#### SU(2|1) supersymmetric mechanics on curved spaces

HJE
SU(2|1) supersymmetric mechanics on curved spaces
Nikolay Kozyrev 0 1 2
Sergey Krivonos 0 1 2
Olaf Lechtenfeld 0 1
Anton Sutulin 0 1 2
Leibniz Universita¨t Hannover,
0 Appelstrasse 2 , 30167 Hannover , Germany
1 141980 Dubna , Russia
2 Bogoliubov Laboratory of Theoretical Physics , JINR
We present SU(2|1) supersymmetric mechanics on n-dimensional Riemannian manifolds within the Hamiltonian approach. The structure functions including prepotentials entering the supercharges and the Hamiltonian obey extended curved WDVV equations specified by the manifold's metric and curvature tensor. We consider the most general u(2)-valued prepotential, which contains both types (with and without spin variables), previously considered only separately. For the case of real K¨ahler manifolds we construct all possible interactions. For isotropic (so(n)-invariant) spaces we provide admissible prepotentials for any solution to the curved WDVV equations. All known one-dimensional SU(2|1) supersymmetric models are reproduced.
Extended Supersymmetry; Field Theories in Lower Dimensions; Space-Time
Symmetries
Examples of n-dimensional mechanics with potentials
HJEP05(218)7
Qa, Qb =
2i δbaH ,
Qa, Qb
= 0 ,
Qa, Qb = 0 .
One of the interesting features of N = 4 supersymmetric mechanics is its relation with the
Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations [1, 2]. The most natural
appearance of the WDVV equations is seen at the component level. As was first demonstrated
in [3], on the (2n+4n)-dimensional phase space
a, b = 1, 2, the simplest ansatz for the N = 4 supercharges Qa and Qa,
xi, pj , ψai, ψ¯jb , with i, j = 1, . . . , n and
Qa = piψia + iFi(j0k)ψibψj ψ¯ka
b
and
Qa = piψ¯ia + iFi(j0k)ψ¯i ψ¯jbψak ,
b
yields the WDVV equations
Fi(j0m) δnmFk(l0n) − Fi(l0m) δnmFk(j0n) = 0
with
Fi(j0k) = ∂i∂j ∂kF (0)(x)
for totally symmetric structure functions Fi(j0k), if one requires the supercharges to obey the
N = 4 super Poincar´e algebra
1 Introduction
Supercharges and Hamiltonian
One-dimensional SU(2|1) mechanics
2
3
4
5
4.2 Isotropic spaces
Conclusions
Introduction
The evaluation of the brackets in (1.3) assumed the standard Dirac brackets between the
basic variables,
2i δbaδij .
The simplest form (1.1) of the supercharges does not produce (classically) any potential
term in the Hamiltonian H. To generate physically interesting systems, the supercharges
have to be extended by terms linear in the fermionic variables. Such linear terms come
with new structure functions, so-called prepotentials, which obey differential equations
extending the WDVV ones. Prepotentials come in two variants, called W and U . The
– 1 –
variables [4]. Examples of such constructions can be found in [3, 5–8] and references therein.
So far we discussed N = 4 supersymmetric mechanics on the Euclidian space Rn.
Recently [9, 10], the structure given by (1.1)–(1.4) was generalized to N = 4 supersymmetric
mechanics on arbitrary Riemannian spaces, rendering it covariant under general coordinate
transformations. In this case, the WDVV equations (1.2) are superseded by the ‘curved
WDVV equations’ [9]
∇iFjkm = ∇j Fikm
and
FikpgpqFjmq − FjkpgpqFimq + Rijkm = 0
(1.5)
= 2i δbaH. This step deforms the N = 4 super Poincar´e algebra to an
su(2|1) algebra [11]. A systematic study of one-dimensional SU(2|1) supersymmetric
mechanics has been conducted in [12–15] using the superspace approach.
Our main goal is to construct n-dimensional SU(2|1) supersymmetric mechanics with
a (2n+4n)-dimensional phase space over an arbitrary Riemannian manifold within the
Hamiltonian approach.1 In section 2 we introduce generalized Poisson brackets which are
general coordinate covariant, write down the most general ansatz for the supercharges
(linear and cubic in the fermionic variables), and analyze the conditions on the structure
functions. These determine the structure functions and the explicit structure of the
Hamiltonian. In section 3 the known solutions [11, 12, 15] for one-dimensional SU(2|1) mechanics
are reproduced. Section 4 specializes on two examples corresponding to real K¨ahler and
isotropic spaces. For the first one, we provide exact supercharges and Hamiltonian for
so-called real K¨ahler spaces, generalizing the results of [18, 19] to SU(2|1) supersymmetry.
