ЯДЕРНАЯ ФИЗИКА, 2013, том 76, № 8, с. 1113-1119
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
N = 4 SUPERSYMMETRIC MECHANICS: HARMONIC SUPERSPACE AS A UNIVERSAL TOOL OF MODEL-BUILDING
©2013 E. A. Ivanov*
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia
Received May 29, 2012
We overview applications of the harmonic superspace approach in models of N = 4 supersymmetric mechanics, with emphasis on some recent results.
DOI: 10.7868/S0044002713080175
1. INTRODUCTION
Supersymmetric Quantum Mechanics (SQM) is the simplest (d = 1) supersymmetric theory. It has a lot of applications. In particular, it displays the salient features of higher-dimensional supersymmetric theories via the dimensional reduction and provides superextensions of integrable models like Calogero— Moser systems and Landau-type models. Extended supersymmetry in d = 1 (with N > 2) exhibits some surprising specific features, such as off-shell dualities between various supermultiplets [1, 2], the existence of nonlinear "cousins" of off-shell linear multiplets [3, 4], one-to-one correspondence with Dolbeault and Dirac differential complexes [5], etc.
An efficient tool to deal with extended supersym-metries is Harmonic Superspace (HSS) [6, 7]. This approach allowed to construct, for the first time, an off-shell formulation of hypermultiplets in N = = 2,d = 4 and N = 1,d = 6 supersymmetries, as well as a formulation of N = 4,d = 4 super Yang-Mills theory with the maximal number N = 3 of offshell supersymmetries.
The N = 4, d = 1 version of HSS was elaborated in [3] as an extension of ordinary N = 4 superspace by SU(2) harmonic variables: (t,d%,dk) ^ ^ (t,d%, dk,u±). The basic advantages of this approach are as follows:
It is the powerful device of N > 4 SQM modelbuilding;
It allows one to understand an interplay between various N = 4 SQM models via the N = 4, d = 1 covariant gauging procedure [8];
It allows one to construct generic N = 4, d = 1 superfield actions of the linear and nonlinear (4,4,0) multiplets [9];
E-mail: eivanov@theor.jinr.ru
It allows one to define new N = 4 superextensions of Calogero-type models [10];
Using the HSS approach, we managed to construct new off-shell SQM models with the Lorentz-force-type couplings to the external non-Abelian gauge fields [11, 12]. The basic trick was to employ the spin (or semi-dynamical) (4,4,0) multiplet. After quantization, its bosonic components become target harmonic variables [10]. Thus, the worldline harmonics u± eventually give rise to target harmonics.
2. HARMONIC N = 4, d = 1 SUPERSPACE
2.1. Generalities We start with ordinary N = 4, d =1 superspace: (t,di,dk), a = 1,2. Its harmonic extension is defined as
(t,ei,9k) ^ (t,ei,0k,u±),
u+ju- = 1, uf e SU(2)Aut,
where SU(2)Aut is one of the two commuting automorphism SU(2) groups of the N = 4, d = 1 Poincare superlgebra.
The most important feature of HSS is the existence of analytic basis:
(tA, e+, e+, u±, e-, d~) = (z, u±, e-,e-), e± = eiuf, e± = eiuf, tA = t + i(e+e- + e-e+).
The coordinate subset
(C,u±) = (tA,e+,e+,u±)
is closed under the action of N = 4 supersymmetry. It is called harmonic analytic superspace.
The closedness of the analytic superspace and the possibility to define N = 4 superfields on it, the analytic superfields, is related to the fact that the projections of the spinor harmonic derivatives D\ D1 on the harmonics u+, i.e. D+ = Dlu+, D + = Dlu+, are short in the analytic basis:
d
D+ =
D+ = -
d
ddd0~ D+$ = D+$ = 0 ^ $ = $((,u±).
An important element of the HSS approach is the harmonic derivatives:
= uf
d
du^,
+ e
±
d
ddt
+ e
±
d_
cm
+
+ 2 ie+ë+J-,
dt a
[D+,D++] = [D + ,D++] =0 ^ ^ D++$((,u±) is analytic.
