An off-shell formulation for internally gauged D=5, N=2 supergravity from superconformal methods

0.41 MB·English

by  Frederik Coomans

MIFPA-12-36
An Off-Shell Formulation for Internally Gauged
D = 5, N = 2 Supergravity from Superconformal
Methods
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N F. Coomans1, M. Ozkan2

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] 1Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven,
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t Celestijnenlaan 200D B-3001 Leuven, Belgium
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email: [email protected]
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[ 2 George and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,

Texas A&M University, College Station, TX 77843, USA
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email: [email protected]
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ABSTRACT
1
2
1 We use the superconformal method to construct a new formulation for pure off-shell
:
v D = 5, N = 2 Poincar´e supergravity and present its internal gauging. The main difference
i
X between the traditional formulation and our new formulation is the choice of the Dilaton
r Weyl Multiplet as the background Weyl Multiplet and the choice of a Linear compensating
a
Multiplet. We do not introduce an external vector multiplet to gauge the theory, but instead
use the internal vector of the Dilaton Weyl Multiplet. We show that the corresponding on-
shell theory is Einstein-Maxwell supergravity. We believe that this gauging method can
be applied in more complicated scenarios such as the inclusion of off-shell higher derivative
invariants.
Contents
1 Introduction 3
2 Superconformal Multiplets 4
2.1 The Standard Weyl Multiplet . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Vector Multiplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 The Linear Multiplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Actions 9
3.1 Density formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Action for the Linear Multiplet . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Action for the Vector Multiplet . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Ungauged D = 5, N = 2 Supergravity 11
4.1 Construction of the Superconformal Action . . . . . . . . . . . . . . . . . . . 11
4.2 Gauge Fixing, Decomposition Rules and the Off-Shell Poincar´e Action . . . . 14
5 Minimal Gauged Supergravity 15
5.1 The Off-shell Internally Gauged Supergravity . . . . . . . . . . . . . . . . . 16
5.2 Einstein-Maxwell Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3 Truncation to Minimal Gauged On-Shell Supergravity . . . . . . . . . . . . . 21
6 Conclusions 22
2
1 Introduction
Pure on-shell five dimensional supergravity with eight supercharges was first introduced
in [1], its gauging was investigated in detail in [3, 4] and general matter couplings and
their geometrical aspects were studied in [5, 6, 7]. To study general matter couplings it is
useful to work in an off-shell superconformal setting and, eventually, gauge fix the redundant
conformal symmetries. To this end, the N = 2 superconformal program was initiated in
[8, 9].
The pure on-shell theory consists of the metric, graviphoton and gravitino. We denote an
off-shell pure theory as a theory with minimal field content on which the super Poincar´e alge-
bra closes off-shell and which reduces, upon going on-shell and decoupling the matter fields,
to the pure on-shell theory. The off-shell nature of the theory implies that, in addition to the
metric, graviphoton and gravitino, they also contain auxiliary fields. Off-shell formulations
are constructed most easily by using the method of superconformal tensor calculus. Different
off-shell formulations can correspond to one physical on-shell theory. They can differ in the
compensator multiplet used to compensate for the redundant conformal symmetries or in
the choice of Weyl multiplet. In 5D superconformal tensor calculus there are two possible
choices for the Weyl multiplet: the Standard Weyl and the Dilaton Weyl Multiplet. These
multiplets contain the same superconformal gauge fields but differ in their matter sector: a
scalar D, an antisymmetric tensor T and a symplectic Majorana spinor χi for the Standard
ab
Weyl Multiplet and a scalar σ, a vector C , a 2-form B and a symplectic Majorana spinor
µ µν
ψi for the Dilaton Weyl Multiplet. In [11] a pure off-shell 5D theory with eight supercharges
is written down using the Standard Weyl Multiplet and a Hypermultiplet compensator, and
in [12], a Nonlinear Multiplet was used as compensator instead of a Hypermultiplet. In this
paper we will write down a different off-shell pure theory using the Dilaton Weyl Multiplet
and a Linear Multiplet compensator.
