Gravitational Edge Mode in Asymptotically AdS 2 : JT Gravity Revisited

: We study the gravitational edge mode of the Jackiw-Teitelboim (JT) gravity and its sl (2 , R ) BF theory description with the asymptotic AdS 2 boundary condition. We revisit the derivation of the Schwarzian theory from the wiggling boundary as an action for the gravitational edge mode. We present an alternative description for the gravitational edge mode from the metric fluctuation with the fixed boundary, which is often referred as “would-be gauge mode”. We clarify the relation between the wiggling boundary and the would-be gauge mode. We demonstrate a natural top-down derivation of P SL (2 , R ) gauging and the path integral measure of the Schwarzian theory. In the sl (2 , R ) BF theory, we incorporate the gravitational edge mode and derive the Schwarzian theory with P SL (2 , R ) gauging. We also discuss the path integral measure from the Haar measure in the Iwasawa decomposition of P SL (2 , R ).

The JT gravity is so simple that, at first glance, no physical degree of freedom is left except for the topological ones.Hence, the "nearly-AdS 2 ", which allows a wiggling boundary, was introduced to account for the fluctuation on the boundary of AdS 2 [12].This seems to make it distinct from the previous approach to higher dimensional gravity theories.However, this "new" approach with a wiggling boundary brought about questions in the derivation of the Schwarzian theory.For example, the path integral of the JT gravity should include the integration over the wiggling boundary because the resulting Schwarzian theory does.In essence, the wiggling of the boundary is treated as a degree of freedom of the theory, and consequently the variation of the JT action must take the variation of the wiggling boundary into account.And the well-posed path integral with a wiggling boundary should have provided a top-down derivation of the P SL(2, R) gauging and the path integral measure of the Schwarzian theory.Furthermore, it is not clear whether the "nearly-AdS 2 " is fundamentally different from the usual approach to the asymptotically AdS [32].
In fact, the wiggling boundary has been discussed to deal with the broken radial diffeomorphism on the boundary of AdS [33][34][35][36][37][38][39][40].One way to resolve this issue on the breaking of the diffeomorphism is to disallow such broken gauge transformations on the boundary.Since a part of the gauge symmetry is prohibited, we regain additional physical degrees of freedom, so-called would-be gauge mode, which would have been gauged away by a radial diffeomorphism if it is not broken.Alternatively, one may introduce a Stueckelberg field to recover the full diffeomorphism invariance on the boundary.And the Stueckelberg field can be "eaten" by the field-dependent coordinate transformation, which leads to the wiggling boundary.Therefore, the would-be gauge mode is resurrected as a wiggling boundary.
Exactly the same phenomenon has been intensively studied in the context of the fractional quantum Hall effect (FQHE) context [41][42][43][44].For example 1 , in the U (1) Chern-Simons theory on a manifold with boundary, the U (1) gauge symmetry is broken.And, the would-be gauge mode from this broken U (1) gauge symmetry is known as the edge mode described by the chiral boson on the boundary.The JT gravity with the asymptotic AdS 2 boundary condition can also be investigated by the sl(2, R) BF theory [28,[57][58][59][60][61][62][63][64].Therefore, the Schwarzian theory can be derived as a gravitational edge mode of the sl(2, R) BF theory.In this paper, we will revisit the derivation of the Schwarzian theory from the sl(2, R) BF theory.In particular, we clarify the boundary condition and demonstrate a natural top-down derivation of P SL(2, R) gauging and the path integral measure.
This paper aims at understanding the Schwarzian theory as a gravitational edge mode of the asymptotically AdS 2 with P SL(2, R) gauging and the path integral measure from the three different points of view: the wiggling boundary, the would-be gauge mode and the sl(2, R) BF theory.This paper is organized as follows.In Section 2, we review the edge mode in the U (1) Chern-Simons theory.In Section 3.1, we derive the Schwarzian theory with a wiggling boundary.In Section 3.2, we also derive the Schwarzian theory from the would-be gauge mode with a fixed boundary, and we present the relation to the wiggling boundary.In Section 4, we discuss the derivation of the Schwarzian theory from the sl(2, R) BF theory.In Section 5, we make concluding remarks.

