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JHEP05(2018)019

Published for SISSA by Springer

Received: March 30, 2018

Accepted: April 20, 2018

Published: May 4, 2018

From N = 4 Galilean superparticle to

three-dimensional non-relativistic N = 4 superﬁelds

Sergey Fedoruk,

Evgeny Ivanov

and Jerzy Lukierski

Bogoliubov Laboratory of Theoretical Physics, JINR,

Joliot-Curie 6, 141980 Dubna, Moscow region, Russia

Institute for Theoretical Physics, University of Wroc law,

pl. Maxa Borna, 50-204 Wroclaw, Poland

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

jerzy.lukierski@ift.uni.wroc.pl

Abstract: We consider the general N = 4, d = 3 Galilean superalgebra with arbi-

trary central charges and study its dynamical realizations. Using the nonlinear realization

techniques, we introduce a class of actions for N = 4 three-dimensional non-relativistic

superparticle, such that they are linear in the central charge Maurer-Cartan one-forms.

As a prerequisite to the quantization, we analyze the phase space constraints structure of

our model for various choices of the central charges. The ﬁrst class constraints generate

gauge transformations, involving fermionic κ-gauge transformations. The quantization of

the model gives rise to the collection of free N = 4, d = 3 Galilean superﬁelds, which

can be further employed, e.g., for description of three-dimensional non-relativistic N = 4

supersymmetric theories.

Keywords: Extended Supersymmetry, Space-Time Symmetries, Superspaces

ArXiv ePrint: 1803.03159

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP05(2018)019

JHEP05(2018)019

Similarly as in the relativistic case, one can study the NR N = 4, D = 3+1 supersym-

metric theories by following several paths:

i) One can start with NR N = 4, d = 3 Galilean superalgebras (for arbitrary N see [12])

and then construct their superspace and superﬁeld realizations. In this way we obtain

the universal tool for constructing NR supersymmetric ﬁeld theories.

ii) The NR ﬁeld theories can be reproduced by performing the non-relativistic contraction

c → ∞ (c is the speed of light) in the known relativistic non-supersymmetric, as well

as supersymmetric ﬁeld theory models (see, e.g., [4, 13, 14]). One of the advantages

of such a method is the possibility to derive the proper NR contractions of relativistic

action integrals.

iii) For a deﬁnite type of (super)symmetric framework one can consider the dynamics as-

sociated with particles, ﬁelds, string, p-branes, etc. An important role in such a list is

played by the free classical and ﬁrst-quantized (super)particle models, with the prop-

erty that their ﬁrst quantization leads to the classical (super)ﬁeld realizations (see,

e.g., [15, 16]). In our case, we will look for the free superparticle models invariant

under N = 4, d = 3 Galilean supersymmetry. One can mention that in the rela-

tivistic case this way of deriving free superﬁelds from the classical and ﬁrst-quantized

superparticles with extended N = 4, D = 4 Poincar´e supersymmetry was already

proposed in [17, 18].

In this paper we will follow the path iii). We will consider the most general NR

N = 4, d = 3 Galilean superalgebra, introduce the corresponding N = 4 Galilean super-

group and its cosets, construct the relevant nonlinear realizations and use the associated

Maurer-Cartan (MC) one-forms to build NR N = 4 superparticle models. They will

be subsequently quantized to obtain the NR superﬁelds providing realizations of N = 4

Galilean supersymmetries. Note that in such a setting the original coset parameters are

treated as the D=1 world-line ﬁelds. However, the whole formalism could be equally ap-

plied along the lines of path i), with the coset parameters treated as independent NR

superspace coordinates.

As a prelude to our considerations, we will describe the Galilean symmetries and their

supersymmetrizations in a short historical survey.

The Galilean theories describe the low energy, non-relativistic dynamical systems,

which can also be obtained as non-relativistic limit (c → ∞) of the corresponding relativis-

tic theories (see, e.g., [28–32]). Such a contraction limit, applied to D = 4 Poincar´e algebra

, M

µν

; P

= (P

, P

), M

µν

= (M

ijk

, N

= M

)), after shifting and rescaling

= m

c +

, P

= P

, N

= c B

, M

= J

, (1.1)

Here we will deal only with those Galilean supersymmetries which, after quantization, leads to the

superﬁelds depending on spinorial Grassmann variables. One could introduce as well relativistic [19–22]

and Galilean [23, 24] world-line supersymmetries with vectorial Grassmann variables. However, this option

is out of the scope of our study.

The Galilean symmetries are used also in the description of light cone quantization of relativistic

theories [25–27]. In this paper, we shall not deal with this application of Galilean symmetries.

– 2 –

JHEP05(2018)019

where H stands for non-relativistic Hamiltonian and B

for the Galilean boosts, yields

“quantum” d = 3 Galilean algebra [33]

, J

] = iε

ijk

, P

] = iε

ijk

, [J

, B

] = iε

ijk

[H, B

] = −iP

, [P

, B

] = −iM δ

(1.2)

The central charge M = m

describes a non-relativistic mass which can be identiﬁed with

the relativistic rest mass.