The second example, which relates to isotropic spaces, extends the solutions found in [10]
as well as gives explicit solutions for spheres and pseudospheres. A few comments and
remarks conclude the paper.
2
Supercharges and Hamiltonian
Our goal is to realize the su(2|1) superalgebra
i a
Qa, Qb = 2 δb H −µI ba +iµI 0δb ,
a
Qa, H = Qa, H = 0 ,
Iab, Icd = −ǫacIbd −ǫbdIac ,
Qa, Qb = Qa, Qb = 0 ,
I0, Qa =
Iab, Qc = −
i a
Q ,
2
I0, Qa = − Qa ,
i
2
(2.1)
2
1 ǫacQb +ǫbcQa ,
Iab, Qc =
2
1 δcaQb +δcbQa ,
1Particular cases of N = 2, 4 supersymmetric mechanics with weak supersymmetry and
(4n+4n)dimensional phase spaces have been considered in [16, 17].
– 2 –
with a constant deformation parameter µ on the (2n+4n)-dimensional phase space given
by n coordinates xi and momenta pi, with i = 1, . . . , n, each of which is accompanied by
four fermionic ones ψia and ψ¯jb = (ψjb)†. On the cotangent bundle over an n-dimensional
Riemannian manifold, the Poisson brackets between the basic variables are defined as
= −2iRijkmψakψ¯am .
2i δba gij ,
metric gij (x) defined in a standard way as
Γikj =
1 gkm (∂igjm + ∂j gim − ∂mgij )
Here, Γijk and Rjkl are the components of the Levi-Civita connection and curvature of the
i
and
Rijkl = ∂kΓijl − ∂lΓijk + Γjml Γimk − Γjmk Γiml .
R-symmetry, combining the two types of prepotentials used in [10]:
Qa = piψia + iWiψia + J acUiψci + iFijkψicψcj ψ¯ka + iGijkψiaψjcψ¯ck ,
Qa = piψ¯ia − iWiψ¯ia − JacUiψ¯ic + iFijkψ¯icψ¯jcψak + iGijkψ¯iaψ¯jcψck .
Here, ǫacWi and J acUi are associated with the U(1) and SU(2) parts of the R-symmetry,
generated by I0 and Iac, respectively. To realize the SU(2) currents J ac, one needs to
adjoin additional bosonic spin variables {ua, u¯a| a = 1, 2} [4] parameterizing an internal
two-sphere and obeying the brackets
{ua, u¯b} = −i δba ,
in terms of which these currents read
i
2
J ab =
uau¯b + ubu¯a
⇒
nJ ab, J cdo = −ǫacJ bd − ǫbdJ ac.
The structure functions Ui, Wi, Fijk and Gijk entering the supercharges (2.4) are, for the
time being, arbitrary functions of the n coordinates xi. In addition, by construction, Fijk
and Gijk are symmetric and anti-symmetric over the first two indices, respectively:
Fijk = Fjik ,
Gijk = −Gjik .
The requirement that the supercharges (2.4) span the su(2|1) superalgebra (2.1) results
in the following equations:
and
Gijk = 0,
Fijk − Fikj = 0
⇒
Fijk = F(ijk) ,
∇iFjkm − ∇j Fikm = 0 ,
FikpgpqFjmq − FjkpgpqFimq + Rijkm = 0
∇iWj − ∇j Wi = 0 and
∇iUj − ∇j Ui = 0 ⇒
Wi = ∂iW
and
Ui = ∂iU ,
∇iUj − UiUj − FijkgkmUm = 0 ,
∇iWj + FijkgkmWm + µ g ij = 0 ,
gij WiUj − µ = 0 or
Uj = 0 ,
– 3 –
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
and
∇iFjkl = ∂iFjkl − Γimj Fklm − ΓimkFjlm − Γiml Fjkm . (2.15)
Finally, the other generators of the su(2|1) superalgebra acquire the form
H = gij pipj + gij ∂iW ∂jW +
2
1 J 2gij ∂iU ∂j U + 4 ǫcd∇i∂jW −iJ cd∇i∂j U ψciψ¯jd
− 4 ∇mFijk+Rijkm ψicψ¯jc ψkdψ¯dm,
Iab = J ab + igij ψiaψ¯jb + ψibψ¯ja
and I0 = gij ψciψ¯jc ,
(2.16)
(2.17)
where the Casimir J 2 = J cdJcd plays the role of a coupling constant. The equation (2.9)
qualifies Fijk as a so-called third-rank Codazzi tensor [20], while (2.10) is the curved WDVV
equations [9], and (2.11)–(2.14) are the deformed analogs of the curved equations considered
in [10] and of the flat potential equations discussed in [6] and [8].
