_{R}^{2}+\mathsf{s}^{2}}\left[\log
Z_{\Delta,\mathsf{s}}\right]_{(2)}$ $\displaystyle=$
$\displaystyle-\frac{48}{5}\mu_{R}^{3}\,e^{-2\pi\mu_{R}}+\left(\frac{24}{\pi}-\frac{16\pi}{3}+32\pi\mathsf{s}^{2}-i16\hat{\delta}\right)\mu_{R}^{2}\,e^{-2\pi\mu_{R}}+\ldots\leavevmode\nobreak\
,$ (4.14)

where in the second line we have written the leading and next to leading terms
in a large mass expansion. Note that the magnitude of the leading term is
$\mathsf{s}$ independent and consistent131313In [1] the renormalization
condition was taken (tacitly) as  $\frac{\big{\langle}\log
Z_{\Delta_{R},\text{scalar}}\big{\rangle}_{\rm
grav}}{\mathcal{Z}_{\text{grav}}}\overset{!}{=}\log
Z_{\Delta,\text{scalar}}\leavevmode\nobreak\ ,$ (4.15) which leads to an
overall minus sign with respect to (4).with the leading mass renormalization
for scalar fields [1].

We note that this is not the only renormalization condition that one may
choose to set. We could instead to renormalize Newton’s constant,
$G_{N}\rightarrow G_{N,R}$, while holding the mass of the spinning field
fixed. To illustrate this, we will set $\hat{\delta}=0$ and consider the
following renormalization condition:

$\displaystyle\frac{\int[\mathcal{D}g]e^{-I_{\text{EH}}[g]}Z_{\Delta,\mathsf{s}}[g]}{Z_{\Delta,\mathsf{s}}[g_{S^{3}}]}\Big{|}_{G_{N}}=$
$\displaystyle\mathcal{Z}_{\text{grav}}\left(1+\frac{\big{\langle}\log
Z_{\Delta,\mathsf{s}}\big{\rangle}_{\rm grav}}{\mathcal{Z}_{\text{grav}}}-\log
Z_{\Delta,\mathsf{s}}[g_{S^{3}}]+\ldots\right)\Big{|}_{G_{N}}$
$\displaystyle\overset{!}{=}\mathcal{Z}_{\text{grav}}\Big{|}_{G_{N,R}}\leavevmode\nobreak\
,$ (4.16)

or equivalently

$\frac{\mathcal{Z}_{\text{grav}}\big{|}_{G_{N,R}}}{\mathcal{Z}_{\text{grav}}\big{|}_{G_{N}}}=1+\left(\frac{G_{N,R}}{\ell_{\text{dS}}}\right)^{2}\left[\log
Z_{\Delta,\mathsf{s}}\right]_{(2)}+\ldots\leavevmode\nobreak\ .$ (4.17)

Note we have normalized by $Z_{\Delta,\mathsf{s}}[g_{S^{3}}]$ in (4) as this
leading term decouples from metric fluctuations. It would be responsible for a
renormalization of the cosmological constant, but does not mediate the
gravitational self-interactions relevant for renormalizing the coupling,
$G_{N}$. Writing

$G_{N}=G_{N,R}\left(1+\delta_{G_{N}}^{(2)}\left(\frac{G_{N,R}}{\ell_{\text{dS}}}\right)^{2}+\ldots\right)\leavevmode\nobreak\
,$ (4.18)

we find from expanding (4) and comparing to the right-hand side of (4.17)

$\displaystyle\delta^{(2)}_{G_{N}}=$ $\displaystyle-\frac{1}{3}\left[\log
Z_{\Delta,\mathsf{s}}\right]_{(2)}$ $\displaystyle=$
$\displaystyle\frac{16\pi}{5}\mu^{5}\,e^{-2\pi\mu}-\left(8-\frac{16\pi^{2}}{9}+\frac{32\pi^{2}}{3}\mathsf{s}^{2}\right)\mu^{4}\,e^{-2\pi\mu}+\ldots\leavevmode\nobreak\
,$ (4.19)

where again we have written the leading and next-to-leading terms in a large
mass expansion.

It is worth noting that the results of this section, both the renormalization
of the field mass, (4), and the renormalization of $G_{N}$, (4.17), are novel.
To our knowledge, corresponding computations have not been carried out in the
metric formulation of dS3.141414In the absence of a cosmological constant, a
three-loop computation involving gravitons was done in [40]. It seems feasible
that one could apply that approach to dS3 and verify (4) and (4.17). To this
end, the results of this section provide concrete and testable predictions of
the Chern-Simons formulation of gravity.

##  5 Spinning spool on AdS3

To illustrate the general utility of the spinning Wilson spool constructed in
Sec. 3.2, as well as bolster the Conditions I & II that lead to it in Sec.
3.1, we can repeat this construction in an AdS3 background. An expression for
the one-loop determinant of massive spinning fields on a BTZ black hole as a
classical Wilson spool was conjectured in [2] based on an extension of the
result for a massive scalar field. In this section we show how to derive the
spinning spool on AdS3 from the principles set out in Sec. 3.1 and demonstrate
that the final expression accords with the general result (1.4).

We wish to compute

$Z_{\Delta,\mathsf{s}}=\det\left(-\nabla_{(\mathsf{s})}^{2}+\ell_{\text{AdS}}^{2}\bar{m}_{\mathsf{s}}^{2}\right)^{-\frac{1}{2}}$
(5.1)

with $\nabla^{2}_{(\mathsf{s})}$ being the Laplace-Beltrami operator acting on
spin-$\mathsf{s}$ STT tensors. The background geometry entering in (5.1) will
be the rotating BTZ black hole [41, 42] with AdS3 radius $\ell_{\text{AdS}}$;
properties of this geometry are described in App. A.2. The effective mass
$\ell_{\text{AdS}}^{2}\bar{m}_{\mathsf{s}}^{2}$ is related to the standard
mass, $m^{2}$, via [43]

$\ell_{\text{AdS}}^{2}\bar{m}_{\mathsf{s}}^{2}=\ell_{\text{AdS}}^{2}m^{2}+\mathsf{s}(\mathsf{s}-3)\leavevmode\nobreak\
.$ (5.2)

We recall that the standard mass is related to the conformal dimension of a
dual primary through [44]

$\ell_{\text{AdS}}^{2}m^{2}=(\Delta+\mathsf{s}-2)(\Delta-\mathsf{s})\leavevmode\nobreak\
.$ (5.3)

$Z_{\Delta,\mathsf{s}}$ is completely captured by this one-loop determinant
over STT tensors: in contrast to the previous section, there is no additional
zero mode product as AdS3 is non-compact.

The isometry group of Lorentzian AdS3 is $SO(2,2)$ with an algebra isomorphic
to $\mathfrak{sl}(2,\mathbb{R})_{L}\oplus\mathfrak{sl}(2,\mathbb{R})_{R}$
which we will take to be generated by $\\{L_{0},L_{\pm}\\}$ and
$\\{\bar{L}_{0},\bar{L}_{\pm}\\}$, respectively.151515Our conventions for the
$\mathfrak{sl}(2,\mathbb{R})$ algebra follow [45, 46]. As done for dS3, it is
useful to briefly establish our conventions regarding the description of
single-particle states living in AdS3 as representations of
$\mathfrak{sl}(2,\mathbb{R})_{L}\oplus\mathfrak{sl}(2,\mathbb{R})_{R}$.
Lowest-weight representations $\mathsf{R}_{j}^{{\scaleto{\text{LW}}{4pt}}}$
are defined by a lowest-weight state,
$|j,0\rangle_{\scaleto{\text{LW}}{4pt}}$, which is annihilated by $L_{-}$ and
labelled by its $L_{0}$ eigenvalue:

$L_{-}|j,0\rangle_{\scaleto{\text{LW}}{4pt}}=0\leavevmode\nobreak\
,\qquad\qquad
L_{0}|j,0\rangle_{\scaleto{\text{LW}}{4pt}}=j|j,0\rangle_{\scaleto{\text{LW}}{4pt}}\leavevmode\nobreak\
.$ (5.4)

All other states in $\mathsf{R}^{{\scaleto{\text{LW}}{4pt}}}_{j}$ are
generated by the action of $L_{+}$ acting on
$|j,0\rangle_{\scaleto{\text{LW}}{4pt}}$. This representation has a character

$\chi_{j,\scaleto{{\scaleto{\text{LW}}{4pt}}}{4pt}}(z)=\mathrm{Tr}_{\mathsf{R}_{j}^{{\scaleto{\text{LW}}{4pt}}}}\left(e^{i2\pi
zL_{0}}\right)=\frac{e^{i\pi z(2j-1)}}{2\sinh(-i\pi z)}\leavevmode\nobreak\ .$
(5.5)

Highest-weight representations $\mathsf{R}^{{\scaleto{\text{HW}}{4pt}}}_{j}$
are defined by a highest-weight state,
$|j,0\rangle_{\scaleto{\text{HW}}{4pt}}$, which is instead annihilated by
$L_{+}$. We will take by convention its $L_{0}$ eigenvalue to be $-j$:

$L_{+}|j,0\rangle_{\scaleto{\text{HW}}{4pt}}=0\leavevmode\nobreak\
,\qquad\qquad
L_{0}|j,0\rangle_{\scaleto{\text{HW}}{4pt}}=-j|j,0\rangle_{\scaleto{\text{HW}}{4pt}}\leavevmode\nobreak\
.$ (5.6)

All other states of $\mathsf{R}^{\scaleto{\text{HW}}{4pt}}_{j}$ are generated
by the action of $L_{-}$ on $|j,0\rangle_{\scaleto{\text{HW}}{4pt}}$. This
representation has a character

$\chi_{j,{\scaleto{\text{HW}}{4pt}}}(z)=\mathrm{Tr}_{\mathsf{R}_{j}^{{\scaleto{\text{HW}}{4pt}}}}\left(e^{i2\pi
zL_{0}}\right)=\frac{e^{-i\pi z(2j-1)}}{2\sinh(i\pi z)}\leavevmode\nobreak\ .$
(5.7)

Our conventions have been chosen so that the Casimir of both highest and
lowest-weight representations is given by

$c_{2}^{\mathfrak{sl}(2,\mathbb{R})}|j,p\rangle_{\scaleto{\text{LW/HW}}{6pt}}=j(j-1)|j,p\rangle_{\scaleto{\text{LW/HW}}{6pt}}\leavevmode\nobreak\
,\qquad\qquad\forall\;\;|j,p\rangle_{\scaleto{\text{LW/HW}}{6pt}}\in\mathsf{R}^{{\scaleto{\text{LW/HW}}{6pt}}}_{j}\leavevmode\nobreak\
.$ (5.8)

