#### Super-BMS3 invariant boundary theory from three-dimensional flat supergravity

Received: November
Super-BMS3 invariant boundary theory from
Glenn Barnich 0
Laura Donnay 0
Javier Matulich 0 1
Ricardo Troncoso 0 1
Open Access 0
c The Authors. 0
0 Casilla 1469 , Valdivia , Chile
1 Centro de Estudios Cient ́ıficos (CECs)
The two-dimensional super-BMS3 invariant theory dual to three-dimensional asymptotically flat N = 1 supergravity is constructed. It is described by a constrained or gauged chiral Wess-Zumino-Witten action based on the super-Poincar´e algebra in the Hamiltonian, respectively the Lagrangian formulation, whose reduced phase space description corresponds to a supersymmetric extension of flat Liouville theory.
Conformal and W Symmetry; Space-Time Symmetries; Gauge-gravity corre-
1 Introduction 2 3 4
Chiral constrained super-Poincar´e WZW theory
Solving the constraints in the action
Symmetries of the chiral WZW model
Super-BMS3 algebra from a modified Sugawara construction
Reduced super-Liouville-like theory
Gauged chiral super-WZW model
A Conventions
Gauged chiral bosonic WZW theory
Introduction
A prime example of duality between a three-dimensional and a two-dimensional theory
is the relation between a Chern-Simons theory in the presence of a boundary and the
associated chiral Wess-Zumino-Witten (WZW) model: on the classical level for instance,
the variational principles are equivalent as the latter is obtained from the former by solving
the constraints in the action [1–3].
In the case of the Chern-Simons formulation of three-dimensional gravity [4, 5], the role
of the boundary is played by non trivial fall-off conditions for the gauge fields. For anti-de
Sitter or flat asymptotics, a suitable boundary term is required in order to make solutions
with the prescribed asymptotics true extrema of the variational principle. Furthermore,
the fall-off conditions lead to additional constraints that correspond to fixing a subset of
the conserved currents of the WZW model [6, 7]. The associated reduced phase space
description is given by a Liouville theory for negative cosmological constant and a suitable
limit thereof in the flat case [8, 9]. This procedure was also implemented in the context of
three dimensional higher spin gravity without cosmological constant, where a flat limit of
Toda theory is recovered [10].
supergravity, whose algebra of surface charges has been shown to realize the centrally
extended super-BMS3 algebra [11]. The non-vanishing Poisson brackets read
i{Jm, Jn} = (m − n)Jm+n +
i{Jm, Pn} = (m − n)Pm+n +
i{Jm, Qn} =
{Qm, Qn} = Pm+n +
2 − n
c1 = µ
c2 =
where the fermionic generators Qm are labeled by (half-)integers in the case of
(anti)periodic boundary conditions for the gravitino, and the central charges are given
Here, G and µ stand for the Newton constant and the coupling of the Lorentz-Chern-Simons
form, respectively.
The resulting two-dimensional field theory admits a global super-BMS3 invariance.
By construction, the associated algebra of Noether charges realizes (1.1) with the same
values of the central charges. We provide three equivalent descriptions of this theory: (i) a
Hamiltonian description in terms of a constrained chiral WZW theory based on the
threedimensional super-Poincar´e algebra, (ii) a Lagrangian formulation in terms of a gauged
chiral WZW theory and (iii) a reduced phase space description that corresponds to a
supersymmetric extension of flat Liouville theory.
Besides the extension to the supersymmetric case, previous results in the purely bosonic
sector are also generalized. This is due to the inclusion of parity-odd terms in the action,
which suitably modifies the Poincar´e current subalgebra, and consequently, turns on the
additional central charge c1 in (1.1).
