Hypothetical particle found in supergravity
In theoretical physics , the dual graviton is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality , as an S-duality , predicted by some formulations of eleven-dimensional supergravity .[3]
The dual graviton was first hypothesized in 1980.[4] It was theoretically modeled in 2000s,[1] [2] which was then predicted in eleven-dimensional mathematics of SO(8) supergravity in the framework of electric-magnetic duality.[3] It again emerged in the E 11 generalized geometry in eleven dimensions,[5] and the E 7 generalized vielbein-geometry in eleven dimensions.[6] While there is no local coupling between graviton and dual graviton, the field introduced by dual graviton may be coupled to a BF model as non-local gravitational fields in extra dimensions.[7]
A massive dual gravity of Ogievetsky–Polubarinov model[8] can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.[9] [10]
The previously mentioned theories of dual graviton are in flat space. In de Sitter and anti-de Sitter spaces (A)dS, the massless dual graviton exhibits less gauge symmetries dynamics compared with those of Curtright field in flat space, hence the mixed-symmetry field propagates in more degrees of freedom.[11] However, the dual graviton in (A)dS transforms under GL(D) representation, which is identical to that of massive dual graviton in flat space.[12] This apparent paradox can be resolved using the unfolding technique in Brink, Metsaev, and Vasiliev conjecture.[13] [14] For the massive dual graviton in (A)dS, the flat limit is clarified after expressing dual field in terms of the Stueckelberg coupling of a massless spin-2 field with a Proca field.[11]
Dual linearized gravity
The dual formulations of linearized gravity are described by a mixed Young symmetry tensor
T
λ λ -->
1
λ λ -->
2
⋯ ⋯ -->
λ λ -->
D
− − -->
3
μ μ -->
{\displaystyle T_{\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu }}
, the so-called dual graviton, in any spacetime dimension D > 4 with the following characters:[2] [15]
T
λ λ -->
1
λ λ -->
2
⋯ ⋯ -->
λ λ -->
D
− − -->
3
μ μ -->
=
T
[
λ λ -->
1
λ λ -->
2
⋯ ⋯ -->
λ λ -->
D
− − -->
3
]
μ μ -->
,
{\displaystyle T_{\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu }=T_{[\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}]\mu },}
T
[
λ λ -->
1
λ λ -->
2
⋯ ⋯ -->
λ λ -->
D
− − -->
3
μ μ -->
]
=
0.
{\displaystyle T_{[\lambda _{1}\lambda _{2}\cdots \lambda _{D-3}\mu ]}=0.}
where square brackets show antisymmetrization.
For 5-D spacetime, the spin-2 dual graviton is described by the Curtright field
T
α α -->
β β -->
γ γ -->
{\displaystyle T_{\alpha \beta \gamma }}
. The symmetry properties imply that
T
α α -->
β β -->
γ γ -->
=
T
[
α α -->
β β -->
]
γ γ -->
,
{\displaystyle T_{\alpha \beta \gamma }=T_{[\alpha \beta ]\gamma },}
T
[
α α -->
β β -->
]
γ γ -->
+
T
[
β β -->
γ γ -->
]
α α -->
+
T
[
γ γ -->
α α -->
]
β β -->
=
0.