Off-shell formulations are of great use when studying higher order curvature extensions of
the theory. Off-shell formulations for curvature squared invariants have been constructed in
5 dimensions in [11, 13]. Unlike an effective supergravity Lagrangian of a compactified string
theory which has higher-order correction terms in α(cid:48) and which is supersymmetric only order
by order in α(cid:48), these invariants can be added to a pure off-shell supergravity theory, and they
are exactly supersymmetric. The invariant constructed in [11] is the supersymmetric com-
pletion of the Weyl tensor squared and makes use of an external Vector Multiplet manifested
in the appearance of a mixed gauge-gravitational Chern-Simons (CS) term A∧tr(R ∧R),
where A denotes the external vector. It is constructed in the background of the Standard
Weyl Multiplet. The invariant constructed in [13] is the supersymmetric completion of the
Riemann tensor squared. It is constructed in the background of the Dilaton Weyl Multiplet
and is based on a map between the Yang-Mills multiplet and the Dilaton Weyl Multiplet.
It does not require an external Vector Multiplet, but rather it has a purely gravitational CS
term C ∧tr(R∧R), where C denotes the internal vector of the Dilaton Weyl Multiplet.
In [8] it was shown that the Dilaton Weyl Multiplet can be obtained by solving the
equations of motion for a Vector Multiplet coupled to the Standard Weyl Multiplet. In
this paper our purpose is to exploit this connection between the Weyl multiplets to obtain
3
an off-shell Lagrangian for 5D minimal supergravity in the background of a Dilaton Weyl
Multiplet in which the U(1) R-symmetry is gauged dynamically by the internal vector C,
i.e. the vector field C has a kinetic term in the Lagrangian. This forms a basis for a future
study of the Weyl squared invariant in a Dilaton Weyl background with the external vector
A replaced by the internal one C.
The paper is built up as follows. In section 2 we summarize the 5D superconformal cal-
culus and introduce the Standard Weyl Multiplet as well as two types of matter multiplets:
the Linear Multiplet and the Vector Multiplet. In section 3 we construct invariant actions
for the Linear and Vector Multiplet. In section 4 we construct 5D pure off-shell supergravity
by coupling the Standard Weyl Multiplet to a Linear compensator. This procedure is very
similar to the one used in 6D in [14]. We then add a superconformal abelian Vector Mul-
tiplet action to the Linear Multiplet action and compute the field equations for the Vector
Multiplet components. These equations allow us to solve for the matter fields of the Stan-
dard Weyl Multiplet (D, T χi) in terms of the fields of the Vector Multiplet1 (σ, C , ψi)
ab µ
plus an additional 2-form (B ). Using these expressions in the Lagrangian we obtain the
µν
action for the Linear Multiplet in the background of the Dilaton Weyl Multiplet. Gauge
fixing the redundant conformal symmetries leads to off-shell Poincar´e supergravity which
has apart from the graviton and the gravitino, a vector field, a 2-form gauge field, a dilaton,
a symplectic Majorana spinor and a number of auxiliary fields.
In section 5 we develop an off-shell method to gauge the theory. We start with an off-
shell action consisting of the Linear Multiplet action, the Vector Multiplet action and a
coupling between the Vector and Linear Multiplet, all in the background of the Standard
Weyl Multiplet. Then we compute, as in the ungauged case, the field equations for the
Vector Multiplet components to obtain expressions for the Standard Weyl matter fields.
We notice that these expressions get deformed by the Vector-Linear coupling. After using
these expressions in the Lagrangian and gauge fixing the conformal symmetries we obtain
an off-shell expression for U(1)-gauged supergravity. After eliminating the auxiliary fields
˜
and dualizing the 2-form B to a vector C , we show that the resulting theory is Einstein-
µν µ
˜
Maxwell supergravity gauged by a linear combination of C and C . This theory agrees
µ µ
completely with the one constructed in [3] via the Noether procedure. Finally, we show that
˜
we can consistently eliminate σ, ψ and C . The resulting on-shell theory is minimal gauged
µ
5D supergravity [3] consisting of the graviton, the graviphoton and the gravitino.
