A Deformation for Curved Spacetimes from 3d Gravity
Abstract
We propose a generalisation of the deformation to curved spaces by defining, and solving, a suitable flow equation for the partition function. We provide evidence it is welldefined at the quantum level. This proposal identifies, for any CFT, the deformed partition function and a certain wavefunction of 3d quantum gravity. This connection, true for any , is not a holographic duality — the 3d theory is a “fake bulk.” We however emphasise that this reduces to the known holographic connection in the classical limit.
Concretely, this means the deformed partition function solves exactly not just one global equation, defining the flow, but in fact a local Wheelerde Witt equation, relating the operator to the trace of the stress tensor. This also immediately suggests a version of the deformation with locally varying deformation parameter.
We flesh out the connection to 3d gravity, showing that the partition function of the deformed theory is precisely a 3d gravity path integral. In particular, in the classical limit, this path integral reproduces the holographic picture of Dirichlet boundary conditions at a finite radius and mixed boundary conditions at the asymptotic boundary.
Further, we reproduce known results in the flat space limit, as well as the large partition function, and conjecture an answer for the finite partition function.
1 Introduction
In flat space, the deformation has been discovered at least five times Zamolodchikov:2004ce ; Smirnov:2016lqw ; Lechner:2006kb ; Dubovsky:2012wk ; Freidel:2008sh . This serves as a testimony to the many deep facts we may hope to learn from a better understanding of the deformation and its generalization to curved space. So far, each incarnation has revealed novel features. Zamolodchikov:2004ce ; Smirnov:2016lqw highlighted its solvability and the way it preserves the seed theory’s integrability. being an irrelevant operator, the flow generated by the deformation provides an exceptional example of what resembles flowing up an renormalization group (RG) trajectory ZamTalk .^{1}^{1}1Though, as emphasized in Komargodski:talk , it is not precisely an RG flow, since we vary only the coupling but keep all other couplings in the theory fixed. Analysing modular transformation properties of the partition function, Datta:2018thy ; Aharony:2018bad ; Aharony:2018ics pointed out how the and related deformation were singled out by very few assumptions. Further, the deformed theory’s energy spectrum indicates it cannot be reduced to a local quantum field theory.^{2}^{2}2For one sign of the deformation, the spectrum is Hagedorn. For the other, infinitely many energy levels become complex. Cardy:2019qao ; Lewkowycz:2019xse also made precise statements regarding its nonlocality at the level of operator algebras. Cardy:2018sdv ; Dubovsky:2017cnj ; Dubovsky:2018bmo showed the deformed theory on a torus was in fact a theory of twodimensional quantum gravity. Further evidence the theory cannot be a simply local field theory came from the connection to string theory exhibited in Dubovsky:2017cnj ; Callebaut:2019omt . First indications of a higherdimensional interpretation of the deformed 2d theory rested upon a connection to holographic RG McGough:2016lol ; Shyam:2017znq ; Kraus:2018xrn ; Hartman:2018tkw . Higherdimensional generalisations of the deformation itself, i.e. deformations of field theories living in , proceeded by replicating this property Hartman:2018tkw ; Taylor:2018xcy . It is also possible to define lower dimensional deformations, one of which reproduces this connection Gross:2019ach ; Gross:2019uxi . Finally, it should also be noted that closely related deformations can be more directly related to string theory Giveon:2017nie ; Giveon:2017myj . A helpful introduction with a more complete set of references can be found in Jiang:2019hxb .
The usual definition of the deformation has been logically predicated upon the result of Zamolodchikov:2004ce . It explains how this bilinear operator is naturally welldefined in any theory with a local conserved stress tensor, global translationinvariance and with a least at one noncompact spacetime direction.^{3}^{3}3Zamolodchikov:2004ce in fact requires one additional, rather technical condition, see his section 4. Once assured the operator exists, we can use it to define a deformation equation,
(1) 
The purpose of the two subscripts is to indicate that it is the expectation value of the stress tensor of the deformed theory in the vacuum of the deformed theory.
