Non-relativistic and Carrollian limits of Jackiw-Teitelboim gravity

We construct the non-relativistic and Carrollian versions of Jackiw-Teitelboim gravity. In the second order formulation, there are no divergences in the non-relativistic and Carrollian limits. Instead, in the first order formalism there are divergences that can be avoided by starting from a relativistic BF theory with (A)dS2$\times\mathbb{R}$ gauge algebra. We show how to define the boundary duals of the gravity actions using the method of non-linear realisations and suitable Inverse Higgs constraints. In particular, the non-relativistic version of the Schwarzian action is constructed in this way. We derive the asymptotic symmetries of the theory, as well as the corresponding conserved charges and Newton-Cartan geometric structure. Finally, we show how the same construction applies to the Carrollian case.


Introduction
The Sachdev-Ye-Kitaev (SYK) model [1][2][3] is a solvable quantum mechanical model of Majorana fermions in one dimension. In the regime of large coupling (low temperature) this model is perturbative in the 1/N -expansion, and shows emergent conformal symmetry, which is the reparametrisation symmetry group Diff(S 1 ). The one dimensional diffeomorphism symmetry is spontaneously broken to SL (2, R), whose low energy dynamics is described by the 1D Schwarzian theory [3,6,7]. The holographic description of SYK model is the Jackiw-Teitelboim (JT) gravity [4,5] in two dimensions and the boundary description of the bulk JT gravity is also given the Schwarzian action [8][9][10]. The SYK model has been generalised to include complex fermions, its low energy dynamics [11][12][13][14][15][16] is described by a generalised Schwarzian theory with symmetry SL(2, R) × U (1). A flat limit of this generalised Schwarzian action was studied in [17]. JT gravity and its boundary description are also useful for the momentum/complexity correspondence and its connection with the non-relativistic Newton's law [18][19][20].
Motivated by the previous ideas and with the goal of finding another physical sector of the SYK model and its holographic description, 1 we will consider the non-relativistic (NR) and Carrollian limit of JT gravity and its boundary action. The starting point of this study consists in writing the JT gravity action as a BF theory with (A)dS 2 gauge group [23][24][25]. Then, as shown in [26], the Schwarzian action can also be obtained via the method non-linear realisations [27] and the inverse Higgs mechanism (IHM) [28]. However, in order to have a finite NR limit we will need extend the analysis and consider a BF theory with gauge group (A)dS 2 × R. The NR and Carrollian bulk actions can also be expressed as BF theories with Newton-Hooke 2 (NH ± 2 ) and Carroll (A)dS 2 gauge algebras, respectively [29][30][31].
To study the NR boundary theory we should consider the conformal basis of (A)dS 2 ×R, and introduce two different contractions of the SL(2, R) × R algebra. The first one leads to [ which is isomorphic to NH − and we will refer to it as Twisted Extended Galilean conformal algebra in 1D dimension. These two algebras are isomorphic as complex algebras. The boundary actions are constructed via the non-linear realisation method and IHM. In the case of a NR limit of the dS 2 × R, the boundary action has Extended Galilean conformal symmetry. We name this action Non-Relativistic Schwarzian. Instead, if we consider the NR limit of the AdS 2 × R algebra, the boundary action has Twisted Extended Galilean conformal symmetry. It is a complex version of the NR Schwarzian which is closely related to the flat Schwarzian action of [17].
The Carrollian limit of the relativistic bulk and boundary actions is obtained from the observation that the Carroll (A)dS 2 algebra admits a central extension in the commutator of the Galilean boost and momentum generator 3 , we name this algebra Extended Carroll (A)dS 2 algebra. One can see that this symmetry follows from the Extended NH ± 2 algebra by interchanging the generators H and P and changing the sign of the cosmological constant. This fact allows one to pass from Galilean to Carrollian symmetries 4 Extended NH + 2 ↔ Extended Carroll AdS 2 Extended NH − 2 ↔ Extended Carroll dS 2 . (1. 3) The Carrollian actions in the bulk and in the boundary can therefore be obtained from the NR ones.
The NR and Carrollian limits of the JT bulk action [4,5] in the second order formalism are also studied. In this case the divergences in the NR and Carrollian limits can be absorbed by rescaling the Newton constant. This is due to the fact that in two dimensions there are no divergent terms in the expansion of the Ricci scalar, which is a remarkably property that is in high contrast with its analogue in three and four space-time dimensions [40][41][42][43].
The organisation of the paper is as follows: in Section 2, we review the main aspects of JT gravity in first and second order formulations. In Section 3, we study the NR limit of the JT action, first in the second order formulation, and subsequently in the first order formulation by considering a BF theory with (A)dS 2 × R gauge algebra. The Section 4 is devoted to the Carrollian limit of the JT gravity theory, which is carried out by using the duality among NR and Carrollian symmetries. Section 5 deals with the construction of the boundary theory of JT gravity. Here we provide a derivation of the known Schwarzian theory and its asymptotic symmetries by means of the non-linear realisation method and IHM. We generalize the procedure to the (A)dS 2 × R case. In Section 6, we define the NR Schwarzian action and its asymptotic symmetries and conserved charges. We show how this is generalized to the Carrollian case. Finally, in Section 7, we give our conclusions and we elaborate on relevant future directions and possible generalizations of our results.
Note added: When this paper was finished, we noticed in the arXiv the article [44], "Limits of JT gravity" by D. Grumiller, J. Hartong, S. Prohazka and J. Salzer, [arXiv:2011.13870 [hep-th]]. Some of the results of this paper coincide with ours. However, the techniques used in the derivations are different.

