Coupling Fields to 3D Quantum Gravity via Chern-Simons Theory

We propose a mechanism that couples matter fields to three-dimensional quantum gravity, which can be used for theories with a positive or negative cosmological constant. Our proposal is rooted in the Chern-Simons formulation of three-dimensional gravity and makes use of the Wilson spool, a collection of Wilson loops winding around closed paths of the background. We show that the Wilson spool correctly reproduces the one-loop determinant of a free massive scalar field on rotating black holes in AdS$_3$ and Euclidean dS$_3$ as $G_N\to 0$. Moreover, we describe how to incorporate quantum metric fluctuations into this formalism.

It has been a long-standing problem to incorporate matter into Chern-Simons gravity while retaining the topological features that make it natural as a theory of quantum gravity.In this letter we address this problem.In short, we introduce a new object we deem the Wilson spool which provides an effective coupling of massive fields to quantum gravity directly as a gauge-invariant operator.We define it precisely below, but intuitively, the spool represents a Wilson loop winding arbitrarily many times around a closed path for which the fields have non-trivial holonomy.This represents a pivotal entry into a dictionary mapping geometric quantities to quantum operators in Chern-Simons gravity [19][20][21][22][23][24].The Wilson spool will dictate how quantum gravity alters the physics of quantum fields in a manner that is quantitatively controlled by the gravitational coupling, G N .
We will show that the Wilson spool is a natural object regardless of the sign of cosmological constant.In the context of AdS 3 we will exactly reproduce, at tree-level (G N → 0), the one-loop determinant of a scalar field on a rotating BTZ black hole background.This computa-tion is done directly "in the bulk," without reference to holography.In fact, it is in the context of dS 3 , where holography is of limited utility, that we can make use of the full power of this proposal.Certain exact results in Chern-Simons theory can be meaningfully adapted to accommodate features necessary for de Sitter gravity and its massive single-particle states.This provides a principled and controlled method to computing G N corrections to one-loop determinants of fields coupled to dynamical gravity.In this letter, we distill key results that are presented in full detail in the companion paper [25], and also cast them in a presentation that is unified for both signs of the cosmological constant.
In the following we will present our proposal of the Wilson spool, and illustrate its efficacy for AdS 3 and dS 3 gravity.We will evaluate it on AdS 3 in Lorentzian signature, where the Chern-Simons gauge group is SL(2, R) × SL(2, R), while for dS 3 it will be treated in Euclidean signature, where the gauge group is SU (2) × SU (2).This choice facilitates making parallels between them, and hence makes evident the robustness of our proposal, while also showing contrast when appropriate.

THE PROPOSAL
To illustrate our proposal it is instructive to cast some portions in the metric formulation.Consider the path integral for a quantum scalar field φ, of mass m, with no self-interactions, and minimally coupled to the metric field, g µν .This would be ,gµν ] . (1) Including the perturbative, quantum fluctuations of the metric leads to For concreteness we wrote here a Euclidean path integral, where I EH is the Euclidean Einstein-Hilbert action plus a cosmological constant term.The gravitational path integral is taken on a fixed topology M , and it includes all perturbative corrections around that topology.
Our proposal for quantifying the coupling of matter to gravity is captured by the following equality, 1 On the right hand side we have the Wilson spool which captures Z scalar by using solely objects in Chern-Simons theory.Starting from the right, we have path ordered exponentials of A L and A R which encode g µν above: this is the portion that captures the information of the geometry.Next, we have traces over a representation R j : here is where we encode the single-particle representations of the field.The mass of the field is related to the Casimir of the representation with Λ the cosmological constant.Finally, in (4) we have an integral over α.The measure and contour C of this integral serve two purposes [25]: First, they make W free of UV divergences.Second, as will become clear in detail, evaluating this contour integral as a sum over poles implements a sum over Wilson loops with arbitrary winding, which is what makes it a "spool."The specific details of C, and allowed deformations, depend on the holonomies of A L,R ; these will be specified for the backgrounds considered in the following sections.
In this letter we want to uphold (4) on two fronts.First, we believe (4) applies to any smooth threedimensional background, including but also extending beyond dS 3 quantum gravity.Second, it is a useful expression to quantify quantum gravity effects: the Chern-Simons formulation allows us to integrate out the scalar field, and obtain an explicit functional in terms of the connections A L,R .In this context, the main appeal of our proposal is the ability to quantify 1 Since both theories, AdS 3 and dS 3 gravity, involve the product of two gauge groups, we will introduce two Chern-Simons connections, (A L , A R ), corresponding to each group.These will encode the metric field gµν as we will review in the subsequent sections.
where S CS [A] is the Chern-Simons action, k L/R are the levels.For brevity, here DA L/R accounts for the measure for the two copies of the gauge group.The brackets mean that one is accounting for gravitational fluctuations around a fixed topology.In the Chern-Simons language, this means that we fix a background holonomy for A L,R in addition to the topology.It is important to stress that ( 6) is a non-trivial function of G N and the mass of scalar field.

