Tweaking one-loop determinants in AdS3

We revisit the subject of one-loop determinants in AdS3 gravity via the quasi-normal mode method. Our goal is to evaluate a one-loop determinant with chiral boundary conditions for the metric field; chirality is achieved by imposing Dirichlet boundary conditions on certain components while others satisfy Neumann. Along the way, we give a generalization of the quasinormal mode method for stationary (non-static) thermal backgrounds, and propose a treatment for Neumann boundary conditions in this framework. We evaluate the graviton one-loop determinant on the Euclidean BTZ background with parity-violating boundary conditions (CSS), and find excellent agreement with the dual warped CFT. We also discuss a more general falloff in AdS3 that is related to two dimensional quantum gravity in lightcone gauge. The behavior of the ghost fields under both sets of boundary conditions is novel and we discuss potential interpretations.


Discussion 33
A BTZ black hole in various coordinates 36 We evaluate the one-loop contribution to the Euclidean path integral expanded as (1.1) Here g should be viewed as a collection of fields including both metric and matter fields, and S[g] is the corresponding Euclidean action for these fields. g ⋆ corresponds to a classical saddle point around which we carry out a perturbative expansion in . S (0) corresponds to a tree level contribution, S (1) is the one-loop contribution and so forth. We will focus on S (1) exclusively. This contribution is controlled by suitable determinants of Laplacian operators (including any Fadeev-Popov determinants arising from gauge fixing); schematically we would write where the ± refers to whether the determinant appears in the numerator (for fermions) or denominator (for bosons). As we mentioned above, there is an extensive literature on evaluating these determinants with Dirichlet boundary conditions, with one of the most canonical methods being the heat kernel. 1 The heat kernel technique can be extended to include Neumann boundary conditions, as recently done in [11,12] for higher dimensional AdS d . The problem we face here is a mixture of both Dirichlet and Neumann, and while there might be a systematic way to adjust the heat kernel method to this setup, we will take a different route.
The different route we will pursue is to tweak the quasinormal mode method proposed by Denef-Hartnoll-Sachdev (DHS) [13]. As we will review in section 2, the original proposal of DHS is based on analyticity: this leads to a concise expression for the functional determinant in a thermal geometry in terms of a product over quasinormal frequencies. The appearance of quasinormal frequencies in the product is directly tied to requiring Dirichlet boundary conditions for each component of the field in question. This feature allows us to tweak the DHS method to our agenda: by imposing instead Neumann boundary conditions on certain components of the graviton we will modify the spectrum of frequencies that enter in the functional determinant. This procedure will give us the control to treat each component of the metric individually as we implement the boundary conditions we are interested in.
There are two types of chiral boundary conditions we will study: CSS [14,15], and sl(2, R) Kac-Moody boundary conditions [16,17]. Both of these boundary conditions are characterised by allowing one piece of the boundary metric to fluctuate, while other components are fixed. In a nutshell the main features of these setups are: CSS: These boundary conditions for AdS 3 are chosen such that the global symmetries inside the resulting asymptotic symmetry group become SL(2, R) L × U (1) R instead of SL(2, R) L × SL(2, R) R . This smaller group of symmetries is motivated by the near horizon symmetries of extremal black holes. Implementating this condition leads to parity-violating boundary conditions, and as a result the sl(2, R) L × sl(2, R) R isometries of AdS 3 are only enhanced to a left-moving Virasoro-U (1)-Kac-Moody algebra, with central charge c and level k. A field theory with such a geometrical Virasoro-Kac-Moody structure is known as a Warped Conformal Field Theory (WCFT) [18,19].
sl(2, R) KM: This is a generalization of CSS that allows for more structure in the leading metric components while still being parity-violating. As a result the sl(2, R) L × sl(2, R) R isometries of AdS 3 are enhanced to a left-moving Virasoro plus an sl(2, R) Kac-Moody algebra at level k = c/6. Unlike in CSS, one can improve the stress tensor such that we have zero central charge, and the Brown-York stress tensor vanishes. Thus this setup in AdS 3 is dual to a two dimensional quantum gravity in lightcone gauge, as elegantly argued in [17], and not a conformal theory.
Since these boundary conditions are chiral (left-moving) in nature, to highlight their features we will need to implement the DHS method for stationary (not static) thermal backgrounds, i.e. for the Euclidean continuation of the rotating BTZ black hole. 2 As we evaluate the determinants in section 3 and interpret them in section 4, the addition of rotation will make evident that our derivations are unambiguously compatible with the dual description.
One of the most unexpected features in our derivations is the role of the ghost determinant contained in the graviton one-loop effective action. The role of the ghost fields is to remove states with zero energy from the path integral, i.e. to remove gauge redundancies. For Dirichlet boundary conditions one can see this explicitly after implementing the DHS prescription, and it is also in complete agreement with the heat kernel method. However, for the chiral boundary conditions we will use, the treatment of ghosts is more subtle: there are a priori two different conditions one can impose on eigenfunctions of the ghost, which dramatically change the resulting determinant for the graviton and its holographic interpretation. We will discuss these differences from the gravitational perspective (see section 3.3), and show how they affect the holographic interpretation in section 4.2.2.
Our emphasis throughout will be on meromophic properties (the pole structure) of the one-loop determinant in AdS 3 . There is in addition an entire function (a function that is holomorphic and 2 The existing literature on using the DHS method is limited to static thermal backgrounds. The addition of angular momentum is not dramatic, but worth showing explicitly. has neither poles nor zeros) which we will not evaluate. Its purpose is to account for zero modes and contribute to the renormalization of various couplings. We will highlight in the main text when and where we are neglecting this piece and explore its role further in the discussion section.
The outline of this paper is as follows. In section 2 we revisit the DHS method with Dirichlet boundary conditions for rotating BTZ, explicitly implementing the method on a stationary background. In section 3 we consider chiral boundary conditions for the graviton, and evaluate the one-loop determinant on stationary backgrounds. The holographic interpretation of these determinants is discussed in section 4. We close with a brief discussion in section 5. Appendix A contains our conventions for the BTZ background, in appendix B we give a detailed study of the spin-2 fluctuations, and in appendix C we describe the ghost spectrum.

