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Maximally supersymmetric backgroundsand partial N = 2→ N = 1 rigid
supersymmetry breaking
Sergei M. Kuzenko
Department of Physics, University of Western Australia
Kavli Institute for the Physics and Mathematics of the UniverseUniversity of Tokyo
25 March, 2019
Outline
1 Supersymmetric backgrounds: Introductory comments
2 (Conformal) symmetries of curved spacetime
3 (Conformal) symmetries of curved superspace
4 Case study: Supersymmetric backgrounds in 3D N = 2 supergravity
5 4D N = 1 maximally symmetric backgrounds
6 Maxwell Goldstone multiplet for partially broken SUSY
7 Nilpotent chiral superfield in N = 2 supergravity
8 Example: The super-Poincare case
Supersymmetric backgrounds: Introductory comments
Supersymmetric backgrounds in supergravity
Supersymmetric solutions of supergravityM. Duff & C. Pope (1982)
F. Englert, M. Rooman & P. Spindel (1983)P. van Nieuwenhuizen & N. Warner (1984)
. . . . . . . . .M. Duff, B. Nilsson & C. Pope, Kaluza-Klein Supergravity (PR, 1986)
Concept of Killing spinors
One of the highlights:All supersymmetric solutions of minimal (gauged) supergravity in 5D
J. Gauntlett, J. Gutowski, C. Hull, S. Pakis & H. Reall (2003)J. Gauntlett & J. Gutowski (2003)
Superspace formalism to determine (super)symmetric backgrounds inoff-shell supergravityI. Buchbinder & SMK, Ideas and Methods of Supersymmetry andSupergravity or a Walk Through Superspace, IOP, 1995 + 1998
Superspace formalism to identify (super)symmetric backgrounds isuniversal since
it is geometric;
it may be extended to any off-shell supergravity theory formulated insuperspace.
This formalism has been applied to construct off-shell rigidsupersymmetric field theories in curved backgrounds:
5D supersymmetric field theories with N = 1 AdS supersymmetry(Eight months before Pestun’s work on N = 2 SYM theories on S4)
SMK & G. Tartaglino-Mazzucchelli (2007)
4D supersymmetric field theories with N = 2 AdS supersymmetrySMK & G. Tartaglino-Mazzucchelli (2008)
(Most general nonlinear σ-models) D. Butter & SMK (2011)D. Butter, SMK, U. Lindstrom & G. Tartaglino-Mazzucchelli (2012)
3D supersymmetric field theories with (p, q) AdS supersymmetrySMK, & G. Tartaglino-Mazzucchelli (2012)
SMK, U. Lindstrom & G. Tartaglino-Mazzucchelli (2012)D. Butter, SMK & G. Tartaglino-Mazzucchelli (2012)
Important developments in the last decade
Exact results (partition functions, Wilson loops etc.)in rigid supersymmetric field theories on curved backgrounds(e.g., S3, S4 ,S3 × S1 etc.) using localisation techniques
V. Pestun (2007, 2009)A. Kapustin, B. Willett & I. Yaakov (2010)
D. Jafferis (2010)· · · · · · · · · · · · · · · · · · · · · · · ·
Necessary technical ingredients:
Curved space M has to admit some unbroken rigid supersymmetry(supersymmetric background);
Rigid supersymmetric field theory on M should be off-shell.
These developments have inspired much interest in the construction andclassification of supersymmetric backgrounds that correspond to off-shellsupergravity formulations.
Classification of supersymmetric backgrounds in off-shellsupergravity
Component approachesG. Festuccia and N. Seiberg (2011)
H. Samtleben and D. Tsimpis (2012)C. Klare, A. Tomasiello and A. Zaffaroni (2012)
T. Dumitrescu, G. Festuccia and N. Seiberg (2012)D. Cassani, C. Klare, D. Martelli, A. Tomasiello and A. Zaffaroni (2012)
T. Dumitrescu and G. Festuccia (2012)A. Kehagias and J. Russo (2012)
· · · · · · · · · · · · · · · · · · · · · · · ·
These results naturally follow from the superspace formalism developed inthe mid 1990s, as demonstrated in several publications:4D N = 1 SMK (2012)4D N = 2 D. Butter, G. Inverso & I. Lodato (2015)3D N = 2 SMK, U. Lindstrom, M. Rocek, I. Sachs
& G. Tartaglino-Mazzucchelli (2013)5D N = 1 SMK, J. Novak & G. Tartaglino-Mazzucchelli (2014)
Component approaches vs superspace formalism
Both the component approaches and superspace formalisms can beused to derive supersymmetric backgrounds in off-shell supergravity.Practically all classification results have been obtained within thecomponent settings.