Jackiw-Teitelboim gravity
In this section we review certain aspects of the Jackiw-Teitelboim (JT) gravity [4,5] which is dual to the SYK model [1][2][3] in (0 + 1)-dimensions. We start considering the second order formulation of JT gravity and subsequently review its first order formulation.

Jackiw-Teitelboim action
The JT action for gravity in (1 + 1)-dimensions is given by where R is the Ricci scalar,Λ is the cosmological constant, Φ a scalar field, andκ a twodimensional coupling constant. The field equations that follow from varying Φ and g µν in (2.1) read where ∇ µ denotes the affine space-time covariant derivative. The second equation can be decomposed into traceless and trace parts as [47] ∇ In the following, we will review how the above geometric dynamics may be presented in a gauge theoretical fashion.

First order formulation of JT gravity
A first order formulation of JT gravity can be defined by gauging the (A)dS 2 symmetry and constructing a BF theory [23][24][25]. The gauge algebra is given by so (1,2) in the AdS case and so(2, 1) for dS. The generators satisfy the following commutation relations 5 whereP a stand for translations andJ is the boost generator. The invariant tensor in (A)dS 2 is given by whereμ is an arbitrary constant. In order to define a BF theory we consider a scalar field B taking values in the (A)dS 2 algebra and a set of one-form gauge fields, E a = E a µ dx µ and Ω = Ω µ dx µ , corresponding to the zweibein form and the dual spin connection Ω ≡ − 1 2 ε ab Ω ab , respectively. The gauge fields define the (A)dS 2 connection one form together with the corresponding curvature two-form F = dA + A 2 , given by 6 The BF action then reads and its field equations are given by From equation (2.11a) we can solve the spin connection in terms of the zweibein as Using this result, one can evaluate the Riemann tensor 14) which indicates that the field equation (2.11b) can be rewritten in terms of the Ricci scalar R = g µν R ρ µρν as Using this equation in (2.11d) yields where the space-time metric is defined in the usual way This reproduces the field equation (2.4) previously found in the second order formulation. Using (2.11a) and (2.15), the BF action takes the form of the JT action (2.1), namely whenμ andκ are properly identified. However, since the spin connection was not solved by means of its own field equation, the reduced action (2.19) is not dynamically equivalent to the original BF action (2.10). Thus, the equivalence between the first order and second order formulations of JT gravity should be considered only at the level of their field equations. The solutions in the secondorder formalism are contained as subset of the solutions of the first-order formalism.

Non-relativistic Jackiw-Teitelboim gravity
In this section, we will consider the NR limit of JT gravity in second and first order formulations. In the first order formulation, the limit is applied to a BF theory with gauge group (A)dS 2 × R.

