Nonlinear \( \mathcal{N}=2 \) global supersymmetry
Received: April
Published for SISSA by Springer
Open Access 0 3 4
c The Authors. 0 3 4
0 Sidlerstrasse 5 , CH3012 Bern , Switzerland
1 LPTHE, UMR CNRS 7589, Sorbonne Universit ́es
2 Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics
3 We focus on the Goldstone de
4 UPMC Paris 6 , 75005 Paris , France
We study the partial breaking of N = 2 global supersymmetry, using a novel formalism that allows for the offshell nonlinear realization of the broken supersymmetry, extending previous results scattered in the literature. grees of freedom of a massive N = 1 gravitino multiplet which are described by deformed N = 2 vector and singletensor superfields satisfying nilpotent constraints. We derive the corresponding actions and study the interactions of the superfields involved, as well as constraints describing incomplete N = 2 matter multiplets of nonlinear supersymmetry (vectors and singletensors).
Extended Supersymmetry; Supersymmetry Breaking

2 global supersymmetry
Singletensor multiplet formulation
Alternative proof
The vacuum
Dual hypermultiplet formulation
Several singletensor multiplets
Nonlinear deformations
Deformations of the Maxwell superfield
Deformations of the singletensor superfield
Constrained multiplets
The infinitemass limit
Solutions of the constraints
The “long” superMaxwell superfield
The chiralantichiral N = 2 superfield
The long and short superMaxwell superfields
Long superfield and nonlinear deformations
1 Introduction 2 Partial supersymmetry breaking with one hypermultiplet 3
6 Interactions 6.1 The ChernSimons interaction 6.2
Constrained matter multiplets
The ChernSimons interaction with deformed Maxwell multiplet
The ChernSimons interaction with deformed singletensor multiplet
The goldstino in the Maxwell multiplet
The goldstino in the singletensor multiplet
A Conventions and some useful identities
More on the Maxwell supermultiplet
More on Im V
The spontaneous breaking of global symmetries is described at low energies by a
nonlinIn the case of supersymmetry, the Goldstone modes are fermions, the goldstini, and the
symmetries, it can be obtained (up to field redefinitions) by a chiral superfield X satisfying
of the goldstino bilinear [2–5]:
X = − 2F
and one obtains (onshell) the VolkovAkulov action [2, 6].
Besides the use of nonlinear supersymmetry as an effective lowenergy theory at
energies below the sgoldstino mass, it can also be realized exactly in particular vacua of type
I string theory, when Dbranes are combined with antiorientifold planes that break the
linear supersymmetries preserved by the Dbranes, while they preserve the other half that
are realized nonlinearly. In such vacua of “brane supersymmetry breaking”,
superpartners of brane excitations do not exist, and supersymmetry is nonlinearly realized with the
presence of a massless goldstino in the open string spectrum [7–11].
2, broken at two different scales, is a challenging and not straightforward problem. An
interesting case is N
supersymmetry, which can be either a vector or a chiral multiplet. In fact, both cases have
to be studied, since they constitute the Goldstone degrees of freedom of a massive spin3/2
multiplet. Indeed, a massless spin3/2 multiplet contains a gravitino and a graviphoton,
while a massive one contains, in addition, a spin1 and a (Majorana) spinor, so that the
Goldstone modes are a vector, two 2component spinors and two scalars [12].
When the second and nonlinear supersymmetry is taken into account, the above two
a Maxwell multiplet and a hypermultiplet. The latter comes with an extra complication
since it has no offshell formulation in the standard N
= 2 superspace.
the presence of bosonic shift symmetries associated with the wouldbe Goldstone bosons
providing the longitudinal components of the spin1 fields, implies that the chiral multiplet
can be dualized to a linear multiplet having an offshell description when promoted to a
metry [13], extending known results in the literature on Maxwell multiplets [13–17] and
supersymmetry is by a (constant) deformation of the supersymmetry transformations of
the fermions that cannot be absorbed in expectation values of the auxiliary fields, unlike
particular relations, guaranteeing the existence of one goldstino associated with a linear
combination of the two supersymmetries. The goldstino superfield of one nonlinear
supersymmetry can then be obtained by imposing a nilpotent (double chiral) constraint, in
analogy with X2 = 0 of N = 1.