Review: Edge Mode in U (1) Chern-Simons Theory
We begin with a brief review of the edge mode in the U (1) Chern-Simons theory on the manifold (2.1) The variation of the action S CS is The boundary condition for the well-defined variational principle is usually chosen to be In this review, we consider more generic boundary condition For this boundary condition, we add a boundary term such that the variational principle is well defined as we will see soon Then, the variation of the total action S tot ≡ S CS + S bdy becomes And we choose the boundary condition 2 of A on ∂M to be where J = J(x 0 , x 1 ) is a fixed function.
It is well-known that the Chern-Simons action is not gauge invariant.Together with the boundary term (2.5), the total action is still not gauge invariant under the gauge transformation of A → A + dλ in general: One way to demand the gauge invariance is to restrict the gauge parameter λ such that Then, the variation of action under the restricted gauge transformation becomes where we used the equation of motion for the gauge field A. In this case, when two configurations A and A ′ are connected by disallowed "gauge transformation" such that A µ A ′ and A are not redundant configurations, but physically distinct ones.This implies that there are more physical degrees of freedom which would have been removed by the boundary gauge transformation that is not allowed in this set-up (See Fig. 1(a)).This new physical degree of freedom is called as a would-be gauge mode or an edge mode.This is the common way to deal with the edge mode in the U (1) Chern-Simons theory.
In this description of the edge mode with physically distinct configurations A µ , A ′ µ in Fig. 1(a), there could be conceptual confusion about the boundary condition.By definition, the action is not invariant under the variation of the gauge field along the direction of the disallowed gauge transformation, which might give rise to a misunderstanding that a fluctuation along the direction of the disallowed gauge transformation is frozen.Hence, as a boundary condition of A µ , one might mistakenly choose one particular configuration A µ among all physical configuration A µ , A ′ µ that one should have included.In fact, the variation along the disallowed direction does not lead to the boundary condition. 3One simple way to clarify this issue is to separate the degree of freedom for edge mode living on the boundary from the bulk gauge field.
where A on the right hand side denotes the "bulk gauge field" which is restricted to the boundary condition while a(x 0 , x 1 ) denotes the edge mode on the boundary.For this, we utilize the boundary "gauge transformation" by ϕ(x 0 , x 1 ) (i.e. a µ = ∂ µ ϕ in Eq. (2.12)), and we promote it as dynamical edge mode.
Then, the path integral measure can be written as where U (1) ′ denotes the restricted gauge symmetry of A with (∂ 0 − v∂ 1 )λ ∂M = 0, and the configuration of the gauge field A is restricted accordingly.Instead of parameterization of the physical configuration by A µ , A ′ µ , • • • , we parametrize them by the edge mode ϕ and a specific gauge configuration A µ (See Fig. 1(b)).Then the variation along the direction of the disallowed gauge transformation in the previous discussion with A µ , A ′ µ can be thought as the variation with respect to the edge mode, which does leads to the boundary condition but the equation of motion of the edge mode.At the same time, one can now impose the boundary condition on A µ along that direction to freeze the fluctuation.
It is not easy to deal with the restricted field configuration with the restricted gauge symmetry U (1) ′ .This restriction can easily be removed by extending the gauge field configuration and the gauge symmetry at the same time: where the gauge parameter λ is not restricted and ϕ plays a role of Stueckelberg field [65].Hence, we have the full unrestricted U (1) gauge symmetry, and the gauge field A is unrestricted, too (See Fig. 1(c)).In the description of Fig. 1(c), the path integral can be written as DA Dϕ U (1) e iStot [A,ϕ]  (2.17) where the total action S tot [A, ϕ] is given by This action can be understood as the action for the chiral boson (edge mode) coupled to the bulk U (1) Chern-Simons theory.Now using the full U (1) gauge symmetry (2.15) and (2.16), we will fix the bulk gauge field to obtain the boundary action for the chiral boson coupled to the background field.It is convenient to use different coordinates, y µ , defined by (2. 19) In the y µ coordinates, the gauge field can be written as (2.20) Accordingly, the total action becomes and, the boundary condition becomes This boundary condition allows us to fix the temporal gauge in the bulk to be A 0 (y) = J(y 0 , y 1 ) for y ∈ M . (2.23) The Gauss constraint ∂ 1 A 2 − ∂ 2 A 1 = 0 can be solved by Using the gauge symmetry, λ can be gauged away to fix the A 1 : Then the action (2.21) (for x 1 ∈ R) becomes Note that when we introduced the edge mode ϕ in Eq. (2.12), there is a trivial redundancy in a constant shift of ϕ.Hence, one has to impose the equivalence relation on ϕ: where c is a constant.This constant shift gauging eliminates the zero mode of the chiral boson.
