#### Asymptotic structure of \( \mathcal{N}=2 \) supergravity in 3D: extended super-BMS3 and nonlinear energy bounds

HJE
Asymptotic structure of N
Oscar Fuentealba
Javier Matulich
Ricardo Troncoso
Centro de Estudios Cient´ıficos (CECs)
Av. Arturo Prat
Valdivia
Chile
The asymptotically flat structure of N = (2, 0) supergravity in three spacetime dimensions is explored. The asymptotic symmetries are found to be spanned by an extension of the super-BMS3 algebra, endowed with two independent affine uˆ(1) currents of electric and magnetic type. These currents are associated to U(1) fields being even and odd under parity, respectively. Remarkably, although the U(1) fields do not generate a backreaction on the metric, they provide nontrivial Sugawara-like contributions to the BMS3 generators, and hence to the energy and the angular momentum. Consequently, the entropy of flat cosmological spacetimes endowed with U(1) fields acquires a nontrivial dependence on the zero modes of the uˆ(1) charges. If the spin structure is odd, the ground state corresponds to Minkowski spacetime, and although the anticommutator of the canonical supercharges is linear in the energy and in the electric-like uˆ(1) charge, the energy becomes bounded from below by the energy of the ground state shifted by the square of the electric-like uˆ(1) charge. If the spin structure is even, the same bound for the energy generically holds, unless the absolute value of the electric-like charge is less than minus the mass of Minkowski spacetime in vacuum, so that the energy has to be nonnegative. The explicit form of the global and asymptotic Killing spinors is found for a wide class of configurations that fulfills our boundary conditions, and they exist precisely when the corresponding bounds are saturated. It is also shown that the spectra with periodic or antiperiodic boundary conditions for the fermionic fields are related by spectral flow, in a similar way as it occurs for the N = 2 super-Virasoro algebra. Indeed, our supersymmetric extension of BMS3 can be recovered from the Ino¨nu¨-Wigner contraction of the superconformal algebra with N = (2, 2), once the fermionic generators of the right copy are truncated.
Conformal and W Symmetry; Space-Time Symmetries; Gauge-gravity corre-
1 Introduction 2
N = (2, 0) Poincar´e supergravity in three spacetime dimensions
3 Asymptotic structure: supersymmetric extension of BMS3 with N =
Canonical generators and their algebra 3.1.1
Asymptotic symmetry algebra from a truncation of the homogeneous
contraction of the superconformal algebra with N = (2, 2)
4 Nonlinear energy bounds from super-BMS3 with N = (2, 0)
5 Bosonic configurations and some of their properties
Regularity conditions
Metric formalism
Cosmological configurations and their thermodynamics
Conical defects with U(1) fluxes
Minkowski spacetime and conical surpluses endowed with U(1) fields
6 Bosonic solutions with unbroken supersymmetries
6.1
Asymptotic Killing spinor equations
6.1.1
6.1.2
6.1.3
6.1.4
6.1.5
Cosmological configurations
Null orbifold with U(1) fields
Conical defects with U(1) fluxes
Minkowski spacetime with U(1) fields Configurations with conical surpluses 6.2 Global Killing spinor equation
7 Spectral flow: from Ramond to Neveu-Schwarz boundary conditions
spanned by the BMS algebra since long ago [1, 2]. More recently, this analysis has been
further developed and expanded in [3–10], and it has led to the proposal of [11, 12], which
– 1 –
might be promising in order to resolve the information loss paradox [13]. Nonetheless,
some open issues still remain to be suitably understood in the four-dimensional case (see
e.g., [14]), which naturally motivates one to explore them in a simplified setup, as it is the
case of General Relativity in three spacetime dimensions. As shown in [4, 15, 16], the
threedimensional version of the BMS algebra (BMS3) describes the asymptotically flat
symmetries. The BMS3 algebra turns out to be isomorphic to the Galilean conformal algebra in
two dimensions, and it has been shown to be relevant in the context of flat holography [17–
20], as well as for the tensionless limit of string theory [21–23] (see also [24]). It is also worth
pointing out that the generators of the BMS3 algebra can be seen to emerge in a unique way
through a twisted Sugawara-like construction made out from composite operators of affine
currents describing the asymptotic symmetries of the “soft hairy” type of boundary
conditions recently discussed in [25–27]. Similar results along these lines have also been found
in the context of near horizon (twisted) warped conformal symmetry algebras in [28–30].
Besides, in the context of N = 1 supergravity in three spacetime dimensions [31–33],
the minimal supersymmetric extension of BMS3 has been shown to arise from a suitable
set of asymptotically flat boundary conditions [34], which are not necessarily given at null
infinity. The superalgebra turns out to be isomorphic to the supersymmetric extension of
the two-dimensional Galilean conformal algebra in [35, 36] (see also [37]), which was found
from a non-relativistic limit of the superconformal algebra, and hence their generators do
not possess the same physical interpretation.
The extension to the case of N = (1, 1) has also been recently explored in [38], where
it was shown that two inequivalent possibilities can be recovered from different flat limiting
processes of N = (1, 1) AdS3 supergravity [39]. The homogeneous or “democratic”
possibility corresponds to the straightforward extension of the case with N = 1, which agrees
with the results found in [22, 23, 40] in the context of Galilean superconformal algebras. In
the “despotic” possibility, the algebra becomes isomorphic to the inhomogeneous Galilean
superconformal algebra [23, 36].
In the next section we make a brief revision of the N = (2, 0) Poincar´e supergravity
theory constructed out in [41]. We show that demanding the action to be parity-invariant
implies that the U(1) field that is minimally coupled to the complexified gravitino is even
under parity, while the remaining U(1) field has to be odd. In section 3 we propose a
set of boundary conditions that includes a generic choice of Lagrange multipliers, which is
strictly necessary in order to accommodate solutions of physical interest. The asymptotic
symmetries are shown to be spanned by a supersymmetric extension of the BMS3 algebra
with N = (2, 0), endowed with two independent affine uˆ(1) currents of electric and
magnetic type. Specifically, the nonvanishing anticommutator of the complexified fermionic
generators acquires a central extension and depends on the supertranslations as well as on
the electric-like affine uˆ(1) current. It is also shown that our supersymmetric extension of
BMS3 can be recovered from the Ino¨nu¨-Wigner contraction of the superconformal algebra
with N = (2, 2), once the fermionic generators of the right copy are truncated. In section 4
we show that for fermionic fields that fulfill antiperiodic boundary conditions, the ground
state is given by Minkowski spacetime, possibly endowed with uˆ(1) charges of electric type.
Remarkably, although the anticommutator of the supercharges is linear in the energy and
– 2 –
in the electric-like uˆ(1) charge, the energy becomes bounded from below by the energy of
Minkowski spacetime in vacuum, shifted by the square of the electric-like uˆ(1) charge. If
the spin structure is even, the same bound for the energy generically holds, unless the
absolute value of the electric-like charge is less than minus the mass of Minkowski spacetime
in vacuum, so that the energy has to be nonnegative.
Bosonic configurations that fulfill our boundary conditions are revisited in section 5,
where we pay special attention to the conditions that ensure their regularity in terms of
gauge fields. We also discuss them in the metric formalism, and carry out a thorough
analysis of the thermodynamic properties of cosmological spacetimes endowed with U(1)
fields. The presence of U(1) fields of electric and magnetic type also unveils some remarkable
spectra with periodic or antiperiodic boundary conditions for the fermionic fields are related
by spectral flow, in a similar way as it occurs for the super-Virasoro algebra N = 2 [42].