The outline of this paper is the following. In section 2, we present a model of
spontaneous partial breaking of N
= 2 → N
admits a special superpotential that allows for partial supersymmetry breaking, in
analogy with the magnetic FayetIliopoulos (FI) term in the Maxwell multiplet model of [13].
field is chiral under both supersymmetries (CC), while the singletensor superfield is chiral
under the first and antichiral under the second (CA). In section 3, we discuss nonlinear
section 4, we consider the infinitemass limit that freezes half of the degrees of freedom,
and derive the constrained multiplets and the corresponding nilpotent constraints. We
then give the solutions of the constraints (offshell) and derive the generalizations of the
goldstino VolkovAkulov action in the presence of a linear supersymmetry, in addition to
the nonlinear one. These are the supersymmetric DiracBornInfeld (DBI) action and a
similar action for the linear multiplet, in agreement with previous results. We then turn to
for the Maxwell and singletensor multiplets with opposite relative chiralities compared
to the “short” ones, namely CA for the Maxwell and CC for the singletensor, so that
one can write a ChernSimons type of interaction that we discuss in section 6. This
interaction leads to a superBroutEnglertHiggs mechansim without gravity, in which the
linear multiplet is absorbed by the vector which becomes massive [19]. In section 6, we
nonlinear supersymmetry (vectors or singletensors), half of the components of which are
projected out. Finally, section 7 contains concluding remarks and open problems, while
there are three appendices with our conventions (appendix A) and the technical details of
the Maxwell multiplet (appendices B and C).
hatted superfields Wc, Zb . . . have 16B + 16F fields. They are chiral with respect to the first
three real scalar fields and two Weyl (or massless Majorana) spinors. In the same manner
that an antisymmetric tensor is dual to a pseudoscalar with axionic shift symmetry, a
singletensor multiplet is equivalent to a hypermultiplet with shift symmetry. In both
cases, the symmetry implies masslessness. In analogy with the YangMills or Maxwell
multiplet but in contrast with the hypermultiplet, the singletensor multiplet admits an
offshell formulation.
can be viewed as the supersymmetrization either of the gauge invariant threeform field
In this section we show the existence of partial supersymmetry breaking in a large class of
tional) isometry allowing for a description in terms of a dual singletensor multiplet which
admits, like the Maxwell multiplet, a fully offshell formulation. We use this formulation to
obtain these theories, dualize back to the hypermultiplet formulation and then display the
strong similarity between partial breaking with a Maxwell (namely the APT model [13])
and partial breaking with a singletensor multiplet.
gauge symmetry
DDL = 0 is solved by
where H is any real function solving the threedimensional Laplace equation
Lkin. =
∂2H
= 0.
Dα˙ L = iϕα˙ − (θσµ )α˙ (vµ + i∂µ C) − θθ(∂µ ϕσµ )α˙ ,
vµ =
(in chiral coordinates), where the real scalar C is the lowest component of L. Note also
that the superpartner of L (under the second supersymmetry) is
Φ = z + √2θψ − θθf .
Singletensor multiplet formulation
symmetry (2.1) is then
with Vb1 and Vb2 real: the gauge transformation of the singletensor multiplet in the
degenerated by Vb2.
It is not a derivative unless W (Φ) ∼ Φ. Since2
the variation can also be written as3
3We usually omit derivatives when comparing lagrangian terms.
δ∗ L = δn∗l L − √ (ηDΦ + ηDΦ),
d2θ WΦ ηDL + h.c. = −Mf2 δ∗ d2θ W (Φ) + h.c.
Lnl depends on two complex numbers, the deformation parameter Mf2 and the quantity
2 Mf2 (θη + θη),
2 Mf2 ηα˙ ,
Consider now the function
H(L, Φ, Φ) = i −L2[WΦ − W Φ] + ΦW − ΦW ,
which is obviously a solution of the Laplace equation, while the action corresponding to
L = i d2θd2θ h−L2[WΦ − W Φ] + ΦW − ΦW
To break spontaneously the second supersymmetry, we first add the generic
superpoLnl =
= i d2θd2θ h−L2(WΦ − W Φ) + ΦW − ΦW
as well as under the nonlinearly deformed second supersymmetry transformations
Alternative proof
Let us consider the N
the partial breaking, we deform the second supersymmetry transformations of the
singlea goldstino; the transformations take then the form (2.18). The deformation induces a new
term in the variation of the lagrangian under the second supersymmetry:
2 Mf2
components must transform as derivatives under the first, unbroken supersymmetry. This
is the case if the highest component of HL is zero or a derivative,
whose solution is
H =
since terms linear in L do not contribute to the integral
of (2.24), it becomes (since terms proportional to L0 do not contribute):
Now let us consider again the derformation (2.20) of the lagrangian. With the use
Consequently, the deformed lagrangian
= M√f2 Z
Ldef,kin. =
is invariant under the first (linearlyrealized) supersymmetry as well as under the second
nonlinearlyrealized one. It is also obvious that the lagrangians corresponding to (2.15)
The vacuum
superfield with the use of a prepotential function G(Z). Let us define4
Z = Φ + √2i θeDL − 41 θeθeh4iMf2 + DD Φi .
= 2
We then obtain
2 Z
2 GΦΦ (DL)(DL) − 4 GΦ DD Φ − i Mf2 GΦ + h.c. (2.28)
Partial supersymmetry breaking is achieved if theory (2.17) has a vacuum state
invariant under the first (linear) supersymmetry. We then analyze the scalar potential, which,
field lagrangian is5
− 2
It generates the scalar potential
W ΦΦ[f ψψ − f ϕϕ] = −V + Lferm..
Lferm. = − 2 f
− 2 e
ϕϕ + h.c.