2.2. Basic N = 4, d = 1 Multiplets
1. The multiplet (4,4,0) is described by the analytic superfield q+a(Z, u) « (xia, xa, Xa),a = 1,2:
D++q+a = 0,
q+a = xiau+ - 2e+xa - 2e+xa - 2ie+e+xiau-,
Sfr(
I dtd4eduq+aD~-q+
dt (xiaXia + iXaXa)
2. The multiplet (3,4,1) is described by the analytic superfield L++((,u) « (£(ik),^j,ipj,F):
D++L++ = 0,
l++ = eiku+u+ + i(e+xj + e+xj )u+ + + e+e+(F - 2iiik u+u-),
Sfree -J dtd4eduL++(D--)2L++ -
dt
ekhk --t^2 I +
3. Gauge multiplet is described by the analytic superfield V++(Z,u):
V++' = V++ + D++A,
A = A(c,u) ^ vW+ = 2ie+e+B(t),
5B = \(t).
4. Gauged (4,4,0) multiplet is represented by
(v+ ,v+), v+' = etAv+ ,v+' = e-tAv+:
(D++ + iV++)v+ = 0, v+ = + e+u! + e+u2 - 2ie+e+(ft + iBft)u-.
2.3. Gauging in N = 4, d = 1 HSS
An instructive example of gauging in N = 4, d = = 1 HSS is as follows. One starts from the free action of the multiplet (4,4,0),
5 = J dtd4eq+aD--q+,
which is invariant under q+a ^ q+a + \u+a, a = = 1,2, and then gauges the shift symmetry as A ^ ^ A((,u). The gauge covariantization goes as follows
D++q+a = 0 ^ V++ q+a = = D++q+a - V++u+a = 0,
5 ^ Sg = j dtd4eq+aV--q+,
V--q+ = D--q+a - V--u+a, [V++, V--] = D0 ^ D++ V-- - D V++ = 0, V-- = V--(V++,u).
-aq+ = 0 ^
Then one makes the gauge choice u-aq ^q+a=u-aL++:
D++q+a - V++u+a = 0 ^ V++ = L++,
D++L++ = 0,
D++V-- - D--L++ = 0 ^
S g = / dtd4 ev--L++ =
y dtd4eL++(D--)2L++.
Thus, as a result of gauging, the free action of the multiplet (3,4,1) is recovered.
Everything works perfectly also for the interaction case and for other N = 4, d = 1 multiplets. These multiplets and their superfield actions can be reproduced as the appropriate gaugings of the multiplet (4, 4, 0) and of some nonlinear generalizations of the latter [8]:
(4,4,0) ^ linear (3,4,1) — via gauging the shift or rotational U(1) symmetries of q+a;
(4,4,0) ^ nonlinear (3,4,1) — via gauging scale
symmetry, q+a' = Aq+a;
(4,4,0) ^ (2,4, 2) — via gauging some two-generator solvable symmetry of q+a;
(4,4,0) ^ (1,4, 3) — via gauging the "Pauli— GUrsey" SU(2)PG symmetry, q+a' = \aq+b;
(4,4,0) ^ (0,4,4) — via gauging the semi-direct product of SU(2)pg and shift ôq+a = \u+a.
Thus all N = 4 SQM systems of interest with four fermionic fields can be obtained from the (4,4,0) SQM models. This property makes it urgent to know the most general form of off-shell actions of the multiplet (4, 4, 0).
3. MOST GENERAL ACTIONS OF (4,4,0) MULTIPLETS
Most general (4,4,0) multiplets are described by an analytic superfield q+a((,u) = f m(t)u+ + ...,a = = 1,..., 2n, with the nonlinear harmonic constraint [9]
D++q+a = L+3a (q+ ,u).
The sigma-model-type action is an integral over the full superspace:
S& ~ J dtd4dduC(q+a, q-b, u±)
(1)
q
-a _ D--n+a
D q+
Swz dud((-2)C+2(q+a,u±).