{\displaystyle T_{[\alpha \beta ]\gamma }+T_{[\beta \gamma ]\alpha }+T_{[\gamma \alpha ]\beta }=0.}
The Lagrangian action for the spin-2 dual graviton
T
λ λ -->
1
λ λ -->
2
μ μ -->
{\displaystyle T_{\lambda _{1}\lambda _{2}\mu }}
in 5-D spacetime, the Curtright field , becomes[2] [15]
L
d
u
a
l
=
− − -->
1
12
(
F
[
α α -->
β β -->
γ γ -->
]
δ δ -->
F
[
α α -->
β β -->
γ γ -->
]
δ δ -->
− − -->
3
F
[
α α -->
β β -->
ξ ξ -->
]
ξ ξ -->
F
[
α α -->
β β -->
λ λ -->
]
λ λ -->
)
,
{\displaystyle {\cal {L}}_{\rm {dual}}=-{\frac {1}{12}}\left(F_{[\alpha \beta \gamma ]\delta }F^{[\alpha \beta \gamma ]\delta }-3F_{[\alpha \beta \xi ]}{}^{\xi }F^{[\alpha \beta \lambda ]}{}_{\lambda }\right),}
where
F
α α -->
β β -->
γ γ -->
δ δ -->
{\displaystyle F_{\alpha \beta \gamma \delta }}
is defined as
F
[
α α -->
β β -->
γ γ -->
]
δ δ -->
=
∂ ∂ -->
α α -->
T
[
β β -->
γ γ -->
]
δ δ -->
+
∂ ∂ -->
β β -->
T
[
γ γ -->
α α -->
]
δ δ -->
+
∂ ∂ -->
γ γ -->
T
[
α α -->
β β -->
]
δ δ -->
,
{\displaystyle F_{[\alpha \beta \gamma ]\delta }=\partial _{\alpha }T_{[\beta \gamma ]\delta }+\partial _{\beta }T_{[\gamma \alpha ]\delta }+\partial _{\gamma }T_{[\alpha \beta ]\delta },}
and the gauge symmetry of the Curtright field is
δ δ -->
σ σ -->
,
α α -->
T
[
α α -->
β β -->
]
γ γ -->
=
2
(
∂ ∂ -->
[
α α -->
σ σ -->
β β -->
]
γ γ -->
+
∂ ∂ -->
[
α α -->
α α -->
β β -->
]
γ γ -->
− − -->
∂ ∂ -->
γ γ -->
α α -->
α α -->
β β -->
)
.
{\displaystyle \delta _{\sigma ,\alpha }T_{[\alpha \beta ]\gamma }=2(\partial _{[\alpha }\sigma _{\beta ]\gamma }+\partial _{[\alpha }\alpha _{\beta ]\gamma }-\partial _{\gamma }\alpha _{\alpha \beta }).}
The dual Riemann curvature tensor of the dual graviton is defined as follows:[2]
E
[
α α -->
β β -->
δ δ -->
]
[
ε ε -->
γ γ -->
]
≡ ≡ -->
1
2
(
∂ ∂ -->
ε ε -->
F
[
α α -->
β β -->
δ δ -->
]
γ γ -->
− − -->
∂ ∂ -->
γ γ -->
F
[
α α -->
β β -->
δ δ -->
]
ε ε -->
)
,
{\displaystyle E_{[\alpha \beta \delta ][\varepsilon \gamma ]}\equiv {\frac {1}{2}}(\partial _{\varepsilon }F_{[\alpha \beta \delta ]\gamma }-\partial _{\gamma }F_{[\alpha \beta \delta ]\varepsilon }),}
and the dual Ricci curvature tensor and scalar curvature of the dual graviton become, respectively
E
[
α α -->
β β -->
]
γ γ -->
=
g
ε ε -->
δ δ -->
E
[
α α -->
β β -->
δ δ -->
]
[
ε ε -->
γ γ -->
]
,
{\displaystyle E_{[\alpha \beta ]\gamma }=g^{\varepsilon \delta }E_{[\alpha \beta \delta ][\varepsilon \gamma ]},}
E
α α -->
=
g
β β -->
γ γ -->
E
[
α α -->
β β -->
]
γ γ -->
.
{\displaystyle E_{\alpha }=g^{\beta \gamma }E_{[\alpha \beta ]\gamma }.}
They fulfill the following Bianchi identities
∂ ∂ -->
α α -->
(
E
[
α α -->
β β -->
]
γ γ -->
+
g
γ γ -->
[
α α -->
E
β β -->
]
)
=
0
,
{\displaystyle \partial _{\alpha }(E^{[\alpha \beta ]\gamma }+g^{\gamma [\alpha }E^{\beta ]})=0,}
where
g
α α -->
β β -->
{\displaystyle g^{\alpha \beta }}
is the 5-D spacetime metric.