Quasinormal mode method: Rotation
In this section we show how to implement the DHS prescription in spacetimes which are rotating, i.e. they are stationary but not necessarily static. We begin with a generic discussion and then provide concrete examples for one-loop determinants of massive and massless fields on the rotating BTZ black hole background.
The main proposition of DHS [13] is a formula for functional determinants in a thermal geometry, written as a product over quasinormal frequencies. Their arguments rely on the assumption of meromorphicity of the determinant in the mass parameter. For example, consider the one-loop determinant on a thermal background, such as a Euclidean AdS black hole. 3 For a complex scalar field we have (2.1) If Z (1) (∆) analytically continued to the complex ∆ plane is a meromorphic function, then it can be characterized by the locations and degeneracies of its poles and zeros, as well as its behavior at infinity. For a scalar, Z (1) ∝ det −1 , so there are no zeros. Poles occur when det = 0, which happens whenever ∆ is tuned such that a zero mode of ϕ exists.
In Euclidean space, zero modes 4 are solutions of the Klein-Gordon equation which are smooth, obey the given asymptotic boundary conditions, and are single-valued in the Euclidean time direction. We denote these solutions by ϕ ⋆,n where n labels the mode number in the Euclidean time direction, and ⋆ labels all other quantum numbers characterizing the solution. A given ϕ ⋆,n will solve the Klein-Gordon equation only when ∆ is tuned to a particular value dependent on these quantum numbers; we call this value ∆ ⋆,n . Thus from the Euclidean perspective, poles in Z (1) occur at all ∆ = ∆ ⋆,n ; if multiple sets of quantum numbers give Klein-Gordon solutions with the same value of ∆ ⋆,n then the pole is accordingly of higher multiplicity.
The key insight of DHS is to relate the Euclidean zero modes ϕ ⋆,n to Lorentzian quasinormal modes via Wick rotation. The Euclidean thermal spacetime Wick-rotates to a black hole spacetime.
From this Lorentzian perspective, (anti)quasinormal modes are solutions to the Klein-Gordon equation satisfying (out)ingoing boundary conditions at the black hole horizon, as well as normalizable asymptotic boundary conditions. These modes can be characterized by their (anti)quasinormal frequencies ω ⋆ (∆), where ⋆ represents the spatial quantum numbers. Importantly these frequencies depend on ∆, and we find For n < 0, the Euclidean modes instead match onto outgoing quasinormal modes (or antiquasinormal modes). For the "constant" modes with n = 0, one can work with either in or outgoing quasinormal modes.
Consequently, if we know all of the quasinormal and antiquasinormal frequencies as a function of ∆, we know the poles in Z (1) (∆) will be located where ∆ is tuned such that ω ⋆ (∆) = ω n . We can now write the determinant for the complex scalar as Here the product is over all quantum numbers that control the (anti)quasinormal frequencies, denoted succinctly by "⋆." We have also included an entire function (that is, a function that is holomorphic and has neither poles nor zeros), via e Pol(∆) where Pol(∆) is a polynomial with only positive powers in ∆. We can determine this polynomial separately. 5 In this section we want to implement (2.3) for stationary backgrounds, and in particular rotating black holes. The minor tweak we need to implement is to revisit the Euclidean regularity condition, which affects the thermal frequencies ω n ; the second equality in (2.2) will change. For a suitable radial coordinate R and Euclidean time coordinate T E , the metric near the horizon will take the form in a similar fashion as for the static solution. However, for a rotating background at temperature T and angular velocity Ω, the Wick rotation to Lorentzian signature is generically of the form where φ is the axis of rotation of the black hole. This implies that regularity of the fields at R = 0 will impose a condition on quantum numbers conjugate to both ∂ t and ∂ φ .
In the following we will work out explicit examples to illustrate this modification.

Example: real scalar field on BTZ black hole
As a warmup, in this subsection we evaluate the one-loop determinant for a massive real scalar field on the rotating BTZ black hole with Dirichlet boundary conditions. This should be contrasted with the static case done in [13]; see [4,5] for a derivation using heat kernel methods.
To start, we impose Dirichlet asymptotic boundary conditions on scalar field solutions for large values of r. Here we have written the Fourier mode with frequency ω and wave number k, as appropriate for the coordinate system (A.1). 6 In addition, periodicity in the φ coordinate restricts the wave number k to take values over all of the integers. Now, let us consider the behavior of the Lorentzian solution for the scalar field near the horizon, The dependence on k T is set by the equations of motion, where k T is defined as the frequency conjugate to the coordinate 7 T as specified in (B.7). For general values of ω and k, solutions satisfying the boundary conditions at r → ∞ will have both of the (r − r + ) ±i k T 2 behaviors near the horizon. Solutions which satisfy only one of the behaviors in (2.6) occur only at specific quantized values of the frequency ω; depending on the sign of k T in (2.6) these are the quasinormal and antiquasinormal frequencies.
Wick-rotating to T E = iT and changing to the regular Euclidean coordinates (A.7) near ξ = 0, the solutions in (2.6) become (2.7) Regularity of these solutions requires that k T = in, where n is any integer. Additionally if n ≥ 0, we must have only the ξ −ik T behavior; if n ≤ 0 we instead have ξ +ik T . 8 Choosing only one of these signs in the Lorentzian solution (2.6) amounts to choosing either ingoing (for n ≥ 0) or outgoing (for n ≤ 0) conditions at the horizon; thus, the solutions we are interested in should be either quasinormal or antiquasinormal modes.
In addition, the requirement to have k T = in forces the (anti)quasinormal frequency ω to take the specific value ω n : Next, the quasinormal frequencies of a real scalar field on the background of a rotating BTZ black hole are [20][21][22] ingoing outgoing The range of k is all integers, and p is a nonnegative integer. Implementing (2.3), the one-loop determinant of a scalar field on the background of rotating BTZ becomes where ω n is given by (2.8). Note that we want the determinant for a real scalar, hence the square on the left hand side of (2.10). The first line in (2.10) corresponds to the ingoing modes hitting thermal frequencies with n > 0, the second line are the outgoing modes and thermal frequencies with n < 0, and the last line corresponds to the zero modes with n = 0. After plugging in ω n and a bit of algebra, we have (2.11) Next, we regulate the product over k by using the formula which, up to a redefinition of Pol(∆), turns (2.11) into 9 e Pol(∆) where we defined 10 q ≡ e −2π(2πT L ) ,q ≡ e −2π(2πT R ) . (2.14) Rewriting (2.13), the answer for the one-loop determinant of a real scalar is in complete agreement with [4,5], and with [13] for the static solution. Note that despite appearances, (2.15) is equal to (2.13). One heuristic way to see this is as follows: the first product in (2.13) corresponds to ℓ > ℓ ′ , the second product is ℓ < ℓ ′ and the last product is ℓ = ℓ ′ . 9 The k = 0 terms in (2.11) conveniently cancel the various terms that appear due to the denominator of (2.12) which are not entire functions of ∆. 10 In terms of the complex structure τ , we would have q = e 2πiτ andq = e −2πiτ , where τ = 2πiTL andτ = −2πiTR. Note that in Euclidean signature (TL) * = TR since r− is purely imaginary.
To fully specify the one-loop determinant one should also determine the e Pol(∆) factor in (2.15).
This term corresponds to a local renormalization of the classical action and can be computed independently in a suitable large-∆ limit, for example by using heat kernel techniques as described in [13]. This result can then be matched to the large-∆ limit of expressions such as (2.15) to determine Pol(∆). In this paper we are specifically interested in the properties of the infinite products that occur in the one-loop determinant, such as that in (2.15). The location of the poles that occur in these products are independent of Pol(∆) and so we will often drop the e Pol(∆) factor completely. In the remainder of this paper, expressions for one-loop determinants should be understood to correspond to the determinant modulo these local renormalization terms. We will only comment on Pol(∆) in cases where determining it may be subtle.

Example: spin-2 fields on BTZ black hole
As a second example we would like to illustrate how to evaluate the one-loop determinant for spin-2 fields, both massive and massless. References [6,23] discuss this evaluation via the quasinormal mode method for the static case, and we follow closely their analysis of the Fronsdal equations. We add the evaluation of the determinants for rotating backgrounds, and an improved discussion on how to identify the set of frequencies ω ⋆ that control the poles of Z(∆).
Following [6], a massive spin-2 excitation h µν in AdS 3 satisfies the first order equation where the sign of m 2 controls the helicity of the field. Using the equations for both helicities, it follows that such a field satisfies the more familiar Fronsdal equations: The determinant we will evaluate is We emphasise that ∇ 2 in (2.18) is acting on a symmetric, traceless and transverse tensor. In this section we evaluate the determinant for standard (Dirichlet) boundary conditions: the leading divergence 11 of the zero modes at the boundary is required to vanish, which is the usual condition for quasinormal modes in AdS. We provide a detailed derivation of the spin-2 quasinormal modes, as well as the mapping to regular Euclidean solutions, in appendix B. Here, we only quote the results for the quasinormal mode spectra; for spin-2 these are in Table 1.
ingoing outgoing 2ik L = −(2p + ∆ 2 + 2) Table 1: Spin-2 quasinormal mode spectrum ω⋆ after imposing standard Dirichlet boundary conditions. When ∆2 = 2 these correspond to the symmetric, transverse, traceless graviton spectrum. Each condition on kR or kL labels a distinct eigenmode and the range of p is over all non-negative integers.
Note that we are parameterizing the quasinormal frequencies in terms of the quantum numbers (k L , k R ) as defined in (B.7), which are conjugate to the coordinates (x L , x R ) in (A.11). In the following we will also use (k T , k Φ ) whose conjugate variables are (T, Φ) in (A.5).
Next, we need to match the quasinormal frequencies to the thermal frequencies, i.e. ω n = ω ⋆ .
Additionally, some of the quasinormal modes with low p and n Wick-rotate to Euclidean modes that diverge at the tip of the Euclidean cigar, so they should be excluded. The relations defining the good Euclidean solutions are enumerated in Appendix B.3, and are reproduced in the following table: As in the previous example, n is defined by the regularity condition at the Euclidean origin, which fixes −ik T = n. Each set of conditions corresponds to a union of the Wick-rotation of a set of ingoing and outgoing states. Ingoing modes correspond to n > 0 and outgoing modes to n < 0, with n = 0 being the zero mode. The frequency k Φ is restricted by the periodicity of the field in the thermal and spatial directions, which is controlled by integers n and k respectively. The relation is We can construct the determinant directly from this information. Consider first the m 2 > 0 states: the conditions from the top row in Table 2 can be written as Relabelling n =ñ − 2 and treating each sign separately, we have Performing similar steps for the m 2 > 0 states on the bottom row in Table 2, using instead n =ñ+2, we find We can compare these conditions with those imposed on the zeros of the expression (2.11). If one makes the replacement , (2.24) where h is the weight of the spin-2 field, given by It is now straightforward to also read off the contribution from the m 2 < 0 states. Since the only difference from the m 2 > 0 case is on the sign of k Φ , the m 2 < 0 result will be the same but with the opposite shift .
Putting it all together we arrive at the entire one-loop massive spin-2 determinant . (2.28) This agrees with the results in [4,5], which were derived using heat kernel methods, and with [6] when the rotation is turned off. We will postpone the holographic interpretation of these determinants to section 4.