Superspace formalism is more useful in order to determine all(conformal) isometries of a given backgrounds.
Superspace formalism is by far more powerful for constructing themost general rigid supersymmetric field theories on a givenbackground.
Superspace formalism is useful to describe partial breaking of rigidSUSY in curved maximally supersymmetric backgrounds.
Partial SUSY breaking: general comments
Is partial N = 2 SUSY breaking posible?Impossible E. Witten (1981)Let’s nevertheless try J. Bagger & J. Wess (1984)Possible J. Hughes, J. Liu & J. Polchinski (1986)
In principle, the formalism of nonlinear realisations may be used toderive models for partial N = 2→ N = 1 SUSY breaking. Inpractice, not a single model was constructed in closed form usingsuch an approach. J. Bagger & A. Galperin (1994)
Superfield techniques were employed to construct four Goldstonemultiplet actions for partial N = 2→ N = 1 breaking of PoincareSUSY, with manifest N = 1 SUSY. They make use of differentN = 1 supermultiplets to which the Goldstino belongs, specifically:Vector multiplet J. Bagger & A. Galperin (1997)Tensor multiplet J. Bagger & A. Galperin (1997)Chiral scalar multiplet J. Bagger & A. Galperin (1997)Complex linear (or non-minimal) scalarmultiplet F. Gonzalez-Ray, I. Park & M. Rocek (1999)
Partial breaking of rigid SUSY in Minkowski space
It was demonstrated that the vector and tensor Goldstone models forpartially broken N = 2 Poincare SUSY can naturally be derived fromconstrained N = 2 superfields.
M. Rocek & A. Tseytlin (1999)
The non-minimal scalar Goldstone multiplet for partially brokenN = 2 Poincare SUSY was also derived from constrained N = 2superfields (the so-called O(4) multiplet).
F. Gonzalez-Ray, I. Park & M. Rocek (1999)
There is no direct way to read off the chiral Goldstone multiplet fromconstrained N = 2 superfields (without intrinsic central charge).
I. Antoniadis, H. Partouche & T. Taylor (1996)E. Ivanov & B. Zupnik (1998)
The above constructions correspond to the case of Poincare SUSY.Minkowski space is one of many supersymmetric spacetimes compatiblewith rigid N = 2 SUSY (8 supercharges).Do such spacetimes allow for models for partial SUSY breaking?
(Conformal) symmetries of curved spacetime
(Conformal) symmetries of curved spacetime
(Conformal) symmetries of a curved superspace may be defined similarlyto those corresponding to a curved spacetime within the Weyl-invariantformulation for gravity (variation on a theme by Hermann Weyl).
S. Deser (1970)B. Zumino (1970)
P. Dirac (1973)Three formulations for gravity in d dimensions:
Metric formulation;
Vielbein formulation;
Weyl-invariant formulation.
I briefly recall the metric and vielbein approaches and then concentrate inmore detail of the Weyl-invariant formulation.
Metric and vielbein formulations for gravity
Metric formulationGauge field: metric gmn(x)Gauge transformation: δgmn = ∇mξn +∇nξmξ = ξm(x)∂m a vector field generating an infinitesimal diffeomorphism.
Vielbein formulation R. Utiyama (1956)Gauge field: vielbein em
a(x) , e := det(ema) 6= 0
The metric is a composite field gmn = emaen
bηabGauge transformation: δ∇a = [ξb∇b + 1
2KbcMbc ,∇a]
Gauge parameters: ξa(x) = ξmema(x) and K ab(x) = −K ba(x)
Covariant derivatives (Mbc the Lorentz generators)
∇a = eam∂m +
1
2ωa
bcMbc , [∇a,∇b] =1
2Rab
cdMcd
eam the inverse vielbein, ea
memb = δa
b;ωa
bc the torsion-free Lorentz connection.
Weyl transformations
Weyl transformationsThe torsion-free constraint
Tabc = 0 ⇐⇒ [∇a,∇b] ≡ Tab
c∇c +1
2Rab
cdMcd =1
2Rab
cdMcd
is invariant under a Weyl (local scale) transformation
∇a → ∇′a = eσ(∇a + (∇bσ)Mba
),
with the parameter σ(x) being completely arbitrary. This transformationacts of the (inverse) vielbein and metric as
eam → eσea
m ⇐⇒ ema → e−σem
a ⇐⇒ gmn → e−2σgmn
Weyl transformations are gauge symmetries of conformal gravity.Einstein gravity possesses no Weyl invariance.