Brief review of (minimal) N
= 1 flat supergravity in 3D
with vanishing cosmological constant admits a Chern-Simons formulation [15]. Different
extensions of this theory have been developed in e.g., [16–29]. Hereafter we consider the
most general supergravity theory with N
flat boundary conditions, and leads to first order field equations for the dreibein, the
supergravity theory is recovered for a particular choice of the couplings (see below). The
spans the super-Poincar´e algebra,
[Ja, Jb] = ǫabcJ c ,
[Ja, Pb] = ǫabcP c ,
[Pa, Pb] = 0 ,
(Γa)βα Qβ , [Pa, Qα] = 0 ,
{Qα, Qβ} = − 2 (CΓa)αβ Pa ,
where C is the charge conjugation matrix (see appendix A for conventions). In these terms,
the action reads
I[A] =
where the bracket h·, ·i stands for an invariant nondegenerate bilinear form, whose only
nonvanishing components are given by
boundary term, the action reduces to
k Z 2Rˆaea + µL (ωˆ) − ψ¯αDˆ ψα ,
supersymmetry transformations
curvature two-form and the covariant derivative of the gravitino are defined as
By construction the action is invariant, up to a surface term, under the following local
decompose as
Ra = 1 γ2ǫabcebec + 1 γψ¯Γaψ , T a = −γǫabcebec − 4
2 4
Dψ = − 21 γeaΓaψ , (2.8)
the general (local) solution is
The asymptotic conditions proposed in [11] imply that the gauge field is of the form
A = h−1ah + h−1dh ,
a =
can be generalized, along the lines of [31], so as to include a generic choice of chemical
potentials [30].
Chiral constrained super-Poincar´e WZW theory
Solving the constraints in the action
Up to boundary terms and an overall sign which we change for later convenience, the
Hamiltonian form of the Chern-Simons action (2.3) is given by
hA˜, duA˜˙i + 2hduAu, d˜A˜ + A˜2i ,
where A = duAu + A˜.
One of the advantages of the gauge choice in (2.10), for which the dependence in the
radial coordinate is completely absorbed by the group element h, is that the boundary can
boundary generically describes a two-dimensional timelike surface with the topology of a
cylinder (R
×S1). We will also discard all holonomy terms. As a consequence, the resulting
action principle at the boundary only captures the asymptotic symmetries of the original
The boundary term in the variation of the Hamiltonian action is given by
the gauge field at the boundary fulfill
so that the boundary term becomes integrable. Consequently, the improved action principle
that has a true extremum when the equations of motion are satisfied is given by
I =
In this action principle Au are Lagrange multipliers, whose associated constraints are locally
I =
dudφ hG−1∂φG, G−1∂uGi − ωφaωaφ
hG−1dG, (G−1dG)2i .
Equivalently, in terms of the gauge field components, the action can be conveniently
dudφ ωφeau + eaφωau − ωφaωaφ + µω φaωau − ψ¯uψφ
a
(3ǫabceaωbωc + µǫ abcωaωbωc − 2
eq. (2.9) allows one to rewrite this expression in terms of a 2 by 2 matrix trace, so that
− 4
−(Λ′Λ−1)2 +µ Λ′Λ−1Λ˙ Λ−1 +
finds that the action reduces to that of a chiral super-Poincar´e Wess-Zumino-Witten theory,
dudφ Tr 2 λ˙λ−1α′ − (λ′λ−1)2 + µλ ′λ−1 λ˙λ−1 +
α + (µ∂ u − ∂φ)(λ′λ−1) − 4
respectively. The general solution of these equations is given by
α = τ a(φ) + δ(u) + uκ′κ−1 − µ [ln τ, ln κ] + 4 ζ1ζ¯2 +
Symmetries of the chiral WZW model
By using the Polyakov-Wiegmann identities, the action (3.10) can be shown to be invariant
under the gauge transformations
Moreover, it is also invariant under the following global symmetries
whose associated infinitesimal transformations read
δγα = 1 νγ¯λ−1 + 1 γ¯λ−1ν1 .