Massive dual gravity
In 4-D, the Lagrangian of the spinless massive version of the dual gravity is
L
d
u
a
l
,
m
a
s
s
i
v
e
s
p
i
n
l
e
s
s
=
− − -->
1
2
u
+
1
2
(
v
− − -->
g
u
)
2
+
1
3
g
(
v
− − -->
g
u
)
3
F
3
F
2
-->
(
1
,
1
2
,
3
2
;
2
,
5
2
;
− − -->
4
g
2
(
v
− − -->
g
u
)
2
)
,
{\displaystyle {\mathcal {L_{\rm {dual,massive}}^{\rm {spinless}}}}=-{\frac {1}{2}}u+{\frac {1}{2}}(v-gu)^{2}+{\frac {1}{3}}g(v-gu)^{3}\sideset {_{3}}{_{2}}F(1,{\frac {1}{2}},{\frac {3}{2}};2,{\frac {5}{2}};-4g^{2}(v-gu)^{2}),}
where
V
μ μ -->
=
1
6
ϵ ϵ -->
μ μ -->
α α -->
β β -->
γ γ -->
V
α α -->
β β -->
γ γ -->
,
v
=
V
μ μ -->
V
μ μ -->
and
u
=
∂ ∂ -->
μ μ -->
V
μ μ -->
.
{\displaystyle V^{\mu }={\frac {1}{6}}\epsilon ^{\mu \alpha \beta \gamma }V_{\alpha \beta \gamma }~,v=V_{\mu }V^{\mu }{\text{and}}~u=\partial _{\mu }V^{\mu }.}
[16] The coupling constant
g
/
m
{\displaystyle g/m}
appears in the equation of motion to couple the trace of the conformally improved energy momentum tensor
θ θ -->
{\displaystyle \theta }
to the field as in the following equation
(
◻ ◻ -->
+
m
2
)
V
μ μ -->
=
g
m
∂ ∂ -->
μ μ -->
θ θ -->
.
{\displaystyle \left(\Box +m^{2}\right)V_{\mu }={\frac {g}{m}}\partial _{\mu }\theta .}
And for the spin-2 massive dual gravity in 4-D,[10] the Lagrangian is formulated in terms of the Hessian matrix that also constitutes Horndeski theory (Galileons/massive gravity ) through
det
(
δ δ -->
ν ν -->
μ μ -->
+
g
m
K
ν ν -->
μ μ -->
)
=
1
− − -->
1
2
(
g
/
m
)
2
K
α α -->
β β -->
K
β β -->
α α -->
+
1
3
(
g
/
m
)
3
K
α α -->
β β -->
K
β β -->
γ γ -->
K
γ γ -->
α α -->
+
1
8
(
g
/
m
)
4
[
(
K
α α -->
β β -->
K
β β -->
α α -->
)
2
− − -->
2
K
α α -->
β β -->
K
β β -->
γ γ -->
K
γ γ -->
δ δ -->
K
δ δ -->
α α -->
]
,
{\displaystyle {\text{det}}(\delta _{\nu }^{\mu }+{\frac {g}{m}}K_{\nu }^{\mu })=1-{\frac {1}{2}}(g/m)^{2}K_{\alpha }^{\beta }K_{\beta }^{\alpha }+{\frac {1}{3}}(g/m)^{3}K_{\alpha }^{\beta }K_{\beta }^{\gamma }K_{\gamma }^{\alpha }+{\frac {1}{8}}(g/m)^{4}\left[(K_{\alpha }^{\beta }K_{\beta }^{\alpha })^{2}-2K_{\alpha }^{\beta }K_{\beta }^{\gamma }K_{\gamma }^{\delta }K_{\delta }^{\alpha }\right],}
where
K
μ μ -->
ν ν -->
=
3
∂ ∂ -->
α α -->
T
[
β β -->
γ γ -->
]
μ μ -->
ϵ ϵ -->
α α -->
β β -->
γ γ -->
ν ν -->
{\displaystyle K_{\mu }^{\nu }=3\partial _{\alpha }T_{[\beta \gamma ]\mu }\epsilon ^{\alpha \beta \gamma \nu }}
.