Graviton determinant
In this section we are interested in the standard Dirichlet boundary conditions for the graviton, which corresponds to allowing only fluctuations which fall off at least as fast as where the denominator is the determinant for symmetric, tranverse and traceless rank-2 tensors and the numerator is the determinant for transerve vector fields. These determinants correspond to fields with physical mass values m 2 2 = 1 for the graviton and m 2 1 = 4 for the ghost; the corresponding conformal dimensions are Let us first evaluate the numerator in (2.30) for spin-1 fields with arbitrary ∆ 1 and standard boundary conditions. The quasinormal mode spectrum of a vector field in AdS 3 is derived in appendix C; the resulting frequencies are listed in the table below. ingoing outgoing Table 3: Spin-1 quasinormal mode spectrum after imposing standard Dirichlet boundary conditions. When ∆1 = 3 these correspond to the spectrum of transverse ghost modes which appear in the graviton one-loop determinant (2.30). Each condition on kR or kL labels a distinct eigenmode and the range of p is over all non-negative integers.
We can derive the spin-1 contribution to the determinant similarly to the spin-2 case (2.28). The general result for the determinant of a massive spin-1 field is .
It is now straightforward to put together the complete graviton determinant in (2.30). The contribution of the spin-2 tensor determinant is given by setting h = 0 in (2.28), giving .
Inserting this value and taking the ratio in (2.30), we find .
There is a simple way to derive this final result without going through the process of constructing each determinant in (2.30) explicitly. In particular, consider the quasinormal mode spectra in Tables  1 and 3. Evaluating the conditions in these tables at ∆ 2 = 2 and ∆ 1 = 3, we see that almost every spin-2 mode has a corresponding spin-1 ghost mode which satisfies the same condition. These modes will cancel when taking the ratio in the graviton determinant (2.30). The only contributions which do not cancel are the spin-2 states at p = 0 which satisfy As described in Appendix B.3, one has to be careful with the Euclidean rotation of these states. In particular, as described in Appendix B.3, in order to ensure that these Euclidean solutions are regular at the origin, the thermal quantum number n should run only over a restricted set of values.
Taking these restrictions into account and performing the sum we can directly recover (2.34). This analysis demonstrates that the physical states that contribute to the graviton determinant come from either purely left-moving or purely right-moving states. This also explains the factorization in (2.34), as the condition 2ik L = 0 yields theq-dependent product in (2.34) while the condition 2ik R = 0 yields the remaining q-dependent part.

Quasinormal mode method: Chiral boundary conditions
We now move on to a further generalization of the DHS prescription, which will be the main focus of this article. The boundary conditions satisfied by quasinormal modes in the asymptotically AdS region correspond to Dirichlet boundary conditions. These are natural as they require fields to fall off in a prescribed way near the boundary such that small on-shell perturbations have a finite energy [29]. However, certain types of fields in asymptotically AdS space-times allow for more general boundary conditions. For example, scalar fields with mass close enough to the Breitenlohner-Freedman bound can be quantized with Dirichlet or Neumann boundary conditions and still yield finite energy excitations [29,30]. 12 Similarly, massless gauge fields, gravitons and higher spin fields can be quantized with Dirichlet or Neumann boundary conditions [11,[31][32][33][34].
Below we will consider particular boundary conditions on the bulk metric which are a mixture of Dirichlet and Neumann boundary conditions.
The goal of this section is to use a simple modification of the DHS argument to construct the one-loop determinant for the three dimensional graviton for cases where certain components of the metric satisfy Neumann boundary conditions while others satisfy Dirichlet. As discussed in the previous section, the assumption that the one-loop determinant is meromorphic as a function of ∆ implies that poles of the one-loop determinant occur whenever a quasinormal mode satisfies equation (2.2). Our application of this method instead requires that we enforce Neumann boundary conditions for certain metric components. Our crucial working assumption is that these new boundary conditions will similarly quantize the frequency of ingoing (and outgoing) solutions such that poles of the determinant will now occur whenever the regularity conditioñ holds. 13 Hereω ⋆ (∆ ⋆,n ) refers to the quantized frequencies associated to ingoing (and outgoing) solutions which satisfy the prescribed Dirichlet-Neumann boundary conditions for each component at infinity. These will in general be different from the standard quasinormal frequencies. That the second equality in (3.1) is unmodified relative to (2.2) follows simply because the near-horizon analysis is independent of the asymptotic boundary conditions. In what follows we will refer to the frequenciesω ⋆ (∆ ⋆,n ) simply as quasinormal and also drop the tilde. In addition, we will utilize the more general prescription discussed in Section 2.1 appropriate to stationary but not necessarily static spacetimes.
We begin by reviewing the details of the various boundary conditions for the metric that we will consider; then we move to a direct calculation of the one-loop determinant of the graviton (including its ghost contributions) following the philosophy discussed above.

Chiral boundary conditions in AdS 3
We consider boundary conditions on metric fluctuations in asymptotically AdS 3 spacetimes which correspond to imposing Dirichlet or Neumann conditions on different components. In three dimensions it is natural to formulate a type of chiral boundary condition in which the left-moving components of the boundary metric are allowed to fluctuate (Neumann), whereas the right-moving components are held fixed (Dirichlet). Such chiral boundary conditions were initially proposed by Compere, Song and Strominger (CSS) in [14]; see also [15]. By additionally restricting the boundary metric to have purely left-moving coordinate dependence, CSS demonstrated that these boundary conditions modify the asymptotic symmetry algebra from a product of left and right-moving Virasoro algebras to a purely left-moving Virasoro plus U (1) Kac-Moody algebra. Following [14], the authors in [16] realized that the left-moving coordinate dependence of the boundary metric in CSS could be relaxed. The resulting boundary conditions enhance the asymptotic symmetry algebra of CSS to an sl(2, R) Kac-Moody; as such we will refer to these simply as "sl(2, R) KM" boundary conditions.
For both boundary conditions, the starting point is pure AdS 3 gravity; the action is given by where the AdS radius is set to one. We consider a class of backgrounds which have the following asymptotic behavior: Here t ± = t ± φ with φ ∼ φ + 2π and m is a fixed constant. The Einstein equations impose some , and L(t + , t − ); the remaining freedom on these functions is controlled by boundary conditions, which we will elaborate on below. In this notation, the BTZ black hole with mass M and angular momentum J corresponds to where L 0 is constant and m > 0. Global AdS also falls into the restrictions in (3.4) upon setting