2 Superconformal Multiplets
In this section, we will recall the elements of N = 2,D = 5 superconformal tensor calculus
constructed in [8, 9]. In the first subsection 2.1 we introduce the gauge multiplet of the N =
2,D = 5 superconformal group: the Standard Weyl Multiplet. In the last two subsections,
2.2 and 2.3, we introduce two types of matter multiplets, the Vector Multiplet and the Linear
Multiplet.
1We suggestively denote the fields of the Vector Multiplet with σ, C and ψi. The Vector Multiplet also
µ
has an auxiliary Yij, but we solve for this auxiliary and use its value in the expressions for D, T and χi.
ab
4
2.1 The Standard Weyl Multiplet
The N = 2,D = 5 superconformal tensor calculus is based on the superconformal algebra2
F2(4) with the generators
P , M , D, K , U , Q , S , (2.1)
a ab a ij αi αi
where a,b,... are Lorentz indices3, α is a spinor index and i = 1,2 is an SU(2) index.
Here M and P are the usual Poincar´e generators, D is the generator for dilatations, K
ab a a
generates special conformal boosts, U is the SU(2) generator and Q and S are the
ij αi αi
supersymmetry and conformal supersymmetry generators respectively.
For each of the generators above we now introduce the following gauge fields
h A ≡ {e a, ω ab, b , f a, Vij, ψi, φi}, (2.2)
µ µ µ µ µ µ µ µ
where µ,ν,... are world vector indices. Using the structure constants f C of the supercon-
AB
formal algebra (given e.g. in appendix B of [8]) and the basic rules
δhA = ∂ (cid:15)A +(cid:15)ChBf A,
µ µ µ BC
R A = 2∂ hA +hChBf A, (2.3)
µν [µ ν] ν µ BC
one can easily write down the linear transformation rules and the linear curvatures R A
µν
of the superconformal gauge fields given in (2.2). The linear transformations given in [8]
satisfy the F2(4) superalgebra, thus resulting in a gauge theory of F2(4) since we have not
related the generators P ,M to the diffeomorphisms of spacetime. This problem can be
a ab
solved by imposing the so-called curvature constraints [8]. These constraints determine the
gauge fields ω ab, φi and f a in terms of the independent gauge fields e a, ψi, b , Vij and,
µ µ µ µ µ µ µ
in addition, achieve maximal irreducibility of the superconformal gauge field configuration.
A simple counting argument shows that the superconformal gauge fields, after imposing
the conventional constraints, represent 21 + 24 off-shell degrees of freedom and therefore
cannot represent a supersymmetric theory. Additional matter fields T (10),D(1) and χi(8)
µν
must be added to the gauge fields in order to obtain an off-shell closed multiplet [8, 9]. This
multiplet is known as the Standard Weyl Multiplet.
Starting from the linear transformation rules of the superconformal fields, the linear
curvatures R A and the matter fields T , D, and χi we can construct the full nonlinear
µν µν
N = 2,D = 5 Weyl Multiplet by applying an iterative procedure (described in detail for 6
dimensions in [15]). The results are [8] (we only give Q, S and K transformations):
δe a = 1(cid:15)¯γaψ ,
µ 2 µ
δψi = (∂ + 1b + 1ω abγ )(cid:15)i −Vij(cid:15) +iγ ·Tγ (cid:15)i −iγ ηi,
µ µ 2 µ 4 µ ab µ j µ µ
δV ij = −3i(cid:15)¯(iφj) +4(cid:15)¯(iγ χj) +i(cid:15)¯(iγ ·Tψj) + 3iη¯(iψj),
µ 2 µ µ µ 2 µ
δT = 1i(cid:15)¯γ χ− 3 i(cid:15)¯R(cid:98) (Q),
ab 2 ab 32 ab
2The notation Fp(4) refers to a compact form of F(4) with bosonic subalgebra SO(7−p,p).
3Weusetheconventionsof[8,19]. Inparticular,thespacetimesignatureis(−,+,+,+,+)andψ¯iγ χj =
(n)
t χ¯jγ ψi with t =t =−t =−t =1. When SU(2) indices on spinors are omitted, northwest-southeast
n (n) 0 1 2 3
contraction is understood.
