Carrollian and Non-relativistic Jackiw-Teitelboim Supergravity

We present non- and ultra-relativistic Jackiw-Teitelboim (JT) supergravity as metric BF theories based on the extended Newton-Hooke and extended AdS Carroll superalgebras in two spacetime dimensions, respectively. The extended Newton-Hooke structure, and, in particular, the invariant metric necessary for the BF construction of non-relativistic JT supergravity, is obtained by performing an expansion of the $\mathcal{N}=2$ AdS$_2$ superalgebra. Subsequently, we introduce the extended AdS$_2$ Carroll superalgebra, and the associated invariant metric, as a suitable redefinition of the extended Newton-Hooke superalgebra. The mapping involved can be seen as the supersymmetric extension of the duality existing at the purely bosonic level between the extended Newton-Hooke algebra with (positive) negative cosmological constant and the extended (A)dS Carroll algebra in two dimensions. Finally, we provide the Carrollian JT supergravity action in the BF formalism. Moreover, we show that both the non-relativistic and the ultra-relativistic theories presented can also be obtained by direct expansion of $\mathcal{N}=2$ JT supergravity.


Introduction
Progresses in the understanding of the quantum nature of black holes have been enabled by Jackiw-Teitelboim (JT) gravity [1,2] (which is a model of 2D dilaton gravity, cf., e.g., [3,4]) and its holographic dual, especially concerning quantum chaos and scrambling, which are indeed believed to be related to the black hole information paradox (for a comprehensive review see, e.g., [5][6][7]).Besides, from the gauge/gravity duality perspective, JT gravity, as a two-dimensional toy model of quantum gravity describing universality in nearly extremal black holes [8,9], is dual to the Sachdev-Ye-Kitaev (SYK) model [10,11] at large N and low energies.The SYK model is an exactly solvable quantum theory of N >> 1 interacting Majorana fermions (cf.[7,12] and references therein) that has received a growing attention since its formulation from both the high energy and the condensed matter physics communities, and it is nowadays considered one of the most notable models for quantum chaos and holography.This is due to the fact that it exhibits some remarkable properties.For instance, it is exactly solvable in the large N and IR limit, where, moreover, it acquires conformal symmetry and the effective action can be approximated by the Schwarzian one.Furthermore, it exhibits a holographic relation with JT gravity, with the latter describing excitations above the near-horizon extremal black hole and, once considered on the clipped Poincaré disk, effectively reducing to the one-dimensional theory with Schwarzian action.In particular, the SYK model resulted to be an excellent toy model for many physical phenomena, including quantum chaos [13], information scrambling [14][15][16], traversable wormholes [17,18], and strange metals [19].
On the other hand, another remarkable feature of JT gravity is that it can be regarded as a topological theory in two spacetime dimensions within the BF formalism, which is reminiscent of the three-dimensional formulation of Chern-Simons theory. 1 The BF formulation of JT gravity [37][38][39] is particularly well-suited to consider other types of theories by extending the underlying symmetries.These symmetries can be either relativistic or ultra/non-relativistic (super)symmetries.Recently, the non-relativistic (c → ∞, where c is the speed of light) and ultra-relativistic (c → 0, also called Carrollian, see, e.g., [40][41][42][43][44][45]) limits of JT gravity have been obtained in both the second-order and the BF formalism in [46,47].Moreover, the BF setup allows one to consider the boundary theory of JT supergravity [48], the latter resulting to be the gravity dual of the supersymmetric extension of SYK model in the low energy limit [48,49].
In the above discussed context, at least to our knowledge, consistent non-and ultra-relativistic JT supergravity theories have not been developed yet, especially within the BF formulation.In this work, we present non-and ultra-relativistic JT supergravities as metric BF theories based, respectively, on the extended Newton-Hooke and extended AdS Carroll superalgebras in two spacetime dimensions.The former superalgebra, together with the associated invariant metric necessary to implement the metric BF construction, is obtained by performing a Lie algebra expansion of the N = 2 AdS 2 superalgebra.Consequently, we introduce the extended AdS 2 Carroll superalgebra and its invariant metric through a suitable redefinition of the extended Newton-Hooke superalgebra.Hence, we construct the Carrollian JT supergravity action in the BF formalism.Moreover, we show that both the non-and ultra-relativistic JT supergravity actions presented in this work can also be obtained by directly expanding JT supergravity formulated as a BF theory based on the N = 2 AdS 2 superalgebra.