Three situations can occur.
and Mf2 6= 0 with
V =
The term depending on L in theory (2.17) does not contribute to the potential. Fermion
mass terms read
The analogy with partial supersymmetry breaking in a N
= 2 Maxwell multiplet
and X, with deformed supersymmetry variations
4 ηαDD X + i(σµ η)α ∂µ X ,
the invariant lagrangian is written as
LMax. =
2 FXX W W − 4 FX DD X + m2X − iM 2FX
+ h.c. + LF.I.,
2 Mf2ηα˙ + √2i
4 ηα˙ DDΦ − i(ησµ )α˙ ∂µ Φ ,
namely the absence of auxiliary fields in L as well as the consequent inexistence of a
Dual hypermultiplet formulation
The duality transformation from the singletensor to the hypermultiplet formulation is a
Lkin. =
d2θd2θ hH(V, Φ, Φ) − (S + S)V i.
hf i = 0.
HV = S + S
(Legendre transformation)
In our case, the Legendre transformation is simply
KV = 0
=⇒
S + S = HV = −2iV (WΦ − W Φ)
HS = 0
=⇒
KS = −V .
are consistent using the field equations for L and S,
HL = S + S ,
KS = −L ,
DDKS = 0 ,
as integrability conditions.
K¨ahler potential K generates a hyperKa¨hler metric, the first term certainly has (onshell)
The dual hypermultiplet theory reads
− 2
The Dterm in the first expression is the K¨ahler potential of a hyperKa¨hler space,
fS of S does not contribute to the potential. Its field equation
(WΦ − W Φ)fS − (S + S)WΦΦfΦ = 0
relations (2.42) and (2.43) with L replacing V ,
derivative) under the variations
where KS = ∂∂S K = − 2 WΦ−W Φ
i S+S . These variations are simply obtained by inserting the
presence of the superpotential Mf2W is then
second duality relation (2.46) in the singletensor offshell variations (2.3). The field
equa
2 Mf2 ηα˙ ,
in agreement with eqs. (2.18) and (2.46).
2 Gab(DLa)(DLb) − 4 GaDDΦa − i(Mfa)2 Ga + me 2a Φa + h.c. ,
Ga =
Gab =
is invariant under the nonlinear second supersymmetry variations
i
δ∗La = √2(Mfa)2(θη + θη) − √ (ηDΦ + ηDΦ),
fields f a:
− ihGabi(Mfb)2 + me2a = 0.
In this vacuum, the kinetic metric 2hRe Gabi must be invertible and the mass matrix of the
The extension to a theory with several singletensor multiplets is straigthforward. Consider
the deformed N = 2 chiral superfields
The lagrangian
L =
Mab = − 2 hRe Ga−c1ihGbcdi(Mfd)2,
controlled by the third derivatives of G.
Nonlinear deformations
2 singletensor and Maxwell multiplets to engineer theories with partial supersymmetry
breaking. As illustrated by eq. (2.27), a nonlinear deformation of the singletensor multiplet
can be introduced as a spurious constant component inserted in a N
= 2 superfield.
In this section, we study general nonlinear deformations of these multiplets, using their
Deformations of the Maxwell superfield
1
W(y, θ, θe) = X + √2i θeW − 4 θeθeDD X,
Wα = −iλα + θαD − 2i (σµ σνθ)αFµν − θθ (σµ ∂µ λ)α,
X = x + √2 θκ − θθ F,
1 DD X = F + √2i θσµ ∂µ κ + θθ ✷x.
superfield W. Defining fermion doublets
and the vector Y~ is in general a complex SU(2)R triplet. But in W, the auxiliary fields
correspond to
Y11 = F,
Y22 = F ,
Y12 = − √
Y~ = Im F, Re F, √
and the SU(2)Rinvariant “reality” condition
≡ Yi∗j = ǫikǫjlYkl
is verified: a complex value of Y~ violating this condition cannot be seen as a background
value of N = 1 superfields X or Wα.
expected for goldstino fermions, should be introduced with
δκα = √2(A2ǫα + Γηα) + . . .
δλα = √2(B2ηα + Γǫα) + . . .
breaking is in any case incompatible with the reality condition (3.7): the auxiliary fields
F and D are not able to induce partial breaking with their background values; in other
words, the deformation parameters cannot be absorbed in the background values of the
While a chiralchiral (CC) superfield is relevant to study deformations of the Maxwell
multiplet, the singletensor multiplet is conveniently described using a chiralantichiral
(CA) N = 2 superfield Z,
with the expansion
in the appropriate coordinates (y, θ, θe), Dα˙ yeµ = Deα yeµ = 0. A particular deformation
e
with partial supersymmetry breaking has been earlier described [eq. (2.28)] and we wish
to generalize it. Since fermion fields are in the components7
of Z, the deformation parameters will add
vector and the deformations are encoded in two complex numbers Ae2 and Be2 only. The
nonlinear variations of the spinors are
and generic values of Ae2 and Be2 break both supersymmetries. Partial breaking occurs if
generate partial breaking on its own.
cludes a tensor with gauge symmetry. A generic lagrangian generated by the CA superfield
L =
d2θeG(Z) + me2 Φ + h.c. = Llin. + Lnl .
where Lnl includes all terms generated by the deformations with parameters Ae2 and Be2.