One can also obtain the same result from the description with A µ , A ′ µ , • • • in Fig. 1(a) as in the literature.But we find it more convenient to explain the JT gravity and BF theory from the description with A µ and ϕ in Fig. 1(c).3 Jackiw-Teitelboim Gravity for Asymptotic AdS 2 The action4 of Jackiw-Teitelboim gravity in a Euclidean manifold M with boundary ∂M is given by where ϕ and g µν are the dilaton and the bulk metric.The boundary ∂M is parameterized by the coordinate u , and h ≡ h uu is the induced metric on the boundary.K is the extrinsic curvature evaluated with the metric g µν , whereas K 0 is the same quantity evaluated with the background metric g 0 µν .Hence, K 0 plays a role of a counter-term which makes the free energy of the background vanish.In this work, for a given background geometry, we will consider geometries which do not change the global properties or topology of the background, which will correspond to the smooth gauge transformation in contrast to the large gauge transformation.Hence, from the beginning of the formulation we choose the background that we are interested in, and the global information of the background is incorporated into K 0 .We will study the gravitational edge mode in the JT gravity as a wiggling boundary in Section 3.1 as in [12] and as a would-be gauge mode in Section 3.2 as in [38].These two descriptions are, in fact, equivalent, and we will explain the relation explicitly.

Description 1: Wiggling Boundary
In this section, we revisit the derivation of the Schwarzian action for the Jackiw-Teitelboim gravity with wiggling boundary.The relation to the edge mode to the wiggling boundary will Figure 3.The variation with respect to the wiggling boundary with the metric and dilaton fixed can be evaluated by the variation of the integrand with respect to the coordinates.be clarified in the next section.Let us begin with the variation of the action: The first line represents the variation of the (bulk) metric and the (bulk) dilaton with the wiggling boundary fixed (Fig. 2 On the other hands, the variation of the wiggling boundary in the second line of Eq. (3.2) can be written as a variation of the integrand with respect to (r, θ) denoting the coordinates of the AdS 2 with the metric and dilaton functions fixed (See Fig. 3).
Unlike the boundary condition for the (bulk) metric and dilaton, this variation gives the equation of motion of the edge mode, i.e., Schwarzian mode.This is analogous to the issue on the variation of the U (1) Chern-Simons theory with the description in Fig. 1(a).By using the diffeomorphism, one can fix the metric and the dilaton to be [66] where r h is the location of the tip of the Euclidean black hole.The Euclidean time θ is periodic with period β, and the smoothness condition around r = r h gives At the cost of trivializing the bulk geometry and the dilaton profile, the non-trivial information is encoded only in the shape of the boundary.The wiggling boundary curve can be parametrized by (r(u), θ(u)) where u ∈ [0, β].One can extend this parametrization of the boundary curve to the AdS 2 .Namely, let us consider the exact EAdS 2 without wiggling boundary, The boundary surface is parametrized by We consider a map from the base AdS (ρ, u) to the target AdS (r, θ) such that the constant ρ = ϵ −1 surface is mapped to the wiggling boundary in (r, θ) space and the boundary metric is identified From the boundary metric matching condition (3.12), one can express r(u) in terms of θ(u) perturbatively.