Weyl-invariant formulation for Einstein gravity
Weyl-invariant formulation for Einstein gravityGauge fields: vielbein em
a(x) , e := det(ema) 6= 0
& conformal compensator ϕ(x) , ϕ 6= 0Gauge transformations (K := ξb∇b + 1
2KbcMbc)
δ∇a = [ξb∇b +1
2K bcMbc ,∇a] + σ∇a + (∇bσ)Mba ≡ (δK + δσ)∇a ,
δϕ = ξb∇bϕ+1
2(d − 2)σϕ ≡ (δK + δσ)ϕ
Gauge-invariant gravity action
S =1
2
∫ddx e
(∇aϕ∇aϕ+
1
4
d − 2
d − 1Rϕ2 + λϕ2d/(d−2)
)Imposing a Weyl gauge condition ϕ = 2
κ
√d−1d−2 = const
reduces the action to
S =1
2κ2
∫ddx e R − Λ
κ2
∫ddx e
Conformal isometries
Conformal Killing vector fieldsA vector field ξ = ξm∂m = ξaea, with ea := ea
m∂m, is conformal Killing ifthere exist local Lorentz, K bc [ξ], and Weyl, σ[ξ], parameters such that
(δK + δσ)∇a =[ξb∇b +
1
2K bc [ξ]Mbc ,∇a
]+ σ[ξ]∇a + (∇bσ[ξ])Mba = 0
A short calculation gives
K bc [ξ] =1
2
(∇bξc −∇cξb
), σ[ξ] =
1
d∇bξ
b
Conformal Killing equation
∇aξb +∇bξa = 2ηabσ[ξ]
Equivalent spinor form in d = 4: ∇(α(αξβ)
β) = 0(∇a → ∇αα and ξa → ξαα)Equivalent spinor form in d = 3: ∇(αβξγδ) = 0
Conformal isometries
Lie algebra of conformal Killing vector fields.
Conformally related spacetimes (∇a, ϕ) and (∇a, ϕ)
∇a = eρ(∇a + (∇bρ)Mba
), ϕ = e
12 (d−2)ρϕ
have the same conformal Killing vector fields ξ = ξaea = ξaea.
The parameters K cd [ξ] and σ[ξ] are related to K cd [ξ] and σ[ξ] asfollows:
K[ξ] := ξb∇b +1
2K cd [ξ]Mcd = K[ξ] ,
σ[ξ] = σ[ξ]− ξρ
Rigid conformal field theories on curved backgrounds.
Isometries
Killing vector fieldsLet ξ = ξaea be a conformal Killing vector,
(δK + δσ)∇a =[ξb∇b +
1
2K bc [ξ]Mbc ,∇a
]+ σ[ξ]∇a + (∇bσ[ξ])Mba = 0 .
It is called Killing if it leaves the compensator invariant,
(δK + δσ)ϕ = ξϕ+1
2(d − 2)σ[ξ]ϕ = 0 .
These Killing equations are Weyl invariant in the following sense:Given a conformally related spacetime (∇a, ϕ)
∇a = eρ(∇a + (∇bρ)Mba
), ϕ = e
12 (d−2)ρϕ ,
the above Killing equations have the same functional form whenrewritten in terms of (∇a, ϕ), in particular
ξϕ+1
2(d − 2)σ[ξ]ϕ = 0 .
Isometries
Because of Weyl invariance, we can work with a conformally relatedspacetime such that
ϕ = 1
Then the Killing equations turn into[ξb∇b +
1
2K bc [ξ]Mbc ,∇a
]= 0 , σ[ξ] = 0
Standard Killing equation
∇aξb +∇bξa = 0
Lie algebra of Killing vector fields
Rigid field theories in curved space
(Conformal) symmetries of curved superspace
(Conformal) symmetries of curved superspace
Weyl-invariant approach to spacetime symmetries has a naturalsuperspace extension in all cases when supergravity is formulated asconformal supergravity coupled to certain conformal compensator(s) Ξ
zM = (xm, θµ) local coordinates of curved superspaceDA = (Da,Dα) = EA + ΩA + ΦA superspace covariant derivativesEA = EA
M(z)∂M superspace inverse vielbeinΩA = 1
2 ΩAbc(z)Mbc superspace Lorentz connection
ΦA = ΦAI (z)TI superspace R-symmetry connection
Supergravity gauge transformation
δKDA = [K,DA] , δKΞ = KΞ , K := ξBDB +1
2K bcMbc + K ITI
Super-Weyl transformation
δσDa = σDa + · · · , δσDα =1
2σDα + · · · , δσΞ = wΞσΞ ,
with wΞ a non-zero super-Weyl weight
Conformal isometries of curved superspace
Let ξ = ξBEB be a real supervector field. It is called conformal Killing if
(δK + δσ)DA = 0 ,
for some Lorentz K bc [ξ], R-symmetry K I [ξ] and super-Weyl σ[ξ]parameters.