Second order formulation
The NR expansion of JT gravity follows in general terms from the approaches studied in [36,[40][41][42][48][49][50][51]. Let us consider the power expansion of the relativistic metric g µν (x) and g µν (x) up to ε 2 -order, as follows where the higher inverse powers of ε = 1/c → 0 correspond to Post-Newtonian corrections that will not be considered here. From the relation g µρ g ρν = δ ν µ , we obtain the following relations for the different terms in the expansion (2) g µν g µν g µν Now we introduce a symmetric affine Levi-Civita connection with the expansion 7 As usual, we demand metric compatibility on the metric and its inverse where the covariant derivative ∇ ρ is defined with respect to the symmetric connection (3.3).
For the covariant metric g µν , this condition implies the following set of equations order by order where we have introduced the covariant derivative Γ . Solving the system yields (see Appendix B.1) Now we write the metric (2.18) in terms of Newton-Cartan fields τ µ and e µ , and the vector field m µ by considering the zweibein expansion [33][34][35][36][37] where m µ is the vector field associated to the central extension of the Bargmann algebra [41,53]. This leads to (2) where we have introduced the spatial metric h µν ≡ e µ e ν . The relations E µ a E a ν = δ µ ν , and E µ b E a µ = δ a b lead to the following inverse zweibeine which involve the new additional fields τ µ and e µ satisfying the relations e µ e ν + τ µ τ ν = δ µ ν , τ µ e µ = 0, e µ τ µ = 0, τ µ τ µ = 1, e µ e µ = 1 . (3.10) Note that the spatial metric h µν is degenerate. With these definitions the inverse metric takes the form (3.11) Therefore, considering terms only up to order ε 2 , we obtain the following form of the inverse temporal and spatial metrics in (3.1), where we have introduced the inverse spatial metric h µν . With these explicit forms for the metric components, the terms in the connection expansion (3.6) take the form We see that in two dimensions there are no divergent terms in the expansion of the Ricci scalar. This is a remarkably property that is in high contrast with three and four space-time dimensions [40][41][42][43].
We consider now the expansion of the vielbein postulate, Implementing (3.3), (3.9) and considering the anti-symmetric part of (3.15) is whereas the symmetric part reads (2) Γ ρ µλ τ ρ = 0 , (3.18a) (2) Γ ρ µλ e ρ = 0 , Solving this system for all the connections, yields which is compatible with (3.13). Using this result, the leading term in the Ricci scalar expansion (3.14) takes the form (see Appendix B for details), where the NR spin connection ω µ is solved algebraically by equations (3.17c) and (3.17d), yielding Finally, we expand the metric determinant, the JT scalar field, the cosmological constant, and the coupling constant as Note that the rescaling ofκ corresponds to a rescaling of the Newton's constant G. Then, in the ε → 0 limit, we find the NR action, This action leads to the second order equation where we have defined the NR Ricci scalar curvature This defines the NR version of the relativistic JT equation (2.2a). Unlike second order gravitational actions in higher dimensions, (3.25) has no divergent terms. As we will see in the next section, the rescalings (3.24b) are compatible with the first-order formulation results. We will also show that the field equations of the second-order formalism are a submanifold of the field equations of the first order formalism.

First order formulation
The NR limit of JT gravity in the first-order formulation can be defined starting from a BF theory with gauge algebra (A)dS 2 × R. Thus we extend the (A)dS 2 symmetry (2.5) by including an Abelian generatorỸ and consider the invariant bilinear form The NR contraction follows from defining NR generators H, P , J and M as Inverting this relation, we find Using the commutation relations (2.5), and defining the NR cosmological constant Λ as we find, in the the limit ε → 0, the centrally Extended Newton-Hooke algebra [29,30] in two space-time dimensions: Using the invariant tensor on (A)dS 2 (2.6) together with (3.28), and defining the contraction (3.29) yields the following NR invariant bilinear form which is non-degenerate for µ = 0. We next consider the NR limit of the JT action in first-order formulation (2.10). With this aim we consider a one-form connection taking values on (A)dS 2 × R algebra expressed in terms relativistic and NR generators [52] A = E aP a + ΩJ + XỸ = τ H + eP + ωG + mM (3.35) where, using (3.29), we find Besides, we extend the definition of the scalar field (2.7) to take values on (A)dS 2 × R as well B = Φ aP a + ΦJ + ΨỸ , (3.37) and define the NR scalar fields {η, ρ, φ, ζ} Using these definitions, the expansion of the relativistic BF action for the (A)dS 2 × R algebra is (3.39) where we have used (3.31) and (3.28), and we have defined the NR curvature two-forms as Using (3.33) and taking the limit ε → 0, we obtain the NR two-dimensional gravity theory It is worth to mention that this action can be alternatively obtained as a BF theory based on the Extended Newton-Hooke algebra (3.32), where the gauge connection A and the B field are given by In this case the curvature two form associated to A takes the form with the components given in (3.40). It is straightforward to see that, using these definitions, the BF action leads exactly to (3.41) when using the NR invariant tensor (3.34) The field equations coming from the action (3.41) when varying with respect to η, ρ, φ and ζ are given by δη : while varying (3.41) with respect to τ , e, ω and m leads to for some zero-form λ, therefore on-shell there is no torsion. This fact implies that the Newton-Cartan structure admit absolute time. Solving the field equations (4.23b) and (4.23d) the spin-connection is given by (3.23). Therefore, the action (3.41) becomes This action matches with (3.25) identifying µ = −κ.
Since the spin connection has not been solved by its own field equation, this action is dynamically inequivalent to the first order action (3.41). This is in complete analogy to the relativistic case.