where the overall sign on the right hand side is chosen in a way that θ(u) is the increasing function of u.Then, the extrinsic curvature can be evaluated to be where we chose the tangent vector t and the (outward) normal vector n to be From Eq. (3.13), the extrinsic curvature can be expanded with respect to ϵ: And we chose the counter term K 0 in the boundary action (3.1) to be the extrinsic curvature of the geometry without wiggling (i.e.θ(u) = u): This counter-term is different from that in the previous literature.It turns out that this new counter-term plays an important role in the derivation of the Schwarzian action.In the action (3.1), the bulk part vanishes while the boundary action becomes where the Schwarzian derivative Sch[θ, u] is defined by First, note that θ ′−1 in front of the Schwarzian derivative comes from the dilaton solution ϕ = r.This is one of the main differences from the previous derivation of the Schwarzian action.The bulk dilaton field is the bulk Lagrangian multiplier which is determined by the bulk equation of motion.And we choose the boundary value of the dilaton which is consistent with the dilaton solution.
The second term in Eq. (3.19) gives a constant r 2 h β = 2πr h to the action.The last term in (3.19) comes from the term of order O(ϵ 2 ) in the counter term K 0 (3.17).Now using the inversion formula for the Schwarzian derivative we have Note that the last term comes from the counter term.This action vanishes for the trivial map u = θ which is expected by the choice of the counter term.Also note that the sign in front of the Schwarzian derivatives is important for the stability of the Euclidean action.
In the semi-classical analysis around the classical solution u(θ) = θ, the quadratic action of the quantum fluctuation ϵ is bounded below with the minus sign in front of the Schwarzian derivative.
In deriving the Schwarzian action, we had to invert θ(u) into u(θ).This inversion naturally leads 5 to the well-known path integral measure of the Schwarzian action: Now, we will explain how the sl(2, R) gauging appears in our formulation.The path integral of the JT gravity is reduced to the path integral of the map from the base AdS 2 to the target AdS 2 6 (ρ, u) −→ (r, θ) . (3.25) However, the base AdS has the sl(2, R) isometry, and the parametrization of the target AdS cannot distinguish the isometry of the base AdS.This leads to the redundancy in the parametrization of the target Ads which should be gauged to avoid overcounting.For example, the isometry of the Poincare coordinates is given by where a, b, c, d are constants with ad − bc = 1.And on the boundary t ′ becomes which is well-known sl(2, R) transformation of the boundary coordinate t.
Using the coordinate transformation from the Poincare coordinates (z, t) to our (base AdS) coordinates (ρ, u) The time on the boundary z = 0 in the Poincare coordinates is identified with the boundary time u as follows.
Therefore, under the isometry of the base AdS, the coordinate u on the boundary is transformed as tan In the next section, we will derive the Schwarzian action from the would-be gauge mode and will discuss the relation to the wiggling boundary.

Description 2: Would-be Gauge Mode
Let us begin with a Euclidean AdS 2 without the wiggling boundary.Namely, for the coordinates X µ = (ρ, u), the boundary of the EAdS 2 is the surface of constant ρ.We consider the Fefferham-Graham gauge given by Figure 5.The would-be gauge mode is "eaten" by the wiggling boundary via the radial and boundary diffeomorphism.
where r h is a constant corresponding to the radius of the black hole that we are interested as a background.The boundary metric is expanded as One can consider a reference AdS 2 geometry with the coordinates Y µ = (r, τ ) which is mapped to X µ = (ρ, u) by radial and boundary diffeomorphism (See Fig. 5): Then, while the boundary of the (ρ, u) is the surface of constant ρ, the boundary of the (r, τ ) is wiggling by the radial diffeomorphism (3.36).One can still demand that the metric g µν of the reference AdS Y µ = (r, τ ) preserves the Fefferham-Graham gauge (3.34) and that the expansion of the boundary metric g τ τ is truncated.
The condition for the Fefferham-Graham gauge (3.34) and (3.38) determines the function f (2) (u) and h (2) (u) in Eq. (3.36) and Eq.(3.37) Then the boundary metric G uu can be expressed in terms of g uu , w(u) and θ(u): Now in addition to the Fefferman-Graham gauge (3.34) and (3.38), we further impose the asympotic AdS condition.i.e.
and we also demand that the diffeomorphism in Eqs.(3.36) and (3.36) preserve the asymptotic AdS condition.