Returning to the functional determinant (5.1), we now want to implement the
steps in Sec. 3.1: the “mass-shell condition” and Conditions I & II. To start,
we write the Laplace-Beltrami operator in terms of the Casimir of the
isometries of AdS3 [44]:

$2c_{2}^{\mathfrak{sl}(2,\mathbb{R})_{L}}+2c_{2}^{\mathfrak{sl}(2,\mathbb{R})_{R}}=\nabla^{2}_{(\mathsf{s})}+\mathsf{s}(\mathsf{s}+1)\leavevmode\nobreak\
.$ (5.9)

In this language, a pole contributing to $Z_{\Delta,\mathsf{s}}^{2}$
corresponds to a state $|\psi\rangle$ in a representation of
$\mathfrak{sl}(2,\mathbb{R})_{L}\oplus\mathfrak{sl}(2,\mathbb{R})_{R}$
satisfying

$\left(2c_{2}^{\mathfrak{sl}(2,\mathbb{R})_{L}}+2c_{2}^{\mathfrak{sl}(2,\mathbb{R})_{R}}\right)|\psi\rangle=\left(\Delta(\Delta-2)+\mathsf{s}^{2}\right)|\psi\rangle\leavevmode\nobreak\
.$ (5.10)

This is the “mass-shell condition” and it is satisfied for pairs of highest
and lowest-weight representations labeled by

$j_{L}=\frac{\Delta\pm\mathsf{s}}{2}\leavevmode\nobreak\ ,\qquad
j_{R}=\frac{\Delta\mp\mathsf{s}}{2}\leavevmode\nobreak\ .$ (5.11)

Note that representations labeled by (5.11) with $\Delta$ replaced by
$\bar{\Delta}=2-\Delta$ share the same Casimir. However, unlike in dS3, we are
forced to choose either $\Delta$ or $\bar{\Delta}$ due to Dirichlet boundary
conditions tacitly imposed on solutions contributing to (5.1).161616Dirichlet
boundary conditions also exclude $\mathfrak{sl}(2,\mathbb{R})$ representations
that are neither highest nor lowest-weight from contributing to the one-loop
determinant. For what follows we will assume, without loss of generality, that
$\Delta$ corresponds to a normalizable massive spin-$\mathsf{s}$
solution.171717In the special cases where $\Delta$ and $\bar{\Delta}$ both
correspond to normalizable solutions, we simply choose $\Delta$, again without
loss of generality.  Fixing $j_{L/R}$ as in (5.11) with the upper sign, we can
then have pole contributions from any representation appearing in

$\mathcal{R}_{\Delta,\mathsf{s}}=\mathcal{R}^{{\scaleto{\text{HW}}{4pt}}}_{\Delta,\mathsf{s}}\cup\mathcal{R}^{{\scaleto{\text{LW}}{4pt}}}_{\Delta,\mathsf{s}}=\left\\{\mathsf{R}_{j_{L}}^{\scaleto{\text{HW}}{4pt}}\otimes\mathsf{R}_{j_{R}}^{\scaleto{\text{HW}}{4pt}},\mathsf{R}_{j_{R}}^{\scaleto{\text{HW}}{4pt}}\otimes\mathsf{R}_{j_{L}}^{\scaleto{\text{HW}}{4pt}}\right\\}\cup\left\\{\mathsf{R}_{j_{L}}^{\scaleto{\text{LW}}{4pt}}\otimes\mathsf{R}_{j_{R}}^{\scaleto{\text{LW}}{4pt}},\mathsf{R}_{j_{R}}^{\scaleto{\text{LW}}{4pt}}\otimes\mathsf{R}_{j_{L}}^{\scaleto{\text{LW}}{4pt}}\right\\}\leavevmode\nobreak\
.$ (5.12)

For scalars, $\mathsf{s}=0$, we have the reduced set

$\mathcal{R}_{\Delta,\text{scalar}}=\mathcal{R}^{{\scaleto{\text{HW}}{4pt}}}_{\Delta,\text{scalar}}\cup\mathcal{R}^{{\scaleto{\text{LW}}{4pt}}}_{\Delta,\text{scalar}}=\left\\{\mathsf{R}_{j}^{\scaleto{\text{HW}}{4pt}}\otimes\mathsf{R}_{j}^{\scaleto{\text{HW}}{4pt}}\right\\}\cup\left\\{\mathsf{R}_{j}^{{\scaleto{\text{LW}}{4pt}}}\otimes\mathsf{R}_{j}^{\scaleto{\text{LW}}{4pt}}\right\\}\leavevmode\nobreak\
.$ (5.13)

We now apply the group theoretic conditions of Sec. 3.1 to the representations
contributing to (5.1). Conditions I & II as they are stated in Sec. 3.1 can be
readily applied to the one-loop determinant on BTZ, and they respectively
imply:

Condition I. Single-valued solutions.

In Euclidean signature, BTZ is a solid torus that has two cycles that
characterize its global properties. Requiring solutions to be single-valued
around the contractible thermal cycle, $\gamma_{\text{th}}$, of the BTZ
geometry requires $\mathsf{s}\in\mathbb{Z}$. Requiring solutions to be single-
valued around the non-contractible spatial cycle, $\gamma_{\text{sp}}$, of the
BTZ geometry requires a weight
$(\lambda_{L},\lambda_{R})\in\mathsf{R}_{L}\otimes\mathsf{R}_{R}$ to satisfy

$\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R}\in\mathbb{Z}\leavevmode\nobreak\
,$ (5.14)

where $\mathsf{h}_{L,R}$ are holonomies around the non-contractible cycle

$\mathsf{h}_{L}=-\frac{1}{\tau}\leavevmode\nobreak\
,\qquad\mathsf{h}_{R}=-\frac{1}{\bar{\tau}}\leavevmode\nobreak\ ,$ (5.15)

and $\tau$ is the modular parameter defining the geometry. See App. A.2 for
details.

Condition II. Globally regular solutions.

A representation
$\mathsf{R}_{L}\otimes\mathsf{R}_{R}\in\mathcal{R}_{\Delta,\mathsf{s}}$ is
required to lift to a group representation of $SL(2,\mathbb{R})\times
SL(2,\mathbb{R})$ . In contrast to Sec. 3.1, in the present case Condition II
is trivial since every representation of $\mathfrak{sl}(2,\mathbb{R})$ lifts
to a (not necessarily unitary) representation of the universal cover of
$SL(2,\mathbb{R})$ [47].

Thus we fix $\mathsf{s}\in\mathbb{Z}$ and the one-loop determinant (5.1) is
then the product of poles in the complex $\Delta$ plane given by

$Z_{\Delta,\mathsf{s}}=\prod_{\mathcal{R}_{\Delta,\mathsf{s}}}\prod_{(\lambda_{L},\lambda_{R})}\prod_{N\in\mathbb{Z}}\left(\absolutevalue{N}-\lambda_{L}\mathsf{h}_{L}+\lambda_{R}\mathsf{h}_{R}\right)^{-1/4}\left(\absolutevalue{N}+\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R}\right)^{-1/4}\leavevmode\nobreak\
.$ (5.16)

Following (3.31), we next implement a Schwinger parameterization of the
logarithm of this expression which reads

$\log(Z_{\Delta,\mathsf{s}})=\frac{1}{4}\int_{\times}^{\infty}\frac{\text{d}\alpha}{\alpha}\frac{\cosh\left(\frac{\alpha}{2}\right)}{\sinh\left(\frac{\alpha}{2}\right)}\sum_{\mathcal{R}_{\Delta,\mathsf{s}}}\sum_{(\lambda_{L},\lambda_{R})}\left(e^{\alpha(\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R})}+e^{-\alpha(\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R})}\right)\leavevmode\nobreak\
,$ (5.17)

where we have performed the sum over $N$ as in (3.36). We will regulate the
$\alpha\rightarrow 0$ divergence of the above expression by combining the two
terms in the bracket into a single contour integral regulated about the origin
by an $i\varepsilon$ prescription. To ensure convergence of the representation
traces, a separate $i\varepsilon$ prescription must be given to the highest
and lowest-weight representations appearing in
$\mathcal{R}_{\Delta,\mathsf{s}}$:

$\log(Z_{\Delta,\mathsf{s}})=\frac{1}{4}\sum_{\mathcal{R}_{\Delta,\mathsf{s}}}\int_{-\infty\pm
i\varepsilon}^{\infty\pm
i\varepsilon}\frac{\text{d}\alpha}{\alpha}\frac{\cosh\left(\frac{\alpha}{2}\right)}{\sinh\left(\frac{\alpha}{2}\right)}\sum_{(\lambda_{L},\lambda_{R})}e^{\alpha(\lambda_{L}\mathsf{h}_{L}-\lambda_{R}\mathsf{h}_{R})}\leavevmode\nobreak\
.$ (5.18)

with the (+/-) sign applying to representations in
$\mathcal{R}_{\Delta,\mathsf{s}}^{\scaleto{\text{LW/HW}}{4pt}}$, respectively.
Recognizing the sum over weights as a representation trace and redefining
$\alpha\rightarrow-i\alpha$ we then can write this suggestively as

$\log(Z_{\Delta,\mathsf{s}})=\frac{i}{4}\sum_{\mathcal{R}^{\scaleto{\text{LW}}{4pt}}_{\Delta,\mathsf{s}}}\int_{\mathcal{C}_{+}}\frac{\text{d}\alpha}{\alpha}\frac{\cos\left(\frac{\alpha}{2}\right)}{\sin\left(\frac{\alpha}{2}\right)}\Tr_{\mathsf{R}_{L}}\left(\mathcal{P}e^{\frac{\alpha}{2\pi}\oint_{\gamma_{\text{sp}}}a_{L}}\right)\Tr_{\mathsf{R}_{R}}\left(\mathcal{P}e^{-\frac{\alpha}{2\pi}\oint_{\gamma_{\text{sp}}}a_{R}}\right)\\\
+\frac{i}{4}\sum_{\mathcal{R}^{\scaleto{\text{HW}}{4pt}}_{\Delta,\mathsf{s}}}\int_{\mathcal{C}_{-}}\frac{\text{d}\alpha}{\alpha}\frac{\cos\left(\frac{\alpha}{2}\right)}{\sin\left(\frac{\alpha}{2}\right)}\Tr_{\mathsf{R}_{L}}\left(\mathcal{P}e^{\frac{\alpha}{2\pi}\oint_{\gamma_{\text{sp}}}a_{L}}\right)\Tr_{\mathsf{R}_{R}}\left(\mathcal{P}e^{-\frac{\alpha}{2\pi}\oint_{\gamma_{\text{sp}}}a_{R}}\right)\leavevmode\nobreak\
,$ (5.19)

where the contours $\mathcal{C}_{\pm}$ are depicted in Fig. 3 and applied
separately to the lowest and highest-weight representations appearing in
$\mathcal{R}_{\Delta,\mathsf{s}}$.