The objective of this paper is to propose a quantummechanically welldefined generalization of the deformation for twodimensional seed theories defined on curved spaces. The reason this is a conceptually distinct problem from the flatspace case is that we no longer have an analogue of the theorem of Zamolodchikov:2004ce ; Smirnov:2016lqw . Indeed, a crucial ingredient there was a certain translationinvariance assumption that has no hope of being true on curved manifolds. In fact, direct investigation suggests that no simple analogue of their theorem can hold in curved space Jiang:2019tcq . Given this fact, the approach of this paper is somewhat backwards. We first posit the form of the deformed theory’s partition function, then show it satisfies a particular differential equation (and initial condition) we deem a reasonable curvedspace generalisation of the flow.
A particular object in 3d gravity — known as a radial wavefunction — provides the expression for the deformed theory’s partition function, which we denote by for reasons to be explained momentarily. This connection relies crucially on the work of Freidel:2008sh , who showed how a certain integral kernel mapped CFT partition functions to these radial wavefunctions. These radial wavefunctions satisfy an infinite set of local constraints known as the radial Hamiltonian constraint, also known as the WheelerdeWitt equation. As such, unexpectedly satisfies not just the global flow equation fixing its dependence on but an infinite set of equations. These equations relate the expectation value of the operator at a point to the one point function of the trace of the stress tensor at the same point. They therefore provide us with partial control on the divergence structure of the operator. We would like to take another sentence to emphasise that we do not know anything general about any reasonable definition of the operator in other theories, including the QFTs that provide the seeds for the flow.
The main claim we make is that for any seed QFT with partition function , we can write the deformed theory’s partition function via^{4}^{4}4We direct the reader unfamiliar with vielbeins to appendix A for a short introduction.
(2) 
where stands for the partition function of the undeformed theory and is the deformation parameter. Crucially, the seed theory lives on a 2d base space with metric parameterized in terms of vielbeins , while the deformed theory resides on a target space with metric , see figure 1. Beyond requiring the base and target share the same topology, we have yet to find further restrictions on the vielbeins . This partition function satisfies the flow equation
(3) 
where we define the coincident double derivative not as a limit but merely by the subtraction of a contact term. It is not obvious that this flow equation, with this definition of the deformation operator, is the curved space generalisation of the deformation. A more principled approach might indeed give rise to a theory under better analytic control and with more desirable properties.
More importantly, it is not even completely obvious so far that our proposal is even a sensible generalisation. To wit, there’s no guarantee that the path integral (2) can be suitably regularised as a functional of the metric. While the precise consistency conditions are not clear at the moment, a minimal one is that the partition function of a compact Euclidean manifold is rendered finite by a single UV cutoff and a finite number of counterterms. This is equivalent to saying point functions of the stress tensor have singularities that can be integrated against metric perturbations. While we will not be able to prove that this is the case for our , we will provide some evidence for it.
One of the major pieces of evidence for this comes from the connection to 3D gravity. This extends to the fully quantum setting that which was already understood by McGough:2016lol ; Shyam:2017znq ; Kraus:2018xrn ; Hartman:2018tkw ; gorbenko2019ds . This connection is rooted in a relation between the twodimensional “trace flow equation” and one of the equations of motion of threedimensional general relativity. Consider 3d Einstein gravity with a negative cosmological constant. Imagine the manifold is sliced with a radial coordinate and 2d slices transverse to it that are parameterized by . In terms of the BrownYork stress tensor,^{5}^{5}5We have added a BalasubramanianKraus counterterm Balasubramanian:1999re , and will keep it around for most of this paper.
(4) 
the component of the Einstein equations can be rewritten as
(5) 
A very similar equation can be obtained by making an assumption about the deformed theory based on dimensional analysis. For simplicity, let us deform a seed CFT by . This deformation parameter is the only mass scale in the theory. We might thus assume the response of the partition function to an overall scaling is given entirely by its response to a change in alongside the contribution from the conformal anomaly,
(6) 
Plugging the deformation equation into this equation, we find
(7) 
This is tantalisingly similar to (5), but not quite the same.
