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JHEP11(2018)077

Published for SISSA by Springer

Received: September 20, 2018

Accepted: November 4, 2018

Published: November 13, 2018

Boundary theories for dilaton supergravity in 2D

Marcela C´ardenas,

a,b,c,e

Oscar Fuentealba,

c,d,e

Hern´an A. Gonz´alez,

c,e

Daniel Grumiller,

c,d,e

Carlos Valc´arcel

f,g

and Dmitri Vassilevich

g,h

APC, Universit´e Paris Diderot & CNRS, CEA, Observatoire de Paris,

Sorbonne Paris Cit´e 10, rue Alice Domon et L´eonie Duquet, F-75205 Paris CEDEX 13, France

Universidad Cat´olica del Maule, Av. San Miguel 3605, Talca, Chile

Erwin-Schr¨odinger International Institute for Mathematics and Physics,

Boltzmanngasse 9A, A-1090 Vienna, Austria

Centro de Estudios Cient´ıﬁcos (CECs), Av. Arturo Prat 514, Valdivia, Chile

Institute for Theoretical Physics, TU Wien, Wiedner Hauptstr. 8-10/136, A-1040 Vienna, Austria

Instituto de F´ısica, Universidade Federal da Bahia,

Cˆampus Universit´ario de Ondina, 40210-340, Salvador, B.A. Brazil

CMCC-Universidade Federal do ABC, Santo Andr´e, S.P. Brazil

Department of Physics, Tomsk State University, Tomsk, Russia

E-mail: cardenas@apc.in2p3.fr, fuentealba@cecs.cl,

hgonzale@hep.itp.tuwien.ac.at, grumil@hep.itp.tuwien.ac.at,

valcarcel.flores@gmail.com, dvassil@gmail.com

Abstract: The osp(2, N)-BF formulation of dilaton supergravity in two dimensions is

considered. We introduce a consistent class of asymptotic conditions preserved by the

extended superreparametrization group of the thermal circle at inﬁnity. In the N = 1

and N = 2 cases the phase space foliation in terms of orbits of the super-Virasoro group

allows to formulate suitable integrability conditions for the boundary terms that render

the variational principle well-deﬁned. Once regularity conditions are imposed, requiring

trivial holonomy around the contractible cycle the asymptotic symmetries are broken to

some subsets of exact isometries. Diﬀerent coadjoint orbits of the asymptotic symmetry

group yield diﬀerent types of boundary dynamics; we ﬁnd that the action principle can

be reduced to either the extended super-Schwarzian theory, consistent with the dynamics

of a non-vanishing Casimir function, or to superparticle models, compatible with bulk

conﬁgurations whose Casimir is zero. These results are generalized to N ≥ 3 by making

use of boundary conditions consistent with the loop group of OSp(2, N). Appropriate

integrability conditions permit to reduce the dynamics of dilaton supergravity to a particle

moving on the OSp(2, N) group manifold. Generalizations of the boundary dynamics for

N > 2 are obtained once bulk geometries are supplemented with super-AdS

asymptotics.

Keywords: 2D Gravity, Extended Supersymmetry, Field Theories in Lower Dimensions

ArXiv ePrint: 1809.07208

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP11(2018)077

JHEP11(2018)077

9.3 Asymptotic symmetries for osp(2, 2) BF gravity 30

9.3.1 Loop group boundary condition 30

9.3.2 Extended superconformal boundary condition 30

9.3.3 Extended warped superconformal boundary condition 30

10 Concluding remarks 31

A Matrix representation of OSp(2, N) 32

1 Introduction

Gauge symmetry is generically linked to the redundancy in the variables used to describe a

ﬁeld theory. This association is nevertheless not entirely true in theories with boundaries:

part of the gauge degrees of freedom becomes dynamical on these boundaries. The working

example of this realization is the fact that three-dimensional Chern-Simons theories in

presence of a boundary can be expressed as a WZW model in two dimensions [1, 2].

Since then, this relation has been extended to topological theories of gravity [3–9] and

more recently to theories with local degrees of freedom [10–13]. Deepening further this

program can help to understand the role played by boundary modes in the context of

quantum gravity. From a more general perspective, this can shed light on the origin of the

holographic behavior of gravity [14, 15].

Here we focus on the study of boundary dynamics associated with two-dimensional

BF theories based on the gauge group OSp(2, N). These lagrangians are supersymmetric

extensions of the so-called Jackiw-Teitelboim (JT) model [16, 17]. The motivation for this

is twofold. On the one hand, the addition of local supersymmetry is a natural way to gen-

eralize the results of [18], where the asymptotic dynamics was analyzed for (bosonic) higher

spin extensions of the JT theory. On the other hand, it is known that supersymmetric BF

models provide gravitational duals for supersymmetric extensions of the SYK model in the

low temperature regime [19–22]. In this sense, the main purpose of this paper is to explore

the asymptotic dynamics of two-dimensional dilaton supergravity and study its connec-

tion with models placed at the conformal boundary of the spacetime; namely the super-

Schwarzian theory and superconformal quantum mechanics [23–25]. This analysis yields a

consistent setup that extends and complements previous analyses in the N = 1 case [26]

and more recent ones in the second order formulation of dilaton supergravity [27, 28] for

N = 1 and N = 2 supersymmetries.

Working with BF models allows to introduce a generalized notion of Euclidean black

holes: they can be regarded as gauge ﬁelds with trivial holonomies along the thermal cycle.

In two dimensions, this condition adds restrictions on the set of conﬁgurations with given

asymptotics. Speciﬁcally, restricting the phase space to globally well-deﬁned black holes

produces an explicit breaking of the asymptotic symmetry group. Notably, this eﬀect has

been observed in dual descriptions controlled by the Schwarzian theory [8, 29, 30].