$\text{Re}(\alpha)$$\text{Im}(\alpha)$$\mathcal{C}_{+}$$\mathcal{C}_{-}$
Figure 3: The integration contours relevant for the Wilson spool in AdS3
gravity. Highest-weight representations are integrated against
$\mathcal{C}_{-}$ (in blue) while lowest-weight representations are integrated
against $\mathcal{C}_{+}$ (in red). The black crosses depict the poles of the
integration measure, $\frac{1}{\alpha}\frac{\cos\alpha/2}{\sin\alpha/2}$. Not
depicted are the poles of the representation traces which lie along the
imaginary $\alpha$ axis.

We have written $\log Z_{\Delta,\mathsf{s}}$ in this form to draw comparison
to the dS3 Wilson spool in Sec. 3.2. In the construction from that section,
all representations are integrated along both $\mathcal{C}_{\pm}$. This is
natural: as was argued in Sec. 3.1, the poles of the one-loop determinant lie
on states of finite-dimensional $\mathfrak{su}(2)$ representations which are
simultaneously highest and lowest-weight. In the present case,
$\mathcal{C}_{\pm}$ appear distinctly because highest and lowest-weight
representations are distinct for $\mathfrak{sl}(2,\mathbb{R})$.

With this comparison noted, we can express $\log Z_{\Delta,\mathsf{s}}$ purely
in terms of lowest-weight representations, which are more standard in the
AdS/CFT dictionary, by returning to (5.18) and recalling that every weight of
a highest-weight representation is the negative of a weight in a corresponding
lowest-weight representation. This allows to write $\log
Z_{\Delta,\mathsf{s}}$ in the form conjectured in the supplemental material of
[2]

$\log(Z_{\Delta,\mathsf{s}})=\frac{i}{4}\sum_{\mathcal{R}_{\Delta,\mathsf{s}}^{{\scaleto{\text{LW}}{4pt}}}}\int_{2\mathcal{C}_{+}}\frac{\text{d}\alpha}{\alpha}\frac{\cos\left(\frac{\alpha}{2}\right)}{\sin\left(\frac{\alpha}{2}\right)}\Tr_{\mathsf{R}_{L}}\left(\mathcal{P}e^{\frac{\alpha}{2\pi}\oint_{\gamma_{\text{sp}}}a_{L}}\right)\Tr_{\mathsf{R}_{R}}\left(\mathcal{P}e^{-\frac{\alpha}{2\pi}\oint_{\gamma_{\text{sp}}}a_{R}}\right).$
(5.20)

providing a principled derivation of that result. It was shown there that
utilizing the holonomies of the background connections corresponding to a
spinning BTZ black hole (see App. A.2 for the explicit form), the right-hand
side of (5.20) evaluates to the correct one-loop determinant of massive
spin-$\mathsf{s}$ fields on that background:

$\log
Z_{\Delta,\mathsf{s}}=-\sum_{\pm}\sum_{l,\bar{l}=0}^{\infty}\log\left(1-q^{\frac{\Delta\pm\mathsf{s}}{2}+l}\bar{q}^{\frac{\Delta\mp\mathsf{s}}{2}+\bar{l}}\right)\leavevmode\nobreak\
,\qquad q=e^{i\frac{2\pi}{\tau}}\leavevmode\nobreak\
,\qquad\bar{q}=e^{-i\frac{2\pi}{\bar{\tau}}}\leavevmode\nobreak\ .$ (5.21)

This agrees with the one-loop determinant of BTZ for spinning fields as
reported in, for example, [48, 49, 43, 50].

We finish this section by reconciling the spinning Wilson spool in AdS3
gravity with our main result (1.4), namely the lack of the $\mathsf{s}^{2}$
correction to the $\alpha$ integration measure. This lack stems from two
places in our derivation: (i) the absence of normalizable zero-modes and (ii)
the triviality of Condition II which leaves the product over the weight spaces
unrestricted. However it is interesting to note that, on-shell,
$\mathrm{Tr}_{\mathsf{R}_{L/R}}\mathcal{P}e^{\pm\frac{\alpha}{2\pi}\oint_{\gamma_{\text{sp}}}a_{L/R}}$
only have poles along the imaginary $\alpha$ axis [2]. Thus this correction
term is completely regular along the real $\alpha$ axis and integrates to
zero. We may then include it into the definition of the spinning Wilson spool
and state succinctly, for both signs of the cosmological constant,

$\log
Z_{\Delta,\mathsf{s}}=\frac{1}{4}\mathbb{W}_{j_{L},j_{R}}[a_{L},a_{R}]\leavevmode\nobreak\
,$ (5.22)

with $\mathbb{W}_{j_{L},j_{R}}$ appearing in (1.4) and the representations
$\mathsf{R}_{j_{L,R}}$, the integration contour, and the background-
connections $a_{L,R}$ chosen appropriately.

##  6 Discussion

In this paper we extended the Wilson spool constructions of [1, 2] to
incorporate one-loop determinants of massive spinning fields in both Euclidean
dS3 and AdS3 backgrounds. This construction was based upon arranging the
quasinormal mode spectra of the respective backgrounds into the representation
theory of the isometry algebra. Along the way we codified two important
principles for evaluating one-loop determinants in a representation-theoretic
framework. These conditions lead naturally to a spinning Wilson spool
expression for the local path integral, $\log Z_{\Delta,\mathsf{s}}$, of a
massive spinning field, (1.4). This expression mimics the scalar Wilson spool:
it is an integral over gauge invariant Wilson loop operators with the integral
providing a mechanism for “wrapping” the loops around cycles of the base
manifold. The spinning Wilson spool only departs from the scalar expression in
its integration measure which cleanly captures the “edge contributions” of
[8]. We posited an off-shell expression for the spinning Wilson spool which
allows its insertion into the Chern-Simons path integral. In the context of a
Euclidean dS3 background, exact methods in Chern-Simons theory provide
efficient methods for evaluating quantum gravity corrections to $\log
Z_{\Delta,\mathsf{s}}$. We discussed two renormalization conditions in which
these quantum corrections can be interpreted as renormalizing either the
particle mass or Newton’s constant. This provides concrete predictions for
testing the correspondence between Chern-Simons theory and three-dimensional
quantum gravity.

There are multiple comments in order about our results, as well as future
directions to which the spinning Wilson spool may prove useful. We briefly
discuss these below.

### Group theoretic perspective on one-loop determinants

One important result of Sec. 3.1 is the restating of the quasinormal mode
method in a manner directly utilizing aspects of the representation theory and
isometries of the background. At the core of that section are Conditions I &
II which precisely isolate the representations and the states of a
representation that contribute poles to massive one-loop determinants. We
regard these conditions to be a significant new asset to the evaluation of
one-loop determinants. For one, Conditions I & II will play an important role
in establishing the Wilson spool for matter on Euclidean saddles beyond
$S^{3}$, which is ongoing work [51]. For instance, our preliminary findings
indicate that these two conditions correctly predict the placement and the
degeneracies of the poles of scalar one-loop determinants on Lens spaces. This
gives us strong credence that a Wilson spool built from Conditions I & II will
accurately describe the physics of matter on those backgrounds.

More broadly, we have taken care to state the content of these conditions in a
manner not specific to spacetime dimension or sign of cosmological constant.
Even outside the context of 3d quantum gravity, it is our expectation that
Conditions I & II will provide a powerful asset to organizing quasinormal
modes and evaluating one-loop determinants for any manifold which possesses a
transitive group action, regardless of dimension.

### On the role of finite dimensional $\mathfrak{su}(2)$ representations

In Sec. 2 we took great care to build $\mathfrak{su}(2)$ representations
corresponding to massive spinning particle states; these representations are
infinite dimensional and non-standard. The reader might then find it
surprising that finite dimensional representations played such important role
in Sec. 3.1 (particularly in the implementation of Condition II). It is worth
disentangling the roles of these two separate series of representations.

We regard the mass-shell condition, (3.12), as always providing a link between
representation theory and the particle mass. For physical values of the mass,
these representations are generically non-standard. The implementation of
Condition II is a separate statement about the analytic structure of
$Z^{2}_{\text{STT}}$, namely it encounters poles on finite dimensional
representations of $\mathfrak{su}(2)$. These however do not (necessarily) lie
on physical values of the mass. In evaluating $\mathbb{W}_{j_{L},j_{R}}$ our
goal is not to land on a pole; our goal is to evaluate a one-loop determinant
for a physical field. As such the representations appearing in (1.4) should be
appropriate for a physical value of $\Delta$ and are, again, generically non-
standard. Obtaining the right hand side of (3.41) is a non-trivial
verification of this fact.

We can further illustrate this distinction by investigating what happens if we
insert finite dimensional representations $j_{L/R}\in\frac{1}{2}\mathbb{Z}$
directly into the Wilson spool (say, for the classical $S^{3}$ background).
For the sake of this illustration we will take a non-spinning case
$j_{L}=j_{R}=j$:

$\left.\mathbb{W}_{j}\right|_{\text{finite dim
rep}}=\frac{i}{2}\int_{\mathcal{C}}\frac{\differential\alpha}{\alpha}\frac{\cos\left(\frac{\alpha}{2}\right)}{\sin\left(\frac{\alpha}{2}\right)}\frac{\sin^{2}\left((2j+1)\frac{\alpha}{2}\right)}{\sin^{2}\left(\frac{\alpha}{2}\right)}\leavevmode\nobreak\
,$ (6.1)

where we have inserted directly the finite dimensional $\mathfrak{su}(2)$
characters $\chi_{j}(z)=\frac{\sin\left(\pi(2j+1)z\right)}{\sin(\pi z)}$ and
utilized the holonomies around $\gamma_{\text{hor.}}$, (3.18). However it is
clear what will happen if we wrap $\mathcal{C}$ as in Fig. 2: the poles at
$\alpha\in 2\pi\mathbb{Z}_{\neq 0}$ are now only first order and now without
any exponential damping. We thus find

$\left.\mathbb{W}_{j}\right|_{\text{finite dim
rep}}=\sum_{n=1}^{\infty}\frac{(2j+1)^{2}}{n}=(2j+1)^{2}\zeta(1)=\infty\leavevmode\nobreak\
,$ (6.2)

where $\zeta$ is the Riemann zeta function. This illustrates the importance of
utilizing the non-standard representations (corresponding to physical masses)
in $\mathbb{W}$.