Let us list out the differences. First, the coupling constants are different. This can easily be dealt with by positing a dictionary relating to . Secondly, (5) is valid pointwise, while (7) resides under an integral sign. Third, (5) is a classical relation, but (7) is a quantum one; we posit the latter as a quantisation of the first, classical relation. More precisely, quantising a theory of gravity requires the imposition of certain equations of motion as constraints identically satisfied by all physical wavefunctions; these constraints are known as the Hamiltonian constraint or the Wheelerde Witt (WdW) equation,
(8) 
It is worth noting the unconventional feature that the “wavefunction” that appears in this equation is a state on a 2d spacetime with the radial direction playing the role of time — this object is known as a radial wavefunction.
The important point about this connection with the WdW equation is that the heuristic derivation works for any CFT, independent of its central charge or field content. While, for a holographic CFT in the stress tensor sector, this deformation appears to define the theory with Dirichlet boundary conditions at finite radius in the dual , the point is that for every CFT the deformation behaves like flowing into a quantum ‘‘fake bulk.’’^{6}^{6}6This point was first explained to us by Aitor Lewkowycz, and it will be expanded upon in Belin:unp . This “fake bulk” is not a quantum holographic dual, but it does agree with holography in the classical limit.
As it happens, a radial wavefunction satisfying (6) – (8) was already found in Freidel:2008sh . Minor rescalings lead immediately to (2). Briefly, Freidel:2008sh proved that, given a 2D CFT partition function as a function of the 2D vielbein that satisfies the conformal anomaly equation
(9) 
(2) satisfies the radial WdW equation (8). This result is exact, and does not depend on any large limit.
The central point of this paper is that this same object that satisfies the Wheelerde Witt (WdW) equation also satisfies the flow equation. Whereas it satisfies the WdW equation only when is a CFT partition function satisfying (9), it satisfies the flow equation for any . In other words, even though the connection to 3d GR is only true when the seed is a CFT partition function, the flow equation is indifferent to the nature of . Thus, the theory (2) is a proposal for the curved space deformed theory for any QFT.
We present the deformed theory first from a purely 2d perspective by exhibiting various properties the kernel satisfies. In particular, we show

the deformed theory reduced to the undeformed one in the limit .

Its stress tensor is conserved and symmetric, which expresses the invariance of under tangent space rotations and diffeomorphisms of the manifold equipped with the vielbeins .

The integral kernel composes. Plugging in a theory already deformed by as the seed results in a theory deformed by .

At leading order in large , where the kernel has a classical limit, it reproduces previously known partition functions and entanglement entropies.

The loop expansion is controlled by a renormalisable coupling , suggesting that the theory can be regulated with a finite number of counterterms.
Further, our analysis immediately suggests an important further generalisation of the deformation. The standard flow equation is only related to the integral of the WdW equation, not its local form. Since solves the local WdW equation, we may in fact easily modify the kernel to promote . This local deformation also satisfies a different local WdW equation.
We then move on to flesh out the connection with 3d gravity, showing precisely how the kernel can be thought of as the path integral on an annular region with a Dirichlet boundary condition on one side and a CFTdependent boundary condition on the other. We also show how many of the properties of the 2d theory are natural from this point of view.
We also, using a particular gaugefixing of the 3d gravity wavefunction, conjecture — up to ignorance parameters — an exact answer for the finite deformed partition function on an . Most interestingly, for a particular value of the ignorance parameter, it can be written as a localised version of the Freidel kernel.
Finally, we discuss some objections to the existence of the curved space deformation. We outline several arguments that it should not exist and highlight to what extent we have shown that the kernel evades those issues. A byproduct of this discussion is the observation that a quantum version of the deformations of Hartman:2018tkw ; Taylor:2018xcy may in fact exist as well.