So the zeroth interaction part, i.e., the third term in the Lagrangian, can be read as
K
α α -->
β β -->
θ θ -->
β β -->
α α -->
{\displaystyle K_{\alpha }^{\beta }\theta _{\beta }^{\alpha }}
so the equation of motion becomes
(
◻ ◻ -->
+
m
2
)
T
[
α α -->
β β -->
]
γ γ -->
=
g
m
P
α α -->
β β -->
γ γ -->
,
λ λ -->
μ μ -->
ν ν -->
∂ ∂ -->
λ λ -->
θ θ -->
μ μ -->
ν ν -->
,
{\displaystyle \left(\Box +m^{2}\right)T_{[\alpha \beta ]\gamma }={\frac {g}{m}}P_{\alpha \beta \gamma ,\lambda \mu \nu }\partial ^{\lambda }\theta ^{\mu \nu },}
where the
P
α α -->
β β -->
γ γ -->
,
λ λ -->
μ μ -->
ν ν -->
=
2
ϵ ϵ -->
α α -->
β β -->
λ λ -->
μ μ -->
η η -->
γ γ -->
ν ν -->
+
ϵ ϵ -->
α α -->
γ γ -->
λ λ -->
μ μ -->
η η -->
β β -->
ν ν -->
− − -->
ϵ ϵ -->
β β -->
γ γ -->
λ λ -->
μ μ -->
η η -->
α α -->
ν ν -->
{\displaystyle P_{\alpha \beta \gamma ,\lambda \mu \nu }=2\epsilon _{\alpha \beta \lambda \mu }\eta _{\gamma \nu }+\epsilon _{\alpha \gamma \lambda \mu }\eta _{\beta \nu }-\epsilon _{\beta \gamma \lambda \mu }\eta _{\alpha \nu }}
is Young symmetrizer of such SO(2) theory.
For solutions of the massive theory in arbitrary N-D, i.e., Curtright field
T
[
λ λ -->
1
λ λ -->
2
.
.
.
λ λ -->
N
− − -->
3
]
μ μ -->
{\displaystyle T_{[\lambda _{1}\lambda _{2}...\lambda _{N-3}]\mu }}
, the symmetrizer becomes that of SO(N-2).[9]
Dual graviton coupling with BF theory
Dual gravitons have interaction with topological BF model in D = 5 through the following Lagrangian action[7]
S
L
=
∫ ∫ -->
d
5
x
(
L
d
u
a
l
+
L
B
F
)
.
{\displaystyle S_{\rm {L}}=\int d^{5}x({\cal {L}}_{\rm {dual}}+{\cal {L}}_{\rm {BF}}).}
where
L
B
F
=
T
r
[
B
∧ ∧ -->
F
]
{\displaystyle {\cal {L}}_{\rm {BF}}=Tr[\mathbf {B} \wedge \mathbf {F} ]}
Here,
F
≡ ≡ -->
d
A
∼ ∼ -->
R
a
b
{\displaystyle \mathbf {F} \equiv d\mathbf {A} \sim R_{ab}}
is the curvature form , and
B
≡ ≡ -->
e
a
∧ ∧ -->
e
b
{\displaystyle \mathbf {B} \equiv e^{a}\wedge e^{b}}
is the background field.
In principle, it should similarly be coupled to a BF model of gravity as the linearized Einstein–Hilbert action in D > 4:
S
B
F
=
∫ ∫ -->
d
5
x
L
B
F
∼ ∼ -->
S
E
H
=
1
2
∫ ∫ -->
d
5
x
R
− − -->
g
.
{\displaystyle S_{\rm {BF}}=\int d^{5}x{\cal {L}}_{\rm {BF}}\sim S_{\rm {EH}}={1 \over 2}\int \mathrm {d} ^{5}xR{\sqrt {-g}}.}
where
g
=
det
(
g
μ μ -->
ν ν -->
)
{\displaystyle g=\det(g_{\mu \nu })}
is the determinant of the metric tensor matrix, and
R
{\displaystyle R}
is the Ricci scalar .