All parameters K bc [ξ], K I [ξ] and σ[ξ] are uniquely determined interms of ξB .
The spinor component ξβ is uniquely determined in terms of ξb.
The vector component obeys an equation that contains all theinformation about the conformal Killing supervector field.Examples:
d = 3 DI(αξβγ) = 0
d = 4 DI(αξβ)β = 0
Isometries of curved superspace
Let ξ = ξBEB be a conformal Killing supervector field,
(δK[ξ] + δσ[ξ])DA = 0 , (?)
for uniquely determined parameters K bc [ξ], K I [ξ] and σ[ξ].It is called Killling if the compensators are invariant,
(δK[ξ] + wΞσ[ξ])Ξ = 0 . (??)
The Killing equations (?) and (??) are super-Weyl invariant in the sensethat they hold for all conformally related superspace geometries.
Using the compensators Ξ we can always construct a scalar superfieldφ = φ(Ξ), which (i) is an algebraic function of Ξ; (ii) nowhere vanishing;and (iii) has a nonzero super-Weyl weight wφ, δσφ = wφσφ.
(δK[ξ] + wΞσ[ξ])φ = 0 .
Super-Weyl invariance may be used to impose the gauge φ = 1, and then
σ[ξ] = 0 .
(Conformal) supersymmetries of curved superspace
Of special interest are curved backgrounds which admit at least one(conformal) supersymmetry. Such a superspace must possess a conformalKilling supervector field ξA of the type
ξa| = 0 , ξα| 6= 0
and describe a bosonic background with the property that all spinorcomponents of the superspace torsion and curvature tensors
[DA,DB = TABCDC +
1
2RAB
cdMcd + RABITI
have vanishing bar-projections,
ε(T······) = 1→ T···
···| = 0 , ε(R······) = 1→ R···
···| = 0 .
These conditions are supersymmetric.At the component level, all spinor fields may be gauged away.
Case study: Supersymmetric backgrounds in 3D N = 2supergravity
N = 2 supergravity in three dimensions
3D N = 2 curved superspace, M3|4, parametrised by local coordinateszM = (xm, θµ, θµ), m = 0, 1, 2 and µ = 1, 2Superspace structure group SO(2, 1)×U(1)RSuperspace covariant derivatives
DA = (Da,Dα, Dα) = EA + ΩA + iΦAJ .
Algebra of covariant derivatives
Dα,Dβ = −4RMαβ , Dα, Dβ = 4RMαβ ,
Dα, Dβ = −2i(γc)αβDc − 2CαβJ − 4iεαβSJ+4iSMαβ − 2εαβCγδMγδ .
Mab = −Mba ←→ Mαβ = Mβα Lorentz generatorsDimension-1 torsion superfields: (i) real scalar S; (ii) complex scalar Rsuch that JR = −2R; (iii) real vector Ca ←→ Cαβ .Bianchi Identities:
DαR = 0 , (D2 − 4R)S = 0 . . .
Conformal isometries
The conformal Killing supervector fields obey the equation
(δK + δσ)DA = 0 ,
where
δKDA = [K,DA] , K = ξCDC +1
2K cdMcd + i τJ
and
δσDα =1
2σDα + (Dγσ)Mγα − (Dασ)J , . . .
It suffices to require (δK + δσ)Dα = 0, which implies
ξα = − i
6Dβξβα , Kαβ = D(αξβ) − D(αξβ) − 2ξαβS
σ =1
2
(Dαξα + Dαξα
), τ = − i
4
(Dαξα − Dαξα
)All parameters ξα, Kαβ , σ and τ are expressed in terms of ξa.
Conformal isometries
The remaining vector parameter ξa satisfies
D(αξβγ) = 0 (?)
and its conjugate.Implication: superfield analogue of the conformal Killing equation
Daξb +Dbξa =2
3ηabDcξc .
Eq. (?) is fundamental in the sense that it implies (δK + δσ)DA ≡ 0provided the parameters ξα, Kαβ , σ and τ are defined as above.The conformal Killing supervector field is a real supervector field
ξ = ξAEA , ξA ≡ (ξa, ξα, ξα) =(ξa,− i
6Dβξβα,−
i
6Dβξβα
)which obeys the master equation (?).If ξ1 and ξ2 are two conformal Killing supervector fields, their Lie bracket[ξ1, ξ2] is a conformal Killing supervector field.
Conformal isometries
Equation (δK + δσ)Dα = 0 implies some additional results that have notbeen discussed above. Define
Υ := (ξB ,Kβγ , τ)
It turns out that
DAΥ is a linear combination of Υ, σ and DCσ;
DADBσ can be represented as a linear combination of Υ, σ andDCσ.