The plan of this paper is as follows. In section 2, we carefully define the theory as a partition function, show some basic properties, and explain the natural way to obtain other deformations. Then, in section 3, we flesh out the picture of the kernel as a radial wavefunction and recast the above properties of the 2d theory in 3d language. We proceed to obtain the CMC gaugefixed genus sphere partition function — which we conjecture to be the exact partition function for a CFT — in section 4. We also comment on the large limit of the ungaugefixed partition function in section 5. Finally, we address the expectation that the curved space theory should not exist and the extent to which we have addressed these concerns in section 6. We conclude in section 7, sketching future directions of work.
Note Added: While this manuscript was in preparation, the preprint Tolley:2019nmm with some overlapping results appeared on the arXiv.
A Note on Notation
Firstly, we will liberally use the firstorder formalism in this paper, since that is the one in which the kernel is simplest. For those unfamiliar with this formalism, we have included a short introduction to vielbeins in appendix A.
Secondly, we have found ourselves in the unfortunate situation of having to deal with both LeviCivita tensor densities as well as LeviCivita symbols. We will consistently use the notation for the tensor density and for the symbol,
(10) 
We will explicitly state which one turns up in an equation when the difference is important.
A Note on A Sign
Throughout this paper, we work with being the holographic sign of the deformation. While this is widely considered the bad sign, we find it useful for two reasons. The first is that much of the story in this paper relates to 3d gravity, which is somewhat more confusing for the other sign. The second is that this is the sign for which the kernel (2) does not obviously have a conformal mode problem.
2 The 2D Story: Flows of Partition Functions
The deformation defines a flow for the partition function. Analysing this flow via an integral kernel, i.e. rewriting the deformed theory partition function as a path integral transform of the seed’s, dates back to Cardy:2018jho . It has since appeared in multiple guises, amongst others in the works of Dubovsky:2017cnj ; Dubovsky:2018bmo ; McGough:2016lol ; Hashimoto:2019wct ; Tolley:2019nmm ; Ireland:2019vvj . In this section, we present the deformation purely in terms of an integral kernel and show what properties it satisfies. It conveniently circumvents several of the issues raised in generalizing the precise operator to curved space. While many of the kernel’s remarkable properties are most easily seen from a 3d gravity perspective, we first present our results in their immediate 2d setting.
2.1 A closer look at the Kernel
To argue for the welldefinedness of the proposed deformation, the path integral transform quoted in the introduction
(11) 
requires a careful specification of the measure and seed partition function .
The measure for the integration over vielbeins, elaborated upon in appendix B, is both diffeomorphism as well as translationinvariant,^{7}^{7}7We thank A. Tolley for pointing out the existence of this measure.
(12) 
The existence of this measure is a special fact about 2d gravity in the firstorder formalism.
This measure ensures that the seed partition function reappears from the kernel in the limit . This choice is also required for the kernel to compose, that is, deforming by and again by is identical to doing a single deformation by .
We take to be defined via its path integral formulation. Even for a CFT, this involves a particular regularization scheme, such as the choice of a localcounter term canceling off any contribution to the cosmological constant (see for example polchinski1986evaluation ). This matches the implicit prescription of Dubovsky:2018bmo for the case of the torus. We will further assume it is invariant under background spacetime diffeomorphism , where parameterizes the diffeomorphism.
2.2 The Flow Equation
In this subsection, we show that the Freidel kernel (11) satisfies a particular generalisation of the deformation equation (1), independent of the nature of the seed. The most important point is that we do not have an a priori definition of the operator that appears in this equation, but simply that the kernel satisfies it.
The flow equation satisfied by the kernel is
(13) 
where the ‘normal ordering’ is defined not by a coincident limit of any sort but simply as
(14) 
This is all the normal ordering one needs to do to get the flow equation (13) to work; if it turns out that there are more divergences on the RHS, they also drive this flow.