WILSON SPOOLS IN AdS3 GRAVITY
As a first example of the utility of our proposal we will focus on Chern-Simons gravity with negative cosmological constant, Λ = −ℓ −2 AdS .This is quantum gravity on spaces that are locally asymptotically Anti-de Sitter (AdS).In Lorentzian signature, the relevant isometry group is SL(2, R) × SL(2, R) and massive particles in this space can be organized into sl(2, R) representation theory.The Lorentzian Einstein-Hilbert action is given by the difference in SL(2, R) Chern-Simons theories where k = ℓ AdS 4GN .The connections are related to the coframe and the spin connection via Above L a and La generate the independent sl(2, R) algebras.Our conventions follow those in [3,23].At the classical level, the background geometries of interest are generated by flat background connections where x ± = t ± ϕ and ϕ ∼ ϕ + 2π.Upon using (8), these are rotating BTZ black hole geometries [26,27] with mass, M , and angular momentum, J, given by In Euclidean signature, we can rotate (x + , x − ) to complex coordinates (z, −z).Periodicity in ϕ plus smoothness of the horizon implies these parameterize a complex torus, (z, z) ∼ (z + 2πm + 2πnτ, z + 2πm + 2πnτ ), m, n ∈ Z, with modular parameter What separates this complex torus from the torus defining the boundary of thermal AdS 3 is how we "fill it in" in the bulk: in particular, the black hole geometry is filled in so that the thermal-cycle is bulk contractible while the spatial-cycle is not.The holonomies of the background connections a L/R around this cycle, γ ϕ , are easily computed to be where u L/R are periodic group elements and Next we will show that the Wilson spool of these background connections, a L/R , reproduces the one-loop determinant of a massive scalar field on the BTZ background.That is, we will test the relation Here j labels a lowest-weight representation of sl(2, R) related to the mass of the bulk field via The expression for the Wilson spool, (4), can be derived for this background.Given (12), we have where h L/R are given by ( 13) and is the character of the lowest-weight representation, R j .The contour, C, is given by twice the contour running up the imaginary α axis to the right of zero: C = 2C + .This follows from the procedure in [25] but assigning an iǫ prescription appropriate for representations and holonomies relevant to AdS 3 .Further detail of this is also described in the supplementary material.Because τ, τ ∈ iR (equivalently, L and L are real and positive), all poles in the α integrand in (16) to the right of zero are simple poles at 2πZ >0 and arise from the measure, cos α/2 sin α/2 .We then deform the contour C to the right where the α integrand is damped, picking up the residues of the simple poles.We can then write This can be easily rewritten into where q = e −i 2π τ and q = e i 2π τ .This matches exactly log Z scalar [BTZ] for a real massive field propagating on the rotating BTZ background [28].This is the tree-level (G N → 0) contribution to log Z scalar grav .Promoting the background connections to dynamical fields, a L/R → A L/R , the Chern-Simons path-integral provides a way forward, in principle, for calculating perturbative G N corrections to (19).The noncompact gauge group and the non-compact background topology make this program still difficult.However, one may still make progress using large-k Chern-Simons perturbation theory similar to [22,29,30].Below we will see that in the context of positive cosmological constant, our ability to calculate G N corrections is under even better control.