### Massive versus massless spinning fields

It is worth commenting on the important distinction between describing massive
spinning fields (the focus of this work) versus massless (higher) spin fields
in the Chern-Simons formalism. Our comments are very much motivated by viewing
Chern-Simons gravity as an effective field theory. This effective field theory
provides a description of the physics below some mass gap; indeed we can
expect that massive degrees of freedom can be “integrated” out, leaving behind
an effective response on the remaining low-energy degrees of freedom. This is
precisely what the spinning Wilson spool encapsulates: the response of massive
degrees of freedom directly in variables natural to the Chern-Simons path
integral.

We contrast this with massless degrees of freedom which cannot be integrated
out. Instead their presence must alter the low-energy effective field theory.
This is an indication that the one-loop determinants of massless higher-spin
fields do not have a Wilson spool description. Instead, we already know they
are described by modifying the effective field theory to a theory of higher-
spin gravity. For example, one simple way of describing massless higher-spin
fields up to spin-$N$ is by replacing $SL(2,\mathbb{R})\rightarrow
SL(N,\mathbb{R})$ in asymptotically AdS3 spacetimes [52, 53, 54] and
$SU(2)\rightarrow SU(N)$ in asymptotically Euclidean dS3 spacetimes [55, 8].

These above comments lead naturally to a line of inquiry: “How does one couple
massive matter to a theory of higher-spin gravity?” The relations between
Wilson lines and particle worldlines in higher-spin gravity have been
explored, e.g., in [56, 57]. However we believe the Wilson spool (generalized
to, e.g., $SU(N)$ or $SL(N,\mathbb{R})$) will be an invaluable tool for a more
complete answer to this important open question.

### Generalized symmetries in 3d gravity

Let us offer a final, speculative, remark on the insights that the Wilson
spool can possibly lend to our understanding of gravity as an effective field
theory. A modern perspective on organizing low-energy physics is the extension
of the Landau paradigm of broken symmetries to higher-form and non-invertible
(what we will collectively call, generalized) symmetries [58, 59]. Attempts to
categorize gravity as a low-energy phase through this lens include [60, 61,
62]. An important conjecture along these lines is the wide-held belief that
all global symmetries are either gauged or explicitly broken in a UV theory of
quantum gravity [63] and these statements extend to generalized symmetries.

Chern-Simons theory provides a natural area for exploring these discussions
applied to three-dimensional gravity: the Wilson line operators of Chern-
Simons theory both generate and are mutually charged under a generalized
symmetry.181818We are being imprecise about the distinction between
topological operators and topological defects here. Because we often have in
mind Euclidean signature, we will call an operator anything that can be
inserted into the path integral to obtain an expectation value. It naturally
follows that the Wilson spool is also charged under this generalized symmetry
and its non-zero expectation value is a direct sign of the explicit breaking
of this symmetry through the inclusion of matter. While the spectrum of
operators in Chern-Simons theory might be too large, we note the special role
non-standard representation theory plays in this statement: as noted above,
the Wilson spool associated to a finite-dimensional $\mathfrak{su}(2)$
representation diverges even classically, (6.2). It would be very interesting
to make more precise the relation between the Wilson spool and the generalized
symmetries of Chern-Simons theory as well as what these relations lend to the
question, “What is 3d gravity?”191919We thank Nabil Iqbal for extended
discussions in this direction.

## Acknowledgements

We thank Eleanor Harris, Eric D’Hoker, Nabil Iqbal, Per Kraus, Albert Law,
Victoria Martin, and Rahul Poddar. This work has been partially supported by
STFC consolidated grants ST/T000694/1 and ST/X000664/1. JRF is additionally
supported by Simons Foundation Award number 620869. JRF thanks the University
of Amsterdam for its hospitality. We would like to thank the Isaac Newton
Institute for Mathematical Sciences, Cambridge, for support and hospitality
during the programme “Black holes: bridges between number theory and
holographic quantum information” where work on this paper was undertaken. This
work was supported by EPSRC grant no EP/R014604/1. This research was supported
in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical
Physics (KITP).

##  Appendix A Chern-Simons gravity dictionary

###  A.1 Euclidean dS3

Three-dimensional de Sitter spacetime is a maximally symmetric spacetime with
two-sphere spatial slices expanding into future and past infinity. Due to this
expansion, inertial observers have access to a finite portion of spacetime
called the static patch. A coordinate patch covering one-half of this patch is
given by $(t,\rho,\varphi)$ with $\rho\in[0,\frac{\pi}{2})$,
$t\in(-\infty,\infty)$, and $\varphi\in[0,2\pi)$ with $\varphi$ a periodic
coordinate. The observer’s origin and causal horizon lie at $\rho=0$ and
$\rho=\frac{\pi}{2}$, respectively. The corresponding metric of the static
patch is

$\frac{\differential s^{2}}{\ell_{\text{dS}}^{2}}=-\cos^{2}\rho\,\differential
t^{2}+\differential\rho^{2}+\sin^{2}\rho\,\differential\varphi^{2}\leavevmode\nobreak\
.$ (A.1)

Above, the “de Sitter radius,” $\ell_{\text{dS}}$, sets the length scale for
this maximally symmetric spacetime. For the purpose of moving between the
following Chern-Simons description and the metric description, as well for
computing one-loop determinants, it will be useful to go to Euclidean
signature through the Wick-rotation, $t=-i\tau$. Under this rotation the
static-patch rotates to a three-sphere in “torus coordinates”

$\frac{\differential
s^{2}}{\ell_{\text{dS}}^{2}}=\cos^{2}\rho\,\differential\tau^{2}+\differential\rho^{2}+\sin^{2}\rho\,\differential\varphi^{2}\leavevmode\nobreak\
.$ (A.2)

Absence of a conical singularity at the horizon, $\rho=\frac{\pi}{2}$, sets
$\tau\sim\tau+2\pi$. The isometry group of this Euclidean space is
$SU(2)_{L}\times SU(2)_{R}/\mathbb{Z}_{2}$ with the $L/R$ denoting left/right
group action. We will denote the generators of these two actions as
$\\{L_{a}\\}_{a=1,2,3}$ and $\\{\bar{L}_{a}\\}_{a=1,2,3}$, respectively.

In accordance with the split structure of this isometry group, we will
describe Euclidean dS3 gravity with a pair of $SU(2)$ Chern-Simons theories

$S=k_{L}\,S_{\text{CS}}[A_{L}]+k_{R}\,S_{\text{CS}}[A_{R}]\leavevmode\nobreak\
,$ (A.3)

with

$S_{\text{CS}}[A]=\frac{1}{4\pi}\mathrm{Tr}\int\left(A\wedge\differential
A+\frac{2}{3}A\wedge A\wedge A\right)\leavevmode\nobreak\ ,$ (A.4)

and the trace taken in the fundamental representation. The basic ingredients
of the dictionary between this Chern-Simons description and the more familiar
“metric description” is given by relating the gauge connections, $A_{L/R}$, to
a dreibein, $e^{a}$, and (dual) spin-connection,
$\omega^{a}=\frac{1}{2}\epsilon^{abc}\omega_{bc}$, via

$A_{L}=i\left(\omega^{a}+\frac{1}{\ell_{\text{dS}}}e^{a}\right)L_{a}\leavevmode\nobreak\
,\qquad\qquad
A_{R}=i\left(\omega^{a}-\frac{1}{\ell_{\text{dS}}}e^{a}\right)\bar{L}_{a}\leavevmode\nobreak\
.$ (A.5)

The levels, $k_{L/R}$, are written as

$k_{L}=\delta+is\leavevmode\nobreak\ ,\qquad k_{R}=\delta-
is\leavevmode\nobreak\ .$ (A.6)

We can rewrite the action (A.3) as

$iS=-I_{\text{EH}}-i\delta I_{\text{GCS}}\leavevmode\nobreak\ ,$ (A.7)

with

$I_{\text{EH}}=-\frac{s}{4\pi\ell_{\text{dS}}}\int\epsilon_{abc}e^{a}\wedge\left(R^{bc}-\frac{1}{3\ell_{\text{dS}}^{2}}e^{b}\wedge
e^{c}\right)$ (A.8)

the Euclidean Einstein-Hilbert term in the first-order formalism202020Here
$R^{ab}$ is the Riemann two-form. with positive cosmological constant,
$\Lambda=\ell_{\text{dS}}^{-2}$. Thus $s\equiv\frac{\ell_{\text{dS}}}{4G_{N}}$
is the (inverse) gravitational coupling; in this paper we will find it
convenient to keep it written as $s$ and keep in mind the classical limit is
$s\rightarrow\infty$. $I_{\text{GCS}}$ is the gravitational Chern-Simons term

$I_{\text{GCS}}=\frac{1}{2\pi}\mathrm{Tr}\int\left(\omega\wedge\differential\omega+\frac{2}{3}\omega\wedge\omega\wedge\omega\right)+\frac{1}{2\pi\ell_{\text{dS}}^{2}}\mathrm{Tr}\int
e\wedge T\leavevmode\nobreak\ ,$ (A.9)

with $T$ the torsion two-form. Under quantization the levels will undergo a
finite renormalization $k_{L/R}\rightarrow r_{L/R}\equiv k_{L/R}+2$ [64]
amounting to a renormalization of the $I_{\text{GCS}}$ coupling,
$\delta\rightarrow\hat{\delta}=\delta+2$. For the rest of this paper we will
always work directly with the renormalized levels.