Dual gravitoelectromagnetism
In similar manner while we define gravitomagnetic and gravitoelectic for the graviton, we can define electric and magnetic fields for the dual graviton.[17] There are the following relation between the gravitoelectic field
E
a
b
[
h
a
b
]
{\displaystyle E_{ab}[h_{ab}]}
and gravitomagnetic field
B
a
b
[
h
a
b
]
{\displaystyle B_{ab}[h_{ab}]}
of the graviton
h
a
b
{\displaystyle h_{ab}}
and the gravitoelectic field
E
a
b
[
T
a
b
c
]
{\displaystyle E_{ab}[T_{abc}]}
and gravitomagnetic field
B
a
b
[
T
a
b
c
]
{\displaystyle B_{ab}[T_{abc}]}
of the dual graviton
T
a
b
c
{\displaystyle T_{abc}}
:[18] [15]
B
a
b
[
T
a
b
c
]
=
E
a
b
[
h
a
b
]
{\displaystyle B_{ab}[T_{abc}]=E_{ab}[h_{ab}]}
E
a
b
[
T
a
b
c
]
=
− − -->
B
a
b
[
h
a
b
]
{\displaystyle E_{ab}[T_{abc}]=-B_{ab}[h_{ab}]}
and scalar curvature
R
{\displaystyle R}
with dual scalar curvature
E
{\displaystyle E}
:[18]
E
=
⋆ ⋆ -->
R
{\displaystyle E=\star R}
R
=
− − -->
⋆ ⋆ -->
E
{\displaystyle R=-\star E}
where
⋆ ⋆ -->
{\displaystyle \star }
denotes the Hodge dual .
Dual graviton in conformal gravity
The free (4,0) conformal gravity in D = 6 is defined as
S
=
∫ ∫ -->
d
6
x
− − -->
g
C
A
B
C
D
C
A
B
C
D
,
{\displaystyle {\mathcal {S}}=\int \mathrm {d} ^{6}x{\sqrt {-g}}C_{ABCD}C^{ABCD},}
where
C
A
B
C
D
{\displaystyle C_{ABCD}}
is the Weyl tensor in D = 6. The free (4,0) conformal gravity can be reduced to the graviton in the ordinary space, and the dual graviton in the dual space in D = 4.[19]
It is easy to notice the similarity between the Lanczos tensor , that generates the Weyl tensor in geometric theories of gravity, and Curtright tensor, particularly their shared symmetry properties of the linearized spin connection in Einstein's theory. However, Lanczos tensor is a tensor of geometry in D=4,[20] meanwhile Curtright tensor is a field tensor in arbitrary dimensions.
See also
References
^ a b
Hull, C. M. (2001). "Duality in Gravity and Higher Spin Gauge Fields" . Journal of High Energy Physics . 2001 (9): 27. arXiv :hep-th/0107149 . Bibcode :2001JHEP...09..027H . doi :10.1088/1126-6708/2001/09/027 .
^ a b c d e
Bekaert, X.; Boulanger, N.; Henneaux, M. (2003). "Consistent deformations of dual formulations of linearized gravity: A no-go result". Physical Review D . 67 (4): 044010. arXiv :hep-th/0210278 . Bibcode :2003PhRvD..67d4010B . doi :10.1103/PhysRevD.67.044010 . S2CID 14739195 .
^ a b
de Wit, B.; Nicolai, H. (2013). "Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions". Journal of High Energy Physics . 2013 (5): 77. arXiv :1302.6219 . Bibcode :2013JHEP...05..077D . doi :10.1007/JHEP05(2013)077 . S2CID 119201330 .
^
Curtright, T. (1985). "Generalised Gauge Fields". Physics Letters B . 165 (4–6): 304. Bibcode :1985PhLB..165..304C . doi :10.1016/0370-2693(85)91235-3 .
^
West, P. (2012). "Generalised geometry, eleven dimensions and E 11 ". Journal of High Energy Physics . 2012 (2): 18. arXiv :1111.1642 . Bibcode :2012JHEP...02..018W . doi :10.1007/JHEP02(2012)018 . S2CID 119240022 .
^
Godazgar, H.; Godazgar, M.; Nicolai, H. (2014). "Generalised geometry from the ground up" . Journal of High Energy Physics . 2014 (2): 75. arXiv :1307.8295 . Bibcode :2014JHEP...02..075G . doi :10.1007/JHEP02(2014)075 .
^ a b
Bizdadea, C.; Cioroianu, E. M.; Danehkar, A.; Iordache, M.; Saliu, S. O.; Sararu, S. C. (2009). "Consistent interactions of dual linearized gravity in D = 5: couplings with a topological BF model". European Physical Journal C . 63 (3): 491–519. arXiv :0908.2169 . Bibcode :2009EPJC...63..491B . doi :10.1140/epjc/s10052-009-1105-0 . S2CID 15873396 .