WILSON SPOOLS IN dS3 GRAVITY
Our next example will be Chern-Simons gravity with positive cosmological constant, Λ = ℓ −2 dS , which describes quantum gravity on de Sitter space.We will be working with a Euclidean action given by two copies of the SU (2) Chern-Simons action; its relation to dS 3 gravity is via where I EH is the Euclidean Einstein-Hilbert action and I GCS is the gravitational Chern-Simons action.The Chern-Simons couplings k L,R are now in general complex, with an imaginary part that is related to Newton's constant, s = ℓ dS 4GN , and a real part, δ, giving the coefficient of a gravitational Chern-Simons action.Quantum effects lead to a renormalization in the coupling constants [31] which amounts to a renormalization of δ → δ = δ + 2. The above matching is facilitated by relating the gauge fields to the vielbein and spin connection via where now {L a } and { La } generate the two independent su(2)'s.We will focus presently on the round S 3 saddle whose classical background geometry is generated by background connections These background connections have a singularity at the causal horizon (ρ = π/2) and the holonomy around that singularity is given by2 with u L/R periodic group elements and Now let us add matter and test our proposal.We will begin with the tree-level check which evaluates the spool on the background connections: The representation, R j , that appears in W j is a highestweight representation of su(2) related to the mass of the scalar field as Note that j is a continuous parameter and can even become complex for large enough masses.Thus these representations do not correspond to the standard finite dimensional representations of SU (2).Instead they correspond to infinite dimensional "non-standard" representations whose weight spaces line up with the dS 3 quasinormal mode spectrum [24,25].Despite not lying in the standard SU (2) representation theory, the representations obeying (28) can be equipped with an inner product such that all states have positive norm [24,25].Additionally they admit well-defined characters: We can then express (30) in terms of the holonomies (26) as Deviating slightly from the procedure for the BTZ computation, the iǫ prescription appropriate for de Sitter results in a contour, C = C − ∪ C + , that is, the union of contours running up the imaginary α axis both to the left and right of zero [25].We pull both C ± towards the positive real axis to pick up residue of the quadruple pole at α = 0 as well as twice the residues of the poles at α ∈ 2πZ >0 along the positive real line; these latter poles are now third-order.Doing so we find where Li q (x) = ∞ n=1 x n n q are polylogarithm functions.Upon using ( 28), this answer matches precisely the finite contribution to the scalar one-loop determinant on S3 ; see for instance [17].This provides a second non-trivial check of the physical relevance of the Wilson spool.
In the context of de Sitter gravity we can actually make substantial progress in discussing the Wilson spool beyond its classical expectation value.This is because there exists a library of "exact techniques" for SU (2) Chern-Simons theories on compact topologies.These techniques reduce the Chern-Simons path-integral, as well as the expectation values of certain operators, to ordinary integrals.In the present context, care is needed to alter these methods to accommodate the features necessary for Chern-Simons gravity (i.e.complex levels, non-zero background connections, and non-standard representations).However several exact methods remain amenable to these alterations [25].For instance, a suitable alteration of Abelianisation [32][33][34] reduces the expectation value of W j [A L , A R ] with dynamical connections to a pair of simple integrals [25]: which we can equate to the quantum gravity corrected one-loop determinant about the S 3 saddle, log Z scalar [S 3 ] grav .The integral appearing in ( 32) is difficult to evaluate analytically, however it can systematically be evaluated in a s −1 Taylor expansion.This provides a controlled procedure to computing G N corrections to log Z scalar [S 3 ].These corrections are naturally identified with a mass renormalization.To extract them one simply computes log Z scalar [S 3 ] grav , normalized by the gravitational path integral, using (32); the resulting ) Above we have kept the leading term in a large mass expansion, m 2 ℓ 2 dS ≫ 1, however (32) provides an expression for the G 2 N renormalization of the mass that can be calculated analytically [25].We emphasize that this is a concrete predictive statement about how dynamical quantum gravity renormalizes quantum field theory.