Carrollian Jackiw-Teitelboim gravity
In this section, we work out the Carrollian limit of the JT gravity. We consider the Carroll contraction of the (A)dS 2 × R algebra. In order to do that we first interchange in P 0 ↔ P 1 , and H ↔ P in (3.29), which gives Secondly, taking the limit ε → 0, and using the rescaling (3.31), we get We shall call this symmetry Extended Carroll (A)dS 2 algebra. One can see that these commutation relations follow from the Extended NH ± algebra (3.32) by interchanging H and P and changing the sign of the cosmological constant. This fact allows one to pass from Galilean to Carrollian symmetries. This relation is a generalisation in two dimensions of a more general duality between of these two types of symmetries [45] previously found in the flat case. This procedure defines the following dualities It is important to remark that in 1 + 1 dimensions, the Carroll algebra (even without cosmological constant) admits the central extension M given in (4.2) in the same way as its Galilean counterpart does. This situation is a unique feature of the two-dimensional case since, unlike the Galilean case, the Carroll algebra does not admit a non-trivial central extension in four dimensions.

Second order formulation
We start carrying out the Carrollian contraction in the second order formulation. To make contact with the Carrollian geometry we first write the relativistic zweibein in terms of the Carrollian zweibein gauge fields The completeness relations E µ a E a ν = δ µ ν , and E µ b E a µ = δ a b imply the inverse relativistic zweibein In analogy with (4.1), these expressions can be obtained by interchanging E 0 and E 1 , and interchanging τ and e in (3.7). The gauge fields τ, e and m verifying the completeness relation (3.10). From the zweibein we can read the metric expansion where (2) Using the expression of the inverse zweibein (4.5), we find the following form of the inverse metric This has the general form where the components read with the inverse degenerate spatial metric defined as h µν = e µ e ν . Note that (4.6) and (4.9) have the same form than the NR expansion given in the expression (3.1). However, in this case, the expansion parameter is defined as ε = c. Therefore, we will understand as Carrollian contraction the limit ε → 0, which corresponds to the ultra-relativistic limit. With this consideration, the relations (3.2)-(3.6) also hold in this case.
The expansion (3.3) of the Carrollian affine connection can be obtained from the NR one (3.19) by interchanging the gauge fields τ and e. In a similar way to the NR secondorder formulation, the first non-vanishing term in the expansion of Ricci scalar (3.14) is given by where the Carrollian spin connection ω µ is As before the Carrollian limit for the JT action (2.1), can be read off from (3.24) by interchanging τ and e. Finally, taking the limit ε → 0, the Carrollian version of JT gravity is given by Notice that as in the NR case the divergent terms has been cancelled. This is a special property of two-dimensions.

First order formulation
In this section, we consider the Carrollian limit of the JT gravity in the first-order formalism.
Let us consider a one-form connection taking values on (A)dS 2 × R algebra expressed in terms relativistic and Carrollian generators where, using (4.1), we find Also we consider the scalar field to take values on (A)dS 2 × R B = Φ aP a + ΦJ + ΨỸ , (4.16) and define the Carrollian scalar fields {η, ρ, φ, ζ} as These expressions can also be obtained from (4.17) by interchanging Φ 0 ↔ Φ 1 , and η ↔ ρ. Using these definitions, the relativistic BF action takes the form Using (3.33), the Carrollian limit is then obtained by taking ε → 0, we find This action can be alternatively obtained as a BF theory based on the Extended Carroll (A)dS 2 algebra (4.2) and the bilinear form which is non-degenerate for µ = 0. The field equations coming from the action (4.20) when varying with respect to η, ρ, φ, and ζ read δη : while varying (4.20) with respect to τ , e, ω and m leads to Note that equation (4.22a) implies for some zero-form λ. This defines an absolute one-dimensional space. Solving the field equations (4.23b) and (4.23d) yields (4.12). The action (4.20) becomes with the Carrollian curvature R Carrollian = given in (4.11).