We conclude with some comments about the extension of our results in section 8. Our
conventions are discussed in appendix A.
Note added: while this manuscript was in the process of typesetting, ref. [43] was
posted in the arxiv, which possesses some overlap with particular cases of our results.
2
N
= (2, 0) Poincar´e supergravity in three spacetime dimensions
As shown in [41], N = (2, 0) Poincar´e supergravity in three spacetime dimensions can
be formulated as a Chern-Simons theory for a suitable extension of the super-Poincar´e
group. The algebra has to be endowed with two additional bosonic U(1) generators that
respectively correspond to an automorphism and a central charge, so that it admits a
non-degenerate invariant bilinear form. The nonvanishing (anti-)commutators read
[Ja, Jb] = ǫabcJ c ,
[Ja, QIα] =
(Γa)βα QIβ ,
1
2
{QIα, QJβ } = − 12 δIJ (CΓa)αβ Pa + CαβǫIJ Z ,
[Ja, Pb] = ǫabcP c ,
[QIα, T ] = ǫIJ QJα ,
where Cαβ and (Γa)αβ stand for the charge conjugation and Dirac matrices, respectively
(for our conventions see appendix A).
The existence of a nontrivial Casimir operator, given by allows to define an invariant bilinear form whose nonvanishing components read
I = 2J aPa − QIαCαβQIβ − 2T Z ,
hJa, Pbi = ηab ,
hQIα, QJβ i = CαβδIJ ,
to a parity even action, once suitable parity properties of the fields are taken into account
The entire field content can be arranged within a single connection for the gauge
A = eaPa + ωaJa + ψIαQIα + B T + C Z ,
so that apart from the dreibein ea, the (dualized) spin connection ωa, and the N = 2
gravitini ψIα, the theory possesses two additional U(1) fields, given by B and C, respectively
being even and odd under parity.
HJEP09(217)3
The supergravity theory can then be described by a Chern-Simons action
I =
k Z
4π
hAdA +
3
which by virtue of (2.3) and (2.4), reduces to
I =
k Z
4π
2eaRa + iψ¯I ∇ψI − 2BdC ,
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
up to a boundary term.
The level and the Newton constant are related as k = 41G , while Ra = dωa + 21 ǫabcωbωc
stands for the dualized curvature two-form. The covariant derivative acting on spinors
reads
The field equations then imply that the field strength
vanishes, where T˜a and F˜C stand for the supercovariant torsion and the supercovariant
U(1) curvature along Z, respectively. They are given by
2
∇ψI = dψI + 1 ωaΓaψI + BǫIJ ψJ .
F = T˜aPa + RaJa + ∇ψIαQIα + dB T + F˜C Z
T˜a = T a − 41 iψ¯I ΓaψI ,
F˜C = dC + 1 iǫIJ ψ¯I ψJ ,
2
with T a = dea + ǫabcωbec.
By construction, the action (2.6) is invariant under local supersymmetries that
correspond to gauge transformations, δA = dλ + [A, λ], spanned by a Lie-algebra-valued
fermionic parameter λ = ǫIαQIα. The nontrivial supersymmetry transformations of the
fields are then given by
1
δea = 2 iǫ¯I ΓaψI ,
δψI = ∇ǫI ,
δC = −iǫIJ ǫ¯I ψJ ,
and therefore, along the lines of [44], the algebra of the local supersymmetries spanned
in (2.11) can be seen to close off-shell according to super-Poincar´e with N = (2, 0) in (2.1),
without the need of introducing auxiliary fields.
In the next section we perform an exhaustive analysis of the asymptotic structure of
the theory.
– 4 –
Asymptotic structure: supersymmetric extension of BMS3 with N
=
in [34]. Following the lines of [45] and [46], it was extended in [47] so as to incorporate
a generic choice of Lagrange multipliers at infinity (for the case of fermionic fields of spin
Here we extend these results to the case of Poincar´e supergravity with N = (2, 0). It is
then useful to change the basis of the extended super-Poincar´e algebra in (2.1) according to
L−1 = −
M−1 = −
G 1 = √
I
√2J0 ,
√
2P0 ,
2QI+ ,
L1 = √2J1 ,
M1 = √
GI1 = √
2
2P1 ,
2QI ,
−
L0 = J2 ,
M0 = P2 ,
HJEP09(217)3
(3.1)
(3.2)
(3.3)
(3.4)
1
where m, n = ±1, 0, and p, q = ± 2 . Thus, the nonvanishing components of the invariant
bilinear form in (2.3) reduce to
hL1, M−1i = hL−1, M1i = −2 ,
hGI− 12 , GJ1 i = 2δIJ ,
2
hL0, M0i = 1 ,
general criteria as the ones spelled out in [48–52]. Once adapted to the theory under
discussion, they are:
(i) The asymptotic symmetries of the set must include BMS3 as well as the two fermionic
(ii) The fall-off of the fields has to be relaxed enough so as to incorporate the bosonic
(iii) The decay must simultaneously be sufficiently fast in order to ensure finiteness of the
(iv) The boundary conditions have to guarantee that the variation of the charges fulfills
ones.
solutions of interest.
variation of the global charges.
suitable functional integrability conditions.
so that the superalgebra now reads
[Lm, Ln] = (m − n) Lm+n ,
[Lm, GIp] =
m
{GIp, GqJ } = δIJ Mp+q − 2 (p − q) ǫIJ Z ,
[Lm, Mn] = (m − n) Mm+n ,
[GIp, T ] = ǫIJ GpJ ,
Taking into account these four requirements, the asymptotic behaviour is proposed to be
of the form
A = h−1ah + h−1dh ,
– 5 –
where as in [53] the group element h entirely captures the dependence on radial coordinate
r, so that the auxiliary connection a depends only on the remaining ones u, φ.
In concrete, we choose h = e r2 M−1 , and the spacelike component of a to be given by
so that deviations with respect to the background configuration, which we assume to be
given by the null orbifold [54], are described by arbitrary functions of u, φ that go along
the highest weight generators. According to [45, 46], the bosonic functions P, J , Z, T ,
and the fermionic ones ψ, S, correspond to the dynamical fields.
Moreover, as required by criterion (ii), in order to accommodate the widest possible
class of bosonic solutions, the Lagrange multipliers associated to the dynamical fields have
to be explicitly incorporated in the asymptotic behaviour. They turn out to be defined
through the lowest weight components of
au = Λ[µ J , µ P , µ ψ, µ S , µ T , µ Z ] ,
iψµ ψ +
π
2k iSµ S
M−1
(3.5)
(3.6)
(3.7)
(3.8)
Λ = µ J L1 +µ P M1 +µ ψG11 +µ S G21 + µ T − k
2 2
µ J Z
2π
− µ ψ′ + k µ J
ψ − k Zµ S
G1− 21 − µ S ′ + k µ J S +
G2 1 ,
4π
P − k Z
π
L−1
2
+
π
2k
2π
k Zµ ψ
where prime denotes ∂φ.
The bosonic Lagrange multipliers µ J , µ P , µ T , µ Z as well as the fermionic ones µ ψ,
µ S, can be assumed to be given by arbitrary independent functions of u, φ, that are held
fixed at the boundary without variation.
The asymptotic symmetries are then described by the subset of gauge transformations
δa = dλ+[a, λ] that preserve the asymptotic form of the auxiliary connection in (3.5), (3.6).