The super Lie algebra of the conformal Killing vector fields on M3|4 isfinite dimensional. The number of its even and odd generators cannotexceed those in the N = 2 superconformal algebra osp(2|4).
Charged conformal Killing spinors
Look for curved superspace backgrounds admitting at least oneconformal supersymmetry. Such a superspace must possessa conformal Killing supervector field ξA with the property
ξa| = 0 , εα := ξα| 6= 0 .
All other bosonic parameters are assumed to vanish, σ| = τ | = Kαβ | = 0.Bosonic superspace backgrounds without covariant fermionic fields:
DαS| = 0 , DαR| = 0 , DαCβγ | = 0 .
These conditions mean that the gravitini can completely be gauged away
Da| = Da := ea +1
2ωa
bcMbc + ibaJ ≡ Da + ibaJ , ea := eam∂m .
Introduce scalar and vector fields associated with the superfield torsion:
s := S| , r := R| , ca := Ca| .
S-supersymmetry parameter: ηα := Dασ|.
Charged conformal Killing spinors
Q-supersymmetry parameter εα := ξα| obeys the equation
Daεα +
i
2(γaη)α + iεabc c
b(γcε)α − s(γaε)α − ir(γaε)
α = 0 ,
which is equivalent to(D(αβ − ic(αβ
)εγ) =
(D(αβ−i(b + c)(αβ
)εγ) = 0 ,
ηα = −2i
3
((γaDaε)α + 2i(γaε)αca + 3sεα + 3ir εα
).
This follows by bar-projecting the equation (Cαβγ = −iD(αCβγ))
0 = Daξα +i
2(γa)α
βDβσ − iεabc(γb)αβCcξβ − (γa)α
β(ξβS + ξβR)
−1
2εabcξ
b(γc)βγ(iCαβγ −
4i
3εα(βDγ)S −
2
3εα(βDγ)R
),
which is one of the implications of (δK + δσ)Da = 0.
Supersymmetric backgrounds
Rigid supersymmetry transformations (in super-Weyl gauge φ = 1)are characterised by
σ[ξ] = 0 =⇒ ηα = 0 ,
The conformal Killing spinor equation turns into
Daεα = −iεabccb(γcε)α + s(γaε)
α + ir(γaε)α .
Da = ea +1
2ωa
bcMbc + ibaJ = Da + ibaJ .
[Da,Db] =1
2Rab
cdMcd + iFabJ = [Da,Db] + iFabJ .
4D N = 1 maximally symmetric backgrounds
4D N = 1 maximally symmetric backgrounds
There are only 5 maximally supersymmetric backgrounds in off-shellN = 1 supergravity theories in four dimensions.
G. Festuccia & N. Seiberg (2011)This classification was stated in the component setting.
Superspace formalism to determine (super)symmetric backgrounds inoff-shell supergravity:I. Buchbinder & SMK, Walk Through Superspace (1995, RE: 1998)
This formalism was used to provide a rigorous derivation of theFestuccia-Seiberg statements.
SMK (2012)
4D N = 1 maximally symmetric superspaces
Minkowski superspace M4|4 V. Akulov & D. Volkov (1973)A. Salam & J. Strathdee (1974)
Anti-de Sitter superspace AdS4|4 B. Keck (1975), B. Zumino (1977)E. Ivanov & A. Sorin (1980)
Dα, Dβ = −2iDαβ ,Dα,Dβ = −4RMαβ , Dα, Dβ = 4RMαβ ,
[Da,Dβ] = − i2 R(σa)βγDγ , [Da, Dβ] = i
2R(σa)γβDγ ,[Da,Db] = −|R|2Mab ,
with R = const.
Superspaces M4|4T , M4|4
S and M4|4N (G 2 < 0, G 2 > 0 and G 2 = 0)
Dα,Dβ = 0 , Dα, Dβ = 0 , Dα, Dβ = −2iDαβ ,[Dα,Dββ] = iεαβG
γβDγ , [Dα,Dββ] = −iεαβGβγDγ ,
[Dαα,Dββ] = −iεαβGβγDαγ + iεαβGγβDγα ,
where Gb is covariantly constant, DAGb = 0
Minkowski superspace M4|4 is a unique maximally supersymmetricsolution of supergravity without a cosmological term.
S = − 3
κ2
∫d4xd2θd2θ E
AdS superspace AdS4|4 is a unique maximally supersymmetricsolution of AdS supergravity.
SSUGRA = − 3
κ2
∫d4xd2θd2θ E +
µκ2
∫d4xd2θ E + c.c.
Superspaces M4|4
T , M4|4S and M4|4
N are maximally supersymmetricsolutions of scale-invariant R2 supergravity.