(2)
The superfunctions L+3a(q+ ,u), L(q+,q-,u), L+2(q+ ,u) are general functions of their arguments. Their 9 = 0 projections are unconstrained potentials specifying the relevant bosonic target space geometry (L+3a| and L|) and coupling to the external Abelian gauge field (L+2|).
3.1. Sigma-ModelAction
The most general target space geometry in the case under consideration is weak hyper-Kahler with torsion (HKT) [13]. N = 4, d =1 HSS formulation solves this geometry in terms of two potentials L+3a(f+,u±) and L(f+,f-,u±), with f±a = q±a. The basic features of HKT geometry is the presence of torsion and of the triplet of covariantly constant (with respect to the connection with torsion) complex structures forming the algebra of quaternions. The basic geometric objects naturally appear in the
component action corresponding to the superfield action (1)
1
La — 2 9iakbf f
1 ï~6
(3)
- Jq (e-V^V^G^]) X-X-X-X--
Here, the underlined indices refer to the tangent space (1 = 1) 2; a = 1,..., 2n) and
Vxk = xk + fkbefb(^)lxà,
where éçb is the vielbein coefficients and (wm)| is the generalized spin connection related to the affine connection with torsion C^¡^ic, such that
Ciakblç = VkaG[çb] + ÊM^Î^M •
In the above relations, the skew-symmetric object G[çd] is the new "symplectic" metric which is related to the potential L(f+, f-,u)
+ f-
L(f ) = j duL(f+,f-,u).
The Wess—Zumino type action is an integral over the analytic subspace
The same object appears in the expression for the target space metric
si lc td
Qiakb = ^]cd\(-Ueiaekb-
The vielbeins efh are related to the analytic potential C+3a(f+,u). The coefficient before the 4th-order fermionic term is just the exterior derivative of the torsion:
idpiakblc\-
If it is vanishing, the torsion 3-form is closed and the weak HKT geometry becomes the strong HKT geometry.
The particular case of strong HKT geometry is the torsionless general hyper-Kahler target space geometry, with the following particular potentials
L+3a(f + ,u) = na] 0+bC+4(f+,u),
C(f,u) = n{ab]f+af -b,
where Q[ab] is a constant symplectic metric.
3.2. Most General N = 4 Wess-Zumino (WZ) Term
The component Lagrangian corresponding to the superfield WZ term ~C+2(q+a, u±) is
¿WZ = Aiafa - 'iakbX~X~,
A
ftŒPHAfl OH3HKA tom 76 № 8 2013
9
*
where
AaU) = egj duu-kV+b_C+2U+,u)
and
Fiakb = ^¿^kb^jdlc, Fjdlc = 9jdAlc ~ 9icAjd-
One can check that
fiakb ~ £ifc,
i.e. the general background gauge field Aia(f) is self-dual,
T,
(ia k)b
0.
= eJa DZв,
ÖZa ÖUa = KDU
These pairs form different (4,4,0) multiplets. The complex structures are
Ji
0 J 0 0
J 0 0 0
0 0 0 K
0 0 K 0 J
J2 =
^ 0 iJ 0 0 ^ -iJ 0 0 0 0 0 0 -iK
0 0 iK 0
while the structure corresponding to the manifest N = 2 supersymmetry reads
Ja =
3.3. Two Sorts of (4,4,0) Multiplets: Weakening Target Geometry
The most general target geometry of the component (4,4,0) models is known to be even weaker than weak HKT [14—16]: the complex structures JM do not form quaternionic algebra and they satisfy the condition D(MJffj = 0, which is weaker than
the standard covariant constancy condition. How to gain this geometry within the off-shell superfield framework?
The answer [9] is that one needs to take into account, along with the (4, 4, 0) multiplets described by q+a(Z,u), also the so-called mirror (4,4,0) multiplets in which bosonic fields are doublets with respect to another SU(2) factor in the full automorphism symmet
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.