The spacelike component of a in (3.5) is maintained for Lie-algebra-valued parameters
of the form
λ = Λ[ǫJ , ǫP , ǫψ, ǫS, ǫT , ǫZ ] ,
– 6 –
δP = 2PǫJ ′ + P′ǫJ − 2π J
ǫ ′′′ − 4ZǫT ′ ,
δJ = 2J ǫJ ′ + J ′ǫJ + 2PǫP ′ + P′ǫP − 2π P
ǫ ′′′
δψ =
δS =
3
2
3
2
+ZǫZ ′ + T ǫT ′ − 2
iψǫψ′ − 2
iψ′ǫψ − 2 iSǫS ′ − 2 iS′ǫS
ψǫJ ′ + ψ′ǫJ − SǫT − Pǫψ +
π ǫψ′′ − 2Z′ǫS − 4ZǫS ′ ,
SǫJ ′ + S′ǫJ + ψǫT − PǫS +
ǫ ′′ + 2Z′ǫψ + 4Zǫψ′ ,
1
k
k
π S
δ
T = −iSǫψ + iψǫS − 2π ǫZ ′ + ǫJ ′T + ǫJ T ′ − 4ǫP ′Z − 4ǫP Z′ ,
k
δZ = − 2π ǫT ′ + ǫJ ′Z + ǫJ Z′ .
Note that Λ in (3.8) is precisely the same as in (3.7), but now depends on arbitrary bosonic
and fermionic functions of u, φ, given by ǫJ , ǫP , ǫT , ǫZ , and ǫψ, ǫS, respectively.
Preserving the form of au then implies that the field equations have to hold at the
asymptotic region, and also provides additional suitable conditions for the parameters that
span the asymptotic symmetries. In the reduce phase space, the field equations then read
provided that the transformation law of the dynamical fields is given by
k
3
k
k
3
k
which can be readily obtained from (3.9) by taking into account that time evolution is
generated by gauge transformations whose parameters correspond to the Lagrange multipliers.
The conditions for the parameters are explicitly given by (3.9)
J = µ J ǫJ ′ − µ J ′ǫJ ,
1
1
1
1
ǫ˙
P = µ P ǫJ ′ + µ J ǫP ′ − µ J ′ǫP − µ P ′ǫJ − iµ S ǫS − iµ ψǫψ ,
ǫ˙ψ = µ J ǫψ′ + 2 µ ψǫJ ′ − 2 µ J ′ǫψ − µ ψ′ǫJ + µ S ǫT − µ T ǫS ,
ǫ˙S = µ J ǫS ′ + 2 µ S ǫJ ′ − 2 µ J ′ǫS − µ S ′ǫJ + µ T ǫψ − µ ψǫT ,
ǫ˙
ǫ˙
T = µ J ǫT ′ − µ T ′ǫJ ,
Z = µ J ǫZ ′ − µ Z ′ǫJ − 4µ P ǫT ′ + 4µ T ′ǫP − 2iµ ψǫS ′ + 2iµ S ǫψ′ − 2iµ S ′ǫψ + 2iµ ψ′ǫS ,
– 7 –
and they can be seen to ensure that the variation of the canonical generators is conserved.
This is discussed next.
Canonical generators and their algebra
In the canonical approach [55], the variation of the surface integrals that define the global
charges is given by
δQ[λ] = − 2π
k Z
hλδaφidφ ,
which by virtue of (3.3), (3.5), (3.8), readily integrates as
Q[ǫJ , ǫP , ǫψ, ǫS, ǫT , ǫZ ] = −
Z (ǫJ J + ǫP P + iǫψψ + iǫS S + ǫT T + ǫZ Z) dφ .
to Xn = R Xe−inφdφ, it is given by
The asymptotic symmetry algebra can then be obtained from the direct evaluation of the
Poisson brackets of the global charges in (3.13). As a shortcut, if one takes into account the
variation of the dynamical fields in (3.9), the asymptotic symmetry algebra can be directly
read from δλ2 Q[λ1} = {Q [λ1] , Q [λ2]}, so that once expanding in Fourier modes according
i{Jm, Jn} = (m − n) Jm+n ,
i{Jm, Pn} = (m − n) Pm+n + km3δm+n,0 ,
m
i{Tm, Zn} = −kmδm+n,0 ,
i{Gp , Tm} = iǫIJ J
I
Gm+p ,
i{Gp , Gq } = δIJ
I J
Pp+q + 2kp2δp+q,0 + 2iǫIJ (p − q) Zp+q ,
where G1 = ψ and G
2 = S.
Here, m, n stand for integers, and p, q are given by (half-)integers when fermions fulfill
(anti)periodic boundary conditions, while the reality conditions of the modes are given by
(Jm)∗ = J−m, (Pm)∗ = P−m, (Tm)∗ = T−m, (Zm)∗ = Z−m, (GmI)∗ = G−Im.
The asymptotic symmetry algebra (3.14) then manifestly contains BMS3 with the
central extension found in [16] as the subalgebra spanned by Jm and Pm. The remaining
part of the bosonic subalgebra1 consists on two commuting independent affine algebras
generated by the spin-one currents Jm± = √12 (Zm ± Tm), whose level is determined by k.
The super-Poincar´e algebra in (3.2) tuns out to be a subalgebra of (3.14) in the
antiperiodic case, being spanned by n
Jm, Pn, Gp(i), Z0, T 0
o provided the labels are restricted
according to m, n = ±1, 0 and p, q = ± 21 , and the supertranslation generator P0 is shifted
1As an interesting remark, it is worth pointing that the bosonic subalgebra of (3.14) coincides with the
one obtained in [56] for pure gravity with a nonstandard set of boundary conditions, which also corresponds
to the vanishing cosmological constant limit of the asymptotic symmetry algebra found in [57].
– 8 –
(3.12)
(3.13)
(3.14)
HJEP09(217)3
G(pi) = Gp(i), T = −T0 and Z = −Z0.
as P0 → P0 + k2 . The explicit matching is recovered provided that Lm = Jm, Mm = Pm,
Asymptotic symmetry algebra from a truncation of the homogeneous
contraction of the superconformal algebra with N = (2, 2)
It is simple to verify that the supersymmetric extension of the BMS3 algebra with N =
(2, 0) in (3.14), can be recovered from a homogeneous (democratic) or ultra-relativistic
Ino¨nu¨-Wigner contraction of the superconformal algebra with N = (2, 2), provided that
the fermionic generators of the right copy are truncated (see e.g., [40]). Indeed, this is so
regardless the truncation is performed before or after the contraction process. This can be
seen as follows.
The superconformal algebra with N = (2, 2) is given by two independent (left and
right) copies of the super-Virasoro algebra with N = 2, which reads
HJEP09(217)3
κ
2
m3δm+n,0 ,
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
i{Lm, Ln} = (m − n) Lm+n +
i{Lm, QIp} =
m
I
2 − p Qm+p ,
i{Lm, Rn} = −nRm+n ,
i{Rm, Rn} = 2κmδm+n,0 ,
i{Qp, Rm} = −iǫIJ
I
J
Qm+p ,
i{Qp, QqJ } = δIJ
I
– 9 –
Lp+q + κp2δp+q,0 + 1 iǫIJ (p − q)Rp+q .
2
It is then useful to change the basis according to
+
Jm = Lm − L−−m ,
+ −
Tm = − Rm − R−m ,
Gp
+I =
r 2
ℓ Qr+I ,
Pm =
Zm =
Gp
−I =
1
1
r 2
ℓ L+m + L−m ,
−
4ℓ R+m + R−m ,
−
ℓ Q−−Ir ,
so that the contraction process is performed through rescaling the level as κ = kℓ, and then
taking the limit ℓ → ∞. It is then clear that if one truncates the fermionic generators of the
right copy, the super-BMS3 algebra with N = (2, 0) in (3.14) is recovered with GpI = Gp+I .