Because bosons and fermions occur in both relativistic and non-relativistic settings,

one can consider the non-relativistic supersymmetry as well. The ﬁrst proposal for super-

symmetrization of Galilei algebra (1.2) was given in [34], where N = 1 and N = 2, d = 3

Galilean superalgebras were presented. The N = 1, d = 3 Galilean superalgebra is an

extension of relations (1.2) by complex NR USp(2) ' SU(2) supercharges S

:= (S

)

†

(A → A

†

denotes Hermitian conjugation) which satisfy the relations

} = M δ

, [J

, S

] = (σ

)

, [J

] = −

(σ

)

. (1.3)

Passing to the N = 2 d = 3 Galilean superalgebra [34] is accomplished by adding to

the N = 1 Galilean superalgebra generators (J

, P

, B

, H; S

, M) the second pair of

complex SU(2) supercharges Q

:= (Q

)

†

, subject to the following relations:

} = 2δ

H ,

} = 2(σ

)

+ 2iYδ

, Q

] = (σ

)

, [B

, Q

] = (σ

)

] = −

(σ

)

, [B

] = −

(σ

)

(1.4)

In the relations (1.4), (1.3), besides the central charge M, there appears the new central

charge Y. The N = 2, d = 3 Galilean superalgebra can be derived from N = 2, D = 4

Poincar´e superalgebra (a = 1, 2)

βb

} = 2(σ

)

, Q

} = 2



αβ

Z ,

{

˙αa

βb

} = −2



˙α

(1.5)

(plus the commutation relations with Poincar´e and internal R-symmetry U(2) generators)

by taking the c → ∞ contraction limit with M = m

. In general, the N = 2, D = 4

Poincar´e superalgebra is endowed with one complex central charge Z or, equivalently, two

The Galilean algebra (1.2) with central charge M 6= 0 is called Bargmann algebra. In d = 2 one can

introduce also second central charge through the modiﬁed commutator [B

, B

] = 

ρ [6–9].

We list only non-vanishing (anti)commutators.

– 3 –

JHEP05(2018)019

real central charges, Z = X + iY . . .

Before taking the NR limit c → ∞, the Galilean

supercharges Q

and S

(see (1.3), (1.4)) should be identiﬁed with the following linear

combinations of two N = 2 Weyl supercharges in (1.5)

= c

1/2

= c

1/2

˙α

, S

= c

−1/2

−

= c

−1/2

−

˙α

, (1.6)

where

√



∓ 

αβ

β 2



˙α

√



˙α 1

∓ 

˙α



(1.7)

and

˙α

= (Q

)

†

. Also, we should postulate the following c-dependence of the central

charges in (1.5)

X = −m

c +

+ O





, Y = Y + O





. (1.8)

If X is ﬁnite in the contraction limit, it merely generates the shift H → H + X in the

relations of the N = 2, d = 3 Galilean superalgebra (see the ﬁrst relation in (1.4)).

The Galilean N = 2 superalgebra and its dynamical realizations were studied in several

papers, but mostly for the case of two (d = 2) space dimensions [2–4, 10, 37, 38].

In the present paper we consider N = 4, d = 3 Galilean superalgebra with all possible

central charges. It will be obtained by the c → ∞ contraction procedure from the general

N = 4, D = 4 relativistic Poincar´e superalgebra [39] which involves 6 complex central

charges Z

= −Z

(A, B = 1, 2, 3, 4). Correspondingly, the N = 4, d = 3 Galilean

superalgebra obtained in the c → ∞ limit involves 12 real central charges.

If these central

charges are numerical, then, using a suitable redeﬁnition of supercharges by an unitary

4 × 4 matrix, one can cast the antisymmetric 4×4 complex matrix of six central charges

= −Z

(A, B = 1, 2, 3, 4) into a quasi-diagonal Jordanian form [40, 41]







0 −Z







←→ Z

= diag(Z

, Z

), (1.9)

Since N = 2 D = 4 Poincar´e superalgebra is covariant under the phase transformation of Weyl super-

charges (Q

˙α

; r = 1, 2)

→ e

Z → e

iα

Z (Z = e

−iα

|z|)

one could think that one real central charge is enough in N = 2 case. However, as was found by studying

concrete dynamical models [35, 36], it is the complex N = 2 central charge Z = X

+ iX

what actually

matters. It amounts to two physical real central charges: the topological magnetic charge X

and the

non-topological electric charge X

. Only if these charges take constant eigenvalues, i.e. are numerical, they

can be rotated to the single central charge by the phase transformations just mentioned.

In fact, the NR N = 4 Galilean superalgebra involves 13 central charges if we take into account the

Bargmann central charge M = m

obtained from the leading terms in the asymptotic expansion of P

and

X in c (see (1.1) and (1.8)).

– 4 –

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