5
δχi = 1(cid:15)iD− 1 γ ·R(cid:98)ij(V)(cid:15) + 1iγabD/T (cid:15)i − 1iγaDbT (cid:15)i
4 64 j 8 ab 8 ab
−1γabcdT T (cid:15)i + 1T2(cid:15)i + 1γ ·Tηi,
4 ab cd 6 4
δD = (cid:15)¯D/χ− 5i(cid:15)¯γ ·Tχ−iη¯χ,
3
δb = 1i(cid:15)¯φ −2(cid:15)¯γ χ+ 1iη¯ψ +2Λ , (2.4)
µ 2 µ µ 2 µ Kµ
where
D χi = (∂ − 7b + 1ω abγ )χi −Vijχ − 1ψiD+ 1 γ ·R(cid:98)ij(V)ψ
µ µ 2 µ 4 µ ab µ j 4 µ 64 µj
−1iγabD/T ψi + 1iγaDbT ψi + 1γabcdT T ψi − 1T2ψi − 1γ ·Tφi ,
8 ab µ 8 ab µ 4 ab cd µ 6 µ 4 µ
D T = ∂ T −b T −2ω c T − 1iψ¯ γ χ+ 3 iψ¯ R(cid:98) (Q). (2.5)
µ ab µ ab µ ab µ [a b]c 2 µ ab 32 µ ab
The relevant modified curvatures are
R(cid:98) ab(M) = 2∂ ω ab +2ω acω b +8f [ae b] +iψ¯ γabψ +iψ¯ γ[aγ ·Tγb]ψ
µν [µ ν] [µ ν]c [µ ν] [µ ν] [µ ν]
+ψ¯ γ[aR(cid:98) b](Q)+ 1ψ¯ γ R(cid:98)ab(Q)−8ψ¯ e [aγb]χ+iφ¯ γabψ ,
[µ ν] 2 [µ ν] [µ ν] [µ ν]
R(cid:98) ij(V) = 2∂ V ij −2V k(iV j)−3iφ¯(iψj) −8ψ¯(iγ χj) −iψ¯(iγ ·Tψj),
µν [µ ν] [µ ν]k [µ ν] [µ ν] [µ ν]
1
R(cid:98)i (Q) = 2∂ ψi + ω abγ ψi +b ψi −2Vijψ −2iγ φi +2iγ ·Tγ ψi .(2.6)
µν [µ ν] 2 [µ ab ν] [µ ν] [µ ν]j [µ ν] [µ ν]
As mentioned before, the dependent fields, which relate the generators P ,M to the diffeo-
a ab
morphisms of spacetime, are completely determined by the following curvature constraints
R a(P) = 0,
µν
eν R(cid:98) ab(M) = 0,
b µν
γµR(cid:98) i(Q) = 0. (2.7)
µν
Notice that our choices for the above constraints are not unique, i.e. one can impose different
constraints by adding further terms to (2.7). However such additional terms only amount
to redefinitions of the dependent fields defined below. Using the curvature constraints we
identify ω ab, φi and f a in terms of the other gauge fields and matter fields
µ µ µ
ω ab = 2eν[a∂ e b] −eν[aeb]σe ∂ e c +2e [abb] − 1ψ¯[bγa]ψ − 1ψ¯bγ ψa,
µ [µ ν] µc ν σ µ 2 µ 4 µ
φi = 1iγaR(cid:98)(cid:48) i(Q)− 1 iγ γabR(cid:98)(cid:48) i(Q), (2.8)
µ 3 µa 24 µ ab
fa = −1R a + 1 e aR,
µ 6 µ 48 µ
where R ≡ R(cid:98)(cid:48) ab(M)e ρe and R ≡ R µ. The notation R(cid:98)(cid:48)(M) and R(cid:98)(cid:48)(Q) indicates that
µν µρ b νa µ
we have omitted the f a dependent term in R(cid:98)(M) and the φi dependent term in R(cid:98)(Q). The
µ µ
constraints imply through Bianchi identities further relations between the curvatures. The
Bianchi identities for R(P) imply [8]
R = R , e aR(cid:98) (D) = R(cid:98) a(M), R(cid:98) (D) = 0. (2.9)
µν νµ [µ νρ] [µνρ] µν
6
The full commutator of two supersymmetry transformations is
[δ ((cid:15) ),δ ((cid:15) )] = δ (ξµ)+δ (λab)+δ (η )+δ (λij)+δ (Λa ), (2.10)
Q 1 Q 2 cgct 3 M 3 S 3 U 3 K K3
where δ represents a covariant general coordinate transformation4. The parameters ap-