As a final remark of this section, it is also worth pointing out that there is a different,
so-called inhomogeneous, or “despotic”, Ino¨nu¨-Wigner contraction of the superconformal
algebra with N = (2, 2), which leads to an inequivalent supersymmetric extension of the
super-BMS3 algebra with N = (2, 0). This alternative possibility has been simultaneously
analyzed in the context of the asymptotic structure of a different theory in [58].
4
Nonlinear energy bounds from super-BMS3 with N
= (2, 0)
Supersymmetric bounds for different definitions of the global charges in the context of the
supergravity theory under consideration, have been previously discussed in [41, 59]. In this
section we show that the Poisson brackets of the fermionic generators of the asymptotic
supersymmetries in (3.14), in spite of being linear in the bosonic generators, yield an
infinite number of nonlinear bounds for the energy. Only a finite number of them are able
to saturate, precisely corresponding to the same number of unbroken supersymmetries. In
order to carry out this task, as explained in [46], generic bosonic configurations can be
brought to the “rest frame” by acting on them with a suitable combination of the
asymptotic symmetries. Therefore, it is enough to focus in the case of bosonic configurations
endowed with zero mode charges, given by P0 = 2πP and Z0 = 2πZ, regardless the value
of the remaining ones (J and T ). The supersymmetric bounds we look for can then be
obtained along the semi-classical reasoning in [60–66]. In particular, we follow a similar
strategy as the one in [67]. The fermionic Poisson brackets in (3.14) are then promoted to
anticommutators. In the case of p = −q = r they become
(2π)−1 GˆrI Gˆ−Jr + Gˆ−JrGˆrI
= δIJ
Pˆ +
so that (BrIJ )† = BrJI .
Therefore, when I = J , the left hand side of (4.1) is a positive definite hermitian
operator for any value of r and I, which implies that in the classical limit, the bosonic
charges have to fulfill the following bounds
In the case of periodic boundary conditions for the fermionic fields, the strongest bound
so that the energy becomes bounded from below according to P ≥ − 4kπ .
in (4.2) corresponds to r = 0, which implies that the energy is nonnegative (P ≥ 0).
1
Analogously, for antiperiodic boundary conditions, the strongest bound is given by r = ± 2 ,
Additional bounds also arise in the case of I 6= J . In order to obtain them, it is useful
to define the following complex fields
BrII =
π
k r2 + P ≥ 0 .
1
2
Gˆr± := √
Gr ± iGˆr2 ,
ˆ1
r2 ± k
4π
rZ +
π
k P ≥ 0 .
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
Gˆr± † Gˆr± ≥ 0. The latter bounds then imply that B11
r ≥ ±iBr12,
which by virtue of (4.1) yields
It is convenient to factorize the bounds in (4.4) according to
with
r ± λ[+]
r ± λ[−] ≥ 0 ,
2π
λ[±] = − k Z ±
s π
k
4π
2
where Λ[±] = iλ[±] correspond to the eigenvalues of the spacelike components of the
sl(2, R) ⊕ u(1) connection ωˆ = ω + B that minimally couples to the fermionic fields (see
section eq. (5.15) below).
4π
case of P − k Z
It is then clear from (4.6) that for periodic or antiperiodic boundary conditions, in the
2 > 0, the bounds in (4.5) are automatically fulfilled and never saturated.
provided that λ[+] − λ[−] ≤ 1, which by virtue of (4.6) implies P ≥ − 4kπ + 4kπ Z2.
4π
In the remaining possibility, P − k Z
2
≤ 0, the bounds in (4.5) can be satisfied only
In sum, taking into account the infinite number of bounds in (4.2) and (4.5), one
deduces that the stronger ones imply that energy has to be bounded from below as follows.
Antiperiodic boundary conditions: P ≥ − 4kπ + 4kπ Z2
Periodic boundary conditions: P ≥
(
0
,
− 4kπ + 4kπ Z2 ,
|Z| < 4kπ
|Z| ≥ 4π
k
As a closing remark of this section, it is worth emphasizing that, although the algebra
of the asymptotic symmetries is a linear one, the lower bounds for energy turn out to be
quadratic in the electric-like uˆ(1) charge.
5
Bosonic configurations and some of their properties
Here we explore some properties of stationary spherically symmetric bosonic solutions that
fit within the asymptotic fall off described in section 3. The solutions are endowed just
with zero-mode bosonic charges, and hence they are described through dynamical gauge
fields given by
with J , P, T , Z, constants.
In the absence of fermionic charges, it is consistent to switch off their corresponding
Lagrange multipliers; i.e., the “chemical potentials” µ ψ, µ S, can be set to vanish. For the
sake of simplicity, the remaining bosonic ones, given by µ J , µ P , µ T , µ Z , are assumed to
be constants and held fixed without variation at the boundary. Therefore, the timelike
component of the gauge field in (3.6) reduces to
2π
au = µ J L1 + µ P M1 + µ T − k
µ J Z
T + µ Z − k
In the case of regular solutions, their smoothness can be established through the fact that
the holonomy of the gauge fields along a contractible cycle C has to be trivial, i.e.,
8π
k
µ P Z
Z
4π
H
C = P eRC aμdxμ = Γ± ,
(5.2)
(5.3)
where the sign of Γ± corresponds to different choices of spin structures (see, e.g, [68]).
Indeed, Γ+ belongs to the center of the group, but this is not the case for Γ−, since for
antiperiodic boundary condition it must anticommute with the fermionic generators, i.e.,
{Γ−, GIp} = 0.
In order to evaluate the regularity condition (5.3), it is worth pointing out that neither
the Poincar´e algebra nor the super-Poincar´e algebra with N
= (2, 0) in (2.1) possess
a suitable standard matrix representation from which neither the invariant bilinear form
in (2.3) nor the Casimir operator in (2.2) can be obtained from the trace of a product of two
generators. Hence, regularity cannot be directly carried out through “diagonalizing” the
holonomy in (5.3). Nonetheless, this task might be carried out by virtue of a nonstandard
matrix representation [69] along the lines of [70, 71]. Hereafter, regularity conditions are
implemented according to [72], which possesses the advantage of being independent of the
existence of a suitable matrix representation for the entire gauge group. Once adapted to
super-Poincar´e with N = (2, 0), one proceeds as follows:
(i) One begins finding a group element of the form g = eλnMn+λZ that allows gauging
away the components of the gauge field along Mn and Z projected over the
contractible cycle, so that the corresponding components of the dreibein e and the U(1)
field C can be consistently set to vanish. Consequently, the Lagrange multipliers of
electric type become generically fixed in terms of the ones of magnetic type and the
global charges.
(ii) Regularity conditions can then be straightforwardly evaluated through the
diagonalization of the holonomy matrix along the contractible cycle for the fundamental
representation of the remaining sl(2, R) ⊕ u(1) connection ωˆ = ω + B.
In order to continue with the analysis of the bosonic solutions under consideration, it is
necessary to identify the suitable contractible cycles along which the regularity conditions
have to be applied. Indeed, these cycles might correspond either to the circles along
Euclidean time or the ones for the angular coordinate. These are the cases of cosmological
spacetimes or solitonic-like solutions, respectively. This is discussed in what follows.