The Noether currents associated to a global symmetry, whose parameters are collectively
denoted by X1, generically read JX1 = −kX1 + ∂∂μLφi δX1φi, with δX1L = ∂μkXμ1. Hence, in
μ μ
the case of global symmetries spanned by (3.15), the corresponding currents are given by
P =
Q =
λ−1α′λ − u(λ−1λ′)′ + µλ −1λ′ − 4
general results guarantee that, in the Hamiltonian formalism, this computation corresponds
{Ja(φ), Jb(φ′)}∗ = ǫabcJ cδ(φ − φ ) − µ
′
{Ja(φ), Qα(φ′)}∗ = 1 (QΓa)αδ(φ − φ′) ,
{Qα(φ), Qβ(φ′)}∗ = − 2 (CΓa)αβPaδ(φ − φ ) − 2π Cαβ∂φδ(φ − φ′) ,
′
which is the affine extension of the super-Poincar´e algebra (2.2).
Super-BMS3 algebra from a modified Sugawara construction
In order to recover the super-BMS3 algebra (1.1) from the affine extension of the
superPoincar´e algebra in (3.19), it can be seen that the standard Sugawara construction has to
be slightly improved. Indeed, let us consider bilinears made out of the currents components
H =
P = − k
P2 Q+ + √
2 P0 Q− ,
for which the current algebra (3.19) implies
{H(φ), Ja(φ′)}∗ = −Pa(φ)δ′(φ − φ′) ,
{P(φ), Pa(φ′)}∗ = Pa(φ)δ′(φ − φ′) ,
{P(φ), Ja(φ′)}∗ = Ja(φ)δ′(φ − φ′) ,
{P(φ), Qα(φ′)}∗ = Qα(φ)δ′(φ − φ′) ,
When expressed in terms of modes, the algebra of the generators H, P corresponds to
the pure BMS3 algebra without central extensions, i.e., the bosonic part of (1.1) with
whose algebra disagrees with the non-centrally extended super BMS3 algebra given in (1.1).
It reflects the fact that the non-constrained super-WZW model (3.10) is invariant under
class constraints
global BMS3 transformations, but not under the full super-BMS3 symmetries, in the sense
that there are no (obvious) superpartners to H, P that would close with them according
to the (non centrally extended) super-BMS algebra (see [7] for an analogous discussion in
the case of the superconformal algebra).
According to the fall-off of the gauge field in (2.11), the remaining boundary conditions
P0 =
Q+ = 0 .
The super-BMS3 invariance of our model with the correct values of the central charges
is recovered only once the constraints (3.23) are imposed. The generators of super-BMS3
symmetry in the constrained theory are given by
which are representatives that commute with the first class constraints (3.23), on the
surface defined by these constraints. Furthermore, on this surface, the Dirac brackets of
the generators are given by
{H˜(φ), P˜(φ′)}∗ = (H˜(φ) + H˜(φ′))∂φδ(φ − φ ) − 2kπ ∂φ3δ(φ − φ′) ,
′
{P˜(φ), P˜(φ′)}∗ = (P˜(φ) + P˜(φ′))∂φδ(φ − φ ) − 2π ∂φ3δ(φ − φ′) ,
′
1 G˜(φ′))∂φδ(φ − φ′) ,
so that, once expanded in modes according to
Pm =
Jm =
Qm =
the super-BMS3 algebra (1.1) with central charges given in (1.2) is recovered.
Reduced super-Liouville-like theory
In order to obtain the reduced phase space description of the action (3.10) on the constraint
surface defined by (3.23), it is useful to decompose the fields according to
and hence, by virtue of (4.1) and (4.2), the reduced chiral super-WZW action (3.10) is
IR =
dudφ ξ′ϕ˙ − ϕ′2 + µϕ ′ϕ˙ + √ χχ˙ ,
super-BMS3 generators (3.24) then reduce to
H˜ =
ϕ′2 −2ϕ′′ ,
P˜ =
G˜= 21/4 k
which generate the following transformations
δξ = 2f ϕ′ + ξ′Y + 2f ′ − 21/4ǫχ ,
Liouville-like theory turns out to be invariant under (4.5), and the mode expansion of the
algebra of Noether charges is again given by (1.1) and (1.2).