The paper is organized as follows: We start by briefly reviewing, in Sections 2 and 3 respectively, the first-order formulation of JT gravity as a BF theory based on the AdS 2 algebra and the extension to the case of N = 2 JT supergravity as a BF theory based on the N = 2 AdS 2 superalgebra.In Section 4 we introduce the extended Newton-Hooke superalgebra as a Lie algebra expansion of the N = 2 AdS 2 one, deriving the associated invariant metric which allows us to consequently develop the non-relativistic JT supergravity action as a BF model.Subsequently, in Section 5 we move on to the construction of the ultra-relativistic counterpart of JT supergravity.
Hence, we derive the supersymmetric extended AdS 2 Carroll algebra, which can be obtained both as a redefinition of the extended Newton-Hooke superalgebra and as an expansion of the N = 2 AdS 2 one.The same applies at the level of the invariant metric, which permits us to construct the Carrollian JT supergravity theory in the BF setup.Section 6 is devoted to concluding remarks and possible future developments of our analysis.In Appendix A we collect our notation and conventions.
2 Review of Jackiw-Teitelboim gravity as a metric BF theory In this section we will consider the first-order formulation of JT gravity as a BF theory based on the AdS 2 algebra [37][38][39].We will follow [46] and briefly review the structure of (metric) BF theories for later purposes.
The BF theory action is given by with where k is a dimensionless constant, X * = X I E I is a scalar transforming in the coadjoint representation of the Lie algebra g on which the theory is based, and the Lie algebra valued 1-form where e I are the generators of g.The dual g * has basis E I obeying E I (e J ) = δ I J .The action (2.1) is invariant under gauge transformations The equations of motions of the theory read If g admits an invariant metric •, • : g × g → R that is a non-degenerate, symmetric, adinvariant bilinear form,2 then one can use this metric to identify elements of the dual g * with elements of the Lie algebra g by means of X , • = X * (•), that is X I = g IJ X J (where g IJ = e I , e J and g IJ g JK = δ I K ).In BF theories having an invariant metric is not a requirement, but in this paper we are interested in a particular subclass of BF theories (see [46]), called metric BF theories, whose construction is in fact based on Lie algebras exhibiting an invariant metric.The action for a metric BF theory is given by and its equations of motion read (2.7) A standard example of metric BF theories can be obtained by considering as a starting point simple Lie algebras where one can use the matrix trace to write L mBF = tr(X F ).
The first-order formulation of (AdS) JT gravity as a metric BF theory was developed by considering the AdS 2 algebra, which is spanned by the Lorentz and spacetime translations generators {J, P A }, with A, B, . . .= 0, 1, and read where Λ = − 1 ℓ 2 is the cosmological constant written in terms of the AdS 2 radius ℓ, and ǫ 01 = −ǫ 10 = 1.This algebra admits an invariant metric given by (2.9) In this setup, the gauge field and the coadjoint scalar take the following form (cf. [46]): and the corresponding covariant curvatures are given by Using (2.10), (2.11), and the invariant metric (2.9), one obtains the BF theory formulation of JT gravity, that is (2.12) The equations of motion of the theory are given by In particular, the equations of motion of the fields X A enforce the two-dimensional torsion constraint for the zweibein E A .Upon solving the latter for the spin connection Ω and plugging it back into the action, one is left with the second-order action for JT gravity, where X turns out to be the dilaton field.In this way, one can derive the JT equations of motion in the second-order formalism (see [47]), R being the curvature scalar.The first of these equations yields locally AdS 2 spacetime, and the second can be considered as a back-reaction of the field X in response to the metric g µν .
3 N = 2 Jackiw-Teitelboim supergravity In this section, we will review the first-order formulation of N = 2 JT supergravity as a BF theory presented in [48], adopting, however, the slightly different notation and conventions of [69] (see also Appendix A).In order to proceed with the construction of the theory, it is convenient to introduce the N = 2 AdS 2 superalgebra as a starting point. 3The N = 2 AdS 2 superalgebra is given by the commutation relations and the anticommutator where C is the charge conjugate matrix, defined as C = iσ 2 , Λ = − 1 ℓ 2 is the cosmological, i, j = 1, 2 label the number of supercharges, and ε ij is the two-dimensional Levi Civita symbol (ε 12 = −ε 21 = 1).The superalgebra (3.1)-(3.2) is endowed with an invariant metric The gauge connection 1-form, the coadjoint scalar, and the curvature 2-form associated with the N = 2 AdS 2 superalgebra are, respectively,4 where now Notice that the λ i in (3.4) are fermionic, while X A , X, Y are bosonic.Our convention for the gamma matrices in two dimensions is given in Appendix A. We can then construct the BF theory of JT supergravity by using (3.4) and (3.5) along with the invariant metric (3.3).The resulting action takes the following form: One can show that the action (3.6) is invariant under the following supersymmetry transformations: where ǫ iα are the supersymmetry parameters.The equations of motion of the theory are , coming from variation of the action with respect to X A , X, Y, λ i , respectively, together with The construction above is the first-order formulation of N = 2 JT supergravity.Let us move on to the study of its non-and ultra-relativistic counterparts.