5.2
Metric formalism
The spacetime metric can be readily constructed from identifying the dreibein from the
components of the full gauge field A in (3.4) along the generators Pa. The dreibein then
reads
(dφ + µ J du) − √ dr +
1
2
√2π
k
4π
e2 = r (dφ + µ J du) ,
so that the line element is recovered in outgoing null coordinates, as (see e.g., [70, 72])
ds2 = − k
4π
πN
2
kr2 − M
µ 2 du2 − 2µ P drdu + r2 dφ + µ J +
P
2πµ P N
kr2
du
2
where the integration constants M, N are related to the global charges according to
eqs. (5.10), (5.11). The entire configuration is then determined by the metric in (5.7)
together with the U(1) fields of electric and magnetic type, given by
2π
2π
It is worth pointing out that the spacetime metric does not acquire a back reaction due
to the presence of the U(1) fields. Indeed, as it can be seen from the action in (2.6), this has
to be so because, as they are described by a “BF”-type of Lagrangian, they do not couple
to the metric and hence they cannot contribute to the stress-energy tensor. Nonetheless,
their presence does not go unnoticed because they manifestly contribute to the energy and
the angular momentum of the configuration, given by
P = M +
J = N − k T Z ,
4π
k Z2 ,
2π
where Z and T stand for the U(1) charges of electric and magnetic type, respectively.
It is also amusing to verify that static configurations, for which N = 0, are able to
carry a nontrivial angular momentum, because in the “dyonic” case the product of the
electric and magnetic U(1) charges manifestly contribute to J in (5.11).
Regularity of the configurations can also be analyzed from demanding smoothness of
the spacetime metric, so that µ P would be related to the inverse of the Hawking
temperature (µ P = −β), and µ J to the chemical potential associated to the angular momentum
J , when it corresponds.
5.3
Cosmological configurations and their thermodynamics
In this subsection we discuss the class of stationary spherically symmetric bosonic solutions
4π
for the case P − k Z
2
≥ 0, which clearly fulfills the energy bounds in section 4. The line
element generically describes the class of locally flat cosmological spacetimes discussed
in [73–75] endowed with a generic choice of chemical potentials as in [70, 72], and reduces
to the null orbifold in [54] for P = 4kπ Z2. The thermodynamics of cosmological spacetimes
in vacuum has been analyzed in [70, 72, 76, 77]. Here we extend the analysis to the case
of cosmological spacetimes endowed with U(1) charges of electric and magnetic type.
As explained in [72] the Euclidean geometry possesses the topology of a solid torus, so
that the circles spanned by the Euclidean time τ = −iu, correspond to contractible cycles,
and the orientation is reversed as compared with the one of a BTZ black hole [78, 79].
Following refs. [45] and [46], the solid torus can be chosen to be parametrized as a “straight”
one, i.e., described by a fixed range of the coordinates, 0 ≤ τ < 1 and 0 ≤ φ < 2π. In
this way, the Hawking temperature and the chemical potential associated to the angular
momentum explicitly appear in the metric through µ P and µ J
, respectively. Thus, all
of the chemical potentials, including the ones for the U(1) charges, given by µ Z and µ T
become treated in the same footing. It is worth pointing out that the chemical potentials of
the U(1) charges have to be switched on, otherwise an interesting class of physical solutions
would fail to be regular.
Regularity of the configurations then implies that the holonomy of the connection along
a thermal circle has to be trivial. As explained in section 5.1, according to criterion (i) we
apply a suitable gauge transformation given by g = eλ0M0 , with λ0 constant, so that the
timelike component of the connection (5.2) transforms as
L−1 + (µ P + λ0µ J ) M1
Therefore, the components of agu along Mn and Z are gauged away for λ0 = − μμJP , provided
that the chemical potentials of electric type, µ P and µ Z , are fixed in terms of the magnetic
one µ J and the global charges.
Note that the gauge transformation spanned by g = e− μμJP M0 depends only on chemical
potentials that are kept fixed at the boundary, and hence it turns out to be a permissible
one in the sense of [46]. The connection agu then reduces to the following one for the
sl(2, R) ⊕ u(1) algebra
agu = µ J
L1 − k
π
The remaining regularity condition (ii) then corresponds to demanding that (5.14)
possesses a trivial holonomy, i.e., H
I
C = ± 2×2, which can be readily performed in the
fundamental representation of sl(2, R) ⊕ u(1) (see appendix A). Therefore, the eigenvalues of
Λ[±] =
1
2
tr [ωˆ] ±
q2tr [ωˆ2] − tr [ωˆ]2
,
I
and odd for − 2×2). The regularity condition then reduces to
with ωˆ = aτg , have to be given by Λ[±] = ±iπm, with m an arbitrary integer (even for I2×2,
2π
r π
± iπm = − µ T − k
µ J Z
± i
k
|µ J | P − k Z
4π
2
1/2
which implies that
2π
k
µ T =
µ J Z ,
|µ J | =
P − k Z
m
√
πk
It is worth noting that the branch that is connected to the standard cosmological spacetime,
so that the Hawking temperature is given by µ P = −β, corresponds to m = 1, which agrees
with what is found from requiring smoothness of the Euclidean metric.
Since we are dealing with a Chern-Simons theory, the entropy associated to the
cosmological horizon can be directly found from the following formula [46, 80–82]
which by virtue of (5.1) and (5.2), evaluates as
S = 2π (2µ J J + 2µ P P + µ T T + µ Z Z) .
Therefore, as required by the action principle, the entropy of regular configurations, by
virtue of (5.18) reduces to
In sum, regularity implies that the chemical potentials become fixed in terms of the global
charges according to
(5.18)
Note that for m = 1, the entropy
S = 2π m
"
s
πk
2π
k T Z| .
#
S = 2π
s
πk
4π 2 |J +
2π
k T Z| ,
precisely gives a quarter of the cosmological horizon area over 4G.
It is amusing to verify that, unlike the case of pure gravity, configurations without
angular momentum (J = 0) can still be stationary and carry a nonvanishing entropy.
It is also worth pointing out that the entropy expressed in terms of the extensive
variables, manifestly depends not only on the energy and the angular momentum, but also
on the electric and magnetic U(1) charges. Indeed, it is simple to verify that the first law
holds in the grand canonical ensemble, since
δS = 2π (µ P δP + µ J δJ + µ T δT + µ Z δZ) ,
(5.23)
provided that the chemical potentials are given by (5.18). In other words, an attempt of
fixing the chemical potentials in a different way as in eq. (5.18), might generate a severe
clash with the first law of thermodynamics.
As an ending remark of this section, it is worth mentioning that the entropy of the
class of cosmological spacetimes with U(1) fields in (5.22), has also been obtained
simultaneously and in an independent way through a different approach in [58], which is certainly
reassuring.
In the case of − 4kπ < P − k Z
4π 2 < 0, the configurations are described by spacetime
metrics (5.7) that generically describe rotating conical defects [83, 84] with nontrivial lapse and
shift functions, endowed with U(1) fields of electric and magnetic type that do not
generate a back reaction. In the case of antiperiodic boundary conditions the energy bounds in
section 4 are clearly satisfied, but this is not necessarily so for the case of periodic
boundary conditions. Indeed, for periodic boundary conditions, if |Z| ≥ 4kπ , the stronger energy
bound is trivially satisfied and never saturates; while for |Z| < 4kπ , the bound is fulfilled only
for configurations with nonnegative energy (P ≥ 0). Note that in vacuum (Z = T = 0), this
class of configurations does not fulfill the energy bounds when the spin structure is even.