cgct
pearing in (2.10) are
ξµ = 1(cid:15)¯ γ (cid:15) ,
3 2 2 µ 1
λab = −i(cid:15)¯ γ[aγ ·Tγb](cid:15) ,
3 2 1
λij = i(cid:15)¯(iγ ·T(cid:15)j),
3 2 1
ηi = −9i(cid:15)¯ (cid:15) χi + 7i(cid:15)¯ γ (cid:15) γcχi
3 4 2 1 4 2 c 1
(cid:16) (cid:17)
+1i(cid:15)¯(iγ (cid:15)j) γcdχ + 1 R(cid:98)cd (Q) ,
4 2 cd 1 j 4 j
Λa = −1(cid:15)¯ γa(cid:15) D+ 1 (cid:15)¯iγabc(cid:15)jR(cid:98) (V)
K3 2 2 1 96 2 1 bcij
(cid:0) (cid:1)
+ 1 i(cid:15)¯ −5γabcdD T +9D Tba (cid:15)
12 2 b cd b 1
(cid:0) (cid:1)
+(cid:15)¯ γabcdeT T −4γcT Tad + 2γaT2 (cid:15) . (2.11)
2 bc de cd 3 1
For the Q,S commutators we find the following algebra
[δ (η),δ ((cid:15))] = δ (1i(cid:15)¯η)+δ (1i(cid:15)¯γabη)+δ (−3i(cid:15)¯(iηj))+δ (Λa ),
S Q D 2 M 2 U 2 K 3K
[δ (η ),δ (η )] = δ (1η¯ γaη ), (2.12)
S 1 S 2 K 2 2 1
with
(cid:0) (cid:1)
Λa = 1(cid:15)¯ γ ·Tγ − 1γ γ ·T η. (2.13)
3K 6 a 2 a
This concludes our review of the Standard Weyl Multiplet.
2.2 The Vector Multiplet
The off-shell abelian D = 5, N = 2 Vector Multiplet contains 8+8 degrees of freedom and
consists of the fields
{C ,σ,Yij,ψi}, (2.14)
µ
with Weyl weights (0,1,2,3/2) respectively. The bosonic field content consists of a vector
field C , a scalar field σ and an auxiliary field Yij = Y(ij), that is an SU(2) triplet. The
µ
fermion field is given by an SU(2) doublet ψi.
TheQ-andS-transformationsoftheVectorMultiplet, inthebackgroundoftheStandard
Weyl Multiplet, are given by [7]
δC = −1iσ(cid:15)¯ψ + 1(cid:15)¯γ ψ ,
µ 2 µ 2 µ
δYij = −1(cid:15)¯(iD/ψj) + 1i(cid:15)¯(iγ ·Tψj) −4iσ(cid:15)¯(iχj) + 1iη¯(iψj),
2 2 2
δψi = −1γ ·G(cid:98)(cid:15)i − 1iD/σ(cid:15)i +σγ ·T(cid:15)i −Yij(cid:15) +σηi ,
4 2 j
δσ = 1i(cid:15)¯ψ. (2.15)
2
4The covariant general coordinate transformations are defined as δ (ξ)=δ (ξ)−δ (ξµhI), where the
cgct gct I µ
index I runs over all transformations except the general coordinate transformations and the hI represent
µ
the corresponding gauge fields.
7
We have used here the superconformally covariant derivatives
D σ = (∂ −b )σ − 1 iψ¯ ψ ,
µ µ µ 2 µ
D ψi = (∂ − 3b + 1 ω abγ )ψi −Vijψ
µ µ 2 µ 4 µ ab µ j
+1γ ·G(cid:98)ψi + 1iD/σψi +Yijψ −σγ ·Tψi −σφi, (2.16)
4 µ 2 µ µj µ µ
and the supercovariant Yang-Mills curvature
G(cid:98) = G −ψ¯ γ ψ + 1iσψ¯ ψ , (2.17)
µν µν [µ ν] 2 [µ ν]
where G = 2∂ C . The supersymmetry transformation rule for G(cid:98) is given by
µν [µ ν] µν
δG(cid:98) = −1iσ(cid:15)¯R(cid:98) (Q)−(cid:15)¯γ D ψ +i(cid:15)¯γ γ ·Tγ ψ +iη¯γ ψ. (2.18)
µν 2 µν [µ ν] [µ ν] µν
This concludes our discussion on the Vector Multiplet.