Before proving the equation, let us define the one and twopoint functions of the stress tensor. The onepoint function is defined by^{8}^{8}8Despite the factor of , this is consistent with (4). The factor is absorbed by the transformation between metrics and vielbeins.
(15) 
which means that
(16) 
Similarly, the twopoint function is defined as
(17) 
where the factors are outside so that the change in the free energy is a double integral of the twopoint function. With this definition, the RHS of (13) (up to the normal ordering) is ^{9}^{9}9Recall that is a tensor density. on the other hand is simply the LeviCivita symbol. Hence, .
(18) 
Thus, it appears a sensible generalisation of the operator.
Moving on to prove the kernel satisfies the flow equation, we see the left hand side of (13) becomes simply
(19) 
Let us first work out the right hand side, without the normalordering, keeping any new contact terms which may arise:
(20) 
We thus define the normalordering by subtracting out the piece proportional to . Note this term exists equally well in flat space. With this prescription, the flow equation(13) holds rather trivially:
(21)  
(22) 
Note, finally, that this proof did not care about the nature of the seed. It can be a CFT, a QFT, or indeed a deformed theory itself. It can also be a physically uninteresting function of the vielbein, but we will ignore this possibility.
2.3 The “Wheelerde Witt” Equation
In this section, for completeness of presentation, we reproduce the main result of Freidel:2008sh , namely that the kernel satisfies not just the flow equation but also a local equation, which in the 3d gravity interpretation is the Hamiltonian Wheelerde Witt equation.
The equation is
(23) 
The first line is the requirement that the seed be a CFT. For a nonCFT, there is a somewhat less tractable generalisation,
(24) 
On the right hand side of the first equality, stands for the undeformed field theory expectation value of the specific operators defining the QFT (away from a fixed point) multiplied by their respective beta functions. It is essentially a local version of the CallanSymanzik equation (see Baume:2014rla for a nice overview). The additional expectation value appearing on the RHS of the second line denotes the fact the field theory is then further averaged over geometries under the path integral. Crucially, the ‘normal ordering’ in these two “Wheelerde Witt” equations matches exactly (14), required for the flow equation to hold.
A further important point is that the flow (13) and WdW equations (23) together give us a fully quantum version of the dimensional analysis equation (6), with a shift in the effective central charge,
(25) 
There are again extra terms for a nonCFT seed. This is a nontrivial fact. It provides the underlying justification as to why the heuristic argument in the introduction makes any sense. An equation of this form is usually expected to be true for relevant deformations. That it also holds for this irrelevantseeming deformation is likely an important part of the solubility of .
We will first derive (23) for a CFT, and then comment on the proof of (24). First consider the action of the single derivative, scaling operator on the partition function:
where in going to the final line, we used (22). All that remains to be shown is
(27) 
The proof of Freidel:2008sh can be repackaged into two SchwingerDysonlike equations satisfied by the full path integral. One is for scalings of the base space vielbein . The other concerns base space diffeomorphisms.
First, we notice that generates local scale transformations. The basic idea is then to integrate by parts in (27). Assuming no boundary contributions in field space, the scaling generator may then act on both the seed partition function, but also on the measure which transforms anomalously. More precisely, we can extract the dependence of the measure and seed partition function on the Weyl mode of the metric, , via a Liouville action
(28) 
In writing this equation, we have used the work of david1988conformal ; distler1989conformal ; d19902 which shows the exponentiated Liouville action provides the Jacobian between the Weyl anomalous measure and the scale invariant one . We show in B their results apply equally to well to translationinvariant measure we have chosen. We also elaborate on the origin of the coefficient 24 in the appendix. The scale dependence of the seed CFT partition function follows immediately upon integrating the conformal anomaly equation
(29) 
We can therefore write
(30) 
This step, where we have used the conformal anomaly equation, is the only one that differs between CFTs and other QFTs. For a QFT, this equation becomes instead
(31) 
None of the following steps will differ between the two sides, and so finishing the proof of the CFT WdW will also furnish a proof of the more general one.