Although this class of configurations fails to be regular due to the presence of sources
at the origin, it turns out to be very interesting. This is because they might admit Killing
spinors provided that the U(1) fluxes are suitably tuned with the angular deficit [41, 59]. In
our terms, these U(1) fluxes correspond to charges of electric type. In section 6, the explicit
form of the Killing spinors is constructed, and we also show that two of our supersymmetry
bounds in section 4 are saturated, so that the configurations correspond to half-BPS states.
5.5
Minkowski spacetime and conical surpluses endowed with U(1) fields
Configurations with P −
4π 2 < − 4kπ correspond to spacetime metrics with conical
surk Z
pluses, so that they possess angular excess. Despite they do not fulfill the energy bounds,
they might be interesting because it can be seen that they could also admit Killing spinors
provided that suitable U(1) fluxes are switched on. Nonetheless, it should be emphasized
that they cannot describe BPS states.
In the case of P −
4π
2 = − 4kπ the bosonic solution is described by the Minkowski
spacetime endowed with electric and magnetic U(1) fields. Noteworthy, the energy bounds
for the antiperiodic boundary conditions are clearly saturated, but this is not necessarily
so for periodic boundary conditions.
Remarkably, in the absence of angular momentum and magnetic U(1) charges, these
configurations can be regarded as regular ones provided that the energy and the electric
U(1) charges take discrete values, according to
k
π
± correspond to (half-)integers in the case of (anti)periodic boundary conditions.
Note that for antiperiodic boundary conditions, the electric U(1) charge is then given by
an integer multiple of the mass of Minkowski spacetime in vacuum (P = − 4kπ ), while for
periodic boundary conditions, it corresponds to an even multiple.
This can be seen as follows. In these cases, regularity of the configurations corresponds
to requiring the holonomy of the connection for a contractible circle along the angular
coordinate to be trivial. Thus, following the criterion (i) in section 5.1, one can show that
there is no group element of the form g = eλnMn+λZ that helps in order to gauge away the
components of aφ in (5.1) along the generators Mn and Z, and hence regularity necessarily
implies that the angular momentum and the magnetic U(1) charge vanish, i.e., J = T = 0.
The connection aφ then reduces to
that takes values on the sl(2, R) ⊕ u(1) subalgebra. Criterion (ii) then implies that the
holonomy of aφ in (5.25) along the angular circle is trivial (HC = ± 2×2). Hence, the
eigenvalues of ωˆ = aφ, can be obtained from (5.15), and they have to be given by Λ[±] =
I
iλ[±] = in±, where n
± stand for integers or half-integers in the cases of even or odd spin
structures, respectively, and λ[±] are given in eq. (4.6). The regularity condition then reads
2π
n
± = − k Z ±
s π
k
4π
2
which implies that the energy and the electric U(1) charge take discrete values given
by (5.24).
n
− + 1, which implies that
In particular, it is worth pointing out that the configurations that correspond to
Minkowski spacetime endowed with electric U(1) charge turn out to be regular for n+ =
k
(2n− + 1) ,
P =
k
π n− (n− + 1) .
is given by even or odd multiples of the mass of Minkowski spacetime in vacuum for
antiperiodic or periodic boundary conditions, respectively; while the total energy of the
configuration is given by
Remarkably, Minkowski spacetime endowed with an electric U(1) charge given by (5.27)
fulfills all of the energy bounds and saturates four of them for periodic or antiperiodic
boundary conditions for the fermions, so that the configurations turn out to be maximally
supersymmetric, admitting four Killing spinors whose explicit form is given in the next
section. This is in stark contrast with what occurs for Minkowski spacetime in vacuum,
which does not fulfill the energy bounds in the case of periodic boundary conditions for
the fermions. In other words, for an even spin structure, Minkowski spacetime can be
brought back into the allowed spectrum provided that it is endowed with an electric-like
U(1) charge given by an odd multiple of the mass of the Minkowski spacetime in vacuum.
As a closing remark of this subsection, it is worth pointing out that similar classes
of solitonic-like objects with conical surpluses have also been disscused in the context of
three-dimensional gravity coupled to higher spin fields in refs. [47, 68, 85–91].
(5.26)
(5.27)
(5.28)
6.1
Bosonic solutions with unbroken supersymmetries
Asymptotic Killing spinor equations
Here we look for the class of bosonic configurations that asymptotically behave as the
stationary spherically symmetric ones described by the gauge fields in (5.1) and (5.2),
that admit Killing spinors being well-defined in the asymptotic region. These unbroken
supersymmetries are spanned by fermionic gauge transformations that leave the bosonic
configuration invariant. Thus, they fulfill δa = dλ + [a, λ] = 0, where λ stands for the
purely fermionic asymptotic symmetries that can be obtained from eqs. (3.7) and (3.8).
The fermionic Lie-algebra-valued parameter then reads
λ [ǫψ, ǫS ] = ǫψG121 −
2π
ǫψ′ − k ZǫS
G1 1 + ǫSG212 −
− 2
ǫS ′ +
2π
k Zǫψ
− 2
G2 1 = ǫIαQIα , (6.1)
so that the explicit form of the spinors ǫIα in terms of the Grasmann-valued functions ǫψ, ǫS ,
can then be obtained by virtue of the change of basis in (3.1). The spinors are then given by
,
ǫψ
ǫ2 = √
2
−ǫS ′ − 2kπ Zǫψ !
.
ǫS
The asymptotic Killing spinor equations can be partially read from the transformation
law of the fields in (3.9) under the asymptotic supersymmetries. The nontrivial ones are
given by
k
k
π S
δψ = −Pǫψ +
π ǫψ′′ − 4ZǫS ′ = 0 ,
δS = −PǫS +
ǫ ′′ + 4Zǫψ′ = 0 .
ǫ˙ψ = µ J ǫψ′ − µ T ǫS ,
ǫ˙S = µ J ǫS ′ + µ T ǫψ .
1
2
ξ = √ (ǫψ + iǫS ) ,
ξ′′ +
4π
π
k iZξ′ − k Pξ = 0 ,
ξ˙ − µ J ξ′ − iµ T ξ = 0 .
The solution of the asymptotic Killing spinor equations in (6.6) and (6.7) can then readily
found. In the generic case P 6= 4kπ Z
2 , the solution reads
ξ = ξ1eiμT ueiλ[+]φˆ + ξ2eiμT ueiλ[−]φˆ ,
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
(6.8)
Analogously, the conditions for the Grasmann-valued parameters can be obtained
from (3.11), so that they read
In order to perform the analysis it is useful to define a single complex Grasmann-valued
parameter defined as
so that the asymptotic Killing spinor equations in (6.3) and (6.4) can be rewritten as
where λ[±] correspond to the eigenvalues of the spacelike components of the sl(2, R) ⊕ u(1)
connection ωˆ = ω + B in (4.6), and φˆ := φ + µ J u. Here ξ1 and ξ2 stand for arbitrary
complex (Grasmann-valued) constants. In the special case of P = 4kπ Z2, which corresponds
to energy of the null orbifold endowed with U(1) fields, the eigenvalues degenerate (λ[+] =
λ[−] = − 2kπ Z), and hence the solution is given by
ξ = ξ1eiμT ueiλ[+]φˆ + ξ2φe(μJ +iμT )ueiλ[+]φˆ .
(6.9)
One is then ready to analyze whether they are well-defined for the different classes of
bosonic solutions discussed in section 5.
Cosmological configurations
In this case, the energy fulfills P > 4kπ Z2, so that the eigenvalues λ[±] in (4.6) turn out to
be complex. Therefore, the solution for the Killing spinor equations in (6.8) is not globally
well-defined because it cannot fulfill neither periodic nor antiperiodic boundary conditions.