Relativistic boundary theory
In the previous sections, we have considered the NR and Carrollian limits of JT gravity, focusing only on bulk dynamics and without paying attention to boundary terms. However, in order for the theory to have a well-defined variational principle as well as to define the holographic dual theory, we need to study the boundary dynamics. To address this problem, we consider general BF action of the form For now we consider the zero-form B and the one-form A as taking values on a generic gauge Lie algebra G with a non-degenerate invariant bilinear form. In order to have a well defined variational principle, we follow the Regge-Teitelboim procedure [54] and supplement the action (5.1) with a boundary term where b is a one-form defined in such a way that, provided suitable boundary conditions for the gauge fields, the variation of the action does not drop boundary terms. A general variation of (5.2) leads to where D = d+[A, · ] is the covariant derivative. Considering the bulk coordinates x µ = (t, r) and the boundary defined at r → ∞, the gauge fields can be decomposed as Thus, the boundary term on the right hand side of (5.3) takes the form For this term to vanish, the boundary term b must be such that with k an arbitrary constant. On the other hand, the field equations coming from (5.3) imply that B is covariantly constant, while A is locally pure gauge, i.e. A = g −1 dg with g an element belonging to a gauge group G. Furthermore, we consider the following gauge fixing condition [32] The gauge connection then takes the form where a = U −1 dU only depends on the coordinate t. Evaluating the action (5.2) on this solution space, the theory is reduced to the boundary and the action reads where the prime stands for derivative with respect to t. An important step in our construction is that at this point we can consider the boundary target coordinate t as a world-line parameter of a particle, and the connection a as the pull-back ⋆ of the left-invariant Maurer- with Ω satisfying the MC equations dΩ + Ω ∧ Ω = 0 . that preserve its form. The corresponding conserved charges can then be computed using the known result for BF theory [55,56] δQ where we have switched to Euclidean time τ E in order to have a periodic coordinate. Imposing the boundary condition (5.7) for the field B and using the usual decomposition λ bulk = b(r) −1 λ(t)b(r), we find the expression In the following, we will apply this procedure in the relativistic case and subsequently in the different NR and Carrollian scenarios.

(A)dS 2 case
As a first example we consider the (A)dS 2 symmetry and the derivation of the Schwarzian action [3,6,7]. The AdS 2 and the dS 2 cases can be treated in a unified way by going to the conformal basis, as both they are isomorphic to the sl(2, R) algebra (see Appendix A) In this basis, the non-degenerate invariant bilinear form reads with γ 0 an arbitrary constant. Then, it is possible to locally parametrize an arbitrary group elementŨ asŨ = eρH eỹK eũD , (5.20) withρ,ỹ, andũ the group manifold coordinates. They will be the Goldstone fields when we take the pull-back to the world-line of the particle. To simplify the computation of the MC one-formΩ, we use the following 2 × 2 representation of the sl(2, R) algebrã Locally we can parametrize the group element in (5.20) as The MC one-formΩ =Ũ −1 dŨ is given in a matrix representation bỹ where we have defined the one-forms, The MC one-forms can be used to construct the boundary action (5.13), where prime denotes derivative with respect to the world-line parameter t. This model is invariant under the global symmetries δρ =α +βρ −γρ 2 , δỹ = (2γρ −β)ỹ −γ , δũ = 2γρ −β , (5.26) where the symmetry parameters can be grouped as We identify in (5.26) the transformation for the Goldstone fieldρ as the infinitesimal version of a SL(2, R) Möbius transformation. We can reduce the number of Goldstone fields in the action by setting some MC forms to zero or defining (invariant)covariant relations among them. This procedure is also know as the inverse Higgs mechanism (IHM) [28]. In the following, we present an example of this method. 8 We consider the constraints with µ a constant. This allows us to expressỹ andũ in terms ofρ as follows, where we have introduced the function L(ρ) := 1 2 µ Sch(ρ, t), with which is the Schwarzian derivative. In this case, the IHM is equivalent to a Drinfeld-Sokolov reduction [59,60]. Plugging (5.29) into the action (5.25), we find This result shows that it is possible to obtain the Schwarzian action using the IHM. 9 8 We do not attempt to classify all possible IHM constraints. 9 The Schwarzian can also be obtained by integrating out the gauge transformations of the particle model with variables x µ , λ and Lagrangian L = 1 2ẋ 2 − 1 2 λx 2 , see [57,58].
In order to obtain the gauge field in the bulk, we can introduce a radial dependence by performing a gauge transformation with the group element b = e r/2 0 0 e −r/2 , (5.34) this leads to the following form for the connection (5.9) where we have used (5.11) that implements the pull-back of the MC forms to the boundary coordinate t. Choosing µ = 1 2 , we find the the gauge field given in [61], namely This shows that IHM relations can be interpreted as boundary conditions on the BF connection A. Using the change of variables that relates sl(2, R) and the (A)dS 2 algebra (see Appendix A), we can write down the explicit form for the gravitational gauge fields in (2.8).
In the AdS 2 case we find In order to write down the zweibein and the spin connection for the dS 2 case, we interchange the coordinates t and r. This is due to the fact that the de-Sitter space has a boundary at time-like infinity. In this particular case, the prime denotes derivative with respect to r. We find E 0 = dt , From the expression of the zweibeine, we find the following the space-time metrics If we consider the leading term of (5.39) for r → ∞ we recover the boundary conditions of [61].
The asymptotic symmetry is the residual symmetry that leaves the conditions (5.30) invariant. These conditions restrict the gauge parameter The resulting gauge parameter is given by whereas the transformation law of the function L leads to where we have considered the following representation for the Kronecker delta δ mn = 1 2π e i(m−n)τ E dτ E .