Appropriate classical flat background connections, $a_{L/R}$, describing the
$S^{3}$ are given by

$\displaystyle a_{L}=$ $\displaystyle
iL_{1}\,\differential\rho+i\left(\sin\rho L_{2}-\cos\rho
L_{3}\right)(\differential\varphi-\differential\tau)\leavevmode\nobreak\ ,$
$\displaystyle a_{R}=$
$\displaystyle-i\bar{L}_{1}\,\differential\rho-i(\sin\rho\bar{L}_{2}+\cos\rho\bar{L}_{3})(\differential\varphi+\differential\tau)\leavevmode\nobreak\
.$ (A.10)

An important aspect of the above connections is that they each possess ring
singularities at $\rho=0$ and $\rho=\frac{\pi}{2}$ where the $\varphi$ and
$\tau$ coordinates degenerate, respectively. These Wick-rotate to the
worldline at the static patch origin and to the causal horizon bifurcation
surface, respectively. There is potential for holonomy around these
singularities which we will write generically as

$\mathcal{P}\exp\oint_{\gamma}a_{L}=g_{\rho}^{-1}e^{i2\pi
L_{3}\mathsf{h}_{L}}g_{\rho}\leavevmode\nobreak\
,\qquad\mathcal{P}\exp\oint_{\gamma}a_{R}=\bar{g}_{\rho}\,e^{i2\pi\bar{L}_{3}\mathsf{h}_{R}}\bar{g}_{\rho}^{-1}\leavevmode\nobreak\
,$ (A.11)

for periodic group elements, $g_{\rho}=e^{iL_{1}\rho}$ and
$\bar{g}_{\rho}=e^{i\bar{L}_{1}\rho}$, and holonomies, $\mathsf{h}_{L/R}$.
Given the solution (A.1), it is easy to deduce that for cycles,
$\gamma_{\text{orig}}$, wrapping the static-patch origin at $\rho=0$

$\gamma_{\text{orig}}\colon\qquad\mathsf{h}_{L}=1\leavevmode\nobreak\
,\qquad\mathsf{h}_{R}=1\leavevmode\nobreak\ ,$ (A.12)

while for cycles, $\gamma_{\text{hor}}$, wrapping the causal horizon at
$\rho=\frac{\pi}{2}$,

$\gamma_{\text{hor}}\colon\qquad\mathsf{h}_{L}=1\leavevmode\nobreak\
,\qquad\mathsf{h}_{R}=-1\leavevmode\nobreak\ .$ (A.13)

Lastly we point out that these singularities give delta function sources of
curvature212121Importantly the metric geometry remains smooth everywhere. that
is important for reproducing the on-shell action

$ir_{L}S_{\text{CS}}[a_{L}]+ir_{R}S_{\text{CS}}[a_{R}]=\frac{\pi\ell_{\text{dS}}}{2G_{N}}\leavevmode\nobreak\
,$ (A.14)

which is the tree-level de Sitter entropy.

###  A.2 Lorentzian AdS3

For Anti-de Sitter space, we will start directly in Lorentzian signature where
the relevant isometry group exhibits a similar split,
$SL(2,\mathbb{R})_{L}\times SL(2,\mathbb{R})_{R}$. Accordingly gravity is
described by two Chern-Simons theories, this time with opposite levels:

$S=kS_{\text{CS}}[A_{L}]-kS_{\text{CS}}[A_{R}]=S_{\text{EH}}\leavevmode\nobreak\
,\qquad k=\frac{\ell_{\text{AdS}}}{4G_{N}}\leavevmode\nobreak\ ,$ (A.15)

where $S_{\text{EH}}$ is the Lorentzian Einstein-Hilbert action and
$\ell_{\text{AdS}}$ is the AdS radius. This matching is facilitated by writing
the connections as

$A_{L}=\left(\omega^{a}+\frac{1}{\ell_{\text{AdS}}}e^{a}\right)L_{a}\leavevmode\nobreak\
,\qquad\qquad
A_{R}=\left(\omega^{a}-\frac{1}{\ell_{\text{AdS}}}e^{a}\right)\bar{L}_{a}\leavevmode\nobreak\
,$ (A.16)

where now $\\{L_{a}\\}_{a=0,\pm}$ and $\\{\bar{L}_{a}\\}_{a=0,\pm}$ generate
the Lie algebras $\mathfrak{sl}(2,\mathbb{R})_{L}$ and
$\mathfrak{sl}(2,\mathbb{R})_{R}$, respectively. We will follow the
conventions set in [45, 46] for the $\mathfrak{sl}(2,\mathbb{R})$ algebra.

The general asymptotically AdS3 vacuum solutions are Bañados geometries [65]
described by background connections

$\displaystyle a_{L}=$ $\displaystyle
L_{0}\,\differential\eta+\left(e^{\eta}\,L_{+}-e^{-\eta}\frac{2\pi\mathcal{L}}{k}\,L_{-}\right)\differential
x^{+}\leavevmode\nobreak\ ,$ $\displaystyle a_{R}=$
$\displaystyle-\bar{L}_{0}\,\differential\eta-\left(e^{\eta}\,\bar{L}_{-}-e^{-\eta}\frac{2\pi\bar{\mathcal{L}}}{k}\,\bar{L}_{+}\right)\differential
x^{-}\leavevmode\nobreak\ ,$ (A.17)

where $\mathcal{L}(x^{+})$ and $\bar{\mathcal{L}}(x^{-})$ are arbitrary
functions. Here $\eta$ is a radial coordinate with the conformal boudary at
$\eta\rightarrow\infty$ and $x^{\pm}=t\pm\phi$ are the lightcone coordinates
of the boundary cylinder. A special case of interest in this paper are the
rotating BTZ black hole geometries described by

$\mathcal{L}=\frac{M\,\ell_{\text{AdS}}+J}{4\pi}\leavevmode\nobreak\
,\qquad\qquad\bar{\mathcal{L}}=\frac{M\,\ell_{\text{AdS}}-J}{4\pi}\leavevmode\nobreak\
,$ (A.18)

where $M$ and $J$ are the mass and spin of the black hole, respectively [42,
41].

In Euclidean signature, $t\rightarrow it_{E}$, smoothness of the horizon
requires that Euclidean time is compact and so the geometry is the interior of
a complex torus

$(w,\bar{w})\sim(w+2\pi m+2\pi n\tau,\bar{w}+2\pi
m+2\pi\bar{\tau})\leavevmode\nobreak\ ,\qquad
m,n\in\mathbb{Z}\leavevmode\nobreak\ ,$ (A.19)

where $w=\phi+it_{E}$ and $\bar{w}=\phi-it_{E}$ and with modular parameter

$\tau=\frac{i}{2}\sqrt{\frac{k}{2\pi\mathcal{L}}}\leavevmode\nobreak\
,\qquad\qquad\bar{\tau}=-\frac{i}{2}\sqrt{\frac{k}{2\pi\bar{\mathcal{L}}}}\leavevmode\nobreak\
.$ (A.20)

Around the two cycles of this torus the connections have holonomy

$\mathcal{P}\exp\oint_{\gamma}a_{L}=u_{L}^{-1}\,e^{i2\pi
L_{0}\mathsf{h}_{L}}\,u_{L}\leavevmode\nobreak\
,\qquad\qquad\mathcal{P}\exp\oint_{\gamma}a_{R}=u_{R}\,e^{i2\pi\bar{L}_{0}\mathsf{h}_{R}}\,u_{R}^{-1}\leavevmode\nobreak\
,$ (A.21)

where $u_{L/R}$ are periodic along both cycles (only depending on $\eta$). For
the contractible, thermal, cycle
$\gamma_{\text{th}}:(w,\bar{w})\rightarrow(w+\tau,\bar{w}+\bar{\tau})$ the
connections have holonomy

$\gamma_{\text{th}}:\qquad\mathsf{h}_{L}=-1\leavevmode\nobreak\
,\qquad\mathsf{h}_{R}=-1\leavevmode\nobreak\ ,$ (A.22)

while for the non-contractible spatial cycle,
$\gamma_{\text{sp}}:(w,\bar{w})\rightarrow(w+2\pi,\bar{w}+2\pi)$, they have
holonomy

$\gamma_{\text{sp}}:\qquad\mathsf{h}_{L}=-\frac{1}{\tau}\leavevmode\nobreak\
,\qquad\mathsf{h}_{R}=-\frac{1}{\bar{\tau}}\leavevmode\nobreak\ .$ (A.23)

##  Appendix B DHS for massive spinning fields on $S^{3}$

In this appendix, we will review DHS [29] as it was originally intended:
first, determine the quasinormal modes from the equations of motion of a
massive spinning field, Wick-rotating them carefully to Euclidean space, and
then evaluate the one-loop as a product of the poles.

Special care is needed when implementing DHS for massive spinning fields. The
quasinormal modes of massive fields on de Sitter are found directly from the
equations of motion of the field in the manner of [66]. However, as shown in
[67], for spin $\mathsf{s}>0$ some care is required in Wick-rotating de Sitter
quasinormal modes into regular modes on $S^{3}$, that is symmetric,
transverse, traceless spherical harmonics. In particular, imposing the naïve
fall-off condition on the frequency $\omega$ is not sufficient. We will review
carefully and explicitly this regularity condition.

We will then construct the one-loop determinant from the quasinormal mode
frequencies of a spin-$\mathsf{s}$ field on $S^{3}$ using the DHS method and
regularise it via zeta function regularisation, as in the case for a scalar
field in [29]. Identities of special functions are then utilised to rewrite
these expressions in the form of polylogarithms to make explicit the
comparison to the one-loop determinants derived in [8] and Sec. 3.3.

The metric for the de Sitter static patch in “torus coordinates” is given by
(A.1). For the sake of finding the quasinormal modes of this background, we
will also make use of the radial coordinate

$r=\sin\rho\leavevmode\nobreak\ .$ (B.1)

###  B.1 Real massive scalar field

To introduce notation and the basic ingredients needed in subsequent sections,
we will start by deriving the quasinormal modes of a real scalar field in dS3
and then review the derivation of the one-loop determinant as done originally
in DHS [29]. This will set up the procedure for massive spinning fields in the
next subsections.

#### Quasinormal modes

First, we construct the solutions to the equations of motion for a scalar
field $\Phi$ on dS3

$\nabla^{2}\Phi=\ell_{\text{dS}}^{2}m^{2}\Phi\leavevmode\nobreak\ ,$ (B.2)

where $\nabla^{2}$ is the Laplacian on (A.1) and as in the main text
$\ell_{\text{dS}}^{2}m^{2}=\Delta(2-\Delta)$. Suitable boundary conditions
must also be imposed on the solutions to ensure that they Wick-rotate to a
regular solution on $S^{3}$. These are:

  * •

The solution must be single-valued in $\varphi$.

  * •

The solution must be single-valued in imaginary time.

  * •

The solution must be regular at the origin of the static patch $(r=0)$.

  * •

The solution must be regular and purely in-going or out-going at the horizon
$(r=1)$.