Two limiting cases are noteworthy. First, putting W = 0 implies via (2.13) that µ = 0,
bringing us back to the standard N = 4, d=1 super Poincar´e algebra — the case considered
in detail in [10]. The converse is not true: µ = 0 admits the simultaneous presence of both
U and W , as long as their gradients are orthogonal to each other. Second, putting U = 0
solves (2.12) and (2.14), and it removes the spin variables together with their currents J ab
from the supercharges, the Hamiltonian and the R-currents.
Summarizing, to construct SU(2|1) supersymmetric n-dimensional mechanics on a
Riemannian manifold with metric gij , one has to
• solve the curved WDVV equations (2.9), (2.10) for the fully symmetric function Fijk,
• find the admissible prepotentials W and U as solutions to the equations (2.11)–(2.14).
In the following we shall use this procedure. To begin with, let us demonstrate how the
known particular cases of one-dimensional SU(2|1) mechanics fit into our scheme. Then we
shall investigate two special geometries allowing for explicit solutions of the curved WDVV
equations.
3
One-dimensional SU(2|1) mechanics
In the distinguished case of a one-dimensional space the metric is always flat and can be
fixed to g11 = 1 without loss of generality. Therefore, the curved WDVV equations become
trivial and put no restrictions on the single remaining component F111.
The n = 1 variant of (2.12)–(2.14) reads
U ′′ − F111U ′ − U ′2 = 0 ,
W ′′ + F111W ′ + µ = 0 ,
W ′U ′ − µ = 0 or
U ′ = 0 , (3.1)
where ′ means differentiation with respect to the single variable x
1 = x. These three
equations are not independent. For U ′ 6= 0, the two second-order equations follow from
– 4 –
each other via W ′U ′ = µ . In this generic situation, we have the freedom to freely dial one
function. The choice of any one structure function determines the other two:
and
U ′ = µ/W ′
U ′ = −eF11 / ∫x eF11
or
with
W ′ = µ/U ′
,
′ 2 + 4 ǫcdW ′′ − iJ cdU ′′ ψcψ¯d − 4F1′11ψcψ¯c ψdψ¯d ,
which may be expressed purely in terms of either W ′, U ′, or F11 via (3.2) or (3.3).
Three different limits can be taken. First, W ′ = 0 yields µ = 0. However, µ = 0
admits two disjoint solutions,
W ′ = 0 and
U ′ = −eF11 / ∫x eF11
or
U ′ = 0 and
W ′ ∼ e−F11 .
Second, U ′ = 0 removes the spin variables, and the Hamiltonian reduces to
which has been constructed in [11, 12]. Third, F111 = 0 leads to
H = p2 +
W ′ 2 + 4 W ′′ψaψ¯a + 4
W ′′ + µ
W ′
′
ψaψ¯a ψbψ¯b ,
W ′ = −µ (x−x0) and
U ′ = −1/(x−x0) ,
Fi(j1k) = Γijk
and
Fi(j2k) = −Γijk .
– 5 –
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(4.1)
(4.2)
which has been found in [15]. In this case the supercharges become linear in the fermions.
4
Examples of n-dimensional mechanics with potentials
Once we start to consider the n-dimensional mechanics, the first problem is to solve the
curved WDVV equations (2.8)–(2.10). The general solution of these equations is unknown,
but in some exceptional cases the solution can easily be constructed. Solving thereafter
a system of differential equations (2.11)–(2.14), one can explicitly find the corresponding
potentials.
4.1
Real K¨ahler spaces
The first example of multidimensional mechanics concerns the so-called ‘real K¨ahler
spaces’ [18, 19], which are defined by a metric of the form
gij =
∂2G
∂xi∂xj
⇒
Γijk =
1
∂3G
2 ∂xi∂xj ∂xk
determined by a scalar function G. It is rather easy to check that two solutions of the
curved WDVV equations for such a metric are
With this input the equations (2.11)–(2.14) drastically simplify and can be solved
explicitly as
W (1) = −µ G
+ λixi
W (2) = −µ xi∂iG − G + λi∂iG
and
and
U (1) = − log σi∂iG ,
U (2) = − log σixi ,
where λi and σj are constants subject to the condition
with r2 = δij xixj
⇒
Γikj = −
xiδjk + xj δik − xkδij
f
′
rf
gij =
1
f (r)2 δij
respect to r. The ansatz
with a positive real function f , where (in this subsection) ′ means differentiation with
Fijk = a(r) xixj xk + b(r) δij xk + δjkxi + δikx
j + f (r)−2Fi(j0k)
extending an arbitrary solution Fi(j0k) of the flat WDVV equations (1.2) obeys the curved
WDVV equations if xiFi(j0k) = δjk,
a =
2f f − rf ′ ± 2f 2 − 3rf f ′ + r2(f ′)2 + r2f f ′′
r4f 3 f − rf ′
and b = −
f ± f − rf ′
r2f 3
. (4.8)
Thus, we have a family of n-dimensional SU(2|1) mechanics defined on any real K¨ahler space.