4 Supersymmetric extended Newton-Hooke algebra sNH 2 and non-relativistic Jackiw-Teitelboim supergravity We will now develop the non-relativistic counterpart of the JT supergravity theory reviewed in the previous section.The theory will be based on the extended Newton-Hooke superalgebra with negative cosmological constant (which we name sNH 2 ) we are going to present in the following.

Extended Newton-Hooke superalgebra sNH 2
The extended Newton-Hooke superalgebra in two dimensions can be obtained by performing a Lie algebra expansion on the N = 2 AdS 2 superalgebra.For a comprehensive treatment of the Lie algebra expansion procedure see, e.g., [70][71][72] and references therein.The supersymmetric extension of the (purely bosonic) extended Newton-Hooke (NH) algebra of [47] requires three fermionic generators, Q ± α and R α .Besides, it requires two extra bosonic generators U 1 and U 2 , both of them acting non-trivially on the fermionic generators in the presence of a cosmological constant Λ (while they become central charges in the limit Λ → 0, that is ℓ → ∞).In particular, the extra bosonic generators allows us to end up with a well-defined invariant metric.
Before implementing Lie algebra expansion on (3.1)-(3.2),we split the Lorentz indices as A = (0, 1), such that In this way, one obtains the following decomposition of the AdS 2 superalgebra: together with In addition, we introduce the following combination of the fermionic generators Q 1 α and Q 2 α : Hence, the decomposed algebra takes the form along with Then, we consider the following expansion of the generators: where η is the expansion parameter.In this way, we end up with the extended Newton-Hooke superalgebra sNH 2 , which has the non-vanishing commutators5 and anticommutators The supersymmetric extended Newton-Hooke algebra above admits the invariant metric We can now move on to the construction of the non-relativistic JT supergravity action as a BF theory based on the sNH 2 structure obtained above.

Non-relativistic Jackiw-Teitelboim supergravity action
Now that we have unveiled the sNH 2 structure, we can assign gauge fields for each Lie algebra generator, and the 1-form A in the case at hand can be written as The transformation rules for these gauge fields can be found by exploiting where ǫ A is the relevant gauge parameter and C JK I are the structure constants, in this case, of sNH 2 .In particular, the (non-trivial) transformations along the parameters ǫ +α , ǫ −α , ǫ α (associated with ψ + , ψ − , ρ) are, respectively, ) Besides, we have where the sNH 2 -covariant curvatures are given by As we have the invariant metric (4.10) and the curvatures (4.16), we are now able to write down the BF action for the non-relativistic JT supergravity theory, based on sNH 2 , which reads The action (4.17) is a supersymmetric generalization of the bosonic one presented in [46,47].This non-relativistic JT supergravity action is invariant under the transformations (4.12)-(4.14) together with the following (non-trivial) transformations of Φ, Π, Ξ, Σ, T 1 , T 2 , λ + , λ − , λ, with parameters, in order of appearance, ǫ +α , ǫ −α , ǫ: Note that the latter can also be derived by exploiting the Lie algebra expansion method on (3.7).The equations of motion of the theory correspond to the vanishing of the sNH 2 -covariant curvatures and coincide with the ones found in [46,47] when we restrict ourselves to the purely bosonic case.Notice that, in the supersymmetric theory, we end up with a non-vanishing torsion that is given, on-shell, in terms of a spinor bilinear.Finally, let us highlight that the same action (4.17) can be obtained by directly performing the expansion procedure on the supersymmetric AdS 2 theory.To do so, one shall start by decomposing the AdS 2 curvatures (3.5) as follows: Correspondingly, we can also implement the same decomposition on the action (3.6), that is Subsequently, we introduce the spinors Thus, the curvatures (4.21) boil down to We can then see that, performing the expansion we precisely end up with the sNH 2 curvatures (4.16).Finally, the expansion of the action yields X R(ω that is, with the identifications (2) the non-relativistic JT supergravity action (4.17).