Consequently, all of the supersymmetries are broken, which goes by hand with the fact the
energy bounds in this case are always satisfied, but never saturated.
6.1.2
Null orbifold with U(1) fields
For this class of configurations, the energy is given by P = 4kπ Z2, so that the solution of
the asymptotic Killing spinor equations is given by (6.9). It is then clear that it can only
be globally defined provided that ξ2 = 0, where the eigenvalue has to be given by λ[+] = n,
with n a (half-)integer for fermions that fulfill (anti)periodic boundary conditions. Note
that the electric U(1) charge is then restricted to take the following values: Z = − k2πn . The
solution of the asymptotic Killing spinor equation then acquires the form
ξ = ξ1eiμT ue−inφˆ ,
so that it is clear that this class of configurations possesses two unbroken supersymmetries.
This is precisely the number of bounds that saturate, which correspond to the ones in (4.5)
for r = ∓λ[+] = ∓n = ± 2kπ Z.
6.1.3
Conical defects with U(1) fluxes
In this case the energy fulfills the condition − 4kπ < P − k Z
4π 2 < 0, which in terms of the
eigenvalues λ[±] in (4.6), reads
λ[+] − λ[−] < 1 .
Therefore, the solution of the asymptotic Killing spinor equation in (6.8) is globally
welldefined provided that ξ2 = 0 and λ[+] = n+ is given by a (half-)integer, or ξ1 = 0 and
λ[−] = n
− a (half-)integer, for (anti)periodic boundary conditions. Therefore, this class
of configurations preserves half of the supersymmetries, which goes by hand with the fact
that the energy bounds are fulfilled, and the number of them that saturate is just two.
They correspond to the ones in (4.5) either for r = ∓λ[+], or r = ∓λ[−].
(6.10)
(6.11)
Therefore, the asymptotic Killing spinors are given by (6.8), and they are globally
welldefined provided that λ[±] are given by (half-)integers for (anti)periodic boundary
conditions. This is the only maximally supersymmetric case that fulfill the bounds in section 4,
which agrees with the fact that four of them in (4.5) saturate, corresponding to the cases
(6.12)
(6.13)
(6.14)
of r = ∓(λ[−] + 1) and r = ∓λ[−].
respectively, with λ[−] = n−.
which corresponds to configurations with J = T = 0.
6.1.5
Configurations with conical surpluses
Note that the energy and the electric U(1) charge are then given by (5.28) and (5.27),
It is worth emphasizing that this case includes the regular one described in section 5.5,
It is worth then mentioning that configurations with conical defects become half-BPS
for even or odd spin structures, provided that the electric U(1) charge and the energy are
suitably tuned according to
k
π
P =
Minkowski spacetime with U(1) fields
− 4kπ + 4kπ Z2, which implies that the eigenvalues λ[±] in (4.6) fulfill
In the case of Minkowski spacetime dressed with U(1) fields, the energy is given by P =
For this class of configurations the energy lies in the range P < − 4kπ + 4kπ Z2, which amounts
As aforementioned, the energy bounds are not fulfilled, but nonetheless this class of
solutions might admit unbroken supersymmetries. In the maximally supersymmetric case
the asymptotic Killing spinors are given by (6.8) provided that λ[±] correspond to
(half)integers for (anti)periodic boundary conditions. Note that in the case of J = T = 0, this
class of configurations turns out to be regular (see section 5.5). The global charges are
then given by (5.24), with λ[±] = n±.
The remaining possibility consists on configurations with angular excess endowed with
suitable electric U(1) charges, which are not regular. The asymptotic Killing spinors are
then given by (6.8), and they are globally well-defined either for ξ2 = 0 and a (half-)integer
λ[+] = n+, or ξ1 = 0 and λ[−] = n− a (half-)integer, for (anti)periodic boundary conditions.
6.2
Global Killing spinor equation
For the class of exact solutions described in section 5, one can also proceed in the standard
way in order to identify the configurations that admit globally well-defined Killing spinors,
as well as finding the explicit form of them. It is reassuring to verify that the results in this
section precisely agree with the ones in section 6.1. Indeed, the results within this section
can be readily reconstructed from the ones in section 6.1 by virtue of eq. (6.2). Nonetheless,
carrying out the analysis in the standard way turns out to be a healthy exercise. In order
to perform this task it is useful to complexify the spinors according to
so that the Killing spinor equation can be obtained from the local supersymmetry
transformations in (2.11) (δψI = 0), and it turns out to be given by
1
2
χ = √ (ǫ1 + iǫ2) ,
δχ = dχ +
2
1 ωaΓaχ − iBχ = 0 .
χ = Pexp −
Z
C
2
1 ωaΓa − iB
χ0 ,
The generic solution of (6.16) is given by
with χ0 a constant Dirac spinor.
As it can be read from section 5, the spin connection and the electric U(1) field read
ω = √
2 J1 +
2π
B = − k Z (dφ + µ J du) + µ T du ,
π
k
4π
and hence, the solution in (6.17) reduces to
χ = ei(− 2kπ Zφˆ+μT u)exp
2 J1 +
π
k
4π
with φˆ = φ + µ J u.
(6.15)
(6.16)
(6.17)
(6.18)
(6.19)
(6.20)
(6.21)
(6.22)
In the generic case of (P 6= 4kπ Z2), the solution in (6.20) then acquires the form
χ = ei(− 2kπ Zφˆ+μT u) cosh − 4 (λ[+] − λ[−])φˆ I
2×2
i
4i√2
(λ[+] − λ[−])2J0 sinh − 4 (λ[+] − λ[−])φˆ
i
)
χ0 ,
For P = 4kπ Z2, the solution is given by
χ = I
2×2 −
√2J1φˆ ei(− 2kπ Zφˆ+μT u)χ0 .
Here we have made use of the fundamental matrix representation of sl(2, R) ⊕ u(1) that
appears in appendix A.
The remaining analysis can then be directly performed as in the previous section.
In the case of cosmological configurations (P > 4kπ Z2) all of the supersymmetries are
broken because the Killing spinors in (6.21) are clearly not globally well-defined.
The null orbifold with U(1) fields (P = 4kπ Z2) possesses half of the supersymmetries
provided that Z = k2πn , so that the Killing spinors can be obtained from (6.22). They are
explicitly given by
χ = ei(−nφˆ+μT u)χ0 ,
where the constant spinor has to fulfill the projection J1χ0 = 0, and n is a (half-)integer
for (anti)periodic boundary conditions.
In the case of conical defects with U(1) fluxes (− 4kπ < P − k Z
4π 2 < 0), the Killing
spinor in (6.21) becomes
1
4
1
χ = ei(− 2kπ Zφˆ+μT u) cos
(λ[+] − λ[−])φˆ I
2×2
√
4 2
which is globally well-defined provided that the spinor χ0 satisfies the projection
The Killing spinor in (6.24) then reduces to
√
4 2
χ = eiμT u exp niλ[±]φˆo χ0 .
(6.23)
(6.24)
(6.25)
(6.26)
HJEP09(217)3
with λ[±] given by (4.6), and the sign of the projection in (6.25) coincides with the choice
of the label of λ[±] in (6.26). The spinors are then consistent with (anti)periodic boundary
conditions provided that λ± is a (half-)integer. Note that the projection condition in (6.25),
preserves half of the supersymmetries.