CSS boundary conditions
The chiral boundary conditions of CSS [14] require that the boundary metric component g ++ depend only on the left-moving coordinate t + , such that On-shell this condition implies similar restrictions on the other metric functions: and L(t + , t − ) = L(t + ). Furthermore, the equations of motion also imply The resulting metric has the asymptotic form The r-dependence of the allowed fluctuations of the metric under diffeomorphisms becomes In other words, the allowed diffeomorphisms leave m and the leading term of g +− fixed, whereas the functions P (t + ) and L(t + ) are allowed to fluctuate. Note that if we do not allow fluctuations of ∂ + P (t + ), this analysis boils down to the holomorphic sector of the Brown-Henneaux boundary conditions.

sl(2, R) KM boundary conditions
A consistent extension of the CSS boundary conditions is to loosen the constraint while still holding m fixed [16]. In particular, by relaxing the falloff of g r+ in (3.8) such that , one finds that the Einstein equation constrains the t − dependence of the function which is solved by where h(t + ), g(t + ), andḡ(t + ) are arbitrary functions of t + and The remaining functions in the metric are constrained by the form of h(t + , t − ). In particular, where L(t + ) is an arbitrary periodic function of t + andL(t + , t − ) is determined by h(t + ), g(t + ), andḡ(t + ). We refer the reader to [16] for the full details. The important piece of information for us is that the radial falloff of the allowed diffeomorphisms for these boundary conditions are In the rest of this section we will use the DHS method to compute the one-loop determinant for both the CSS and sl(2, R) KM boundary conditions. We will in particular focus on imposing the radial falloff conditions in (3.8) and (3.14) and will then analyze the consistency with the chirality conditions on h(t + , t − ) given in (3.5) and (3.11).

Modified spin-2 determinant
In order to implement the DHS procedure for the boundary conditions discussed in section 3.1, we will first understand how the Neumann boundary conditions for δg ++ translate to boundary conditions on a massive spin-2 field and compute the corresponding determinant. For the massless case, we will also add a detailed discussion of the ghosts for both CSS and sl(2, R) KM boundary conditions, highlighting subtleties that appear relative to the standard scenario in section 2.2.1.

Massive spin-2 with chiral boundary conditions
Our starting point is to specify how the chiral boundary conditions in (3.8)  Near the boundary, a massive spin-2 field has the expansion where for conciseness we are only considering the relevant components to understand the chiral boundary conditions. It is worth mentioning that, according to the standard AdS/CFT dictionary, for m 2 = 1, A ++ acts as the source for the right-moving stress tensor T −− , whereas for m 2 = −1, Dirichlet B.C. : A ++ = 0 for m 2 > 0 , and To implement chiral boundary conditions we require that metric perturbations, δg µν = h µν with |m 2 | = 1, have right-moving components that fall off faster than a constant with while allowing for δg ++ to grow near the boundary. Comparing to the behavior in (3.15), the natural extension of these boundary conditions away from the massless value corresponds to For m 2 < 0, this is the same boundary condition as in the standard Dirichlet situation. However, for m 2 > 0, we are imposing Neumann boundary conditions, as we are holding T −− fixed and allowing the source to fluctuate.
The quasinormal modes 14 associated with these boundary conditions are derived in Appendix B.2.2. The end result for the quasinormal spectrum with chiral boundary conditions is given in the following table: ingoing outgoing Table 4: Spin-2 quasinormal mode spectrum after imposing chiral boundary conditions. When ∆2 = 2 these correspond to the symmetric, transverse, traceless graviton spectrum. Each condition on kR or kL labels a distinct eigenmode and the range of p is over all non-negative integers.
Here we have again organized the modes into "ingoing" and "outgoing" based on their behavior at the horizon. Notice that since the m 2 < 0 states still satisfy Dirichlet boundary conditions, the quasinormal modes in this sector are precisely the same as they were in the previous section. It is also interesting to note that the conditions on the new m 2 > 0 states in table 4 are the same conditions as those on the m 2 < 0 states upon sending ∆ 2 → 2 − ∆ 2 . This suggests that both sets of states have the same chirality and we will see this feature in the final result for the one-loop determinant. Finally, the swapping of ∆ 2 with 2 − ∆ 2 for m 2 > 0 naturally follows from the alternative (Neumann) quantization of these states.
Enumerating the Euclidean solutions in this case is very similar to the situation with Dirichlet boundary conditions. We summarize the conditions on the Euclidean spectrum in the following table. We can now compute the contribution to the one-loop determinant from all of the m 2 > 0 states in Table 5. This gives , (3.20) where h ′ = −∆ 2 /2. Putting this together with the Dirichlet result for m 2 < 0, we have .
Before moving on, we would like to comment on the e Pol(∆ 2 ) factor that we have dropped in the expression for the one-loop determinant above. In this case, the determination of this factor is potentially subtle. In particular, consider the ∆ 2 → ∞ limit of (3.21). For the second factor, which arises from the m 2 < 0 Dirichlet contribution, taking ∆ 2 → ∞ is straightforward. However, in the first (Neumann) factor it appears that one should instead take ∆ 2 → −∞ in order for the limit to commute with the product over (ℓ, ℓ ′ ). Perhaps this could be expected to be the case since the alternative quantization is naturally phrased in terms of ∆ − = 2 − ∆ 2 , and taking ingoing Additional spin-1 ghost states that are consistent with the Neumann conditions on g ++ , but are not contained in the Brown-Henneaux states. The first two lines correspond to new m 1 > 0 states, whereas the k R = 0 states arise both in the m 1 > 0 and m 1 < 0 sectors.

The graviton one-loop determinant
We now construct the graviton one-loop determinant for CSS and sl(2, R) KM boundary conditions from the results for the massive spin-2 determinants. As in (2.30), we need to evaluate The denominator is straightforward to obtain from the massive case: we just set ∆ 2 = 2 in (3.21).
The numerator, which is the contribution from the ghost fields, is more delicate: results vary depending on whether we impose the boundary conditions on the vector field itself or on the metric perturbation they induce as we will show in the following.

The ghost contribution
It turns out that we have already determined most of the ghost contribution to (2.30). In particular, as detailed in Appendix C, for the ghost fields the standard Dirichlet boundary conditions are already consistent with the new chiral boundary conditions. This means that the states in Table  3 will contribute just as they had in the case with Dirichlet boundary conditions. There are, however, several additional sets of quasinormal modes which satisfy chiral boundary conditions, but not Dirichlet. These are given in Table 6. As we will discuss, whether or not we include these extra modes will play an important role in what follows.
As explained in Appendix C, when considering the spin-1 states at the value of the ghost mass, corresponding to ∆ 1 = 3, there are special states that appear in the second and third rows of Table   6 that are actual zero modes of the ghost Laplacian, which locally satisfy the Killing equation. 15 In particular, these occur for |k E | = 1 in the Euclidean solutions for the m 1 = 2 states in the second row as well as the m 1 = −2 states in the third row of Table 6. Since these are zero modes of the ghost Laplacian they will only appear in Pol(∆ 1 ) but not in the poles of Z (1) . Taking into account all of the ghost states that induce a pole, we find where the first product corresponds to the contribution which is also included in the Brown-Henneaux analysis and the second product is from the new states in Table 6.
Before proceeding, there is an important point to make regarding which ghost modes we allow to contribute to the physical graviton determinant. The ghost is a vector field V µ that induces the gauge transformation  Table 6. Whether or not we choose to keep these modes depends on how we implement boundary conditions: 1. We could impose that V µ cannot induce a metric fluctuation (3.24) as leading as those allowed by the asymptotic symmetry group; 2. Or we could impose Neumann boundary conditions on V µ , and hence allow for large induced metric fluctuations compatible with the asymptotic symmetry group.
In what follows we will be agnostic about these ghost contributions and present the determinant for both situations. We will elaborate on the meaning of the subsequent results when we discuss the holographic interpretation of the various boundary conditions in Section 4.

sl(2, R) KM determinant
In order to distinguish between the sl(2, R) KM boundary conditions and the CSS boundary conditions, we need to consider the boundary falloff of the δg r+ components. For the sl(2, R) KM boundary conditions in (3.14), the boundary condition on δg r+ coincides with the generic behavior of a solution with Neumann conditions on δg ++ and so all of the spin-2 states enumerated above contribute to the sl(2, R) KM determinant. Next, for the ghost fields, if we require that (3.24) is subleading relative to (3.14), only the first line of (3.23) contributes. Combining these two contributions in (3.22), we find the following result for the graviton determinant . We will comment on the interpretation of this result and the expression in (3.25) in Section 4.