The curvature term appearing above is that relative to the dynamical vielbein . The final step requires showing . A SchwingerDyson type argument expressing diffeomorphism invariance provides the missing piece. Recall both the measure and seed partition function are invariant under the transformation Infinitesimally, we may write , where is the covariant derivative, , and we have used .
Since , the following path integral identity trivially holds
(32) 
Under such an infinitesimal transformation,
(33) 
These two equations together imply^{10}^{10}10One further lesson here is that we can integrate by parts in the integral, without any contact terms.
(34) 
In particular, using the diffinvariance of the seed , we may write
(35)  
(36) 
Since we can solve for the spin connection from the torsionlessness condition , this implies that
(37) 
Finally we can rewrite the Ricci scalar in terms of the spin connection as . This concludes the proof that solves the WheelerdeWitt equation, provided the solves the conformal anomaly equation.
2.4 Intuitive Derivation of the Kernel
This section attempts to provide an intuitive understanding for the form of the kernel. The recursive nature of the deformation begs the question as to simplicity of the kernel’s dependence.
A fruitful perspective interprets the flow equation
(38) 
as a Schrœdingerlike equation for a wavefunction in the position basis , where the deformation parameter plays the role of “time” and the Hamiltonian
The main inspiration the argument below takes from the connection to 3D gravity is the identification of the deformed partition function with a radial wavefunction. Specifically, while recursive deformations like the deformation are somewhat novel , they are familiar as evolving wavefunctions.
The important observation to make is that above Hamiltonian is itself independent of . We can therefore formally write down a solution to this flow equation as
(39) 
As far as we know, this equation seems to have first appeared in cottrell2019comments . To make this expression tractable, we should work in a basis which diagonalizes the Hamiltonian.
As a warmup, consider the toy example from singleparticle quantum mechanics.
(40) 
we can straightforwardly write the time evolved state by working in the momentum basis:
(41)  
(42) 
The representation of the deformed theory exactly analogous to this exponential form has been used successfully in Cardy:2018jho ; Cardy:2018sdv to understand many of its properties. This parallel naturally leads on to ask what is the analog of momentum space for the deformation?
It turns out that this is a wellknown object: the Legendre transformation that takes the partition function to exponential of the stress tensor effective action. This point of view was already advocated for in cottrell2019comments . This Legendre transformation may be written as
(43) 
With this transform of the partition function, the deformed object is given merely by
(44) 
Performing the inverse Legendre transformation to obtain again a partition function, we find that
(45) 
Finally, integrating out gives us
(46) 
This is exactly the Freidel kernel (2). ^{11}^{11}11In going to the second line, we dropped a dependent prefactor, which could naively affect the flow equation. This constant may be renormalized away via a cosmolomogical constant counter term polchinski1986evaluation .
In two dimensions, the above manipulations mainly serve to gain intuition about the deformation. In section 3, we show the classical equations of motion of the form of the Kernel give precisely the relation between Dirichlet boundary conditions at a cutoff surface and mixed boundary conditions at the asymptotic boundary of found in Guica:2019nzm .
2.5 Further Properties of the Kernel
In this section, we show some simple but important properties of the kernel. The first corresponds to the conservation and symmetry of the stress tensor, the second to the composition property of the integral kernel.
We first show the local stress tensor is conserved and symmetric by exhibiting the partition function is invariant under diffeomorphisms and local rotations,
(47) 
It will be crucial for these proofs that the measure is invariant under both diffeomorphisms and local rotations. Let us first show that the partition function is invariant under diffeomorphisms.
(48) 
where in going to the second line, we simply relabeled the integration variable . The third equality, however, requires diffeomorphism invariance of the measure and the action (what appears in the exponents), along with the fact that .^{12}^{12}12 This "trick" might be most familiar from the FadeevPoppov procedure used for gauge theory path integrals. The second line in (47) can be shown similarly, using rotational invariance of the measure.