Gauged chiral super-WZW model
The super-Liouville-like action (4.3), that has been shown to be equivalent to the chiral
super-WZW model (3.10) on the constraint surface given by (3.23), can also be described
through a gauged chiral super-WZW model. Here we follow the procedure given in [36],
where it was shown that Toda theories can be written as gauged WZW models based on a
Lie group G. The action is endowed with additional terms involving the currents linearly
coupled to some gauge fields that belong to the adjoint representation of the subgroups of
G generated by the step operators associated to the positive and negative roots.
Hence, we consider the following action principle
dudφ Tr Au λ−1α′λ − u λ−1λ′ ′
− 4
One can then show that the action (5.1) is invariant (up to boundary terms) under
the transformations given in (3.15), where a subset of the symmetries has been gauged
δσA˜u = − (σ˙ + [Au, σ]) , δγΨ¯ = −∂uγ¯ .
δϑAu = −(ϑ˙ + [Au, ϑ]) , δϑA˜u = −[A˜u, ϑ].
Therefore, the reduced theory described by the action in (4.3) is equivalent to the one
in (5.1), which corresponds to a WZW model in which the subgroup generated by the first
multipliers for these currents, so that the variation of the action with respect to these
nonpropagating fields sets them to zero. In other words, solving the algebraic field equations
for the gauge fields into the action amounts to imposing the first class constraints (3.23),
which shows the equivalence of both descriptions.
Acknowledgments
We thank M. Ban˜ados, O. Fuentealba, G. Giribet, M. Henneaux, P.-H. Lambert, B. Oblak,
A. P´erez, and very especially H. Gonz´alez for enlightening discussions. J.M. and R.T.
wish to thank the Physique th´eorique et math´ematique group of the Universit´e Libre de
Bruxelles, and the International Solvay Institutes for the warm hospitality. This work is
partially funded by the Fondecyt grants N
◦ 1130658, 1161311, 3150448. The Centro de
Estudios Cient´ıficos (CECs) is funded by the Chilean Government through the Centers of
Excellence Base Financing Program of Conicyt. L.D. is a research fellow of the “Fonds
pour la Formation `a la Recherche dans l’Industrie et dans l’Agriculture”-FRIA Belgium.
She thanks the group of the Centro de Estudios Cient´ıficos (CECs) for the hospitality. The
work of G.B. and L.D. is partially supported by research grants of the F.R.S.-FNRS and
IISN-Belgium as well as the “Communaut´e francaise de Belgique - Actions de Recherche
0 −1
The matrices fulfill the following useful properties:
(Γa)αβ(Γa)γδ = 2δδαδβγ − δβαδδγ ,
and it has the following Noether symmetries
According to (3.23), we are interested in gauging the subset of these symmetries involving
−1 0
Gauged chiral bosonic WZW theory
Let us describe here a way to construct a gauged chiral iso(2, 1) WZW model associated
k Z dudφ Tr h−Au λ−1α′λ − u λ−1λ′ ′
One can check that the action
is invariant under
the suitable final action is
δϑλ = −λϑ(u, φ) , δϑα = −uλϑ′λ−1 ,
δϑAu = −(ϑ˙ +[Au, ϑ]) , δϑA˜u = −[A˜u, ϑ] , (B.4)
δσα = λσ(u, φ)λ−1 , δσAu = 0 ,
Since the constraints we want to implement set some current components to a constant,
k Z dudφ Tr h−Au λ−1α′λ − u λ−1λ′ ′
is indeed still gauge invariant since, as noticed in [36], the variation of Tr[µ M A˜u] under a
gauge transformation is a boundary term.
Finally, in order to see how the constraints are explicitly implemented, it is useful to
parametrize the fields according to
that, taking µ = 1, the reduced action is
I =
found in [9].
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