CSS determinant
We will now move on to construct the determinant for CSS boundary conditions (3.8 the spectrum is given by the intersection of states in Table 4 with those in Table 1 together with the k R = 0 state. When ∆ 2 = 2, we find that the resulting states are 2ik R = 2p + 4 , with p = −2, 0, 1, 2, 3, . . . , (3.28) and ik L = p , with p = 1, 2, 3, . . . . (3.29) The restricted set of states in (3.28), compared to those in Table 4, means we should remove from .
That these states are removed could have been anticipated as they correspond to fluctuations of the boundary metric function h(t + , t − ) that violate the chirality condition in (3.5). The condition (3.29) is as stated in Table 4 for ∆ 2 = 2, so no further modification of the spin-2 contribution to Notice also that there is a difference in the exponent of q in the two factors in (3.31). The origin of this can be seen by noticing that the condition in (3.32) for m 2 > 0 corresponds to the state at the p = 1 level in Table 4, as opposed to the p = 0 level as is the case for the other states in (2.35) and (3.32). As detailed in Appendix B, regular Euclidean solutions at the p = 0 and p = 1 levels allow for only a restricted set of thermal frequencies. For p = 0, the excluded thermal frequencies lead to the shift of ℓ → ℓ + 2 in the exponents of q andq in the graviton partition functions. For p = 1, the exponent is only shifted to ℓ + 1, giving the additional (1 − q) −1 relative to the other cases.
Finally, as in the sl(2, R) KM case, one can in principle include the ghost states which induce metric variations that have radial falloffs on par with the boundary gravitons. In this case this amounts to including the k R = 0 modes in Table 6. Doing so, we again find that the one-loop determinant trivializes We will elaborate on the meaning of the one-loop determinants computed here in the next section.

Holographic interpretation
We will now gather all the determinants we have evaluated in previous sections and discuss their holographic interpretation. Our aim is to highlight how to write the determinants as traces over

Standard boundary conditions
This subsection will serve mostly as review, since the interpretation was already discussed in [2,4].
The punchline in this case is that for standard (Dirichlet) boundary conditions we can interpret each determinant as the character of the two dimensional conformal group. This is in perfect agreement with the statement that these boundary conditions are precisely those behind AdS 3 /CFT 2 . The discussion here should be contrasted with the results in the following subsections.
Real Scalars: In section 2.1 we found that the one-loop determinant of a real scalar field is (4.1) As in [4], it is useful to digest a bit this answer and view it as a trace, i.e. we want to interpret where L 0 andL 0 are each elements of an sl(2) algebra, which we parametrize as and similarly forL i . Rewriting (4.1) as ℓ,ℓ ′ =0 makes the holographic interpretation of (4.1) quite straight forward. The scalar field of mass m 2 = ∆(∆ − 2) corresponds to a primary in a CFT 2 with conformal dimensions (∆/2, ∆/2).
We denote a single particular state associated to the scalar field as |h, h , with ∆ = 2h; multi-particle states correspond to multiple insertions of the operator at the origin. The state |h, h is annihilated by L 1 andL 1 and a descendent of conformal weight (ℓ + h, ℓ ′ + h) is given by The interpretation of the partition in (4.4) is now clear: the contribution for fixed (ℓ, ℓ ′ ) corresponds to the trace of multi-particle configurations of a given descendent state of |h, h .
Note that the states of the scalar operator are organized as a character of sl(2) × sl(2); the full Virasoro algebra will only be evident for the graviton determinant.

Massive Vectors & Tensors:
The result for a massive vector field in AdS 3 was derived in (2.32) and for a massive spin-2 field we found in (2.28) The conformal dimension is ∆ s = 2h + s, and a massless field has h = 0.
The trace interpretation of (4.6) and (4.7) works very similarly to the scalar case. The only difference is that the vector and tensor have two polarization states: (h, h + s) and (h + s, h). Additionally, for each polarization state we have a tower of descendants of sl(2) × sl(2) and the multi-particle state configurations.

Graviton:
We now turn to the determinant of the graviton with standard (Dirichlet) boundary conditions; the answer in (2.34) reads . (4.8) Here the interpretation deviates slightly from our previous examples. Interestingly, because it is dual to the CFT stress tensor, the graviton captures the full structure of the Virasoro group, in contrast to the global sl(2) × sl(2) as seen above. If we denote the vacuum state as |0 , the one-loop determinant (4.8) is counting descendants

Chiral boundary conditions
In the following we will give an interpretation of the graviton one-loop determinants which involved chiral boundary conditions. There are two types of falloff that we considered in section 3. As we will see below their interpretation is dramatically different and will depend on how we choose to implement the ghost determinant with Neumann boundary conditions.

CSS boundary conditions
The analysis of the asymptotic symmetry group with boundary conditions (3.8) suggests that its dual description should be in terms of a warped conformal theory (WCFT). These theories all have the following symmetry features: Given a coordinate system (x + , x − ), a WCFT is classically invariant under the transformations which is a Virasoro-Kac-Moody algebra with central charge c and level k. Here P n generate diffeomorphisms along x + in (4.11) [18,19]: this is what distinguishes a WCFT from other realisations of the Virasoro-Kac-Moody algebra. It is important to stress that this is a chiral algebra (there is noL n sector), and this chirality will be crucial as we interpret our results.
To start, let us review a few facts about unitary representations of (4.12); the discussion here is based on results in [19,38]. A primary state is defined as a state |p, h that is an eigenstate of the zero modes and is annihilated by (L n , P n ) with n > 0. Descendants are created by acting with L −n and P −n (n > 0). The trace that counts the descendants of a single primary reads Tr q L 0q P 0 = q hqp φ(q) −1 χ h (q) . (4.14) The descendants created by acting with P −n 's on |h, p are accounted by the Euler phi function while the descendants arising from the action of L −n 's are counted by an ordinary Virasoro character, χ h (q), with central charge c. We note that a descendant state does not shift the eigenvalue of P 0 and hence the character in (4.14) is holomorphic in q (up to the overall dependence ofq p ).
Finally, the global part of (4.12) is simply sl(2) × u(1): characters of this algebra will be just labelled by the sl(2) piece.
With this background, we can now proceed to interpret the determinants we evaluated in section 3.3. For the graviton we found in (3.31) the following . (4.16) This is in perfect agreement with (4.14) when the primary state is the vacuum state: the first product is counting the P −n descendants, and the second product is the Virasoro character for c > 1 with the L −1 state removed. 17 It is remarkable that the final result is holomorphic as expected from (4.14). We stress that in a WCFT, suitable warped modular transformations also relate thermal AdS and BTZ [19,39]. This relationship explains why we obtain a vacuum character when evaluating the determinant on BTZ.
As for Dirichlet boundary conditions, it is also interesting to interpret the determinant of massive fields. For instance, the massless spin-2 determinant with CSS boundary conditions is given by . (4.17) Note that this determinant does not fit with the global part in (4.12): theq dependence cannot be accounted for by the Virasoro-Kac Moody algebra. The graviton respects the symmetries expected from ASG analysis, but matter in this theory is not organized by the same principle. It is possible to obtain a result compatible with sl(2)× u(1) representations, but this requires fixing the quantum number associated to P 0 in the quasinormal mode spectrum. We find this requirement strange; for the graviton in (4.16) we did not have to implement such a constraint.
It is worthwhile to compare our result with prior literature. The original derivations [14,15] do not obtain (4.12); they obtain a non-canonical form of the algebra where the commutator of L n and P n is shifted and the level depends on the vev of P 0 (which is m in (3.7)). However, [39] argue that there is a non-local transformation that brings the algebra to the form (4.12), where k is independent of state, and modular invariance in the WCFT is restored (since P 0 can now vary). Our derivations are compatible with (4.12) and modular invariance, hence we are indirectly justifying the non-local transformation advocated in [39]. light-cone gauge [40]. This is in seeming conflict with the WCFT interpretation of the CSS boundary conditions that we have just discussed since in order to gauge the Virasoro U (1) KM symmetry the level k must be negative. The ability to treat the asymptotic symmetries as either global or gauge symmetries appears to be related to the fact that in gravity one finds the non-canonical form of the WCFT algebra with the WCFT description only emerging once one allows for the non-local transformations described in [39]. It would be worthwhile to understand this point more completely. As we will discuss in the next section, the interpretation in terms of induced gravity will be much more transparent in the theory with sl(2, R) KM boundary conditions.