SMK, arXiv:1606.00654
S = α
∫d4x d2θd2θ E RR + β
∫d4x d2θ E R3 + c.c.
Scale-invariant R2 supergravity theories were studied by several groups:C. Kounnas, D. Lust & N. Toumbas (2015)
A. Kehagias, C. Kounnas, D. Lust & A. Riotto (2015)L. Alvarez-Gaume, A. Kehagias, C. Kounnas, D. Lust & A. Riotto (2015)
S. Ferrara, A. Kehagias & M. Porrati (2015)
Conformal supergravity in U(1) superspace
U(1) superspace P. Howe (1982)Structure group: SL(2,C)× U(1)R
Algebra of supergravity covariant derivatives
Dα, Dα = −2iDαα ,Dα,Dβ = −4RMαβ , Dα, Dβ = 4RMαβ ,[Dα,Dββ
]= iεαβ
(R Dβ + Gγ
βDγ − (DγG δβ)Mγδ + 2Wβ
γδMγδ
)+i(DβR)Mαβ −
i
3εαβX
γMγβ −i
2εαβXβJ .
The superfields R, Gαα, Wαβγ , and Xα obey the Bianchi identities:
DαR = 0 , DαXα = 0 , DαWαβγ = 0 ;
Xα = DαR − DαGαα , DαXα = DαX α .
The U(1)R generator J is normalised by
[J,Dα] = Dα , [J, Dα] = −Dα .
Grimm-Wess-Zumino superspace geometry
R. Grimm, J. Wess & B. Zumino (1978, 1979)Structure group: SL(2,C)
This superspace geometry is obtained from U(1) superspace by setting
Xα = 0
Conformal superspace D. Butter (2010)
Maximally supersymmetric spacetimes
For any (off-shell) supergravity theory in d dimensions, all maximallysupersymmetric spacetimes correspond to those supergravity backgroundswhich are characterised by the following properties:
all Grassmann-odd components of the superspace torsion andcurvature tensors vanish; and
all Grassmann-even components of the torsion and curvature tensorsare annihilated by the spinor derivatives.
SMK, J. Novak and G. Tartaglino-Mazzucchelli (2014)SMK (2015)
Maximally supersymmetric backgrounds in 4D N = 1 supergravity
Xα = 0 , Wαβγ = 0 ;
DαR = 0 −→ DAR = 0 ;
DαGββ = DαGββ −→ DAGββ = 0 .
Integrability condition
0 = Dα, DβGγγ = 2R(εγαGγβ + εγβGγα) =⇒ R Ga = 0 .
Isometries of curved superspace
Consider a background superspace (M4|4,D). A supervector field
ξ = ξBEB = ξbEb + ξβEβ + ξβEβ on (Md|δ,D) is called Killing if
δKDA = [K,DA] = 0 , K := ξB(z)DB +1
2K bc(z)Mbc + iτ(z)J ,
for some Lorentz (K bc) and R-symmetry (τ) parameters.All parameters ξβ , K bc , τ are determined in terms of ξb, in particular
ξB =(ξb, ξβ , ξβ
)=(ξb,− i
8Dγξγβ ,−
i
8Dγξγβ
).
Vector component ξββ obeys the equation (supersymmetric analogue of
the conformal Killing equation ∇aζb +∇bζa = 2d ηab∇cζ
c)
D(αξβ)β = 0 =⇒ (D2 + 2R)ξαα = 0 .
I. Buchbinder & SMK, Walk Through Superspace (1995)Let (M4|4,D) be a maximally supersymmetric background. ThenG b is covariantly constant, DαGββ = 0, hence
ξb = G b defines a Killing (super)vector field, hence R Ga = 0.
4D N = 1 maximally symmetric superspaces
Spacetimes supported by the superspaces M4|4T , M4|4
S and M4|4N are:
• R× S3;• AdS3 × S1 or its covering AdS3 × R;• a pp-wave spacetime isometric to the Nappi-Witten group.
M4|4T is the universal covering of superspace M4|4 = SU(2|1).
The bosonic body of M4|4 is U(2) = (S1 × S3)/Z2.The isometry group of M4|4 is SU(2|1)× U(2).
M4|4S is the universal covering of superspace M4|4 = SU(1, 1|1).
The bosonic body of M4|4 is U(1, 1) = (AdS3 × S1)/Z2.
The isometry group of M4|4 is SU(1, 1|1)× U(2).
M4|4N . The spacetime metric looks like
ds2 = eu(2du dv + δijdxidx j) , i , j = 1, 2 ,
with G aea ∝ ∂/∂v .C. Nappi & E. Witten (1993)
Maxwell Goldstone multiplet for partially broken SUSY
Maxwell Goldstone multiplet for partially broken SUSY
SMK & G. Tartaglino-Mazzucchelli (2016)
Let M4|4 any of the superspaces M4|4T , M4|4
S and M4|4N .