Proceeding via separation of variables we encounter a second order ODE for
$r$, resulting in two solutions. Imposing regularity at the origin removes one
of these solutions leaving us with the modes

$\Phi=e^{-i\omega
t+il\varphi}\left(r^{2}\right)^{|\frac{l}{2}|}\left(1-r^{2}\right)^{-\frac{i\omega}{2}}{}_{2}F_{1}\left(a,b;c;r^{2}\right)\leavevmode\nobreak\
,$ (B.3)

where

$a=\frac{\Delta+|l|-i\omega}{2}\leavevmode\nobreak\ ,\qquad
b=\frac{\bar{\Delta}+|l|-i\omega}{2}\leavevmode\nobreak\ ,\qquad
c=1+|l|\leavevmode\nobreak\ ,$ (B.4)

and $l,i\omega\in\mathbb{Z}$ are constants constrained to the integers by the
single-valuedness of $\Phi$ around $\varphi$ and imaginary time respectively.

To consider regularity at the horizon we will expand the solution to lowest
order as $\frac{\varepsilon^{2}}{2}=1-r^{2}$ approaches zero. Taking care as
we are in a non-generic case for the hypergeometric function when
$c-a-b=i\omega\in\mathbb{Z}$, the connection formulae for the hypergeometric
function about $r^{2}$ and $1-r^{2}$ [68] imply that a purely ingoing or
outgoing solution will require at least one of
$a,b,(c-a),(c-b)\in\mathbb{Z}_{\leq 0}$. Noting that $a\to b$ simply takes
$\Delta\to\bar{\Delta}$ which has no effect on the form of the mode $\Phi$, we
need consider only two (not necessarily distinct) cases:

Case 1:

$a=-n\in\mathbb{Z}_{\leq 0}\implies\Delta=-2n-|l|+i\omega$.

Case 2:

$c-b=-n\in\mathbb{Z}_{\leq 0}\implies\Delta=-2n-|l|-i\omega$.

In both cases, the hypergeometric function takes the form of a Jacobi
polynomial in terms of Pochhammer symbols (for the rising factorial). For Case
1, $a=-n$ gives

${}_{2}F_{1}\left(-n,b;c;1-\frac{\varepsilon^{2}}{2}\right)=\sum_{d=0}^{n}\left(\frac{\varepsilon^{2}}{2}\right)^{d}\frac{(b)_{d}(c-b)_{n-d}}{(c)_{n}}\leavevmode\nobreak\
,$ (B.5)

and for Case 2 we make use of the identity
${}_{2}F_{1}\left(a,b;c;u\right)=(1-u)^{c-a-b}{}_{2}F_{1}\left(c-b,c-a;c;u\right)$.
Substituting for $a,b,c$, we then find that to lowest order in $\varepsilon$
the following:

Case 1

: $\Delta=-2n-|l|+i\omega\leavevmode\nobreak\ ,\qquad n\in\mathbb{Z}_{\leq
0},i\omega\in\mathbb{Z}\leavevmode\nobreak\ ,$

$\Phi\sim\begin{cases}e^{-i\omega t}\varepsilon^{+i\omega}\qquad\text{(anti-
quasinormal)}&0\leq i\omega\leq n\leavevmode\nobreak\ ,\\\ e^{-i\omega
t}\varepsilon^{-i\omega}\qquad\text{(quasinormal)}&\text{else}\leavevmode\nobreak\
,\end{cases}$ (B.6)

Case 2

: $\Delta=-2n-|l|-i\omega\leavevmode\nobreak\ ,\qquad n\in\mathbb{Z}_{\leq
0},i\omega\in\mathbb{Z}\leavevmode\nobreak\ ,$

$\Phi\sim\begin{cases}e^{-i\omega
t}\varepsilon^{-i\omega}\qquad\text{(quasinormal)}&0\leq-i\omega\leq
n\leavevmode\nobreak\ ,\\\ e^{-i\omega
t}\varepsilon^{+i\omega}\qquad\text{(anti-
quasinormal)}&\text{else}\leavevmode\nobreak\ .\end{cases}$ (B.7)

For quasinormal modes regularity demands $i\omega\leq 0$, for anti-quasinormal
modes regularity demands $i\omega\geq 0$. Note that for $i\omega=0$ the modes
are equivalent.

For $i\omega\geq 0$ Case 2 is a subset of Case 1 and the possible regular
anti-quasinormal modes are

$\displaystyle\Delta$
$\displaystyle=-2\tilde{n}-|l|-i\omega\leavevmode\nobreak\ ,$
$\displaystyle\qquad
n-i\omega\equiv\tilde{n}\in\mathbb{N}_{0}\leavevmode\nobreak\ .$ (B.8)

For $i\omega<0$ Case 1 is a subset of Case 2 and the possible regular
quasinormal modes are

$\displaystyle\Delta$
$\displaystyle=-2\tilde{n}-|l|+i\omega\leavevmode\nobreak\ ,$
$\displaystyle\qquad
n+i\omega\equiv\tilde{n}\in\mathbb{N}_{0}\leavevmode\nobreak\ .$ (B.9)

These two expressions can be combined by allowing $i\omega\equiv
k\in\mathbb{Z}$ to range over all integers (and relabelling $\tilde{n}\to n$
for convenience). The Euclidean regular (anti-)quasinormal modes take the
values

$\Delta=-2n-|l|-|k|\leavevmode\nobreak\ .$ (B.10)

Note that the modes with $\Delta\leftrightarrow\bar{\Delta}$ are equivalent
due to the ambiguity in defining $\Delta$ from the mass. This was seen above
by the invariance of $\Phi$ to the exchange, and we have hence counted these
modes only once. In using the DHS method we consider the one-loop determinant
as a meromorphic function of $\Delta$. Since the determinant also cannot
distinguish between $\Delta\leftrightarrow\bar{\Delta}$ there must be a pole
at both $\Delta$ and $\bar{\Delta}$. The poles in the one-loop determinant
thus appear at

$\Delta,\bar{\Delta}=-2n-|l|-|k|\leavevmode\nobreak\ .$ (B.11)

It can also be confirmed that the modes we have obtained are (expectedly) the
scalar spherical harmonics on $S^{3}$. More simply the degeneracy of each
value of $\Delta=-p,p\in\mathbb{Z}_{\geq 0}$ can be counted as $(p+1)^{2}$
matching the degeneracy of the spherical harmonics fitting into
representations of $\mathfrak{so}(4)$.

#### One-loop determinant

Following [29] the one-loop determinant can be constructed from the location
of the quasinormal frequencies, treating the functional determinant as a
meromorphic function of $\Delta$. For convenience in this section the notation
of [29] is also followed such that $\Delta\equiv\Delta_{+}$, and
$\bar{\Delta}\equiv\Delta_{-}$.

Up to a holomorphic factor, which in this case is trivial, the partition
function is simply a product of poles at the (anti-)quasinormal mode
frequencies:

$Z_{\text{scalar}}=\prod_{\begin{subarray}{c}\pm,n\in\mathbb{N}\\\
l,k\in\mathbb{Z}\end{subarray}}\left(|k|+2n+|l|+\Delta_{\pm}\right)^{-\frac{1}{2}}\leavevmode\nobreak\
.$ (B.12)

Noting that $|k|+|l|+2n=p$ has degeneracy $(p+1)^{2}$, we can rewrite (B.12)
as

$\displaystyle\begin{split}\log(Z_{\text{scalar}})&=-\frac{1}{2}\sum_{\pm,p\in\mathbb{N}}(p+1)^{2}\log\left(p+\Delta_{\pm}\right)\\\
&=\frac{1}{2}\sum_{\pm}\zeta^{\prime}(-2,1\pm i\mu)\mp
2i\mu\zeta^{\prime}(-1,1\pm i\mu)-\mu^{2}\zeta^{\prime}(0,1\pm
i\mu)\leavevmode\nobreak\ ,\end{split}$ (B.13)

where $\Delta_{\pm}=1\pm i\mu$ for the principle series and $\zeta$ is the
Hurwitz zeta function. This is the formula derived in [29] and it will be
convenient to write in terms of polylogarithms. The two key the identities are

$\displaystyle\begin{split}&\zeta^{\prime}(-2,\pm i\mu)\mp
2i\mu\zeta^{\prime}(-1,\pm i\mu)-\mu^{2}\zeta^{\prime}(0,\pm i\mu)\\\
=&\zeta^{\prime}(-2,1\pm i\mu)\mp 2i\mu\zeta^{\prime}(-1,1\pm
i\mu)-\mu^{2}\zeta^{\prime}(0,1\pm i\mu)\leavevmode\nobreak\ ,\end{split}$
(B.14)

and222222The identity (B.15) can be derived by adapting the proof from [69] to
the region of the complex plane where $x\in i\mathbb{R}$.

$\zeta^{\prime}(-n,x)+(-1)^{n}\zeta^{\prime}(-n,1-x)=-\pi
i\frac{B_{n+1}(x)}{n+1}+e^{\frac{\pi
in}{2}}\frac{n!}{(2\pi)^{n}}\text{Li}_{n+1}(e^{-2\pi ix})\leavevmode\nobreak\
,$ (B.15)

where $B_{n}$ are the Bernoulli polynomials. Substituting these into (B.13) we
find

$\log(Z_{\text{scalar}})=\sum_{\pm}\left(-\frac{\mu^{2}}{4}\text{Li}_{1}(e^{\pm
2\pi\mu})\pm\frac{\mu}{4\pi}\text{Li}_{2}(e^{\pm
2\pi\mu})-\frac{1}{8\pi^{2}}\text{Li}_{3}(e^{\pm
2\pi\mu})\right)\leavevmode\nobreak\ ,$ (B.16)

in agreement with [8].

###  B.2 Massive spin-1 field

Next we work out in detail the quasinormal modes of a massive spin-1 field on
dS3, and then its one-loop determinant via DHS. Many results follow from the
analysis of a scalar field. However, it is important to highlight the
subtleties of the regularity condition in this case, and how it reproduces the
appearance of the “edge” partition function [67].

#### Quasinormal modes

The equation of motion is

$\left(\nabla^{2}_{(1)}-\ell_{\text{dS}}^{2}m^{2}-2\right)\Phi_{\mu}=0\leavevmode\nobreak\
,$ (B.17)

accompanied by a transversality condition

$\nabla^{\mu}\Phi_{\mu}=0\leavevmode\nobreak\ .$ (B.18)

These equations of motion are solved in [50] for an AdS3 background using a
first-order formalism, we present here the corresponding results in
dS3.232323One method to derive these solutions is the analytic continuation of
$\ell_{\text{dS}}\to i\ell_{\text{AdS}}$ at which point the results of [50]
can be used directly.