sl(2, R) KM boundary conditions
The sl(2, R) KM boundary conditions are distinguished from those of CSS by relaxing the chirality condition on the boundary metric in (3.5). The asymptotic symmetry analysis for these boundary conditions was performed in [16], where the asymptotic symmetry algebra was shown to be a Generically, k and c do not have to be related: it is a feature of the gravitational setup that relates them. And in particular, this feature that in AdS 3 the level and the central charge are related in this way will play an important role in the following discussion.
As we did for the other examples, it is instructive to discuss unitary representations of the algebra. A primary of (4.18) is defined as a state |m, h; j that is an eigenstate of the zero modes J 0 0 |m, h; j = m|m, h; j , L 0 |m, h; j = h|m, h; j , (4.20) in addition to the quadratic Casimir of sl(2, R) (4.21) and is also annihilated by L n and J a n with n > 0. Descendent states are now created by acting with L −n and J a −n (n > 0). In addition, discrete representations of sl(2, R) typically fall into two classes; these are D for a representation of the type D (+) j . Comparing this to (3.25), we indeed find the appropriate structure, modulo the first factor in (4.22) which corresponds to the J + 0 descendent contribution. We recall that the product in (3.25) corresponds to the graviton determinant where the ghost spectrum is treated with Dirchlet boundary conditions (i.e. the ghost fluctuations are strictly subleading relative to the spin-2 modes).
There is, however, a problem with the above analysis, which can be seen most easily by considering the sign in front of the sl(2, R) level k in the [J 0 n , J 0 m ] commutator. Since k = c/6 is positive, representations of the current algebra (4.18) necessarily contain negative norm states. This is however not a problem: there is a natural interpretation as to why k must appear precisely as in (4.18).
The boundary theory dual to sl(2, R) KM boundary conditions is a theory of induced gravity [17].
In particular, this theory is described by a two-dimensional induced gravity in light-cone gauge as originally formulated in [43,44].
As discussed in [17], the appropriate boundary stress tensor includes the twisted Sugawara term, which amounts to a shift of the form (4.23) 18 Note that, because of the non-compact nature of sl(2, R), the representations D (±) j contain an infinite set of states generated by the zero modes of J + 0 or J − 0 . Since, in thermal AdS3, the J ± 0 correspond to global elements of the symmetry algebra these sl(2, R) descendents will not be seen in the gravity analysis for the graviton determinant.

This introduces the following shift in the Virasoro generators 19
In terms ofL n one can check that the Virasoro algebra becomes where the shift by J 0 n has lead to a cancellation between the bare central charge in (4.18) and a central term induced by the sl(2, R) level k = c/6. Note that it was crucial that the sign in front of k in (4.18) is as written, otherwise the induced central charge would not have canceled the bare central term.
Since the twisted generators satisfy a Virasoro algebra with vanishing central charge, we can gauge the diffeomorphisms on the boundary. Now, we can see that the extra ghost states were necessary in order to arrive at the result in (3.26). These extra ghost states correspond precisely to the boundary diffeomorphisms, which remove all of the Virasoro sl(2, R) KM descendent contributions, as expected when the dual 2d theory is a theory of gravity.

Discussion
In this work we have computed the pole structure of the graviton one-loop determinant in three dimensional AdS gravity with the aim of quantifying how chiral boundary conditions affect the determinant. In the following we discuss some important features of our results and some possible future directions.

Extensions of the DHS method
We extended in three directions the quasinormal mode method first developed in [13].
The first extension is the treatment of stationary, as opposed to static, spacetimes, which is required in order to distinguish between holomorphic and anti-holomorphic contributions to the graviton determinant. In the static case, poles in the one-loop determinant arise when the quasinormal mode frequencies are tuned to be proportional to the Euclidean thermal mode number.
Our primary result here is (2.8), which shows that in the rotational case the quasinormal mode frequencies must instead be tuned to a particular combination of the Euclidean thermal mode 19 We have also included a zero mode shift of − c 24 which can be thought of as arising from mapping the Virasoro generators Ln on the plane to those on the cylinderLn. number and the angular frequency. Although this particular expression is specific to the BTZ black holes we study, we expect that the derivation procedure will be similar for other stationary spacetimes in any number of dimensions.
Our second extension concerns an improved treatment of fields with spin in the quasinormal mode method. Although fields with spin have been studied previously in e.g. [6,25], in appendices B.3 and C.2 we provide a comprehensive discussion of the adjusted integer ranges required in those prior works. We show that for fields with spin, not every quasinormal mode Wick-rotates to a normalizable Euclidean mode. For quasinormal modes with quantum number at or below the field's spin, the thermal mode number may have a restricted range in order to achieve normalizability at the tip of the Euclidean cigar and thus a pole in the one-loop determinant. And although we have studied a particular example, we expect that this subtlety will generalize to any scenario where DHS is applicable.
The most obvious extension required in our work is to apply the quasinormal mode method to the case of chiral boundary conditions. These impose Neumann boundary conditions on the left-moving graviton components, while the right-moving components remain Dirichlet, as detailed in section 3.1. We applied the DHS procedure to this situation, and found reasonable results.
For massive spin fields determining the modes that contribute to the determinant requires some work but is straightforward; this is done in section 3.2 for the spin-2 field, and generalizations should follow naturally. The more interesting feature appears for massless fields and their ghost contribution which we discuss below.

Holography going wild at the boundary
We focused on two types of chiral boundary conditions, distinguished by a particular functional constraint on the boundary metric. Allowing the left-moving components of the boundary metric to vary as an arbitrary function of the boundary coordinates, one finds the asymptotic symmetry algebra contains an sl(2, R) current algebra [16]. As argued in [17] the holographic dual of these boundary conditions corresponds to two-dimensional gravity in a chiral light-cone gauge as in [43,44]. Our results for the graviton determinant in (3.26) confirm these expectations by demonstrating that the Virasoro and sl(2, R) descendants are removed from the spectrum.
The second type of boundary conditions we considered are the more stringent ones of CSS [14].
These conditions require the fluctuating boundary metric to depend only on the left-moving coordinate t + ; they produce a Virasoro U (1) Kac-Moody asymptotic symmetry algebra. As proposed in [14] the holographic interpretation in this case is in terms of a warped conformal field theory and as described in Section 4.2.1 our result in (4.16) for the graviton determinant reinforces this idea.
There are two interesting directions to explore here. One direction is to complement our analysis with the recent work in [45,46]. There a deformation of the action can be interpreted as a modification of the boundary conditions in AdS, which also provides an interesting holographic interpretation. Another direction is to explore the behaviour of one-loop determinants for other boundary conditions in AdS 3 such as those discussed recently in [47,48] and references within.