M4|4 allows the existence of covariantly constant spinors,
DAεβ = 0 .
Consider the N = 1 supersymmetric BI theory on M4|4 with action
SSBI[W , W ] =1
4
∫d4x d2θ E X + c.c. ,
where chiral superfield X is a unique solution of the constraint
X +1
4X D2X = W 2 .
Wα is the field strength of an Abelian vector multiplet
DαWα = 0 , DαWα = DαW α .
Maxwell Goldstone multiplet for partially broken SUSY
The action is invariant under a second supersymmetry given by
δεX = 2εβWβ , DAεβ = 0 .
Of course, this transformation should be induced by that of Wα. Thecorrect supersymmetry transformation of Wα proves to be
δεWα = εα +1
4εαD2X + iεβDαβX−ε
βGαβX .
It has the correct flat superspace limit, Gαα → 0,J. Bagger & A. Galperin (1997)
and respects the Bianchi identity
DαδεWα = DαδεW α .
Nonlinear self-duality and partial SUSY breaking
Supersymmetric Born-Infeld action in curved superspace
SSBI[W , W ] =1
4
∫d4x d2θ E X + c.c. ,
X +1
4X (D2 − 4R)X = W 2
is an example of U(1) duality invariant models for supersymmetricnonlinear electrodynamics.
Im(W ·W + M ·M
)= 0 , iMα := 2
δ
δW αS [W , W ] .
SMK & S. Theisen (2000)SMK & S. McCarthy (2003)
Nonlinear self-duality fixes a unique locally N = 1 supersymmetricextension of the BI action.Otherwise there exist a one-parameter family of models:
S. Cecotti and S. Ferrara (1987)
Nilpotent chiral superfield in N = 2 supergravity
Nilpotent chiral superfield in N = 2 supergravity
SMK & G. Tartaglino-Mazzucchelli (2016)
DiαZ = 0 ,(
Dij + 4S ij)Z −
(Dij + 4S ij
)Z = 4iG ij ,
Z2 = 0 ,
where Dij := Dα(iDj)α , Dij := D(i
αDαj).G ij is a linear multiplet constrained by G ijGij 6= 0.
SU(2)× U(1) superspace P. Howe (1982)
Diα,D
jβ = 4S ijMαβ + 2εijεαβY
γδMγδ + 2εijεαβWγδMγδ
+2εαβεijSklJkl + 4YαβJ
ij ,
Diα, D
βj = −2iδij (σ
c)αβDc + 4
(δijG
δβ + iG δβ ij
)Mαδ
+4(δijGαγ + iGαγ
ij
)M γβ + 8Gα
βJ i j
−4iδijGαβklJkl − 2
(δijGα
β + iGαβ i
j
)J .
Comment for superspace practitioners
SU(2) superspace R. Grimm (1980)is obtained from SU(2)× U(1) superspace by setting
G ijαα = 0 .
SU(2) superspace is suitable to describe general off-shell matter couplingsin N = 2 supergravity.
SMK, U. Lindstrom, M. Rocek & G. Tartaglino-Mazzucchelli (2008)
N = 2 linear multiplet is described in curved superspace by a real SU(2)triplet G ij subject to the covariant constraints
P. Breitenlohner & M. Sohnius (1980)
D(iαG
jk) = D(iαG
jk) = 0 .
These constraints are solved in terms of a chiral prepotential ΨP. Howe, K. Stelle & P. Townsend (1981)
G ij =1
4
(Dij + 4S ij
)Ψ +
1
4
(Dij + 4S ij
)Ψ , Di
αΨ = 0 ,
which is invariant under Abelian gauge transformations
δΛΨ = iΛ ,
with Λ a reduced chiral superfield,
DiαΛ = 0 ,
(Dij + 4S ij
)Λ−
(Dij + 4S ij
)Λ = 0 .
Field strength of every U(1) vector multiplet is a reduced chiral superfield.R. Grimm, M. Sohnius and J. Wess (1978)
Deformed reduced chiral superfield in N = 2 supergravity
Constraints
DiαZ = 0 ,
(Dij + 4S ij
)Z −
(Dij + 4S ij
)Z = 4iG ij
define a deformed reduced chiral superfield. It may be introduced asfollows.
Start with the massless (improved) tensor multipletB. de Wit, R. Philippe & A. Van Proeyen (1983)
STM = −∫
d4x d4θ E ΨW + c.c. , W := −G
8(Dij + 4Sij)
(G ij
G 2
)Introduce a massive deformation SMK (2009)
Smassive = −∫
d4x d4θ E
ΨW +1
4µ(µ+ ie)Ψ2
+ c.c. ,
with µ and e real parameters, µ 6= 0 (mass m =√µ2 + e2).