spaces. Let us remind that in [9] a large class of solutions to the curved WDVV
equations (2.8)–(2.10) has been constructed on isotropic spaces. The metric of such a manifold
is SO(n) invariant, i.e. it admits 12 n(n−1) Killing vectors and can be written in the form
U = log
µ f 2
µ r 2 − 2αf 2
If we choose the minus sign in the above expressions, i.e. for
a =
f f ′ − r(f ′)2 − rf f ′′
r3f 3 (f − rf ′)
and b = −
f
′
rf 3
,
then a prepotential W solving (2.13) is easily constructed,
W = w(r, µ ) + W (0)
with
w′(r, µ ) =
α(f 2)′ − µ r
2 f (f − rf ′) ,
where α is some constant and W (0) obeys the flat equation
∂i∂j W (0) + Fi(j0m) δmn∂nW (0) = 0
subject to
xi∂iW (0) = α .
This extends the prepotential solution found in [10] to µ 6= 0. To this configuration one
may add a simple solution to (2.12) for a prepotential U respecting also (2.14),
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.9)
(4.10)
(4.11)
(4.12)
The prepotentials W and U above generate in the Hamiltonian the bosonic potential
V = f 2 ∂iW (0)∂iW (0) + (µ r −2αf f ′)(µ r 2 −4αf 2 +2αrf f ′)
4 r (f −rf ′)2
+2J 2 r2µ 2(f −rf ′)2
(µ r 2 −2αf 2)2
.
An interesting case is the (pseudo)sphere, f = 1 + ǫ r2 with ǫ = ±1. For this manifold,
the potential reads
Vsphere = (1+ǫr2)2 ∂iW (0)∂iW (0) +
with the Higgs-oscillator potential [
21, 22
]
(µ −8ǫα)2
16ǫ
VHiggs −
For J 2 = 0 or µ = 8αǫ, simplifications occur,
Vsphere µ =8αǫ = (1+ǫr2)2 ∂iW (0)∂iW (0) − 4α2ǫ + 8ǫ J 2 (VHiggs−1) .
5
Conclusions
We extended the previous analysis [10] of N = 4 supersymmetric mechanics on arbitrary
Riemannian spaces to systems from N = 4, d=1 super Poincar´e symmetry to SU(2|1)
supersymmetry. The extension is parametrized by a deformation parameter µ , which only enters
in the equation determining the prepotential W and relating it with the prepotential U .
All other equations, in particular the curved WDVV equations [9], and the form of the
supercharges, R-currents and Hamiltonian are unchanged.
A novel feature in our consideration is the presence of both types of prepotentials, W
and U , associated with the U(1) and SU(2) parts of the R-symmetry, respectively.2
Two special geometries have been considered in detail. Real K¨ahler spaces admit an
explicit solution for all structure functions. On isotropic spaces, we constructed admissible
structure functions for any conformally invariant solution to the flat structure equations.
As an application, a Hamiltonian potential for SU(2|1) supersymmetric mechanics on a
(pseudo)sphere was presented. All known one-dimensional systems enjoying SU(2|1)
supersymmetry [12, 15] can be easily reproduced in our framework.
One future task even on flat space is a classification of admissible potentials when both
prepotentials, W and U , are present. At the moment we can do this only for the special
case when one of them depends on r only. Another interesting question is whether there
exist other geometries besides the real K¨ahler case which admit a fully explicit solution.
Since the real K¨ahler spaces unambiguously arise in the superfield approach [18, 19], it
seems compelling to perform a superspace description of the mechanics presented here. To
this end, it is unclear whether the standard superspace is sufficient or whether we have to
employ the deformed one introduced and advocated in [12, 15].
2This is actually also possible in the super Poincar´e limit, but requires their gradients to be mutually
orthogonal.
– 7 –
This work was partially supported by the Heisenberg-Landau program. The work of N.K.
and S.K. was partially supported by RSCF grant 14-11-00598, the one of A.S. by RFBR
grants 18-02-01046 and 18-52-05002 Arm-a. This article is based upon work from COST
Action MP1405 QSPACE, supported by COST (European Cooperation in Science and
Technology).
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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