Next, we show that the kernel composes, that is
(49) 
The dependent part of the action in (49) is
(50) 
We see that the dependence on the intermediate vielbein is quadratic; in gravitational path integrals, this does not always mean that one can integrate it out in the usual way. However, because of the existence of a translationinvariant measure, in this case one can.^{13}^{13}13Yet again, we thank A. Tolley for pointing out the existence of this measure to us; this was a bit of a puzzle. One then finds, as expected,
(51) 
This also explains why this form of the deformation looks like the simplest way to “exponentiate” the Cardy argument, which gives for an infinitesimal deformation
(52) 
Since this form composes, it naturally retains its form at finite deformation as well.
2.6 Spatially Varying Deformations
One generalisation comes from merely increasing the number of flow equations to match the number of WdW equations. Allowing to vary spatially, . the kernel becomes
(53) 
By the same logic as above, it solves the local flow equation
(54) 
The WdW equation is no longer as simple, because of the spatial dependence of .
Indeed, the proof of the WdW equation proceeds similarly, except that the diffeomorphism SchwingerDyson equation now gives
(55) 
An interesting point about this deformation is that the spacetime dependence of causes the deformed stress tensor to not be conserved.
(56) 
All the manipulations in (LABEL:eqn:wdwcheckpolyterms) still hold, with as do (30) and (31). Therefore, the only change to (23) and (24) is to shift the curvature piece as
(57) 
2.7 Relation to Flat Space “JT’Gravity” Proposal
The beautiful work of Dubovsky et al. Dubovsky:2012wk ; Dubovsky:2017cnj ; Dubovsky:2018bmo provided the impetus for our own. The authors of Dubovsky:2018bmo in particular, whose initials abbreviate to DGHC, precisely recast the deformation in flat space as the coupling of the undeformed theory to 2d topological gravity. They explicitly computed the torus path integral. It reproduced the dressing of the energy levels known from solving the relevant inviscid Burger’s equation. The narrative there focused around a variant of JackiwTeitelboim gravity, dubbed JT’, with an action of the form
(58) 
The dilaton serves as Lagrange multiplier to ensure . The cosmological constant term is taken to be inversely proportional to the deformation parameter. Switching to a first order formalism requires Lagrange multipliers enforcing the torsionlessness constraints, leading to the action
(59) 
The important insight was the Lagrange multipliers could be viewed as providing a map from the base space to the target space on which the deformed theory lived. However, to arrive at the final form of the path integral they computed, they needed i) to neglect any possible holonomies of the spin connection on the torus (the dilaton sets , but not ), ii) supplement the action by an additional term to match the deformed energies,^{14}^{14}14In their notation, a term proportional to , which does vanish as total derivative since winds. and iii) limit the field range of the Lagrange multiplier zero modes giving rise to an important factor of the target space torus’ area. This suggests the coupling to JackiwTeitelboim gravity is not the actual root of the DGHC integral kernel.^{15}^{15}15We should point out that we are by no means the first people to notice this fact, see for example VictorsTalk
The better way to think of it is as a particular form of the kernel. We showed in section 2.5 that ; in other words, that the deformed theory’s partition function is invariant under target space diffeomorphisms (generated by the vector field ). We can therefore average over all target space diffeomorphism as long as we also divide out by the volume of diffeomorphism group
(60) 
When , or alternatively , as in DGHC, diffeormorphisms are rather simple. Under a general coordinate transformation , the vielbein transforms as
(61) 
Hence, while this may at first look like an infinitesimal version of a diffeomorphism, we maybe in fact alternatively parameterize the full as . This are related to the full diffeomorphism via . We may then rewrite (60) as
(62) 
which is none other than the DGHC kernel (cf. their Eqn. 28, with ).^{16}^{16}16Technically, their measure differs from ours. In appendix A, we show how the two ultimately give rise to the same path integral.