Then, the asymptotic AdS condition gives And one can express the boundary metric in terms of θ Now we go back to the JT action in the X µ = (ρ, u) coordinate space with the metric G µν : This diffeomorphism is broken on the boundary of AdS, and therefore the would-be gauge mode by this diffeomorphism becomes the gravitational edge mode as in the U (1) Chern-Simons theory with a boundary.As in Section 2, we promote the function θ(τ ) parameterizing the broken diffeomorphism to be a dynamical edge mode.Inserting the metric relation (3.46) into the action, we have the JT action of the metric g coupled to the gravitational edge mode θ: where the counter term K 0 is chosen to be the extrinsic curvature of the background geometry in Eq. (3.17).By the definition of the edge mode θ, the action (3.48) is invariant under the following transformation which corresponds to the restored (residual) broken gauge symmetry by introducing the edge mode.
Using this gauge symmetry (3.49)∼(3.51),one can fix the boundary metric g (2) to be a constant, and the constant value of g (2) is determined to have the smooth geometry without conical defect.
In addition, the (bulk) equation of motion for the dilaton is simpler in the (r, τ ) space where we can determine the dilaton solution to be Inserting them into the action (3.48), the bulk action vanishes while the boundary action becomes As before, we can invert θ(u) into u(θ) to get the Schwarzian action The path integral measure for the Schwarzian action is induced by the inversion. u After we fix the boundary metric g (2) (3.52) by using the gauge symmetry (3.49)∼(3.51),there is still the residual gauge symmetry in which g (2) is invariant, which exactly corresponds to the isometry transformation of the background.
tan To see this, one can consider the gauge transformation of g (2) induced by the coordinate transformation in (3.49) and (3.50).
Therefore, the function λ which does not change g (2) = −r 2 h is given by and this leads to the P SL(2, R) gauge symmetry (3.57) of the Schwarzian theory.Now we explain the origin of the map from the base AdS 2 to the target AdS 2 with wiggling boundary in Section 3.1.Let us start with the base AdS 2 in the bottom left of Fig. 6.The base AdS 2 has the exact AdS 2 metric (3.8) without the edge mode θ.This corresponds to one particular gauge configuration A µ in Fig. 1(a).From the base AdS 2 , we introduce the edge mode θ, for example, via the gauge transformation by θ (See (a) in Fig. 6).Hence, the figure on the top left of Fig. 6 is analogous to the description by A µ and θ with U (1) ′ gauge symmetry in Fig. 1(b).After introducing the edge mode, one can extend the gauge symmetry as well as the metric configuration, which corresponds to the description by A µ , A ′ µ and ϕ with full U (1) gauge symmetry in Fig. 1(c).And this gauge symmetry leads to the generic asymptotic AdS 2 metric (See (b) in Fig. 6).Finally, one can consider a radial and transverse diffeomorphism similar to Eqs (3.36) and (3.37) to reach the target AdS 2 with the exact AdS 2 metric (See (c) in Fig. 6).In this transformation, the edge mode θ is "eaten" by the wiggling boundary.And the composite of those procedures corresponds to the map from the base AdS 2 to the target AdS 2 in Section 3.1.

sl(2, R) BF Theory for Asymptotic AdS 2
The framelike formulation of the Jackiw-Teitelboim gravity for AdS 2 can be described by sl(2, R) BF theory [57][58][59].The action for sl(2, R) BF theory for the two-dimensional manifold M is given by where Φ is 0-form sl(2, R) field, and the 2-form field strength F is defined by Note that unlike Chern-Simons action, S BF without any additional boundary term is gauge invariant.The variation of the action reads Hence, if we impose the boundary condition δA = 0, the variational principle is also welldefined.In this case, there is no dynamical degree of freedom left, and the theory is trivial.
To introduce an edge mode, we add boundary term which breaks the gauge symmetry on the boundary.Here, (r, θ) denotes the coordinates for the two-dimensional manifold, and the asymptotic boundary ∂M is the surface of constant r = ∞.The variation of the total action now becomes and we impose the following boundary condition.