2.3 The Linear Multiplet
The off-shell D = 5,N = 2 Linear Multiplet contains 8+8 degrees of freedom and consists
of the fields
{Lij,Ea,N,ϕi}, (2.19)
with Weyl weights (3,4,4,7/2) respectively. The bosonic fields consist of an SU(2) triplet
Lij = L(ij), a constrained vector E and a scalar N. The fermion field is given by an SU(2)
a
doublet ϕi. The Q and S supersymmetry transformations of the Linear Multiplet in the
background of the Standard Weyl Multiplet are given by
δLij = i(cid:15)¯(iϕj),
δϕi = −1iD/Lij(cid:15) − 1iγaE (cid:15)i + 1N(cid:15)i −γ ·TLij(cid:15) +3Lijη ,
2 j 2 a 2 j j
δE = −1i(cid:15)¯γ Dbϕ−2(cid:15)¯γbϕT −2η¯γ ϕ,
a 2 ab ba a
δN = 1(cid:15)¯D/ϕ+ 3i(cid:15)¯γ ·Tϕ+4i(cid:15)¯iχjL + 3iη¯ϕ, (2.20)
2 2 ij 2
where we used the following superconformally covariant derivatives
D Lij = (∂ −3b )Lij +2V (i Lj)k −iψ¯(iϕj),
µ µ µ µ k µ
D ϕi = (∂ − 7b + 1ω abγ )ϕi −Vijϕ + 1iD/Lijψ + 1iγaE ψi
µ µ 2 µ 4 µ ab µ j 2 µj 2 a µ
−1Nψi +γ ·TLijψ −3Lijφ ,
2 µ µj µj
D E = (∂ −4b )E +ω Eb + 1iψ¯ γ Dbϕ+2ψ¯ γbϕT +2φ¯ γ ϕ. (2.21)
µ a µ µ a µab 2 µ ab µ ba µ a
Finally, we note that the superconformal algebra closes if the following constraint is satisfied
DaE = 0. (2.22)
a
The solution for E in terms of a 3-form E is
a µνρ
Ea = − 1 e ae−1εµνρσλD E (2.23)
12 µ ν ρσλ
8
and E has the following gauge invariance
µνρ
δ E = 3∂ Λ . (2.24)
Λ µνρ [µ νρ]
Also, for the dual 2-form field we have
Ea = e aD Eµν ,
µ ν
E = eε Eσλ,
µνρ µνρσλ
δEµν = −1i(cid:15)¯γµνϕ− 1ψ¯iγµνρ(cid:15)jL −∂ Λ˜µνρ,
2 2 ρ ij ρ
D Eµν = ∂ Eµν + 1iψ¯ γµνϕ+ 1ψ¯iγµνρψjL . (2.25)
ν ν 2 ν 4 ρ ν ij
This concludes our discussion on the Linear Multiplet.
3 Actions
In this section, we will construct the action for a Linear Multiplet and present the action
for Vector Multiplet coupled to the Standard Weyl Multiplet [7]. Our starting point is a
density formula for the product of a Vector Multiplet and a Linear Multiplet. This will
be presented in subsection 3.1. In subsection 3.2 we will use this formula, after embedding
the Linear Multiplet into a Vector Multiplet, to construct the superconformal action for the
Linear Multiplet. In the last subsection 3.3 we present the action for the Vector Multiplet.