In the function G(Z), a term linear in Z is irrelevant (it contributes with a derivative) and
the component expansion of the lagrangian depends on the second and higher derivatives
1 G′′′(z)[f ψψ + f ϕϕ] − me2 f + h.c.
Lnl = −G′′(z) Be f + Ae2 f + Ae2Be2i
h 2
− 21 G′′′(z)hBe2 ψψ + Ae2 ϕϕi + h.c.
and the scalar potential and the fermion bilinear terms read respectively
V (z, z) =
Lferm. =
2 Re G′′ Be2 G′′ + Ae2 G′′ + me
2 Re G′′ (Be G′′ + Ae2 G′′ + me 2) − Be G′′′ + h.c.
G′′′ 2 2
G′′′ 2 2
2 Re G′′ (Be G′′ + Ae2 G′′ + me2) − Ae G′′′ + h.c.
The kinetic metric of the multiplet is 2 Re G′′(z). Notice that these formulas do not depend
on the real scalar C in L, which always leads to a flat direction.
2hRe G′′i MΦMΦ zz − 2 MΦψψ − 2 MΦ ψψ ,
2hRe G′′i MΦMΦ zz − 2 MΦϕϕ − 2 MΦ ϕϕ ,
2 [Re G′′(z)] f = G′′(z)Be + G′′(z)Ae + me2
− 2
1 G′′′(z)ϕϕ − 2
signs) breaks both supersymmetries, assuming that V has a ground state hzi.
Constrained multiplets
When supersymmetry is partially broken in the Maxwell or singletensor (hypermultiplet)
the field equation of this superfield is a constraint which allows for the elimination of
the massive chiral superfield. One is then left with a nonlinear realization of N
= 2
in terms of which the lagrangian is
L =
1 Z
1 Z
+ h.c. + LF.I.
+ h.c. + LF.I.
deformed superfield
W = X + √2i θeW + θeθe B2
− 4
DD X ,
− 4 hFXXX i χχ − 4 hF XXX i χχ
MX =
2 RehFXX i
Since the auxiliary fields f and D vanish in the ground state, the mass terms of the fermion
and, since the kinetic metric is RehFXX i, the mass of X is
the metric of the scalar manifold), thus disproving the claim made in [25]. Expanding the
field equation of X and retaining only the term in hFXXX i leads to the constraint
and the constraint (4.4) is then equivalent to [16]
W W − 2
XDDX + 2 B2 X = 0 ,
2 = 0 .
We now turn to the partial breaking in a singletensor theory. Again, the two options
chiralantichiral superfield
which induces the nonlinear deformation
L =
− 4
2 GΦΦ(Φ)(DL)(DL) + me2Φ + h.c. ,
The lowest component is the field equation for the auxiliary field f ,
Gzz(z) (f − Be ) = me 2
kinetic metric normalization 2 Re Gzz(hzi).
can be sent to infinity keeping Gzz(hzi) finite as in the Maxwell case. In this limit,
and the field equation becomes8
which does not depend on the function G and which was first given in [18]. This equation
=⇒
n = 0 (n ≥ 2).
The second supersymmetry variation of the constraint (4.9) is
The invariance of the constraint then follows from the results (4.10). Moreover, since
= −2 2 ∂µ (ησµ DLΦ) .
2 = 0 .
Solutions of the constraints
The solution of (4.4), and thus of (4.5), was first given in [14]. In our conventions, it is
The bosonic part of lagrangian (4.2) then takes the form
X = − 2B2 1 − D
4B4 + a + 4B4
a =
(D2W 2 + D2W 2) , b =
− D2W 2) .
Lbos = 8m2B2 1 −
1 − B14 (−Fµν F µν + 2D2) − 4B1 8 (Fµν F˜µν )2
The equation of motion for D is then
and, substituting back into (4.16), one arrives at [14, 16]
Lbos = 8m2B2 1 −
= 8m2B2 1 −
D = 0 ,
It is also possible to add the FI term
to the lagrangian (4.16). Solving the equation of motion for D then gives
− B4
D2 = − ξ2 + 2 · 162m4
and substituting back to (4.16), we find that the latter takes the form
Lbos = 8m2B2 1 −
= 8m2B2
1 −
1 − B4 (−Fµν F µν + 2D2) − 4B8 (Fµν F˜µν )2
which means that the addition of the FI term only changes the prefactor of the BornInfeld
lagrangian included in L.
or equivalently of (4.13). In our conventions, it is
(DL)2(DL)2
where we have assumed that Be is real for simplicity and
a =
Due to the constraint (4.13), only if G has linear dependence on Z will it contribute
to (3.15). However,
e − 4
+ h.c. = derivative .
