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JHEP10(2015)087

Published for SISSA by Springer

Received: August 26, 2015

Accepted: September 20, 2015

Published: October 14, 2015

New realizations of the supergroup D(2, 1; α) in

N = 4 superconformal mechanics

S. Fedoruk

and E. Ivanov

Bogoliubov Laboratory of Theoretical Physics, JINR,

141980 Dubna, Moscow Region, Russia

E-mail: fedoruk@theor.jinr.ru, eivanov@theor.jinr.ru

Abstract: We present new explicit realizations of the most general N = 4, d = 1 super-

conformal symmetry D(2, 1; α) in the models of N = 4 superconformal mechanics based

on the reduc i bl e multiplets (1, 4, 3) ⊕ (0, 4, 4), (3, 4, 1) ⊕ ( 0, 4, 4) and (4, 4, 0) ⊕ (0, 4, 4).

We start from the manifestly supersymmetric superﬁeld actions for these systems and

then descend to the relevant oﬀ- and on-shell component actions from which we derive

the D(2, 1; α) ( super )charges by the Noether procedure. Some peculiarities of these real-

izations of D(2, 1; α) are discussed. We also construct a new D(2, 1; α) invariant system

by joining the multiplets (3, 4, 1) and (4, 4, 0) in such a way that they interact with each

other through an ext r a (0, 4, 4) multiplet. New fermionic conformal coupl i ngs appear as

the result of elimination of the appropriate auxiliary ﬁelds.

Keywords: Exte nd ed Supersymmetry, Superspaces, Conformal and W Symmetry

ArXiv ePrint:

1507.08584

On leave of absence from V.N. Karazin Kh a r kov National University, Ukraine.

Open Access,

 The Authors.

Article funded by SCOAP

doi:

10.1007/JHEP10(2015)087

JHEP10(2015)087

in the models of supersymmetric mechanics were a subjects of many works (see,

e.g., [

8, 9]–[14], [19]–[24, 25], [26–28, 31] and references therein).

As a rule, the realiza-

tions on one or another ﬁxed type of the irreducible N = 4, d = 1 supermultiplet were

considered. Recently, the study of SU(1, 1|2) superconformal systems including some pairs

of such multiplets was initiated in ref. [

30]. Some interesting links with the N = 2, d = 5

supergrav i ty were established there. One of the basic points of the construction in [

30]

was the inclusion of couplings with the fermionic N = 4 multiplets (0, 4, 4) which do not

enlarge the dimen si on of the target bosonic manif ol d.

In the present pap e r we study the analogous realizations of D(2, 1; α) as distinct

from [ 30] , where the particular SU(1, 1|2) case was treated. Another new point of our

consideration is that in all cases we start from the manifestly N = 4 supersymmetr i c

oﬀ-shell superﬁeld description of th e relevant multiplet pairs and write down the oﬀ-shell

component Lagr angi ans , while in [

30] only on-shell versions of the latter were addressed.

Keeping the relevant auxiliary ﬁelds in the combined actions of diﬀerent pairs allows one

to get more general on-shell component actions after elimination of thes e ﬁelds by their

algebraic equations of motion.

Since the natural oﬀ-shel l description of the multiplets considered in [30] and in the

present paper is achieved in the framework of N = 4, d = 1 harmonic superspace [

36], we

start in section

2 with recalling the basics of this approach. Then, in section 3, we present

the superﬁeld and component descriptions of the fermionic multiplet (0, 4, 4) , which is

the common p art of al l systems considered in [

30] and here . In section 4 we describe

the D(2, 1; α) invariant syst e m of interacting (1, 4, 3) and (0, 4, 4) multiplets, both in the

superﬁeld and the component formulations, and give the precise form of the D(2, 1; α)

generators for this case. In sections

5 and 6 we do the same f or the multiplet pair (3, 4, 1)

and (0, 4, 4) , as well as for the pair (4, 4, 0) and (0, 4, 4) . In s ec t i on

7, as an example

of the power of the oﬀ-shell approach, we present a new superconformal system involving

the triple of the multiplets (1, 4, 3), (4, 4, 0) and (0, 4, 4). The elimination of the auxiliary

ﬁelds in the corresponding action yields new fermionic terms which are absent in the actions

of the relevant isolated pairs. Section 8 contains conclusions and outlook.

2 N = 4, d = 1 harmonic superspace

The harmonic analytic N=4, d = 1 superspace [36–38] as the one-dimensional versi on of

the general harmoni c superspace [

39–41] is deﬁned as the following coordinate set

(ζ, u) = (t

, θ

, u

) , u

−

= 1 . (2.1)

These coordinates are related to the standard N=4, d= 1 superspace (central basis) coor-

dinates z = (t, θ

), (

) =

= t + i(θ

−

+ θ

−

), θ

= θ

. (2.2)

For implications of D(2, 1; α) in string theory and AdS/CFT correspondence see, e.g., [

34, 35].

As dist in c t from the D(2, 1; α) invariant systems with the so-called N = 4 spin multiplets [

20, 2 2 ], in

our case all boson ic ﬁelds of physical dimension are dynamical.