(A)dS 2 × R case
We turn now our attention to the (A)dS 2 × R algebra where the gauge connection is given by In the conformal basis (see Appendix A), this algebra corresponds to sl(2, R) × R, with its relations given in (5.18). The non-degenerate invariant bilinear form is given (3.28) which in the conformal basis can be written as whereỸ ≡Ỹ . Now, let us consider the local parametrisation of a group elementŨ associated to (5.18) as followsŨ = esỸ eρH eỹK eũD , (5.50) wheres,ρ,ỹ, andũ are the group manifold coordinates. Using the matrix representation (5.21), and consideringỸ = 1 2×2 , the group element in (5.50) can be written as The MC one-formΩ =Ũ −1 dŨ is given bỹ The MC one-forms can be used to construct the boundary action (5.13) In the following, we consider the following inverse Higgs constraints [26] ΩH = µΩỸ ,ΩD = −2νΩỸ , (5.57) where µ and ν are arbitrary constants. We can expressỹ andũ in terms of the independent Goldstone fieldsρ ands as In this case, the MC one-form (5.52) reduces tõ a ≡Ω ⋆ IHM = µs ′H +s ′Ỹ − 2νs ′D + 1 2µs ′ Sch(ρ, t) − Sch(s, t) + ν 2s′2 K . (5.59) and therefore the action (5.54) is reduced to Note that, if we impose the conditions − t = 0 in (5.60), we recover the Schwarzian action (5.32).
To construct the gauge field in the bulk, we can introduce a radial dependence by performing a gauge transformation (5.9) with the group element Using the matrix representation (5.21), this leads to the following form for the connection (5.48) from which we can read off the zweibein and spin connection. In the AdS 2 case they read (5.63) whereas, for the dS 2 case we find As done in (5.38), we have interchanged the coordinates t and r in the dS 2 case (5.64), since the boundary of the de-Sitter space is at time-like infinity (in this case the prime denotes derivative with respect to the radial coordinate). From these expressions we can construct the space-time metrics Analogously to the previous section, one can ask what are the asymptotic symmetries associated with the theory defined by (5.60). For simplicity let us consider the case ν = 0, and rewrite the boundary field (5.59) as follows where we have defined the functions The asymptotic symmetries correspond to the set of gauge transformations (5.14) preserving the functional form of (5.69). Considering a gauge parameter of the form one can solve (5.14) to find λ 0 = µT ǫ , (5.72a) The functions T and L transform under the asymptotic symmetries as Using the explicit form of the gauge parameters, the charge can be integrated to give where we have defined The Poisson algebra of these charges can be computed using (5.17), which yields two Virasoro algebras Introducing the Fourier modes The central terms have the same form but with opposite sign, which is compatible with the form of the double Schwarzian boundary action (5.60). Finally, it is worth to mention that from the constraintΩH = µΩỸ we can expressρ ′ in terms ofs ′ andũ, ρ ′ = e −u s ′ , (5.81) resulting in the action In the next section, we will show that this last model admits a finite NR limit on its fields, which leads to a NR conformal model invariant under a NR conformal symmetry isomorphic to the Extended Newton-Hooke algebra.

Non-relativistic boundary theory
In this section, we construct the boundary theory associated to the NR JT gravity and compute the the asymptotic symmetries and its associated conserved charges. We show how these results can be extended to the Carrollian case.