Consequently, (3.15) takes the form
L = me2
= − e
4Be4 + a + 4Be4
+ h.c. (4.25)
The “long” superMaxwell superfield
In section 6 we will construct supersymmetric interactions of deformed or constrained
singletensor and Maxwell supermultiplets. We will find it useful to describe the Maxwell
multiplet in terms of a chiralantichiral superfield, with 16B + 16F components, as an
alternative to the 8B +8F chiralchiral superfield (3.1). In the present and technical section,
To begin with, both types of superfields exist for the singletensor multiplet. In
particular, the latter can be described either by the “short” (8B + 8F ) chiralantichiral (CA)
superfield (3.11),
(and its AC conjugate), or by a “long” chiralchiral (CC) superfield [19]
Zb = Y + √
are related by9
= me2Be2 1 −
1 − B4 6 Hµνρ Hµνρ + ∂µ C∂µ C
− 9Be8 (ǫµνρσ Hνρσ∂µ C)2
Z = − 2 De De Zb +
Moreover, using (2.10), we find
Lbos = me2Be2 1 −
(DL)2bos = θ2 (υµ υµ + 2iυµ ∂µ C − ∂µ C∂µ C) ,
eabos = 4 υ2 − (∂C)2 ,
− B8
and the real linear superfield L is
There is a gauge invariance acting on the long CC superfield. According to eqs. (5.1)
Hence, Z is invariant under
9Identities in Apprendix A may help.
Ωα˙ = ωα˙ + (θσµ )α˙ Vµ − θθ λα˙ .
L(x, θ, θ) = Φ(x, θ, θ) − θω − θσµ θ Vµ + θθθλ − 2
θθ θσµ ∂µ ω + θθθθ ∂µ Vµ
Upon defining the chiralchiral superfield
Wc = U + √
W = − 2 DeDe Wc +
W = X + √2i θeαDα˙ DαΩα˙ +
where W is a Maxwell (chiralchiral) superfield (3.1). This gauge invariance eliminates
superfield for the superMaxwell theory.
Wc = U + √
V = 2(L + L)
other words,
instead of V being simply a real superfield. This new condition follows from
with a real linear L. In other words, W is invariant under
1
Y = U + √2i θeDL − θeθe 4 DD U .
Eq. (5.1) indicates that this gauge variation is induced by a singletensor supermultiplet in
a “short” chiralantichiral superfield.
The long and short superMaxwell superfields
To summarize, to describe the singletensor and the Maxwell multiplet, we have obtained
and one short (8B + 8F ) superfield:
Long, 16B + 16F
Short, 8B + 8F
Gauge variation, 8B + 8F
of L. This is discussed in appendix B.
Counting offshell degrees of freedom in the “long” Maxwell multiplet is interesting. Firstly,
X and U include 8B + 8F fields while the complex linear L has 12B + 12F components.10
The variation (5.16) is not the gauge transformation of the superMaxwell theory: it
which is a symmetry of Wc.11 A comparison of 2(L + L) with the standard expansion of the
are respectively
Aµ = −4 Re Vµ ,
D = −4 ∂µ Im Vµ .
Replacing the scalar D by the divergence of a vector field has nontrivial consequences which
integration constant appearing when solving the field equation of Im Vµ and a welldefined
procedure for the elimination of Im Vµ shows that the theories formulated with either D
or Im Vµ are physically equivalent.
11See appendix B.
According to relation (5.12), the nonlinear deformation Wnl can be transferred to a
deforterm would be
Wcnl = − 2
This is the case if the deformation can be viewed as a background value of the auxiliary F
in X, which never leads to partial breaking. A similar argument holds for the singletensor
superfield with relation (5.3). Then, to consider a general deformation and in particular
if the interest is in partial supersymmetry breaking, the deformed short version of the
superfields must be used. Since these short superfields have different chiralities, writing an
interaction of two deformed supermultiplets is problematic.
The ChernSimons interaction
to its counterpart with linear N = 2.
Maxwell multiplets respectively, the ChernSimons interaction with (real) coupling g can
be written as a N = 1 Dterm [16, 19]:
LCS = −g
It is invariant under the second supersymmetry variations (2.3) and (B.1) and it is also
and some partial integrations:
LCS = g
X =
The expressions (6.1) and (6.2) differ by a derivative term. The chiral form can be extended
the Maxwell and singletensor multiplets respectively [19]:12
LCS = ig
2 Z
integral). This expression is also invariant under the gauge transformation (5.6) of Zb,
since, for any pair of (short) Maxwell multiplets W1 and W2,
2 Z
are derivative terms.
Finally, one can also write the ChernSimons lagrangian using the chiralantichiral
superfields Z (short) and Wc (long) for the singletensor and the Maxwell multiplet
respecLCS = ig
2 Z
This can be verified either by direct calculation or by using relation (5.12) and partial
up to a derivative term under the gauge transformation (B.13) of Wc, since, for any pair of
(short) singletensor multiplets Z1, Z2,
2 Z
are derivative terms.