Now, the total action S tot = S BF + S bdy is not invariant under the gauge transformation Therefore, for the gauge invariance of the system, we restrict the gauge parameter h on the boundary by At the cost of the gauge parameter on the boundary, we have more physical degrees of freedom which would have been gauged by the gauge transformation violating Eq. (4.8).For example, let us consider two gauge fields A and A which are related by an "illegal" gauge transformation: Without the condition (4.8),A and A would have been the identical configuration.But, because we disallowed such a gauge transformation, they are distinct physical configurations.Note that the boundary term that we added breaks whole boundary gauge symmetry.Hence, we have to restrict boundary gauge symmetry completely.This implies that all configurations with distinct boundary profiles becomes physical, which gives more physical degrees of freedom than the asymptotic AdS solutions that we want to study.Therefore, we revive a part of gauge symmetry by introducing boundary gauge field c.
The sl(2, R) BF theory for the asymptotic AdS 2 is defined by where c belongs to the nilpotent subgroup The gauge field c retrieve a boundary U (1) gauge symmetry.Namely, the action S AdS is invariant under the gauge transformation where h belongs to the nilpotent subgroup: The recovered U (1) boundary gauge symmetry reduces the boundary physical degree of freedom.As in the Chern-Simons theory, we consider the full boundary gauge transformation h, and we promote h to be a physical degrees of freedom g.Then, the action for the SL(2, R) BF theory for asymptotic AdS coupled to the gravitational edge mode is This can be understood as a decomposition of A into the boundary edge degree of freedom and the "bulk degrees of freedom".As before, the full boundary gauge symmetry revives at the cost of the introducing the edge mode g: where Λ ∈ SL(2, R) is defined in the bulk without any restriction on the boundary.This extension of the full SL(2, R) gauge symmetry is analogous to the U (1) Chern-Simons theory with the description in Fig. 1(c).And the boundary gauge symmetry (4.12)∼(4.14)now becomes the gauge symmetry of g and c where h belongs to nilpotent subgroup.Note that though g and c were introduced in a similar manner, g becomes physical degree of freedom on the boundary while c is to be fixed.Since c belongs to nilpotent subgroup, ∂ θ λ in c −1 ∂ θ c appears in the action linearly.Hence, λ plays a role of Lagrangian multiplier.
The variation of the action with the edge mode (4.16) is found to be This is the main reason why we fix all the component of the matrix A θ and why we introduced the edge mode g separately on top of the A and Φ.
Using the full gauge symmetry (4.17)∼(4.19), one can fix a θ to be the following constant sl(2, R) matrix.
where L 0 is a constant.Note that Φ is the Lagrangian multiplier of the bulk.Therefore, it can be fixed by the equation of motion for Φ: 8 In the literature, the sl(2, R) BF theory does not separate the edge mode from the bulk field.Hence, it corresponds to the description in Fig. 1(a).In this case, as we have pointed out in Section 2, one should be careful in connecting the variation of the action with the boundary condition. 9The sl(2, R) generator Ln (n = 0, ±1) is defined by -20 - The solution for Φ can be found to be where ϕ 0 is a constant sl(2, R) matrix.i.e.
The constraint imposed by c −1 ∂ θ c reads Here, (M ) a (a = 0, ±1) denotes each component of a sl(2, R) element M : The constant (4.36) is fixed to be κ because g includes the identity matrix.
Using the residual gauge symmetry given in (4.20) and (4.21), we choose the gauge condition To see the remaining physical degrees of freedom for g, we parametrize SL(2, R) matrix g by where y(θ) and f (θ) is periodic with period β, and u(θ) has winding number 1. i.e.
In this work, we demand that g The global structure can also be seen in the following holonomy along the Euclidean time θ.