Non-relativistic limit of the SL(2, R) × R boundary action
Here, we derive a NR limit of the relativistic model in (5.82). In order to do that, we consider the following redefinition of the SL(2, R) × R generators D = Taking the limit ε → 0, we get which we name Extended Galilean conformal (EGC) algebra in 1D. This algebra is isomorphic to the NH + algebra. In fact, using the change of basis we find the commutation relations (3.32) with cosmological constant Λ = 1/ℓ 2 Using the transformation (6.1) and its inverse (6.2), one can relate relativistic Goldstone fields in terms of NR ones as Therefore, applying the field transformations (6.6a) and (6.6d) on the action (5.82), we find where we have set γ 1 = −γ 0 in order to cancel a divergent term. Now, defining γ 0 := α 0 /ε 2 , in the limit ε → 0, the NR model becomes We shall refer to this action as the Non-Relativistic Schwarzian. Note that this boundary action appears at order in ε 2 in the same way that the NR gravity theory on the bulk does in (3.39). It is important to remark that if we perform the NR limit of the boundary action when expressed in terms of Schwarzians (5.60), the divergent term is not cancelled. In order to obtain the NH − algebra, we consider the following redefinition of the sl(2, R) × R generators Inverting these relations leads to In the limit ε → 0, we find the following NR contraction of sl(2, R) × R Therefore, one can consider the analogue change of variables between the relativistic Goldstone fieldss ,ũ and the ones of the Extended NH − algebraŝ ,û, i.e.
Applying these field transformations into (5.82), we get again setting γ 1 = −γ 0 , and taking the definition γ 0 := α 0 /ε 2 , in the limit ε → 0, we get the complexified version of NR Schwarzian (6.8), namely This action is closely related to the one obtained in [17]. In the following, we will derive these same actions using an IHM.

Extended Newton-Hooke + 2 case
Following the steps outlined in Section 5, we construct the MC one-form associated to the Extended Galilean conformal algebra (6.3) by considering a local parametrisation of a group element U as U = e sZ e ρH e yK e uD , (6.18) where s, ρ, y, and u correspond to the set of NR coordinates of the group manifold. We consider a faithful 3 × 3 matrix representation of (6.3) given by (6.19) so that the group element U reads (6.20) The left-invariant MC one-form is then given by with components The invariant bilinear form in this case reads (6.3) where c 0 and c 1 are arbitrary constants. This bilinear form is non-degenerate for c 1 = 0. Using these expressions, the boundary action (5.13) takes the form and is invariant under the following global symmetry transformations where the infinitesimal parameters can be grouped as In order to reduce the number of Goldstone fields, we will impose a particular IHM constraint. Let us consider the constraints with α a constant. The boundary theory (6.24) then reduces to S[u, s] = dt c 0 u ′2 + 2 c 1 s ′′ + s ′ u ′ . (6.27) Note that, performing the change of variables (6.28) and setting c 0 = 0 in the bilinear form (6.23), we recover the action (6.8) In this case the boundary connection takes the form where we have defined the functions The gauge transformations preserving these boundary conditions are obtained by solving (5.14), which gives whereas the variations of L and T read The charge (5.16) can be integrated, leading to where The Poisson algebra (5.17) in this case takes the form of the Twisted Warped Virasoro algebra [16,17,62], which after expanding in modes  We can introduce the dependence on a radial coordinate r by applying a gauge transformation with an r-dependent group element of the form [17] b = e −rH = which satisfy the field equations (3.44). In this case, the Newton-Cartan geometric structure is described by the asymptotic spatial metric which is degenerate with null vector 6.3 Extended Newton-Hooke − 2 case Now we consider the boundary action associated to the Twisted Extended Conformal algebra (6.11), isomorphic to the NH − 2 symmetry. The NR analysis in the case of negative cosmological constant can be done in the same way as the previous case by using the the Extended Galilean conformal algebra (6.3) and the complex change of basis (6.1). This relates the coordinates of the coset (6.18) to new coordinates {ŝ,ρ,ŷ,û} associated to the algebra (6.11), with symmetry parameterǫ =θẐ +βĤ +γK +αD . The action (6.45) matches the model found in [17] as the flat limit of the Schwarzian action.
To perform the asymptotic analysis, we can use (6.30) and the change of basis (6.1) to write the boundary connection as Then, the same analysis as the previous section leads to the variations (6.33) for the functions L(t) and T (t) and to the asymptotic symmetry (6.38).