3.1 Density formula
We need an expression constructed from the components of the Linear and Vector Multiplet
that can be used as a superconformal action. In [10] a density formula is given for the
product of a Vector Multiplet and a Linear Multiplet
e−1L = YijL +iψ¯ϕ− 1ψ¯iγaψjL +C Pa
VL ij 2 a ij a
+σ(N + 1ψ¯ γaϕ+ 1iψ¯iγabψjL ), (3.1)
2 a 4 a b ij
where P , the pure bosonic part of the supercovariant field E , is defined as
µ µ
Pa = Ea + 1iψ¯ γbaϕ+ 1ψ¯iγabcψjL . (3.2)
2 b 4 b c ij
Using (2.23), we can express Pa as
Pa = − 1 e ae−1εµνρσλ∂ E . (3.3)
12 µ ν ρσλ
Using (3.3) and (2.25), one can rewrite L as
VL
e−1L = YijL +iψ¯ϕ− 1ψ¯iγaψjL + 1G Eµν
VL ij 2 a ij 2 µν
+σ(N + 1ψ¯ γaϕ+ 1iψ¯iγabψjL ). (3.4)
2 a 4 a b ij
9
3.2 Action for the Linear Multiplet
We want to use the density formula (3.1) to construct an action for the Linear Multiplet.
Hence, we start with embedding the components of the Linear Multiplet (L ,ϕi,E ,N)
ij a
into the components of the Vector Multiplet (Yij,C ,σ,ψi). Such embeddings are already
µ
considered in 4 and 6 dimensions [14, 15, 18] and here we will follow the same procedure.
As described in 2.3 the Linear Multiplet consists of a triplet of scalars L , a constrained
ij
vector E , a doublet of Majorana spinors ϕi and a scalar N. One starts the construction
a
of the Vector Multiplet with the identification σ = N, where σ is the scalar of the Vector
Multiplet. Thereis, however, amismatchbetweentheWeylweightsofthesefields. Therefore
one needs another scalar field to compensate for this mismatch. For this we will use
L2 = L Lij. (3.5)
ij
We can then identify the scalar of the Vector Multiplet as σ = 2L−1N + iϕ¯ ϕ LijL−3.
i j
This identification has the correct Weyl weight, and the second term is the supersymmet-
ric completion that is determined by the S-invariance of σ. Upon applying a sequence of
supersymmetry transformations, we obtained the following embeddings
σ = 2L−1N +(ϕi-terms),
ψ = −2iD/ϕ L−1 +16L−1L χj +(ϕi-terms),
i i ij
Y = L−1(cid:50)CL −D L DaL LkmL−3 −N2L L−3 −E EµL L−3
ij ij a k(i j)m ij µ ij
+8L−1T2L +4L−1DL +2E L DµL kL−3 +(ϕi-terms),
3 ij ij µ k(i j)
G(cid:98) = 4D (L−1E )+2L−1R(cid:98) ij(V)L −2L−3LlD LkpD L +(ϕi-terms). (3.6)
µν [µ ν] µν ij k [µ ν] lp
Here, we did not write the fermionic terms proportional to ϕi explicitly because in the
following section we will set ϕi to zero to fix the S-gauge.
After plugging the components (3.6) into the density formula (3.1) we end up with a
superconformal action for the Linear Multiplet
e−1L = L−1L (cid:50)cLij −LijD L DµL LkmL−3 −N2L−1 −E EµL−1
L ij µ k(i j)m µ
+8LT2 +4DL− 1L−3EµνLl∂ Lkp∂ L +2Eµν∂ (L−1E +VijL L−1)
3 2 k µ ν pl µ ν ν ij
+iL−1L ψ¯iγµD/ϕj +4Lψ¯ γµχ+ 1iL−1Nψ¯iγµνψjL +(ϕi-terms), (3.7)
ij µ µ 2 µ ν ij
where the superconformal d’Alembertian is given by
L (cid:50)cLij = L (∂a −4ba +ω ba)D Lij +2L V i DaLjk +6L2f a
ij ij b a ij a k a
−iL ψ¯aiD ϕj −6L2ψ¯aγ χ−L ϕ¯iγ ·Tγaψj +L ϕ¯iγaφj . (3.8)
ij a a ij a ij a
3.3 Action for the Vector Multiplet
A Vector Multiplet can be embedded into the Linear Multiplet to construct an action using
the invariant action formula 3.1. The action for the abelian Vector Multiplet up to 4-fermion
10