As in the scalar case we discard one set of solutions to the radial part of
the PDE as non-regular at the origin, and impose $l,i\omega\in\mathbb{Z}$ to
ensure single-valuedness. The resulting expressions are

$\displaystyle\begin{split}\Phi_{L}&=e_{L}e^{-i\omega
t+il\varphi}\left({r^{2}}\right)^{|\frac{l}{2}|}\left(1-r^{2}\right)^{-\frac{i\omega}{2}}{}_{2}F_{1}\left(\frac{\bar{\Delta}+1+|l|-i\omega}{2},\frac{\Delta-1+|l|-i\omega}{2};1+|l|;r^{2}\right)\leavevmode\nobreak\
,\\\ \Phi_{R}&=-e_{R}e^{-i\omega
t+il\varphi}\left(r^{2}\right)^{|\frac{l}{2}|}\left(1-r^{2}\right)^{-\frac{i\omega}{2}}{}_{2}F_{1}\left(\frac{\bar{\Delta}-1+|l|-i\omega}{2},\frac{\Delta+1+|l|-i\omega}{2};1+|l|;r^{2}\right)\leavevmode\nobreak\
,\\\
\Phi_{r}&=-\frac{i}{(\Delta-1)}\frac{1}{r(1-r^{2})}\left[(\omega+il)\Phi_{L}+(\omega-
il)\Phi_{R}\right]\leavevmode\nobreak\ ,\end{split}$ (B.19)

where $\Phi_{L/R}$ are related to the components of the vector field via

$\displaystyle\begin{split}i\Phi_{\varphi}&=\Phi_{L}+\Phi_{R}\leavevmode\nobreak\
,\\\ \Phi_{t}&=\Phi_{L}-\Phi_{R}\leavevmode\nobreak\ .\end{split}$ (B.20)

In (B.19) $e_{L/R}$ are constants of integration and describe the polarisation
vector. These constants are further constrained by (B.17), (B.18) to satisfy
one of two conditions depending on the sign of $l$.

$\displaystyle\begin{split}(-\omega+il+i(\Delta-1))e_{R}&=(-\omega-
il+i(\Delta-1))e_{L}\,,\qquad l\leq 0\leavevmode\nobreak\ ,\\\
(-\omega+il-i(\Delta-1))e_{R}&=(-\omega-il-i(\Delta-1))e_{L}\,,\qquad l\geq
0\leavevmode\nobreak\ .\end{split}$ (B.21)

Next, we look for those modes which Wick-rotate into regular Euclidean modes
on $S^{3}$. In order for a mode to be single-valued and regular a necessary
condition is for $\Phi_{L},\Phi_{R}$ themselves to be single-valued and
regular at the horizon. Applying the results from the scalar case (B.10) to
(B.19) with $\Delta\to\Delta\pm 1$ implies that regular modes potentially
occur at

$\displaystyle\Phi_{L}:\qquad\Delta$ $\displaystyle=1-2N_{L}-|l|-|k|$ (B.22)
$\displaystyle\text{or }\bar{\Delta}$
$\displaystyle=1-2(N_{L}^{\prime}+1)-|l|-|k|$
$\displaystyle\Phi_{R}:\qquad\Delta$
$\displaystyle=1-2(N_{R}^{\prime}+1)-|l|-|k|$ (B.23) $\displaystyle\text{or
}\bar{\Delta}$ $\displaystyle=1-2N_{R}-|l|-|k|\leavevmode\nobreak\ .$

Unlike in the scalar case there is no symmetry of $\Delta\to\bar{\Delta}$ in
$\Phi_{\mu}$ and so the two cases result in distinct physical modes.

For $\Phi_{\mu}$ to have a mode at some value of $\Delta$ or $\bar{\Delta}$ it
is necessary for both $\Phi_{L}$ and $\Phi_{R}$ to support a mode. We will
suppose that $k,l$ are fixed and consider two242424There is also a single
exceptional case where $\Delta=\bar{\Delta}=1$ and we can choose a mode in
each component by taking $N_{L}=N_{R}=l=k=0$. This also automatically
satisfies the constraints of (B.21). It can be checked using the regularity
conditions below that this solution cannot be regular at both the origin and
horizon. families of solutions:

  * •

A mode with $N_{L/R}\geq 1$ in one of the two components and
$N_{R/L}^{\prime}=N_{L/R}-1\geq 0$ in the other. We denote these the ‘general
modes’ valid for all $N^{\prime}\geq 0$:

values | ranges
---|---
$\Delta=-1-2N^{\prime}-|l|-|k|$ | $k,l\in\mathbb{Z},N^{\prime}\in\mathbb{Z}_{\geq 0}$
$\bar{\Delta}=-1-2N^{\prime}-|l|-|k|$ | $k,l\in\mathbb{Z},N^{\prime}\in\mathbb{Z}_{\geq 0}$

  * •

If $N_{L/R}=0$ in one component it may be possible to still have a consistent
mode if the other component is identically $0$. For example setting $N_{L}=0$
in (B.22) it may be permissible from (B.21) to set $e_{R}=0$. A unique list
(accounting for overlaps where $k,l=0$) of these ‘special modes’ is

values | ranges
---|---
| $k<0,l\geq 0$
$\Delta=1-|l|-|k|$ | $k=0,l\in\mathbb{Z}$
| $k>0,l\leq 0$
| $k<0,l\leq 0$
$\bar{\Delta}=1-|l|-|k|$ | $k=0,l\in\mathbb{Z}\setminus\\{0\\}$
| $k>0,l\geq 0$

Since the regularity of $\Phi_{L},\Phi_{R}$ is only a necessary condition it
is still important to check the regularity of the Euclidean norm
$|\Phi|^{2}=\Phi_{\mu}\Phi^{\mu}$. This will determine regularity in a
coordinate-independent manner. Consider first regularity as $r\to 0$ at the
origin of the static patch. The only cases which can cause divergences are
those for which $l=0$. In this case, to lowest order in $r$ we find

$|\Phi|^{2}\sim\frac{1}{r^{2}}\left(\frac{|\omega|^{2}}{(\Delta-1)^{2}}+1\right)\left|e_{L}-e_{R}\right|^{2}\leavevmode\nobreak\
.$ (B.24)

The first bracket is always non-zero due to the reality of $\Delta$ and so
regularity implies $e_{L}=e_{R}$. For $l=0$ general modes this is always true
from (B.21). For the special modes one of $e_{L/R}$ is zero and the other is
non-zero and so these modes are irregular.

As in the scalar case, looking at the horizon we can expand in small
$\varepsilon$ for $r^{2}=1-\frac{\varepsilon^{2}}{2}$. Again using the
expansion (B.5) the only potentially problematic case is $k=0$, in which case
this expansion gives to leading order

$|\Phi|^{2}\sim\frac{1}{\varepsilon^{2}}\left(1+\frac{l^{2}}{(\Delta-1)^{2}}\right)\left|e_{L}\frac{(c_{L}-b_{L})_{n_{L}}}{(c_{L})_{n_{L}}}+e_{R}\frac{(c_{R}-b_{R})_{n_{R}}}{(c_{R})_{n_{R}}}\right|^{2}\leavevmode\nobreak\
,$ (B.25)

where again the constants $a=-n,b,c$ are defined as coefficients in the
hypergeometric function for each component. The general modes with $k=0$
satisfy

$\Delta=-1-2N^{\prime}-|l|\qquad\text{ or
}\qquad\bar{\Delta}=-1-2N^{\prime}-|l|\leavevmode\nobreak\ .$ (B.26)

Inserting the first of these into the coefficients of the hypergeometric
function and noting that from (B.21) it is possible to choose

$\displaystyle\begin{split}e_{L}&=+|l|-\Delta+1=2(N^{\prime}+1+|l|)\leavevmode\nobreak\
,\\\ e_{R}&=-|l|-\Delta+1=2(N^{\prime}+1)\leavevmode\nobreak\ ,\end{split}$
(B.27)

it therefore follows252525The identities $(-n)_{n}=(-1)^{n}n!$ and
$(1+|l|)_{N^{\prime}+1}=(1+|l|)_{N^{\prime}}(1+|l|+N^{\prime})$ are useful
here. that

$\displaystyle\begin{split}&e_{L}\frac{(c_{L}-b_{L})_{n_{L}}}{(c_{L})_{n_{L}}}+e_{R}\frac{(c_{R}-b_{R})_{n_{R}}}{(c_{R})_{n_{R}}}\\\
=&2(N^{\prime}+1+|l|)\frac{(-N^{\prime}-1)_{N^{\prime}+1}}{(1+|l|)_{N^{\prime}+1}}+2(N^{\prime}+1)\frac{(-N^{\prime})_{N^{\prime}}}{(1+|l|)_{N^{\prime}}}=0\leavevmode\nobreak\
.\end{split}$ (B.28)

The same process works in the $\bar{\Delta}$ case. All the general modes
therefore are regular.

In the special cases with $k=0$ and $\Delta,\bar{\Delta}=1-|l|$ either
$\Phi_{L}=0,\Phi_{R}\neq 0,n_{R}=0$ or $\Phi_{R}=0,\Phi_{L}\neq 0,n_{L}=0$.
Noting that $(x)_{0}\equiv 1$ the leading order term does not vanish and the
special modes with $k=0$ are not regular. Combining the results of regularity
at the origin and horizon the final list of regular Euclidean modes is

values | ranges
---|---
$\Delta=-1-2N^{\prime}-|l|-|k|$ | $k,l\in\mathbb{Z},N^{\prime}\in\mathbb{Z}_{\geq 0}$
$\bar{\Delta}=-1-2N^{\prime}-|l|-|k|$ | $k,l\in\mathbb{Z},N^{\prime}\in\mathbb{Z}_{\geq 0}$
| $k<0,l>0$
$\Delta=1-|l|-|k|$ |
| $k>0,l<0$
| $k<0,l<0$
$\bar{\Delta}=1-|l|-|k|$ |
| $k>0,l>0$

It can be confirmed that the degeneracy of the modes at $\Delta=-p,p\geq 1$ is
$p(p+2)$. This aligns with the expected degeneracies of transverse spherical
vector harmonics on $S^{3}$, which fit into representations of
$\mathfrak{so}(4)$.

As in the scalar case we note that while each of these modes is distinct, the
one-loop determinant has an ambiguity under
$\Delta\leftrightarrow\bar{\Delta}$ and so we must include a pole for each.
Since the (anti-)quasinormal modes listed above take the same form for
$\Delta$ and $\bar{\Delta}$ this effectively doubles the degeneracy of the
poles in the one-loop determinant relative to the modes listed in the table.