In the case of Minkowski spacetime dressed with U(1) fields (P = − 4kπ + 4kπ Z2),
equation (6.21) reduces to
"
χ = cos
ei(− 2kπ Zφˆ+μT u)χ0 ,
(6.27)
so that the Killing spinors are globally well-defined provided the electric U(1) charge is fixed
by even or odd multiples of − 4kπ for antiperiodic or periodic boundary conditions,
respectively. Noteworthy, this case is maximally supersymmetric for even and odd spin structures.
For the class of conical surpluses endowed with U(1) fluxes (P < − 4kπ + 4kπ Z2) there
are two possibilities.
In the maximally supersymmetric case the Killing spinors are given by (6.24), where
λ[±] = n
± stand for (half-)integers in the case of (anti)periodic boundary conditions.
In the remaining possibility the configurations possess two independent Killing spinors,
now given by (6.26), so that χ0 fulfills the projection in (6.25). The Killing spinors fulfill
(anti)periodic boundary conditions when λ± is a (half-)integer.
As shown in [42], the spectrum spanned by the super-Virasoro algebra with N = 2 in
the case of periodic boundary conditions for the fermionic fields relates to the one for
antiperiodic boundary conditions through spectral flow. These results were generalized for
Here we show that this is also the case for the super-BMS3 algebra with N = (2, 0) in
eq. (3.14). This occurs by virtue of a field redefinition that is induced by a suitable U(1)
gauge transformation. The field redefinition then amounts to a precise change of basis in
the canonical generators that turns out to be an automorphism of the super-BMS3 algebra
with N = (2, 0). This can be seen as follows.
The Dirac spinors ξ± = √12 (ψ1 ± iψ2) are such that ξ+ is the hermitian conjugate of
ξ−, and vice versa. Therefore, for generic “anyonic” boundary conditions characterized by
some parameter η that is held fixed at the boundary, the spinors fulfill
ξ± (φ + 2π) = e±2iπηξ± (φ) ,
where η = 0, 21 corresponds to Ramond (periodic), and Neveu-Schwarz (antiperiodic)
boundary conditions, respectively.
Besides, under U(1) gauge transformation spanned by g = efT , the connection
transforms as Af = g−1Ag + g−1dg, so that the nontrivial transformations in terms of the
components of the gauge fields read
Bf = B + df ,
ξ±f = e±if ξ± .
Therefore, a generic choice of boundary conditions labelled by η, can be obtained from the
one of periodic boundary conditions (η = 0) if one chooses f = ηφ. For this choice, the
dynamical fields that describe the asymptotic structure in (3.5) then transform as
J
η = J + ηT ,
Zη = Z − 2π
η ,
k
Pη = P − 4ηZ +
G±η = e±iηφG± ,
k 2
η ,
π
with T
η = T . In terms of Fourier modes, eq. (7.3) reads
(7.1)
(7.2)
(7.3)
(7.4)
Jmη = Jm + ηTm ,
Pmη = Pm − 4ηZm + 2η2kδm,0 ,
Zmη = Zm − ηkδm,0 ,
Tmη = Tm ,
Gpη±±η = Gp± ,
and it is then simple to verify that the new “η” generators fulfill the super-BMS3 algebra
with N
= (2, 0) for an anyonic choice of boundary conditions. In this sense (7.4) can
be regarded as an automorphism of the asymptotic symmetry algebra (3.14) for generic
boundary conditions for the fermions as in (7.1). Therefore, in particular, the algebras with
Ramond and Neveu-Schwarz boundary conditions become related by spectral flow. One can
then say that a generic choice of boundary conditions for the fermions can be “gauged away”
by virtue of the U(1) automorphism of the super-BMS3 algebra with N = (2, 0). Thus,
different theories characterized by inequivalent choices of boundary conditions, actually
coincide after a suitable field redefinition that is induced through an appropriate U(1)
gauge transformation.
8
Extension of the results
One of the advantages of formulating supergravity in terms of a Chern-Simons action
as in [39, 41, 93], is that the theory can be extended to include parity odd terms in
the Lagrangian in a straightforward way along the lines of [94] (see also [34, 47, 95]).
The procedure amounts to introduce an additional coupling through a simple modification
of the invariant bilinear form of the gauge group, as well as further couplings through
shifting the magnetic-like gauge fields by the corresponding ones of electric type. Therefore,
as explained in [34, 47, 96], the global charges become suitably modified, so that the
asymptotic symmetry algebra is able to acquire additional central extensions. It would
then be interesting to explicitly construct this kind of extension for the supergravity theory
with N = (2, 0) in [41]. It is also worth pointing out that this procedure has recently been
applied in [58] for the case of the inhomogeneous (despotic) theory with N = (2, 0), so that
an additional central extension manifestly shows up in the asymptotic symmetry algebra,
as well as through an additional contribution to the entropy of cosmological spacetimes.
Another possibility that deserves to be explored is to consider the extension of our
analysis for N > 2, since it is natural to expect that the extended super-BMS3 algebra
has to be nonlinear, as it can be directly seen from the flat limit of the superconformal
algebra in two spacetime dimensions. Indeed, nonlinear extensions of the BMS3 algebra
have been shown to arise in the context of hypergravity [47, 96] as well as for higher spin
gravity without cosmological constant in [70, 72, 97, 98].
As a closing remark, it is worth pointing out that induced representations of BMS3, as
well as its extensions that include higher spin generators have been discussed in [99–102].
In the case of (higher spin) fermionic fields, this has been done so far for N = 1. It would
then be interesting to explore the properties of this sort of representations of super-BMS3
for an extended number of fermionic generators, as in the case discussed in this work.
Acknowledgments
We thank Daniel Grumiller, Wout Merbis, Alfredo P´erez and David Tempo for
enlightening discussions and comments. O.F. and R.T. wish to thank Daniel Grumiller and the
organizers of the ESI Programme and Workshop “Quantum Physics and Gravity” hosted
by the Erwin Schr¨odinger Institute (ESI), during June of 2017 in Vienna, for the
opportunity of presenting this work. Special thanks to Max Riegler for letting us know and kindly
explaining his interesting work in collaboration with Rudranil Basu and Stephane
Detournay in [58] during the workshop. This research has been partially supported by Fondecyt
is funded by the Chilean Government through the Centers of Excellence Base Financing
Program of Conicyt.
A
Conventions
We choose the orientation to be such that ε012 = 1. The Minkowski metric ηab is assumed
to be non-diagonal, so that η01 = η10 = η22 = 1, while the remaining components vanish.
In the spinorial representation, the generators of SO(2, 1) are given by (Ja)βα = 21 (Γa)β ,
α
where the matrices Γa fulfill the Clifford algebra, {Γa, Γb} = 2ηab. They are chosen in
terms of the Pauli matrices σi, so that
,
σ3 =
1 0 !
0 −1
.
(A.1)
(A.2)
(A.3)
ψ¯α = ψβCβα, where Cαβ stands for the charge conjugation matrix, given by
Spinors ψα are labeled according to α = +, −, and we define the Majorana conjugate as
Cαβ = Cαβ =
0 −1 !
,
so that Cαβ is the inverse. The charge conjugation matrix then fulfills CT = −C, and
In the fundamental representation of SL(2, R) × U(1), the generators of SL(2, R) and
the U(1) generator are given by Lm, with m = −1, 0, 1, and T , respectively. They are
0 −1 !
L0 =
2
L1 =
0 0 !
T =
(A.4)
Therefore, the nonvanishing components of the trace of quadratic products of them become
tr (L1L−1) = −1, tr L20 = 21 and tr T 2 = −2.
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. with
(CΓa)
T = CΓa.
chosen as
L−1 =
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