– 1 –

JHEP11(2018)077

In this work, we propose two diﬀerent sets of boundary conditions with ﬂuctuating

dilaton multiplets. The ﬁrst one reproduces the SO(N)-extended superreparametrization

group around the thermal circle at inﬁnity, SDiﬀ(S

). The second one is preserved by

the loop group of OSp(2, N). For the former set of asymptotic conditions, the reduced

action principle can be determined by two diﬀerent boundary models related to speciﬁc

coadjoint orbits of SDiﬀ(S

). Concretely, we show that the asymptotic dynamics described

by the extended super-Schwarzian theory is consistent with a nonvanishing Casimir function

while the dynamics compatible with zero Casimir entails the existence of a supersymmetric

particle model at the boundary. In both cases, however, the phase space is written in

terms of orbits of the super-Virasoro group, which is only accessible for N = 1 and N =

2. This is due to quadratic contributions of aﬃne so(N ) currents in the inﬁnitesimal

transformation laws N ≥ 3.

In order to access the asymptotic dynamics for N ≥ 3,

we make use of the second set of boundary conditions, consistent with the loop group

of OSp(2, N), which allows to reformulate the problem in terms of a particle moving on

the group manifold. Further constraints on the latter reduced model are introduced when

requiring to have asymptotically super-AdS

solutions in the bulk. This procedure gives

rise to supersymmetric extensions of particle models without self-interacting potentials and

extended super-Schwarzian theories with N > 2.

The paper is organized as follows. In the next section, we present the BF formulation of

minimal dilaton supergravity in two dimensions with gauge group OSp(2, 1). In section 3, a

consistent set of asymptotic conditions is proposed together with a superspace formulation

that describes in a super-covariant way the action of the asymptotic symmetries on the

ﬁelds. The latter approach turns out to be particularly useful in order to foliate the phase

space of the theory in terms of the adjoint and coadjoint actions of the super-Virasoro

group. Using this prescription, we demand suitable boundary conditions that render the

variational principle well-deﬁned. This is discussed in section 4. Section 5 is devoted to an-

alyze regularity conditions on solutions describing Euclidean black holes. It is shown that

conﬁgurations with trivial holonomies are solutions of the supersymmetric Hill equation,

which must obey (anti)periodic boundary conditions. In section 6, the reduced action is

obtained by evaluating the variational principle on the constraint surface, which for our set

of asymptotic conditions makes the bulk contribution vanish. Therefore, the regularized

action is only determined by a boundary term that leads — after restricting to regular

solutions — to either the N = 1 super-Schwarzian theory, or to a supersymmetric particle

model. In section 7, we present asymptotic conditions extending the previous results but

mainly focusing on the N = 2 case, arriving at analogous conclusions. Section 8 deals with

the boundary dynamics of N > 2 extended dilaton supergravity, where supersymmetric

extensions of conformal mechanics and Schwarzian theories are found. In section 9 we go

beyond the highest-weight ansatz to consider more general supersymmetric asymptotically

AdS

boundary conditions. Finally, section 10 is dedicated to discuss possible applica-

tions and extensions of this work. Conventions associated to the explicit representation of

osp(2, N) generators can be found in appendix A.

This kind of non-linearities also appears in the case of extended AdS supergravity in 3D [31].

– 2 –

JHEP11(2018)077

Before starting we mention that we work in Euclidean signature throughout the whole

paper. Thus, when speaking about “time” we always mean “Euclidean time”.

2 N = 1 dilaton supergravity in two dimensions

The ﬁrst order action for the minimal supersymmetric extension of the JT model [32, 33]

reads

I =

4π





+

ωe

−

ψγ



+X dω+





ψγ



λDψ+

λe



(2.1)

where the manifold M has the topology of a disk, endowed with a time τ identiﬁed as

τ ∼ τ + β and a non-compact direction r > 0.

The ﬁeld content of this theory is the following: X

is a pair of Lagrange multipliers

enforcing the supercovariant torsion constraint, X stands for the dilaton ﬁeld, λ

is the

dilatino ﬁeld, e

corresponds to the one-form zweibein, ω is the dualized spin connection and

is the gravitino one-form. The covariant derivative acting on the gravitino is deﬁned as

Dψ = dψ +

ωγ

ψ . (2.2)

The indices are contracted with the ﬂat metric δ

and 

= −1, where a, b = 0, 1. The

Majorana conjugate is deﬁned as

= ψ

βα

, where C

αβ

= 

αβ

(

+−

= −1) is the

charge conjugation matrix. The matrices γ

satisfy the two-dimensional Cliﬀord algebra,

{γ

, γ

} = 2δ

, and γ

= −γ

, then γ

= −I

It is possible to show that action (2.1), apart from being invariant under local Lorentz

transformations and conformal boosts, is invariant under the following local supersymmetry

transformations

δe

= −

ψγ

 δω =

ψγ

 δψ = D +

 (2.3)

δX

= −

λγ

 δX = −

λγ

 δλ =

 −

Xγ

 . (2.4)

Minimal dilaton supergravity (2.1) does not propagate local degrees of freedom. This

property is made manifest by writing the action in terms of a BF-theory for OSp(2, 1) [34].

Its bulk action reads

I =

2π

Str





dA + A



(2.5)

where the zero-form dilaton multiplet X = X

+ λ

and the one-form connection

A = A

+ ψ

, are valued in the osp(2, 1) algebra

, J

] = 

IJK

, Q

] =

(Γ

)

, Q

} = −

(CΓ

)

αβ

. (2.6)

The form of the action implies that we have adopted the convention (ψ

)

∗

= ψ

for the complex

conjugate of the product of two real Grassmann variables.

– 3 –

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