Hol gA g −1 − gd g ≡ P exp and we have Therefore the BF action with edge mode (4.16) becomes where we used the single-value condition (4.44) with n = 1.The measure of the edge mode can be derived from the Haar measure of P SL(2, R) [31].We use the Iwasawa decomposition (4.41),where the Haar measure is given by

.52)
Together with the constraint and the gauge condition, we find and hence recover the result of [31] which has been obtained in the Gauss decomposition.The parameterization of g −1 a θ gg −1 ∂ θ g by k in Eq.This equivalence relation becomes the P SL(2, R) gauging of the edge mode action (4.51).We have seen in Section 3 that the P SL(2, R) gauging comes from the isometry of the (exact) AdS 2 .In the BF formulation, we can also confirm that it corresponds to the isometry of the (background) bulk field a θ .To see this, we first obtain the relation10 between k and g from Eq. (4.40).
k = e a θ θ g , where a θ is fixed to be a constant element in Eq. (4.32).Then, the equivalence relation (4.56) leads to the equivalence relation of g: where Υ(θ) is defined by Υ ≡ e −a θ θ Υ 0 e a θ θ .(4.60) One can easily see that this is a general solution of the following equation.
This equation means the redundant parametrization of g by Υ, and therefore, we can again confirm that the equivalence relation of g (4.59) should be imposed.Moreover, in Eq. (4.61), Υ can be interpreted as a gauge transformation parameter which does not change the background bulk gauge field a θ , in other words, the isometry of the background AdS 2 .

Conclusion
In this work, we have studied the gravitational edge mode in the JT gravity and the sl(2, R) BF theory with the asymptotic AdS 2 boundary condition.We revisited the derivation of the Schwarzian action from the wiggling boundary of the JT gravity.We introduced a new counter term K 0 which plays an important role in obtaining the correct Schwarzian action.We discussed the variation of the action which is involved not only with the boundary condition of the bulk metric and the bulk dilaton but also with the equation of motion of the gravitational edge mode.Introducing the target and the base AdS 2 , we showed that the inversion between the base and the target AdS 2 gives the Schwarzian action.In addition, we demonstrated that this inversion naturally leads to the path integral measure for the Schwarzian theory.We explicitly showed that the redundant description of the base AdS 2 corresponding to the isometry induces the P SL(2, R) gauging of the Schwarzian action.With the boundary of constant AdS radial coordinate without wiggling, we showed that the broken radial diffeomorphism leads to the would-be gauge mode, in other words, the gravitational edge mode.We demonstrated that this gravitational edge mode in the asymptotically AdS 2 can be described by the Schwarzian action.We also presented the relation between the gravitational edge mode and the wiggling boundary.
In the sl(2, R) BF theory, we incorporated the edge mode in the BF theory based on the U (1) Chern-Simons example.We clarified the variation of the action and the corresponding boundary condition of the bulk field.We demonstrated that the Haar measure of the Iwasawa decomposition of the SL(2, R) gives the path integral measure of the Schwarzian theory.We also showed that the redundancy in the Iwasawa decomposition, which is equivalent to the isometry of the AdS 2 background, brings about the P SL(2, R) gauging of the Schwarzian action.
The overall sign of our JT gravity action (3.1) is opposite to that of the literature for "nearly-AdS".We chose this sign convention to have the positive dilaton solution 11 ϕ and the stability of the Schwarzian action.One can easily expect the opposite sign from the inversion formula of the Schwarzian derivative (3.21).From the point of view of the higher dimensional near extremal black hole [68,69,69], the dilaton of the JT gravity originates from the transverse area of the fixed radial hypersurface.After the dimensional reduction to AdS 2 , the JT action with our convention (3.1) can be obtained by expanding the dilaton Φ coming from the higher dimension around the constant ϕ 0 , which is related to the entropy of the higher dimensional extremal black hole, as follows.
Φ 2 = ϕ 0 − ϕ . (5.1) As a result, the physical requirement for the stability of the near extremal black hole translates to Φ 2 becoming smaller as the black hole deviates from extremality.
It would be interesting to see if there are any interesting consequences of the inversion of degree of freedom that we encountered, namely from θ(u) to u(θ).Perhaps the natural place to look for is in how the Schwarzian modes couples to other matter fields.In this context, one might have to revisit the calculation of correlation functions.Nevertheless, we expect that some perturbative calculations would still hold, because the perturbative expansion around the classical solution u cl (θ) = θ of the Schwarzian action gives the same perturbative expansion of the inverse function up to order O(ϵ) (and up to sign).i.e. u(θ) = θ + ϵ(θ) −→ θ(u) = u − ϵ(u) + O(ϵ 2 ) . (5.2) For example, the leading contribution of the Schwarzian mode to the four point function of the matter might not be able to see the difference.We leave this issue for future works.