DISCUSSION
We have introduced a new object, the Wilson spool, which allows one to have matter fields in the Chern-Simons formulation of three-dimensional quantum gravity while keeping manifest the key topological aspects of gravity.To test this object, we have shown that, for G N → 0, it reproduces correctly one-loop determinants of massive scalar fields on a curved background.The proposal works for spacetimes that are widely different, such as the spinning BTZ black hole and Euclidean de Sitter spacetime.
If our proposal merely provided a way to match onto known one-loop determinants, it would have limited utility.However, it is also possible to use this techniques to make a prediction for quantum corrections to log Z grav , having allowed for quantum fluctuations of the metric around a classical background.For certain AdS geometries (see, e.g., [22]), the 1/c corrections to Wilson lines have been computed in a holographic setting.Our techniques give us a way to extend this beyond the holographic setting to quantum gravity in spacetimes like de Sitter, which are more realistic approximations of our observed universe but where techniques of holography are largely out of reach.
There are two future directions in this area that we would like to highlight.A more in depth discussion is presented in [25].
Massive spinning fields.We have focused on massive scalar fields in this letter, for which we have a direct construction of the spool.But both AdS 3 and dS 3 admit massive fields of arbitrary integer spin.We can ask if the Wilson spool is of utility in coupling these excitations to dynamical gravity.We do not presently have a firstprinciples construction for W applied to spinning fields (as we do for scalars).However, in the context of AdS 3 , the definition of the Wilson spool, (4), can be intuitively extended to reproduce the one-loop determinant for a massive spinning field on the rotating BTZ background.The details of this can be found in the supplementary material.This is a strong indication that W is useful for the physics of spinning fields.We can also make a first pass application of W to spinning fields in de Sitter; this relies on further guess work since the appropriate non-standard su(2) representations have not yet been identified.In this case, while we can reliably reproduce certain contributions to the one-loop determinant, W currently misses an important "edge subtraction" arising from properly subtracting normalizable transverse-traceless zero modes on S3 [17].A proper derivation of the spool to capture the physics of spinning fields will be addressed in future work.
Sum over topologies.Here we have kept the topology of the background, and holonomies of the connections, fixed.It is of great interest to allow for these non-perturbative contributions in (2), albeit it comes with difficulties: if only metric degrees of freedom are incorporated, both the AdS 3 thermal partition function [35,36] and the Euclidean dS 3 path integral [16] suffer from pathologies.But it is also expected that adding matter can fix some of these problems [37].It would interesting to investigate the fate of the Wilson spool under these pathologies, i.e., loop in matter in the sum over manifolds and quantify its imprint.This is specially of interest in dS 3 , where a holographic dictionary is still nascent, but the Wilson spool gives a concise path to quantify the effect of fields at all orders in G N .
We now take the log.We will implement this with a Schwinger parameter log M = − ∞ × dα α e −αM , where in the lower limit we indicate a need to regulate the UV divergence at α ∼ 0. Additionally we will need to regulate the sum over representation weights (λ L , λ R ).Both of these can be taken care of simultaneously through an iǫ prescription which we implement now.Noting Note that h L is equal to i times a positive quantity, while h R is i times a negative quantity.Also, since R j is a lowest-weight representation of sl(2, R), all λ L/R appearing in the sum are real and positive.It then follows that the first (second) sum converges by giving α a small positive (negative) imaginary part.Under the replacement of integration variable, α → −α, in the second sum, we can write the regulated log Z scalar succinctly as We recognize the sum over representation weights times holonomies as trace path-ordered exponentials of the background connections.Finally, we make the integration variable replacement α → iα to write Thus appears the Wilson spool.The contour C + runs upwards along the imaginary α axis to the right of zero.We write it conventionally with the factor of two to uniformize the notation with the de Sitter spool (where the ingoing vs. outgoing periodicity constraints give rise to distinct contours [25]).We derived C + by ensuring representation sums converge, however, in the body of the letter we verify a posteriori that it also regulates the UV divergence in log Z scalar .

B. Massive spinning fields
We provide additional details on a "first-pass" attempt to incorporate spinning fields into the Wilson spool.This is easiest to address in the context of AdS 3 , where massive particles of spin s are already included in sl(2, R) L ⊕ sl(2, R) R representation theory through lowest-weights Noting that in [25] that the construction of W arises from a sum of Casimirs, c 2,L + c 2,R , it is natural to conjecture that both contributions, (j L , j R ) = (j + , j − ) and (j L , j R ) = (j − , j + ), should be included the spool since both choices have the same sum of Casimirs: Indeed doing so, we can calculate for the BTZ background, (9), that the tree-level contribution to the spinning spool is 1 n q n 2 (∆±s) q n 2 (∆∓s) which is the correct one-loop determinant for a massive spinning field on the rotating BTZ background [39][40][41].We stress that, as of present, we do not have a first principles construction for W j + ,j − to justify the above calculation.However in the context of AdS 3 we have reason to trust the above: massive spinning fields in AdS 3 have only two real physical polarizations [40] which we can package into sl(2, R) lowest-weights (j + , j − ).If the one-loop determinant continues to depend only on the summed Casimirs, as it does for scalar fields, then is hard to imagine any other appropriately group theoretic object to construct besides (46).
Trying to extend this reasoning to de Sitter gravity is more subtle.Part of the issue is that we already need novel su(2) representations to capture the physics of scalar excitations; as of yet we have not constructed the non-standard representations necessary to accommodate s = 0. We view this as technical hurdle as opposed to a conceptual no-go.As a first pass, we can analytically continue the su(2) weights to accommodate spin via Investigating (46) for the classical S 3 background, (24), we can implement this analytic continuation at the level of the characters.We find that W j + ,j − [a L , a R ] is, in fact, s independent and essentially doubles the scalar answer, (31).This almost reproduces one piece of the one-loop determinant of massive spinning fields on S 3 , log Z ∆,s however it misses an additional "edge" subtraction log Z edge = s 2 log e iπ(∆−1) 1 − e i2π(∆−1) , (50) which arises from properly subtracting normalizable transverse-traceless zero modes [17].We hope that a suitable modification of W can capture this edge term, which we leave for future work.