Ultra-relativistic Jackiw-Teitelboim supergravity
In this section we will introduce a supersymmetric extension of the extended AdS 2 Carroll algebra which will allow us to develop the corresponding Carrollian, that is ultra-relativistic, JT supergravity action as a metric BF theory.
5.1 Supersymmetric extended AdS 2 Carroll algebra as a redefinition of sNH 2 The purely bosonic extended AdS 2 Carroll structure first appeared in [46] and [47].We shall now present a supersymmetric extension of the latter as a redefinition of sNH 2 .To this aim, let us therefore perform the following redefinition on the sNH 2 structure previously obtained: By doing so, we end up with the supersymmetric extended AdS 2 Carroll algebra, which reads together with Note that this construction is reminiscent of the duality existing at the purely bosonic level in two spacetime dimensions between the extended NH ± and the extended (A)dS 2 Carroll algebras.Let us mention that the same extended AdS 2 Carroll superalgebra (5.2)-(5.3)can also be obtained from the N = 2 AdS 2 superalgebra (3.1)-(3.2),once decomposed as in (4.2)-(4.3),by introducing the fermionic charges and performing the redefinition U → iU with the subsequent expansion (5.5) The extended AdS 2 Carroll superalgebra (5.2)-(5.3) is endowed with the invariant metric which, in particular, can be obtained from (4.10) by means of the mapping in (5.1).

Carrollian Jackiw-Teitelboim supergravity
We shall now proceed with the development of the ultra-relativistic BF JT supergravity theory based on the Carrollian superalgebra just unveiled.The 1-form A has the same form of (4.11), and also the implicit form of X and R is formally the same as in (4.15). 6On the other hand, now the (non-trivial) transformations along the parameters ǫ +α , ǫ −α , ǫ α are respectively given by ) ) (5.9) The extended super-AdS 2 Carroll-covariant curvatures read Therefore, using the invariant metric (5.6) together with the curvatures (5.10), we end up with the following ultra-relativistic JT supergravity action as a metric BF theory based on the extended AdS 2 Carroll superalgebra (5.2)-(5.3): The ultra-relativistic action (5.11) is a supersymmetric generalization of the purely bosonic Carrollian one in [46,47].It is invariant under the transformations (5.7)-(5.9)together with the following (non-trivial) transformations of the fields Φ, Σ, Ξ, Π, T 1 , T 2 , λ + , λ − , λ, along the parameters ǫ +α , ǫ −α , ǫ, in order of appearance: ) The equations of motion of this Carrollian theory correspond to the vanishing of the curvatures (5.10), and they boil down to the ones obtained in [46,47] if we restrict ourselves to the purely bosonic case.In the Carrollian supersymmetric theory above we end up with an on-shell nonvanishing torsion given in terms of a fermion bilinear.Let us conclude by observing that the same ultra-relativistic action (5.11) can be derived by expanding the N = 2 super-AdS 2 relativistic one.More precisely, starting from the decomposed expressions (4.21) and (4.22) and introducing the (new) spinors we get the following "pre-expanded" curvatures: Then, performing the expansion E 0 = ητ , E 1 = e + η 2 m , Ω = ηω , T = r 1 + η 2 r 2 , a theory that, in general, is not dynamically equivalent to its first-order counterpart.However, the equivalence is obtained by considering the non-relativistic and ultra-relativistic sectors dτ = 1 2 ψ+ γ 0 ∧ ψ + and de = − 1 2 ψ+ γ 1 ∧ ψ + , respectively.
The expansions we have performed in this work also enables us to consider BF theories beyond the supersymmetric Newton-Hooke and AdS Carrollian ones, and this would be particularly interesting from the Post-Newtonian expansion and large c expansion point of view [47,75,76].Furthermore, the procedure we have presented in this paper could be useful to obtain the next order boundary Schwarzian actions (which are dual boundary actions of next order gravity actions) by expanding the related Maurer-Cartan forms.
Another possible future direction consists in considering supersymmetric extensions of the Carrollian and non-relativistic boundary Schwarzian actions in [46,47] along the lines of [77] (some work is currently in progress on this point).Besides, the non-relativistic JT supergravity theory may serve as a starting point to consider the supersymmetric extension of the flat space boundary action that appeared in [78].It would also be interesting to consider ultra-and nonrelativistic limits (and associated supersymmetric extensions) of the most general deformation of JT gravity studied in [3,4].