Deformed reduced chiral superfield in N = 2 supergravity
Consider a Stuckelberg-type extension of the massive model
Smassive = −∫
d4x d4θ E
ΨW +1
4µ(µ+ ie)(Ψ− iW )2
+ c.c. ,
where W is the field strength of a vector multiplet.Smassive is gauge invariant provided δΛW = Λ.
Chiral superfieldZ := W + iΨ
obeys the constraint(Dij + 4S ij
)Z −
(Dij + 4S ij
)Z = 4iG ij
Constraints
DiαZ = 0 ,(
Dij + 4S ij)Z −
(Dij + 4S ij
)Z = 4iG ij ,
Z2 = 0
imply that, for certain supergravity backgrounds, the degrees offreedom described by Z are in a one-to-one correspondence withthose of an Abelian N = 1 vector multiplet.
The specific feature of such N = 2 supergrvaity backgrounds is thatthey possess an N = 1 subspace of the full N = 2 superspace.
This property is not universal. In particular, there exist maximallyN = 2 supersymmetric backgrounds with no truncation to N = 1.
D. Butter, G. Inverso and I. Lodato (2015)
Action
S = − i
4
∫d4x d4θ E ΨZ + c.c.
describes partial N = 2→ N = 1 SUSY breaking on certainmaximally supersymmetric backgrounds (with Ψ being background).
Maximally N = 2 SUSY backgrounds & partial SUSYbreaking
Consider a maximally supersymmetric background M4|8 with the propertythat the chiral prepotential Ψ for G ij may be chosen such that
Complex linear multiplet
G ij+ :=
1
4
(Dij + 4S ij
)Ψ
is covariantly constant and null,
DAG ij+ = 0 , G ij
+G+ij = 0 .
Ψ may be chosen to be nilpotent,
Ψ2 = 0 .
G ij = G ij+ + G ij
− is covariantly constant, DAG ij = 0.Then, action
S = − i
4
∫d4x d4θ E ΨZ + c.c.
is N = 2 supersymmetric.
Example: The super-Poincare case
Example: The super-Poincare case
N = 2 Minkowski superspace, R4|8, is the simplest maximallysupersymmetric background.in R4|8 every constant real SU(2) triplet G ij is covariantly constant,
DAGij = 0 ,
where DA = (∂a,Diα, D
αi ) are the flat superspace covariant derivatives.
Let Ψ be a chiral prepotential for G ij , D iαΨ = 0. We represent
G ij = G ij+ + G ij
− , G ij+ =
1
4D ijΨ , G ij
− =1
4D ijΨ .
In R4|8 the constraints on Z turn into
D iαZ = 0 , D ijZ − D ij Z = 4iG ij , Z2 = 0 .
The N = 2 supersymmetric action becomes
S = − i
4
∫d4x d4θZΨ + c.c.
Example: The super-Poincare case
Grassmann coordinates of N = 2 Minkowski superspace
θαi , θjα , i , j = 1, 2
Without loss of generality we can choose
G ij+ = −iδi2δ
j2 ≡ qiqj , Ψ = iθα2 θα2
such that G ij+G+ij = 0 and Ψ2 = 0.
N = 2→ N = 1 superspace reductionGiven a superfield U(x , θi , θ
i ) on N = 2 Minkowski superspace, weintroduce its bar-projection
U| := U(x , θi , θi )|θ2=θ2=0 ,
which is a superfield on N = 1 Minkowski superspace withGrassmann coordinates θα = θα1 and θα = θ
1α
& spinor covariant derivatives Dα = D1α and Dα = Dα
1 .
Example: The super-Poincare case
N = 1 components of Z:
X := Z| , Wα := − i
2D2αZ| .
The original N = 2 constraints, D iαZ = 0 & D ijZ − D ij Z = 4iG ij , imply
DαX = 0 , DαWα = 0 , DαWα = DαWα .
Wα is the field strength of U(1) vector multiplet.Taking into account the nilpotent constraint Z2 = 0 gives
X +1
4XD2X = W 2 , W 2 := W αWα .
The original N = 2 action, S = − i4
∫d4x d4θZΨ + c.c., turns into
S =1
4
∫d4x d2θX +
1
4
∫d4x d2θ X .
Bagger-Galperin solution
The Bagger-Galperin solution to the constraint
X +1
4XD2X = W 2
is as follows:
X = W 2 − 1
2D2 W 2W 2(
1 + 12A +
√1 + A + 1
4B2) ,
A =1
2
(D2W 2 + D2W 2
), B =
1
2
(D2W 2 − D2W 2
).