^ Ogievetsky, V. I; Polubarinov, I. V (1965-11-01). "Interacting field of spin 2 and the einstein equations". Annals of Physics . 35 (2): 167–208. Bibcode :1965AnPhy..35..167O . doi :10.1016/0003-4916(65)90077-1 . ISSN 0003-4916 .
^ a b Alshal, H.; Curtright, T. L. (2019-09-10). "Massive dual gravity in N spacetime dimensions". Journal of High Energy Physics . 2019 (9): 63. arXiv :1907.11537 . Bibcode :2019JHEP...09..063A . doi :10.1007/JHEP09(2019)063 . ISSN 1029-8479 . S2CID 198953238 .
^ a b Curtright, T. L.; Alshal, H. (2019-10-01). "Massive dual spin 2 revisited". Nuclear Physics B . 948 : 114777. arXiv :1907.11532 . Bibcode :2019NuPhB.94814777C . doi :10.1016/j.nuclphysb.2019.114777 . ISSN 0550-3213 . S2CID 198953158 .
^ a b Boulanger, N.; Campoleoni, A.; Cortese, I. (July 2018). "Dual actions for massless, partially-massless and massive gravitons in (A)dS". Physics Letters B . 782 : 285–290. arXiv :1804.05588 . Bibcode :2018PhLB..782..285B . doi :10.1016/j.physletb.2018.05.046 . S2CID 54826796 .
^ Basile, Thomas; Bekaert, Xavier; Boulanger, Nicolas (2016-06-21). "Note about a pure spin-connection formulation of general relativity and spin-2 duality in (A)dS". Physical Review D . 93 (12): 124047. arXiv :1512.09060 . Bibcode :2016PhRvD..93l4047B . doi :10.1103/PhysRevD.93.124047 . ISSN 2470-0010 . S2CID 55583084 .
^ Brink, L.; Metsaev, R.R.; Vasiliev, M.A. (October 2000). "How massless are massless fields in AdS". Nuclear Physics B . 586 (1–2): 183–205. arXiv :hep-th/0005136 . Bibcode :2000NuPhB.586..183B . doi :10.1016/S0550-3213(00)00402-8 . S2CID 119512854 .
^ Basile, Thomas; Bekaert, Xavier; Boulanger, Nicolas (May 2017). "Mixed-symmetry fields in de Sitter space: a group theoretical glance". Journal of High Energy Physics . 2017 (5): 81. arXiv :1612.08166 . Bibcode :2017JHEP...05..081B . doi :10.1007/JHEP05(2017)081 . ISSN 1029-8479 . S2CID 119185373 .
^ a b c
Danehkar, A. (2019). "Electric-magnetic duality in gravity and higher-spin fields" . Frontiers in Physics . 6 : 146. Bibcode :2019FrP.....6..146D . doi :10.3389/fphy.2018.00146 .
^ Curtright, Thomas L. (2019-10-01). "Massive dual spinless fields revisited". Nuclear Physics B . 948 : 114784. arXiv :1907.11530 . Bibcode :2019NuPhB.94814784C . doi :10.1016/j.nuclphysb.2019.114784 . ISSN 0550-3213 . S2CID 198953144 .
^
Henneaux, M.; Teitelboim, C. (2005). "Duality in linearized gravity". Physical Review D . 71 (2): 024018. arXiv :gr-qc/0408101 . Bibcode :2005PhRvD..71b4018H . doi :10.1103/PhysRevD.71.024018 . S2CID 119022015 .
^ a b Henneaux, M., "E 10 and gravitational duality"
https://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_workshops_conferences/jointerc_2014/henneaux.pdf
^
Hull, C. M. (2000). "Symmetries and Compactifications of (4,0) Conformal Gravity". Journal of High Energy Physics . 2000 (12): 007. arXiv :hep-th/0011215 . Bibcode :2000JHEP...12..007H . doi :10.1088/1126-6708/2000/12/007 . S2CID 18326976 .
^ Bampi, Franco; Caviglia, Giacomo (April 1983). "Third-order tensor potentials for the Riemann and Weyl tensors". General Relativity and Gravitation . 15 (4): 375–386. Bibcode :1983GReGr..15..375B . doi :10.1007/BF00759166 . ISSN 0001-7701 . S2CID 122782358 .