Ghosts are scary
For massless fields, such as the graviton, gauge invariance requires the introduction of ghost fields in the path integral. The Neumann nature of the chiral boundary conditions brings a subtlety to the ghost determinant as detailed in section 3.3, which we summarize here.
What are the appropriate boundary conditions for the vector ghost? We can either allow ghosts whose metric variations are on par with the allowed graviton modes, or instead require them to be purely subleading. This crucial distinction arises because allowing Neumann conditions for the ghost eigenfunctions opens up the possibility of including ghost states that actually gauge away the physical boundary gravitons.
The choice of ghost boundary conditions for the sl(2, R) KM case is rather natural. Unitarity of the boundary theory requires the inclusion of ghost states which grow at the boundary in order to cancel negative norm descendent states arising from the non-compact sl(2, R) current algebra.
For CSS, the choice of ghost boundary conditions is more subtle. An interpretation in terms of a WCFT requires that we do not allow for ghost modes which grow at the boundary, as in our result in (4.16). However, within our framework, it is apparently just as valid to include some of the Neumann ghost states which, as in the sl(2, R) case, remove the descendent states from the spectrum as we found in (3.33). A full understanding of the holographic interpretation of this case is still lacking, although the chiral Liouville gravity of [40] will likely play a role.
All of these cases illustrate the importance in defining the physical states corresponding to the boundary gravitons and identifying the appropriate conditions on the ghost eigenfunctions. For this purpose, it would be very useful to develop a gauge invariant procedure for constructing the one-loop determinant which does not require the introduction of ghost modes, but we leave this for future work.

The entire function Pol(∆)
We have chosen to study only the pole structure of the one-loop determinant and thus have ignored the e Pol(∆) factor in (2.15). The purpose of the polynomial factor is to account for zero modes and renormalization effects, including the multiplicative anomaly [49]. This choice to focus on the pole structure of the one-loop determinant alone does mean we cannot compute, e.g., the Casimir energy as noted in footnote 17.
The choice of chiral boundary conditions complicates the calculation of the function e Pol(∆) . In the case of Dirichlet conditions, this factor can be found by comparing the large ∆ behavior of the pole structure to, e.g., the large ∆ behavior required by the heat kernel curvature expansion as in [13]. In the case of pure Neumann conditions, a similar result could be found by instead studying the ∆ → −∞ limit; however, in the chiral conditions we consider, we have both Dirichlet and Neumann modes, so neither limit is easy to study. It might be possible to divide the infinite product into definite helicity sectors similar to the approach taken in [50] which factored the determinant into fixed momentum sectors, but we leave any such consideration to future work.

Acknowledgements
It is a pleasure to thank P. Betzios, F. Larsen, G. Sárosi, W. Song, and D. Vaman for useful discus-

A BTZ black hole in various coordinates
In this appendix we compile several useful coordinate systems to describe the BTZ black hole; all equations have the AdS radius set to one. We begin with the more traditional Boyer-Lindquist type coordinates where as usual we have φ ∼ φ + 2π. The inner and outer horizons are related to the mass and angular momentum via In Fefferman-Graham coordinates the BTZ takes the form where we have defined When performing the Euclidean continuation, it is most natural to make the following coordinate transformation: In these coordinates, the metric is We refer to these as regular coordinates because in terms of the Euclidean time coordinate, T = −iT E , the metric becomes simply and regularity at ξ = 0 naturally fixes the periodicity of T E to be Note that the Euclidean continuation in the coordinates (A.1) implies that r − is purely imaginary and t = −it E . We will also find it occasionally useful to further transform the radial coordinate by in which case the metric is Finally, when analyzing the massive spin-1 and spin-2 equations it is further useful to define dimensionless left-moving and right-moving coordinates in terms of which the various tensor components in the equations become diagonal. In these coordinates the metric is given by

B Linearized graviton equations and quasinormal modes
In this appendix we will derive the massive spin-2 quasinormal spectrum for Dirichlet boundary conditions, which was originally done in [6], and for chiral boundary conditions. We will also elaborate on the restrictions imposed on the Euclidean solutions, which affects the modes contributing to the determinants and are non-trivial for spin-s fields (but trivial for scalar fields).

B.1 Massive spin-2 equations
As in equations (2.16) and (2.17), a massive spin-2 excitation h µν in AdS 3 satisfies the first order which is equivalent to where we have set the AdS radius to one. To avoid cluttering, in this appendix we are dropping the subscript in m (in the main text it is denoted as m 2 ).
Using the tracelessness condition and the first order equations of motion, one can solve algebraically for the components h ξξ , h ξT , h ξΦ , and thus express the equations of motion solely in terms of the components of h µν along the 20 Our notation differs from that used in [6]. In comparison to the coordinates used there, we have (x1, x2) there = (xL, xR) here , and (x + , x − ) there = (T, Φ) here . boundary directions. It is at times useful to express the remaining spin-2 tensor components in the (x L , x R ) basis, whereas at other times it is convenient to express them in the (T, Φ) basis. We will use both often, and the relation between them reads We Fourier expand the spin-2 field as Just as we use either the (x L , x R ) or (T, Φ) basis for the spin-2 components above, we will find it useful below to express the momentum with respect to the several different choices of coordinates.
The relation between the various definitions follows from which implies the relations In addition, because φ parameterizes a circle, regularity of the solutions implies k ∈ Z.
In the (x L , x R ) basis, the equations of motion for the radial wave functions become diagonal.
In particular, one has [6] z(1 − z) The solutions to these equations are given by where we have written the solutions such that the the functions R in ij (z) and R out ij (z) become unity at the horizon z = 0. The sign of the exponent of z indicates that the "in" and "out" superscripts naturally refer to ingoing and outgoing solutions. e in ij and e out ij are polarization constants and the other constant parameters are given by For ingoing solutions (with e out ij = 0), the polarization constants are constrained by the first-order equations to satisfy whereas for the outgoing solutions (with e in ij = 0) one has Note that the ingoing (radial) wave-function is simply related to the corresponding outgoing one by sending (k L , k R ) → −(k R , k L ).

B.2 Determining the spectra
From now on we focus on the ingoing solutions. Writing them out explicitly, we have In order to relate the ingoing wave-function to an expansion at the boundary we use the connection identity Near the boundary, z → 1 and r −2 ∼ 1 − z → 0. Using the connection formula (B.24) to expand the solutions (B.21) for large r, assuming m > 0 we find the following behavior: Notice that the expansion of R RR appears different in structure from the other components. This is because we have assumed that 21 which contains the value m = 1, corresponding to the graviton. For positive values of m with |m| > 2, the two series in the expansion of R RR swap dominance. When m is negative, a similar statement applies. In particular, for m < 0, the relevant expansion is in which case we see that the two series in the expansion of R LL swap dominance for m < −2.

B.2.1 Quasinormal Boundary Conditions
Assuming the condition the standard quasinormal boundary conditions correspond to enforcing that the leading divergence in the boundary expansions in (B.25) or (B.29) vanish; this ensures that the perturbation is normalizable as r → ∞. For m > 0 (m < 0), this corresponds to demanding the leading term in R LL (R RR ) vanish. 21 The condition |m| ≥ 1 corresponds to the unitarity bound ∆ ≥ 2 for spin-2 operators in the dual CFT.
For m > 0 we find the ingoing quasinormal spectrum to be 2ik R = 2p + ∆ + 2 where we have defined ∆ = |m| + 1. We will refer to p as the radial quantum number. Almost all of these modes arise by ensuring the Γ-functions in the denominator of (B.25) acquire poles which set the leading term in R LL to zero. This vanishing occurs when either a in LL or b in LL becomes equal to zero or a negative integer. However, there are two special solutions, corresponding to p = 0, 1 in the k L series. These solutions instead have parameters set such that the polarization tensor The outgoing solutions can also be handled similarly. In the end we arrive at the quasinormal mode spectrum displayed in Table 1.