One might worry that what we have introduced in the denominator is not the volume of the base space diffeomorphisms but those of the target space; further, the measure for the integral over target space diffeomorphisms in the numerator is defined with respect to the target space metric. It turns out that these two subtleties cancel out. There are two ways the measures can have metric dependence: the range, and the inner product the measure is defined with respect to. Since the s are coordinate redefinitions, their range is defined by the coordinate range and so doesn’t depend on the metric. The measures in (60), however, are defined with respect to the inner product,
(63) 
However, this dependence cancels between the numerator and the denominator. Any one form in two dimensions can be written in terms of two scalars and zeromodes valued in the first homology group of the manifold,
(64) 
For the nonzeromodes that can be parametrised by the scalars, it is shown in appendix B that there is no metric dependence leftover. For the zeromodes, the metric dependence cancels between the numerator and denominator. Thus, we may as well write both integrals with respect to the base space and recover the flat space kernel of Dubovsky:2017cnj ; Dubovsky:2018bmo . This provides an important check on our proposal. It shows that gives the right answer for the case of the torus. In particular, our kernel therefore reproduces the correct dressing of the energy levels known otherwise from solving the inviscid Burger’s equation.
3 The 3D Story: Radial Wheelerde Witt Wavefunctions
In this section, we explore the deep connection between and 3d gravity. The crux of the story is that solutions of a modified Wheelerde Witt equation also furnish solutions to the flow equation.^{17}^{17}17The other way is not obvious at all. This equation is best viewed as the Schrœdinger equation in a 2+1 decomposition of the gravity theory. The wavefunction depends on the metric on each twodimensional slice in the same way the partition function depends on the geometry on which it is defined.
We give a highlevel overview of the difference in interpretation between the 3d and 2d understandings of the same objects in this table.
2d Object  3d Gravity Interpretation 

Partition function  Radial Wavefunction 
Trace Flow equation  Zeromode of Hamiltonian WdW equation 
Deformation kernel  Annular path integral 
Expectation value of stress tensor  Action of operator conjugate to metric 
  Expectation value of operators 
Global diff and Lorentz symmetries  Gauge constraints 
Legendre Transform  Change of basis 
3.1 2+1 Decomposition in FirstOrder Formalism
To keep this paper rather selfcontained, we briefly review the 2+1 decomposition of 3d gravity in firstorder formalism. The brief section V of Freidel:2008sh informed much of the discussion below. We also found Carlip:2004ba helpful for a more detailed treatment.
Instead of the secondorder metric variables, the local frame fields or vielbeins form the fundamental degrees of freedom. We explain how this quantum mechanical problem reduces to a system of constraints which the wavefunction must satisfy. Each constraint has a clear geometrical interpretation. The Hamiltonian WheelerdeWitt equation is often also called the refoliation constraint. It points out the arbitrariness in our 2+1 split of the 3d geometry and dictates how the wavefunction must transform under different slicings of the full geometry. The two other relevant constraints correspond to diffeomorphism invariance on the 2d slice and, because we are using firstorder variables, the freedom to make local 2d Lorentz transformations (rotations in our Euclidean setup).
We can write the 3d metric in terms of the vielbeins via the standard relation
(65) 
The spin connection is defined as
(66) 
A peculiarity of 3d is that we may define a one indexed spin connection using the LeviCivita symbol . In terms of these variables, the Einstein Hilbert action for a 3d spacetime with negative cosmological constant reads (after a rescaling of the vielbeins):
(67) 
where is Newton’s constant and is the radius of curvature.
Looking towards a Hamiltonian analysis, consider now a foliation of the 3d geometry by 2d submanifolds (the equal “time” slices of canonical quantisation), . Using a locally adapted coordinate system with normal direction labeled by a coordinate and coordinates on the 2d slice, we decompose the vielbeins and spin connections as:
(68) 
In terms of these, the action becomes