– 2 –

JHEP10(2015)087

The N=4 covariant spinor derivatives and their harmonic projections are deﬁned by

∂

∂θ

− i

∂

− iθ

∂

, (D

) = −

, {D

} = −2i δ

∂

, (2.3)

= u

, {D

−

} = −{D

−

} = −2i ∂

. (2.4)

In the analytic basis z

= (t

, θ

, u

±i

), the derivatives D

and

are short,

∂

∂θ

−

= −

∂

−

. (2.5)

The analyticity-preserving harmonic derivative D

and its conjugate D

−−

are given by

= ∂

+ 2iθ

∂

+ θ

∂

∂θ

−

∂

−

−−

= ∂

−−

+ 2iθ

−

∂

+ θ

−

∂

∂θ

−

∂

, ∂

±±

= u

∂

∂u

∓

, (2.6)

and become the pure partial derivatives ∂

±±

in the central basis. They satisfy the relations

, D

−−

] = D

, [D

, D

±±

] = ±2D

±±

, (2.7)

where D

is the operator counting external harmonic U(1) charges. The integration mea-

sures in the full harmonic superspace and its analytic subsp ace are deﬁned as

= dudtd

θ = dudt

−

)(D

) = µ

(−2)

) ,

(−2)

= dudζ

(−2)

= dudt

dθ

= dudt

−

) . (2.8)

The analytic subspace (ζ, u) is closed under the action of the most general N = 4, d = 1

superconformal group D(2, 1; α) and its degener ate D(2, 1; α = 0) and D(2, 1; α = −1) cases

which are reduced to the semi-direct pr oducts PSU(1, 1|2) ⋊ SU(2)

ext

. In what follows, we

will need the transformation properties of some relevant quantities under the “Poincar´e”

and conformal supersymmetry. The invar i anc e under these transformations is suﬃcient for

ensuring the complete D(2, 1; α) invariance since the rest of the D(2, 1; α) transformations

is contained in the closure of the conformal and manifest Poincar´e N=4, d=1 supersym-

metries.

In the N=4 superﬁeld approach, the invariance under the ordinary d = 1 supertrans-

lations (¯ε

(ε

))

δt = i(ε

− θ

¯ε

), δθ

= ε

, δ

= ¯ε

(2.9)

and

δt

= 2i(ε

−

− ¯ε

−

), δθ

= ε

, δ

= ¯ε

, (2.10)

where ε

= ε

, ¯ε

= ¯ε

, is automatic.

– 3 –

JHEP10(2015)087

The coordinate reali z ati on of the superconformal D(2, 1; α) boosts is as follows:

′

t = it(θ

¯η

) − (1 + α) θ

(θ

¯η

) , (2.11)

′

= −η

t − 2iα θ

(θ

¯η

) + 2i(1 + α) θ

(

) − i(1 + 2α) η

(θ

) , (2.12)

′

= −¯η

t − 2iα

(

) + 2i(1 + α)

(θ

¯η

) + i(1 + 2α) ¯η

(θ

) , (2.13)

′

= α

−1

, δ

′

= Λ

−

, (2.14)

′

= −η

+ 2i(1 + α)η

−

, δ

′

= −¯η

+ 2i(1 + α)¯η

−

, (2.15)

′

(dtd

θ) = −(dtd

θ) Λ

, δ

′

= µ

(2Λ

− (1 + α)Λ

) , δ

′

(−2)

= 0 , (2.16)

′

(−2)

= 0 , δ

′

du = du D

−−

, (2.17)

′

= −Λ

, δ

′

= 0 . (2.18)

Here η

= η

, ¯η

= ¯η

, ¯η

(η

) , and

= 2iα(¯η

−

− η

−

) , Λ

= 2iα(¯η

− η

) , D

= 0 , ( 2. 19)

= α

−1



2Λ

− D

−−



= 2i(θ

¯η

) , D

= 0 . (2.20)

The symbol ∼ means the generalized tilde-conjugation [

39–41]. With such deﬁniti ons,

all the coordinate transformations contain no singularities in the degenerate α = 0 or

α = −1 cases.

3 The fermionic multiplet (0, 4, 4)

This multiplet is the fermionic analog of the multi p l et (4, 4, 0). It is described oﬀ shell

by the fermionic analytic superﬁel d Ψ

, A = 1, 2 ,

(Ψ

) = Ψ

, satisfying the con-

straint [

36]:

= 0 ⇒ Ψ

= φ

+ θ

− 2iθ

−

, (3.1)

where (

) = −φ

, (F

) =

. On the index A, the appropriate SU(2)

P G

group

acts. Its generators commute with the N = 4 supersymmetry and D(2, 1; α) generators.

The r e q ui r e ment of superconformal covariance of the constrai nt (

3.1) uniquely ﬁxes the

superconformal D(2, 1; α) transformation rule of Ψ

, for any α, as

= Λ

. (3.2)

Oﬀ-shell transformations of component ﬁelds are the following

δφ

= −



+ ¯ω



δF

= 2i ¯ω

+ 2iα ¯η

, δ

= 2i ω

+ 2iα η

(3.3)

where

= ε

− t η

, ¯ω

= ¯ε

− t ¯η

– 4 –

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