#### One-loop determinant

We recall from Sec. 3 that for spinning fields we must account for both a set
of zero modes (3.7), due to the compact nature of $S^{3}$, as well as the
functional determinant of a symmetric, transverse, traceless field (3.2). The
DHS method can only reproduce this second part of the calculation. Repeating
the procedure in the scalar case above for the spin-1 field, the relevant
functional determinant for a transverse vector field is

$Z_{\text{STT}}=\prod_{\begin{subarray}{c}\pm,n\in\mathbb{N}\\\
l,k\in\mathbb{Z}\end{subarray}}\left(|k|+2n+|l|+\Delta_{\pm}+1\right)^{-1}\prod_{\begin{subarray}{c}\pm,\\{l>0,k<0\\}\\\
\cup\\{l<0,k>0\\}\end{subarray}}\left(|k|+|l|+\Delta_{\pm}-1\right)^{-1}$
(B.29)

and hence

$\displaystyle\log(Z_{\text{STT}})$
$\displaystyle=-\sum_{\begin{subarray}{c}\pm,n\in\mathbb{N}\\\
l,k\in\mathbb{Z}\end{subarray}}\log\left(|k|+2n+|l|+\Delta_{\pm}+1\right)-\sum_{\begin{subarray}{c}\pm,\\{l>0,k<0\\}\\\
\cup\\{l<0,k>0\\}\end{subarray}}\log\left(|k|+|l|+\Delta_{\pm}-1\right)$
(B.30)
$\displaystyle=-\sum_{\pm,p=0}^{\infty}\log(\Delta_{\pm}+1+p)(p+1)^{2}-\sum_{\pm,p=1}^{\infty}\log(\Delta_{\pm}+p)(2p)$
$\displaystyle=-\sum_{\pm,p=0}^{\infty}\log(\Delta_{\pm}+p)(p+1)^{2}+\sum_{\pm,p=0}^{\infty}\log(\Delta_{\pm}+p)\leavevmode\nobreak\
.$

We now identify the first of these sums as exactly $2\log(Z_{\text{scalar}})$
in (B.12), and for the second term it is convenient to define

$\displaystyle\log(Z_{\text{edge}})$
$\displaystyle\equiv-\sum_{\pm,p=0}^{\infty}\log(\Delta_{\pm}+p)$ (B.31)
$\displaystyle=\sum_{\pm}\zeta^{\prime}(0,\Delta_{\pm})\leavevmode\nobreak\ .$

We refer to this part of the partition function by the term ’edge’ in
reference to [8], where this term is viewed as an ’edge character integral’.
Similarly to before the expansion of the zeta function implies that

$\zeta^{\prime}(0,\pm i\mu)=\zeta^{\prime}(0,1\pm i\mu)-\log(\pm
i\mu)\leavevmode\nobreak\ ,$ (B.32)

and hence

$\displaystyle\begin{split}\log(Z_{\text{edge}})&=\frac{1}{2}\sum_{\pm}\zeta^{\prime}(0,1\pm
i\mu)+\zeta^{\prime}(0,\pm i\mu)+\log(\pm i\mu)\\\ &=\frac{1}{2}\sum_{\pm}-\pi
iB_{1}(\pm i\mu)+\text{Li}_{1}(e^{\pm 2\pi\mu})+\log(\pm
i\mu)\leavevmode\nobreak\ .\end{split}$ (B.33)

Using the definition of the Bernoulli polynomial and $\text{Li}_{1}$ and
inserting this expression for $\log(Z_{\text{edge}})$ into the main
calculation, we arrive at

$\log(Z_{\text{STT}})=2\log(Z_{\text{scalar}})-\left(-\pi\mu-\log(1-e^{-2\pi\mu})\right)-\frac{1}{2}\log(\mu^{2})\leavevmode\nobreak\
.$ (B.34)

Finally, we need to account for the zero modes. Following [8], the
contribution is given by (3.7), and with this the full one-loop determinant is

$Z_{\Delta,\mathsf{s}=1}=Z_{\text{STT}}Z_{\text{zero}}=Z_{\text{STT}}\prod_{\pm}(\Delta_{\pm}-1)^{\frac{1}{2}}\leavevmode\nobreak\
,$ (B.35)

and so

$\log(Z_{\Delta,\mathsf{s}=1})=2\log(Z_{\text{scalar}})-\left(-\pi\mu-\log(1-e^{-2\pi\mu})\right)\leavevmode\nobreak\
.$ (B.36)

###  B.3 Massive spin-$\mathsf{s}$ fields

It is now clear how the story should continue for massive spin-$\mathsf{s}$
fields, when $\mathsf{s}\geq 2$. We solve the equations of motion for a
spin-$\mathsf{s}$ field

$\nabla^{2}\Phi_{\mu_{1}...\mu_{\mathsf{s}}}=\ell_{\text{dS}}^{2}\bar{m}_{\mathsf{s}}^{2}\Phi_{\mu_{1}...\mu_{\mathsf{s}}}\leavevmode\nobreak\
,$ (B.37)

where $\Phi$ is symmetric and subject to the usual constraints

$\nabla^{\mu_{1}}\Phi_{\mu_{1}...\mu_{\mathsf{s}}}=0,\qquad\Phi^{\nu}_{\
\nu\mu_{3}...\mu_{\mathsf{s}}}=0\leavevmode\nobreak\ .$ (B.38)

Applying (anti-)quasinormal mode boundary conditions we can extract the
(anti-)quasinormal mode frequencies. A careful treatment of these quasinormal
modes will show that (as expected) the quasinormal modes which Wick-rotate to
regular Euclidean modes on $S^{3}$ are the symmetric, transverse, traceless
spherical harmonics. These lie in representations of
$\mathfrak{so}(4)\cong\mathfrak{su}(2)\times\mathfrak{su}(2)$ labelled by
highest weights $(\frac{n}{2}+\mathsf{s},\frac{n}{2})$ and
$(\frac{n}{2},\frac{n}{2}+\mathsf{s})$.

The corresponding degeneracies for $\Delta=-p,p\geq-\mathsf{s}$ are
$(\mathsf{s}+1+p)(1-\mathsf{s}+p)$ and so

$\displaystyle\begin{split}\log(Z_{\text{STT}})=&-\sum_{\pm,p=\mathsf{s}}^{\infty}\log(\Delta_{\pm}+p)(p+1+\mathsf{s})(p+1-\mathsf{s})\\\
=&-\sum_{\pm,p=0}^{\infty}\log(\Delta_{\pm}+p)(p+1)^{2}+\mathsf{s}^{2}\sum_{\pm,p=0}^{\infty}\log(\Delta_{\pm}+p)\\\
&+\sum_{\pm,p=0}^{\mathsf{s}-2}\log(\Delta_{\pm}+p)(p+1+\mathsf{s})(p+1-\mathsf{s})\leavevmode\nobreak\
.\end{split}$ (B.39)

The first of these sums reproduces again $2\log(Z_{\text{scalar}})$ in (B.13)
and the second sum is proportional to $\log(Z_{\text{edge}})$ as defined in
(B.31). We can directly insert our previous expression for this.

$\displaystyle\begin{split}\log(Z_{\text{STT}})=&2\log(Z_{\text{scalar}})-\mathsf{s}^{2}\left(-\pi\mu-\log(1-e^{-2\pi\mu})\right)-\frac{\mathsf{s}^{2}}{2}\log(\mu^{2})\\\
&+\sum_{\pm,p=0}^{\mathsf{s}-2}\log(\Delta_{\pm}+p)(p+1+\mathsf{s})(p+1-\mathsf{s})\leavevmode\nobreak\
.\end{split}$ (B.40)

As in the spin-1 case, to compute the full local partition function we also
need to account for a set of zero modes (3.7). Including these, we obtain the
final result

$\log(Z_{\Delta,\mathsf{s}})=\log(Z_{\text{STT}})+\log(Z_{\text{zero}})=2\log(Z_{\text{scalar}})-\mathsf{s}^{2}\left(-\pi\mu-\log(1-e^{-2\pi\mu})\right)\leavevmode\nobreak\
.$ (B.41)

This formula is again in agreement with the calculation in [8], and it matches
with the Wilson spool in (3.41) (up to overall phase).

## References

  * [1] A. Castro, I. Coman, J. R. Fliss, and C. Zukowski, “Keeping matter in the loop in dS3 quantum gravity,” JHEP 07 (2023) 120, arXiv:2302.12281 [hep-th].
  * [2] A. Castro, I. Coman, J. R. Fliss, and C. Zukowski, “Coupling Fields to 3D Quantum Gravity via Chern-Simons Theory,” Phys. Rev. Lett. 131 (Oct, 2023) 171602, arXiv:2304.02668 [hep-th].
  * [3] A. Achucarro and P. K. Townsend, “A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories,” Phys. Lett. B180 (1986) 89.
  * [4] E. Witten, “(2+1)-Dimensional Gravity as an Exactly Soluble System,” Nucl. Phys. B 311 (1988) 46.
  * [5] M. Banados, T. Brotz, and M. E. Ortiz, “Quantum three-dimensional de Sitter space,” Phys. Rev. D 59 (1999) 046002, arXiv:hep-th/9807216.
  * [6] T. R. Govindarajan, R. K. Kaul, and V. Suneeta, “Quantum gravity on dS(3),” Class. Quant. Grav. 19 (2002) 4195–4205, arXiv:hep-th/0203219.
  * [7] J. Cotler and K. Jensen, “A theory of reparameterizations for AdS3 gravity,” JHEP 02 (2019) 079, arXiv:1808.03263 [hep-th].
  * [8] D. Anninos, F. Denef, Y. T. A. Law, and Z. Sun, “Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions,” JHEP 01 (2022) 088, arXiv:2009.12464 [hep-th].
  * [9] S. Collier, L. Eberhardt, and M. Zhang, “Solving 3d gravity with Virasoro TQFT,” SciPost Phys. 15 no. 4, (2023) 151, arXiv:2304.13650 [hep-th].
  * [10] S. Carlip, “The Sum over topologies in three-dimensional Euclidean quantum gravity,” Class. Quant. Grav. 10 (1993) 207–218, arXiv:hep-th/9206103.
  * [11] E. Guadagnini and P. Tomassini, “Sum over the geometries of three manifolds,” Phys. Lett. B 336 (1994) 330–336.
  * [12] E. Witten, “Three-Dimensional Gravity Revisited,” arXiv:0706.3359 [hep-th].
  * [13] A. Castro, N. Lashkari, and A. Maloney, “A de Sitter Farey Tail,” Phys. Rev. D 83 (2011) 124027, arXiv:1103.4620 [hep-th].