LCS = g
In components, using expansions (2.10) and (5.10), we find that (under a spacetime
− g ∂µ ϕσµ ω − g ωσµ ∂µ ϕ ,
The ChernSimons interaction with deformed Maxwell multiplet
The nonlinearlydeformed Maxwell multiplet is described by the CC superfield W, including
the deformation terms (3.8). This leads to the ChernSimons interaction
Lnl = ig
2 Z
= LCS + ig
12See eqs. (3.1) and (5.2).
13Eqs. (3.11) and (5.11).
Lnl = g
tent. One can get around this problem by using l > 1 deformed Maxwell multiplets (namely
one “long” singletensor and at least two “short” and deformed Maxwell multiplets), as
then the relevant equation of motion would take the form of a tadpolelike condition
gaBa2 = 0 ,
a = 1, . . . , l ,
where ga would be the coupling of each ChernSimons interaction. This is in agreement
with the claim made in [26] and [27], namely that one cannot couple hypermultiplets to a
single Maxwell multiplet in a theory with partial breaking induced by the latter.
The ChernSimons interaction (6.8) can be combined with the kinetic lagrangian
Lkin. =
2 Z
for the two multiplets, as well as with an FI contribution
LFI = ξ d2θd2θ V2 +
The theory depends then on a function H solving the Laplace equation and on an
arbitrary holomorphic function F . Imposing the constraint W
2 = 0 (where W is deformed)
eliminates X, which becomes a function X(W W ) of W W and its derivatives. Moreover,
due to the constraint, the lagrangian no longer depends on F and it reduces to
has been analyzed in [19].
The ChernSimons interaction with deformed singletensor multiplet
In the analogous procedure for the nonlinear singletensor multiplet, the CA
superfield (3.11) with deformation (3.13) is coupled to the long Maxwell CA superfield (5.11):
Lnl = ig
2 Z
= LCS + ig
Lnl = g
L = Lnl +
2 Z
2 Z
where Z is deformed and we have added an FI term for V2. Upon imposing the
constraint (4.13), G does not contribute to (6.15), since
2 Z
2
d2θ D Φ + h.c. = deriv. term
1 Z
and the bosonic part of (6.15) becomes
Lbos =
d2θeF (W)bos + h.c. − 2ξ ∂µ Im Vµ
2 2
δ∗ iBe U = √2i Be ηD L ,
ever, the equation of motion of U is inconsistent as that of Y of the previous subsection
— this problem can be solved by coupling the “long” Maxwell multiplet(s) to at least two
“short” and deformed singletensor multiplets.15
The complete theory has then lagrangian
− 26 ǫµνρσ HνρσAµ + C ∂µ Im Vµ − Be2 Im FU
· 1 −
1 − B4 6 Hµνρ Hµνρ + ∂µ C∂µ C
− 9Be8 (ǫµνρσ Hνρσ∂µ C)2
′′ is a real superfield.
15Note that there is no reason to identify the imaginary part of the auxiliary field of U with a
fourform field as was done for Y in [19]. In particular, the variation of Y under the gauge transformation
whose solution is
14See appendix B.
analogue, the BornInfeld lagrangian, does in ref. [19].
where Be has been assumed to be real and FU is the auxiliary field of U . Notice that
the lagrangian (4.27) has acquired a fielddependent coefficient (g Re x + 2 me2)Be2 as its
the equation of motion for the auxiliary field Im Vµ is
∂µ 16 Re Fxx ∂ν Im Vν + 2g C
= 0 ,
16 Re Fxx ∂ν Im Vν + 2g C = −λ ,
make the identification
In this vacuum, x corresponds to a flat direction of the potential and is massless. The
chiral multiplet X; the ChernSimons coupling results in the vector multiplet W absorbing
the goldstino multiplet, while X remains massless. Consequently, we observe a mechanism
analogous to the superBroutEnglertHiggs effect without gravity [19], which is induced
by the ChernSimons coupling of the previous subsection (6.1.1).
Constrained matter multiplets
that couples a Maxwell to a singletensor multiplet, where one of the two contains the
goldstino. In both cases, upon imposing a nilpotent constraint on the goldstino multiplet,
the ChernSimons interaction generates a superBroutEnglertHiggs phenomenon without
massless chiral multiplet remains in the spectrum.
Here, we discuss generalisations of the nilpotent constraint in order to describe, besides
the goldstino, incomplete matter multiplets of nonlinear supersymmetry in which half
of the degrees of freedom are integrated out of the spectrum, giving rise to constraints.