Extended Carroll (A)dS 2 case
Due to the duality (4.3), the Extended Conformal Galilean algebra (6.3) and its twisted version (6.11) are isomorphic to Carrollian symmetries. Indeed starting from (6.3) and using the change of basis This means that the boundary actions (6.27) and (6.45) can also be interpreted as a boundary action of Carrollian JT gravity. Thus, the asymptotic symmetry analysis done in Section 6.2 also holds here and leads to the Warped Virasoro symmetry (6.38).
Similarly to the NR case, the change of variables (6.48) for the Carroll AdS 2 case can be used to define a Carrollian geometry. Indeed, the asymptotic form of the Carrollian gravitational gauge fields read  In this case, the Carrollian geometric structure is described by the temporal metric τ µν ≡ −τ µ τ ν = −ℓ 2 (rT + α + L) 2 dt 2 − 2(rT + α + L)drdt + dr 2 , (6.53) which is degenerate and has a null vector given by A similar construction for the Carroll dS 2 can be done by means of the change of variables (6.50).

Conclusions and Outlook
We have constructed the NR and Carrollian limits of JT gravity theory and the corresponding boundary actions. The analysis in the bulk is done first by considering the NR and Carrollian limits of a BF action with gauge algebra (A)dS 2 × R. Next, we have constructed BF theories with gauge algebras given by NH ± and Carroll (A)dS 2 . The Carrollian analysis is easily done using the duality (4.3) between NR and Carrollian symmetries. The he NR and Carrollian second order theories in the bulk are constructed by considering a suitable expansion of the metric tensor up to order 1/c 2 and c 2 , respectively. The actions are such that no divergent terms appear in the expansion if the Newton's constant is properly rescaled. This property is due to the fact that in two dimensions there are no divergent terms in the expansion of the Ricci scalar. This is a remarkably property that is in high contrast with its analogue in three and four space-time dimensions [40][41][42][43].
The boundary actions associated to the NR and Carroll JT gravities have been obtained using the method of non-linear realisations and IHM. In particular, we have shown that this reproduces the Drinfeld-Sokolov reduction of SL(2, R) [59,60], which has been previously used in [61] to define boundary conditions of JT gravity in analogy to the three-dimensional gravity case [63]. The procedure has been generalised to the case of the SL(2, R)×R algebra and the conformal descriptions of NH ± 2 and Extended Carroll (A)dS 2 . This leads to the NR Schwarzian action (6.8) and its complexification (6.17). We have also derived the corresponding conserved charges and the asymptotic symmetry of these theories, which is given by the twisted Warped Virasoro algebra [16,17,62]. Using the explicit form of the boundary gauge fields, we have reconstructed the fields in the bulk, which define Newton-Cartan [33][34][35][36][37] and Carrollian structures [38,39].
Further studies can be done, for example: The relation between the Carroll geometry (6.52) and Carrollian structures previously constructed in [38] and [39]. The analysis here presented could be useful to find the asymptotic symmetries of general Newton-Cartan and Carroll geometries.
Given the relation between IHM and Drinfeld-Sokolov reduction, it would be useful to do a systematic classification of all possible IHM and to study all possible boundary actions.
An important future direction of this work is also the formulation of the post-Newtonian as well as post-Carrollian corrections of two-dimensional gravity and their boundary theories following recent results that have been obtained in three and four-dimensional gravity [42,43,[64][65][66]. This is a work in progress.
Other possible future directions are the study of the relation of the NR Schwarzian with a novel regime of the SYK model, the quantisation of the NR JT action and its Carrollian counterpart, their relation to matrix models along the lines of [61], and their supersymmetric extensions.

Acknowledgments
We thank Glenn Barnich, Jaume Gomis, Marc Henneaux, Axel Kleinschmidt, Mauricio Valenzuela and Jorge Zanelli for enlightening comments and discussions. We specially acknowledge Luis Avilés for taking active part in the initial stages of this work. JG has been supported in part by MINECO FPA2016-76005-C2-  for which the bilinear form can be expressed simply as ·, · = 2µ Tr( ).

B Building blocks for the NR second order formulation
In this appendix, we compute the NR second-order of JT gravity. The starting point is to consider a ε-expansion for the metric g µν [36,[40][41][42][48][49][50][51] and then from the metric compatibility, we find the Levi-Civita connections at the order in ε which we are interested.

B.1 NR affine connections
Let us start with the expansion (3.1) of the metric and its inverse From the relation g νρ g ρµ = δ µ ν , we get order-by-order the identities (2) g µρ g νρ g νρ g µρ Now we demand the metric compatibility condition for the Lorentzian metric, namely The condition (B.3) implies the following set of equations In order to solve the affine connections, we proceed in the standard way by permuting indexes in (B.4), we find Using the relations (B.2) in the last equations, we get the following expressions for the terms in the expansion Levi-Civita affine connection