B.2.2 Chiral boundary conditions
We will now consider the chiral boundary conditions relevant for the analysis in section 3. In conditions. This means that for m > 0 we require c in RR − a in RR or c in RR − b in RR to be zero or a negative integer. In addition, there are again two special conditions arising from setting e in RR = 0 in (B.17). The entire spectrum satisfying chiral boundary conditions is presented in Table 4.

B.3 Regularity of Euclidean solutions
The mode functions defined in (B.5) have a natural continuation to Euclidean signature. In this section we use the coordinates (A. 7), where regularity at the origin ξ = 0 is made most manifest.
At the level of the solutions to the wave equation, the Euclidean continuation is implemented by making the replacements Correspondingly, the negative values of k E are assigned to the outgoing solutions, with In addition, one can consider the zero modes as arising from either sector.
Finally, for some specific states, there is an additional restriction on the allowed values of n.
This restriction arises from demanding square-integrability of the Euclidean solutions near ξ = 0.
In particular, we demand that the Euclidean solutions h (λ) µν satisfy [51] where λ is an eigenvalue and the asterisk denotes complex conjugation. In order to avoid a nonintegrable singularity at ξ = 0 in the integrand of (B.41), we must further restrict the range of n for Euclidean solutions with certain low-lying values of the radial quantum number p.
One can see that a potential problem exists by considering the component h ξξ which, since the inverse metric component g ξξ = 1, shows up squared with only the metric determinant as prefactor 22 Note that small ξ corresponds to small z, where z ∼ ξ 2 .

in (B.41). The tracelessness condition (B.3) implies that near the origin
where h EE and e EE are the Euclidean rotation of h T T and e T T , which are related to the (L, R) basis by the matrix equation (B.4). This means that, for small ξ, one has Therefore, for k E = 0, 1 there is a potential non-integrable singularity at ξ = 0.
The potential singularity at ξ = 0 is avoided for most values of p because e EE vanishes for k E = 0 or k E = 1 in generic solutions. 23 However, there are a finite number of states where this is not satisfied. In particular, focusing on the m > 0 states, we find that the Euclidean continuations of states belonging to the k L series in Table 1 and to the k R series in Table 4 with mode numbers given by do not satisfy e EE = 0. These modes correspond to wave-functions that are not square-integrable and should be discarded. We could also argue these states should be eliminated because they correspond to the special values of p where components of the polarization tensors e ij vanish, as described following (B.33), for which e EE = 0.
The states with quantum numbers (B.44) should also be discarded from the Euclidean continuation of the k R series with m < 0 in Table 1 and Table 4. By shifting p for the specific case of n = −1 in this series, we can combine the m < 0, n < 0 k R series with the m < 0, n ≥ 0 k L series; this combination results in the bottom row of the m < 0 column in Table 2, now valid for all integers n, k and p ≥ 0. By similarly shifting p to exclude the rest of the singular solutions, we are left with the entire set of possible conditions on allowed Euclidean solutions which are presented for standard Brown-Henneaux boundary conditions in Table 2 and for the chiral boundary conditions in Table 5. 23 One can check this by noticing that the polarization tensors satisfy the same matrix equation as the tensor components in (B.4), where hT T and hT Φ are related to hEE and hEΦ by analytic continuation. Imposing the relations (B.17) or (B.19), one sees that indeed eEE vanishes where kE = 0, kE = 1 (for ingoing) or kE = −1 (for outgoing).

B.4 Checking the δg r+ behavior
Finally, we need to understand the consequences of the δg r+ condition in the two sets of boundary conditions in (3.8) and (3.14). We begin by solving the first order equations (B.1), finding 24 Inserting the generic solutions (B.12) into this expression and using that near the boundary e ξ ∼ r and h ξL ∼ rh r+ we find that for large r and m > 0 C Analysis of ghost contributions to the gravitational path integral In this appendix we present a detailed analysis of the a massive spin-1 field in AdS 3 , and the ghost determinant that appears in the graviton one-loop path integral.

C.1 Spin-1 equations
The massive spin-1 modes can be solved similarly to the massive spin-2 modes. In first order form, the equation of motion is Again we drop indices on m to avoid clutter (in the main text it would be m 1 ). The solutions of this equation satisfy the massive vector equations of motion The specific value of m which corresponds to the spin-2 ghost is then m 2 = 4, i.e. m = ±2. For this value of the mass, we interpret V µ as variation of the metric: 3) It will be useful to switch between the various coordinates: the vector components in the (z, x L , x R ) coordinates are related to those in the (ξ, T, Φ) coordinates by In components, equation (C.1) reads We will again look for solutions of the form V µ (z, T, Φ) = e −i(k L x L +k R x R ) R µ (z) . (C.8) Equation (C.5) can be thought of as a constraint on R ξ , and the remaining equations imply  F (a in R , b in R ; m; 1 − z) Computing the induced metric perturbations in (C.3), we find that Dirichlet boundary conditions require where p is a non-negative integer and the shift by one in the second line arises for the mode where m > 0 m < 0 2p + ∆ + |n + 1| + ik Φ (n, k) = 0 2p + ∆ + |n − 1| − ik Φ (n, k) = 0 2p + ∆ + |n − 1| + ik Φ (n, k) = 0 2p + ∆ + |n + 1| − ik Φ (n, k) = 0 we demand e in L = 0 in (C.16).
In order to hold δg −− fixed, chiral boundary conditions require one of the constraints The final results for the set of ghost states consistent with chiral boundary conditions on the metric are given in Table 3 and Table 6.

C.2 The Euclidean Solutions
All that remains now is to understand the Euclidean solutions into which the spin-1 states Wickrotate. The Euclidean rotation on the momentum is again given by where ingoing solutions require that n > 0 for regularity, outgoing solutions require that n < 0, and the zero modes can again be obtained from either ingoing or outgoing conditions with n = 0.
For the Brown-Henneaux states in Table 3, the process is almost identical to the spin-2 discussion and we compile the conditions in Table 7.
We now analyze the new states given in Table 6 that are consistent with the chiral boundary conditions. First, consider the states in the first row of Table 6. After the Euclidean rotation, we can write the set of states as We need to check that all of these states are regular at the origin. In particular, we require that V ξ is smooth as ξ → 0; from (C.5) we have (C.28) For |k E | > 0 regularity at the origin is guaranteed, because V L ∼ V R ∼ ξ |k E | for small ξ. The contributions from the second and third rows of Table 6 are more subtle. The Wick rotation of the modes in the second row gives: Only a subset of these modes correspond to admissible ghost states. To see this we need to evaluate the induced gauge transformation of the metric from each ghost state. In particular, the pure-gauge metric perturbations can be written in terms of a solution to the spin-1 equations with m = ±2.
The induced gauge transformations are given by The states in the second row of Table 6 have e R = 0, which means that the condition e R = −e L cannot be satisfied (except for the trivial solution e R = e L = 0) and so the k E = 0 states are obviously not regular. Second, the |k E | = 1 contribution needs to be analyzed carefully. The wave-functions for these states are particularly simple; for generic k E , we have Since V R = 0 for these states it is fairly straightforward to write out the induced metric variations.
From (C.32) through (C.37) we have Evaluating these on (C.38) and using (C.30), we find that all of the induced metric variations vanish when we set ∆ = 3 and k E = 1. Therefore, the solutions satisfying (C.30) and (C.31) at k E = ±1 and ∆ = 3 correspond (locally) to Killing vectors of the BTZ background. These are modes with zero eigenvalue of the ghost Laplacian in (C.2) and should be excluded from the pole contribution to the determinant. The same phenomena also occurs for the k R = 0 states with |k E | = 1 and m 1 = −2, in which case V L = 0 because the polarization vectors e L in (C.16) vanish.
One can check that these also give trivial metric variations. The appearance of Killing vectors in the ghost determinant of massless gauge fields in the bulk is generic when performing alternative quantization [11]. As explained in [11], since these are zero modes, they must be treated separately and generate N −n 0 /2 contributions to the partition functions as opposed to poles.