The scalar potential of the theory is then
whose supersymmetric vacuum is at
V =
< C >=
2
canonically normalized mass MC,can that C aquires is then
4 Re Fxx 2g Re x + 4m2
2 = 0.
with the goldstino being part of either a nilpotent (deformed) Maxwell multiplet W with
The goldstino in the Maxwell multiplet
Consider the case in which the goldstino is in a deformed Maxwell multiplet W0, given
W0 = X0 + √2i θeW0 + θeθe B2
− 4
X0 = −2
− DDX0
W0W1 = 0 ,
We now use (6.28) and the identity
to solve the second of equations (6.31), which yields
where h is a chiral superfield. This expression verifies the first eq. (6.31) for all h and the
third eq. (6.31) if
where W1 is an undeformed (and short) Maxwell multiplet given by (3.1):
The constraint (6.29) then yields the following set of equations
X0X1 = 0 ,
− 4
DD(X0X1 + X1X0) + W0W1 = 0 .
(W0W1)W0α = − 2 (W0W0)W1α
X1 = −4
− DDX0
+ h W0W0 ,
h = −2
− DDX0)2
X1 = −4
− DDX0
− 2
− DDX0)2 W0W0 .
One may further use the solution (4.14) for X0 and solve (6.35) to obtain X1 as a function
of W0, W1 and their derivatives; the constraint (6.29) eliminates X1.
dabcWbWc = 0 ;
a, b, c = 1, . . . , l
introduced in [28, 29] to obtain coupled DBI (DiracBornInfeld) actions. In eqs. (6.36), all
Wa are in general deformed with different deformation parameters Ba and the constants
W0Zb = 0 ,
where Zb is a “long”16 singletensor multiplet given by (5.2). Equation (6.37) then leads to
X0Y = 0 ,
− 2
DD(X0Y + Y X0) − i W0χ = 0 ,
which, following the same steps as before, yield
W0Wg = 0 ,
Y = 4i
− DDX0
− 2
− DDX0)2 W0W0 ,
Zb + Wg ,
One can also check if the expression (6.39) is covariant under the gauge variation (5.6)
Under (6.41), the expression (6.39) becomes
Xg = −4
− DDX0
− 2
− DDX0)2 W0W0 ,
which, as was previously shown, is actually the consequence of
leads to an overconstrained system of equations.
that is the variation of (6.37) under (6.40). The expression (6.39) is thus invariant only
under the reduced gauge transformations (6.40) subject to the constraint (6.43). These are
not sufficient to eliminate all unphysical components of Zb.
In the physicallyrelevant linear superfield L however, Wg disappears:
− DDX0)
Z0, given by
which satisfies (4.13)
or equivalently eq. (4.10) [18]:
since Wg verifies the Bianchi identity.
The goldstino in the singletensor multiplet
Now let us consider the case in which the goldstino is in a deformed singletensor multiplet
Z02 = 0 ,
Z0Z1 = 0 ,
where Z1 is an undeformed (and short) singletensor multiplet given by (3.11)
Following the same steps as before, as well as the identity
(DL0DL1)Dα˙ L0 = − 2
DL0DL1
− 2
DL0DL0 ,
deabcZbZc = 0 ;
a, b, c = 1, . . . , l ,
in analogy with the system (6.36), where deabc are totally symmetric constants, in order to
obtain a coupled action of nonlinear (deformed) singletensor multiplets.
Finally, we consider the constraint
Z0Wc = 0 ,
as before, we obtain
U = 4i
− 2
2iX + DDU
DL0DL0 ,
which eliminates U . Using the same reasoning as before, one can show that the
solution (6.54) is invariant under the reduced gauge variation (5.16)
Following the same procedure as for the solution of the constraint (6.37), one can use the
terms of the N = 1 chiral superfield X:17
Wc + Zg ,
Z0Zg = 0 .
This result defines L up to the addition of an arbitrary chiral field: as expected, the
constraint equation (6.57) is invariant under the Maxwell gauge transformation
gauge ambiguity (6.55).
17Since L is real linear, SL is complex linear for any chiral S.
vanishing expectation value for the Fcomponent of the goldstino superfield arising a priori
through a nontrivial deformation that cannot be obtained by an expectation value of the
auxiliary fields.
supersymmetry (vectors and singletensors that have offshell descriptions), as well as
generalisations of the nilpotent constraints describing incomplete matter multiplets. The
interactions are of the ChernSimons type and describe a superBroutEnglertHiggs
phemultiplet. The constraints describe, in the case of a goldstino in a Maxwell multiplet,
a goldstino in a linear multiplet, the constraints describe either incomplete singletensors
It would be interesting to study the interactions of the Goldstone degrees of freedom of
It is not clear whether our results are sufficient to provide a description of such a system.
Another open but related question is the coupling to supergravity realising partial breaking
of N = 2 supersymmetry and its rigid limit.
Acknowledgments
J.P. D. wishes to thank the LPTHE at UPMC, Paris and CNRS for hospitality and
support. The work of J.P. D. has been supported by the Germaine de Sta¨el francoswiss
bilateral program (project no. 201517). C.M. would like to thank the Albert Einstein
Center for Fundamental Physics of the Institute for Theoretical Physics of the University
of Bern for very warm hospitality and for financially supporting her stay there.
Conventions and some useful identities
The notation [. . .] in (2.1) is used for antisymmetrization with weight one. Specifically,
6 ∂µ Bνρ ± 5 permutations .