Figure 1 .
Figure 1.Three equivalent delineations of gauge field configurations and gauge symmetry.(a) A µ , A ′ µ , • • • , which are connected by disallowed gauge transformation, are independent physical degrees of freedom with the restricted U (1) ′ gauge symmetry.(b) Physical gauge configurations A µ , A ′ µ , • • • can be parametrized by the edge mode ϕ (would-be gauge mode) and a (reference) gauge configuration A µ with the restricted U (1) ′ gauge symmetry.(c) With full U (1) gauge symmetry, we have A µ , A ′ µ , • • • and the edge mode ϕ (Stueckelberg field).

Figure 2 .
Figure 2. The variation of the JT action can be split into two part.(a) The variation of the metric and the dilaton with the wiggling boundary fixed leads to the boundary condition for the bulk metric and the bulk dilaton.(b) The variation of the wiggling boundary with the metric and the dilaton fixed gives the equation of motion of the edge mode.
(a)) while the second line corresponds to the variation of the wiggling boundary with the metric and the dilaton fixed (Fig. 2(b)).For the variation of the first line of Eq. (3.2), we impose the boundary condition that the boundary metric and the boundary dilaton is fixed for a given wiggling boundary.i.e. δh uu = δϕ = 0 for a fixed wiggling boundary .(3.3)

. 33 )
Note that the Schwarzian action in(3.22)  is invariant under the P SL(2, R) transformation of u (3.33) which comes from the isometry of the base AdS.And this P SL(2, R) should be gauged because of the redundant description of the target AdS.It is important to note that the isometry of the target AdS, which can change the dilaton solution, should not be gauged in the Schwarzian action.

d where ad − bc = 1 . 2 Figure 6 .
Figure 6.(a) From the base AdS 2 (the exact AdS 2 metric, no edge mode), one can one can generate the edge mode on the boundary.(b) Using the (extended) gauge symmetry, one can have generic asymptotic AdS 2 metric.(c) The metric can be fixed to be the exact AdS 2 metric by the coordinate transformation eating the edge mode.

. 40 )
Note that g −1 a θ g + g −1 ∂ θ g = k −1 ∂ θ k does not distinguish k and −k.Hence, k belongs to P SL(2, R) by identifying k with −k.To solve the constraint and the gauge condition, we will use the Iwasawa decomposition of k given by

. 43 )
This gives us the relation between β and Ω: Ω 2 = nπ β where n ∈ Z .(4.44)And the periodic condition for k(θ) reads k(θ + β) = (−1) n k(θ) .(4.45) (4.40) is invariant under k → Υ 0 k where Υ 0 is a constant P SL(2, R) element.Υ 0 ≡ d c b a where ad − bc = 1 .(4.55) Therefore, this redundant description should be eliminated by imposing the equivalent relation Υ 0 k ∼ k .(4.56)This equivalence relation can be translated into the equivalence relation of u(θ) via the Iwasawa decomposition (4.41): a tan πu(θ) β + b c tan πu(θ) Figure 4.The wiggling boundary can be understood as a map from S 1 to the wiggling boundary of AdS 2 .It is convenient to extend the circle S 1 to the exact AdS 2 to define base AdS 2 .We can consider a map from the base AdS 2 to the target AdS 2 .
[67]The single-value condition (4.44) makes the holonomy trivial.Namely, it belongs to the center subgroup {±I} of SL(2, R).Furthermore, it determines L 0 in a θ in terms of β.This trivial holonomy condition can be interpreted as the smoothness of the geometry[67].From now on, let us focus on n = 1 case where k(θ) is a map from S 1 to P SL(2, R) with winding number 1.Using the Iwasawa decomposition (4.41), one can determine y(θ) and f (θ) from the constraint (4.38) and the gauge condition (4.39), respectively.