{Dα, Dα˙ } = −2i(σµ )αα˙ ∂µ :
As a consequence,
[Dα, DD] = −4i(σµ D)α∂µ ,
The Maxwell fieldstrength chiral superfields are defined as
where V is a real superfield. In addition,
DD DD Y = −✷Y.
DD DαL = −2 Dα˙ DαDα˙ L = 4i(σµ D)α∂µ L ,
where L is a complex linear superfield.
More on the Maxwell supermultiplet
superfields V1 and V2 with second supersymmetry variations
δ∗V2 = √2i(ηD + ηD)V1.
δ∗Λc = √2i ηDΛℓ. (B.3)
X =
Variations (B.1) imply:
4 ηαDD X + i(σµ η)α ∂µ X ,
4 ηα˙ DD X − i(ησµ )α˙ ∂µ X .
These are the second supersymmetry variations of the components of the “short”
chiralchiral superfield (3.1):
To go to the “long” Maxwell multiplet, one introduces the complex linear L with
and variations (B.1) suggest to write
V2 = 2(L + L),
close18 and the number of offshell fields is not a multiple of 8B + 8F .
To find the complete multiplet, we rely upon the chiralantichiral superfield written in
its two forms (5.7) and (5.11):
Wc = U + √
Wc = U + √
2 θeΩ − θeθe 2i X + 41 DD U ,
2 θeD L − θeθe 2i X + 41 DD U .
Since the first expression is a chiralantichiral superfield with 16B + 16F components,19 the
second supersymmetry variations
18See below.
supplementary components which are actually invisible in Wc: the gauge variation (B.6)
a supplementary condition on the chiral X. This is where
X =
helps by firstly adding 4B + 4F fields to reach 24B + 24F with U and L and secondly by
turning the second supersymmetry variations (B.9) into
δ∗L = √ ηD V1 + √ (ηD U + ηD U ) ,
δ∗L = √ ηD V1 − √ (ηD U + ηD U ) ,
24B + 24F fields. Since
δ∗ V2 = √2i (ηD + ηD)V1 ,
the 16B + 16F multiplet with superfields V1 and V2 is included in the long representation.
The long multiplet has two gauge variations generated by two independent singletensor
which is a short chiralantichiral multiplet similar to eq. (5.1). This is the gauge
trans[δ1∗, δ2∗] L = 2i (η1σµ η2 − η2σµ η1) ∂µ L − iΛℓ
20Verifying explicitly the closure of the algebra is relatively easy.
and the multiplet is not complete without U .21
The two sets of gauge variations (B.12) remove 16B + 16F components in the long
supermultiplet, to obtain the 8B + 8F physically relevant components of the superMaxwell
(1B), the two complex scalars in X (4B) and two Majorana gauginos (8F ).
More on Im V
as is usually the case, a component of a real superfield V , but it appears in the expansion
auxiliary scalar field D in the expansion of V is replaced by the divergence of a vector field.
Comparing expansion (5.10) of L with
with the auxiliary scalar D, its lagrangian is quadratic in D:
LD =
A D2 + (B + ξ)D,
In particular, A would be the gauge kinetic metric in superMaxwell theory (hence the
positivity condition). To integrate over D, it is legitimate to solve the field equation
This theory does not have any symmetry and the (supersymmetric) ground state is at
variables of A and B is of course given by
LD = −
= −V.
∂zLD = −∂z
= −∂zV.
quadratic lagrangian for the divergence of a vector field,
L =
A(∂µ Vµ )2 + (B + ξ) ∂µ Vµ ,
21In this gauge, variations (B.7) hold.
23They do not depend on derivatives of fields. These scalar fields are collectively denoted by z.
instead of expression (C.1). Now, the FI term is a derivative which does not contribute to
the dynamical equations and the field equation for Vµ is
Its solution
righthand side of the solution is not a derivative of offshell fields.
This situation is not new in the literature. Redefine
Vµ =
the lagrangian (C.4) becomes
∂µ Vµ =
(∂µ Vµ )2 = − 24
1 F µνρσ Fµνρσ ,
LF = − 48
A F µνρσ Fµνρσ + (B + ξ) ǫµνρσ Fµνρσ .
been studied as a potential source for a cosmological constant [30]. Another example is the
massive Schwinger model [31]24 where the Maxwell lagrangian
24As also explained in ref. [30].
L =
Returning to our lagrangian (C.4) and solution (C.6), if we substitute the solution
field equation of z. We obtain
L = −
= −V
Comparing with expression (C.3), equivalence is obtained if we identify the integration
except if A is constant (the superMaxwell theory has then canonical kinetic terms), in
which case the second constant term in the potential is irrelevant. With this procedure,
of the superMaxwell theory.
Notice that a derivative term may in general contribute to currents. The canonical
Tµν = ξ [∂ν Vµ − ηµν ∂ρVρ]
energymomentum is zero, assuming the absence of boundary contributions):
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