CC BY
35
UDC 530.1; 539.1
Landau models with both worldline and target super symmetries: N = 2 and N =4 examples
E. Ivanov
Bogoliubov Laboratory of Theoretical Physics, JLNR, 141980 Dubna, Moscow Region, Russia.
E-mail: [email protected]
We present a synopsis of superextended Landau models possessing both worldline supersymmetry and graded internal symmetry acting in the target space. The main focus is on the recently constructed model with the worldline N = 4 supersymmetry and the target ISU(212) symmetry.
Keywords: supersymmetrv, Landau levels, superspace.
1 Introduction
The original quantum Landau model fl] describes
a charged particle moving on a plane orthogonal to a
constant uniform magnetic flux. Its generalization is
the spherical Landau-type model [2] which describes a
charged particle on the 2-sphere S2 ~ SU(2)/U(1) in
the Dirac monopole background.
Superextensions of the Landau and Haldane models
deal with non-relativistic particles moving on super-S2
S2
1. Landau problem on the (2 + 2)-dimensional supersphere SU(2|1)/U(1|1) [3,41;
2. Landau problem on the (2+4)-dimensional superflag SU(2|1)/[U(1) x U(1)] [4,5].
S2
dau models [6-9]. Most surprising feature of the super planar Landau problems is the hidden world-line N =2 supersymmetry. Thus, the super planar Landau models simultaneously provide a class of super-symmetric quantum mechanics (SQM) models. SQM models [10] have a plenty of applications in diverse domains.
A natural approach to constructing super planar Landau models is as follows. One takes the notion of N = 2
reproduces the planar Landau model and its most genN = 2
superfield formalism [9,11]. Recently, this “bottom-up” approach was applied for constructing the first exam-
N = 4
supersymmetry [12]. We found the target space supersymmetry ISU(2|2) as a natural generalization of the ISU(111) symmetry of the N =2 case. The present talk is devoted to a short account of this construction, with collecting, as a prerequisite, some salient facts
N = 2
dau models.
Note that sigma models with the supergroup target spaces received much attention for the last years, in particular, in connection with superbranes (see, e.g., [13-15]).
2 Bosonic Landau models
The Lagrangian and Hamiltonian of the planar bosonic Landau model are given by the following expressions
Lb = |z|2 - in (zz — zz) = |z|2 + (Azz + Asz) , (1)
where
Az = -
and
,A,
d?Az — dz A¡f = — 2ík
1
(2)
Hb = — (a^a + aa^ = a) a + k , where
a = i(dz + kz), a^ = i(dz — kz), [a, a^] = 2k .
The invariances of the model are the “magnetic translations” and 2D rotations generated by
Pz = —i(dz + kz), Pz = —i(dz — kz), Fb = zdz — zdz,
[Pz, Pz] = 2k, [H, Pz] = [H, Pz] = [H, Fb] = 0.
The n-th Landau level (LL) wave function is defined
as
z) = [i(dz - Kz)]ne K|z|2^(n)(z),
Htf(n) = K(2n +1)tf(„) .
Each LL is infinitely degenerate due to (Pz ,Pz) invari-
ance.
S2
SU(2)
1 |z|2 + is 1 2 (zz — zz), (3)
Lb
(1 + |z|2)2
1 + |z|2
where the 2nd term is the d =1 Wess-Zumino term on the coset SU(2)/U(1) - S2.
The relevant wave functions are finite-dimensional SU(2) irreps, s, s +1, s + 2,... s + £ being their “spins”. The energy spectrum is determined by the formula
and OG is mother operator such that [H, Og] = 0. The symmetry generators that do not commute with G thus generate, in general, additional “hidden” symmetries.
In our case G commutes with all ISU (111) generators, except Q Q*, hence
Ee = i(2s + i + 1) + 2s , i = 0,1, 2,... .
(4)
Each LL is finitely degenerated since wave functions are SU(2) irreps. Redefining z ^ rz, H ^ Hr2, sr2 = k, where r is the “inverse” radius of S2, in the limit r ^ 0, with k fixed, one recovers the planar Landau model.
3 Planar super Landau models
Planar super Landau models are the large radius limits of the supersphere and superflag Landau models. In this limit, the supersphere SU(2|1)/U(111) goes into an (2 + 2)-dim. superplane.
The Lagrangian and Hamiltonian of the superplane Landau model read:
l = Lf + Lb = izi2 + cc— *k (zc — Cz + CC + CC
H = a^a — a^a = dz — dz dz + k (c<9z + Cd; — zdz — Ç0() + k2 (zc + Cc) .
The invariances are generated by Pz, Pz, n = dz + kC, n^ = + kZ and the new generators
Q = zdz — , Qf = -z^z + Çdz
C = zdz + C^z — C<9Z — (J’dz .
(5)
They form the algebra of the supergroup ISU(111), SU(2|1)
{Q, Qt} = C, [Q, P] = in, {Qt, n} = iP . (6)
The natural ISU(1|1)-invariant inner product
< ^|0 >= jd^ ^ (z, Z; Z, C) 0 (z, Z; Z, C) , d^ = d2zd2Z,
leads to negative norms for some component wave functions. All norms can be made positive by introducing the “metric” operator:
G = _ [dc+ k2Cz + k (Zdc — Cdz)] ,
Q* = Q*------S, S = i (dzdz + k2zZ — kz^z — kZ^z)
S = a*a , S* = aa* .
The operators S, S*, H form N = 2, d =1 superalgebra
{S, S*} = 2kH, {S, S} = {S*,S*} = 0,
[H,S] = [H,S *]=0.
The LLL ground state is annihilated by ^d S*:
S0(o) = S *0(o) = 0,
N = 2 N= 2
is unbroken and all higher LL form its irreps.
N = 2
One can recover the planar super Landau model
N = 2
worldline supersymmetry as an input [9].
The starting point is N=2, d=1 superspace in the left-chiral basis, (0, z, t = t + i#). The basic objects are N=2, d=1 chiral bosonic and fermionic superfields $ = z(t) + 0x(t), C = C(t) + 0h(T), with x(t) and h(T) being auxiliary fields. The superfield action yielding the superplane model action is:
S = — kJ dtd20 {$$ + CC + p [$DC — $DC]} ,
where p = 1/(^v/k), k = 0. On shell, the fields ^d x are expressed as x = i/V^Z, h = i/v/«Z. In terms of physical fields:
S ^ I dt îk fzz — zz + CC — cc) + fzi + CC
This approach triggered the idea [11] to construct N = 2 N = 2
Sgen = / dtd2^K($, $) + V($, $)C’Z
<< ^|0 >>- J d^ (G^) 0 .
Note that H commutes with G, so H = H* = H*. However, the hermitian conjugation properties of the
G
O be a symmetry generator, such that [H, O] = 0. Then
O* = GO*G = O* + GOG , OG = [G, O],
+ p ($DC — $DC)
dtC.
(7)
This action involves two independent superfield potentials, K($, $), V($, $). Eliminating auxiliary fields in
$ = z + ... , C = 0 + ..., one gets
Lcomp = V-1 zZ + i (zKz — ZKz) + (0 — terms). (8)
VK produces a background super gauge field.
N= 4 N = 4
supersymmetry
N = 4 N= 2
tion is written in bi-harmonic superspace (bi-HSS) [16]: respects “internal” supersymmetry which is realized by
■ , r differential operators in the target (4 + 4) superspace
Sn=4 = — J M-2’0q(1’0)Aq(1’0)B Cab (/*a,x“b )• Their set involves:
— i J ^0’-20(0’1)A0(0,1)B A- “Magnetic” supertranslations:
. 1 [ -2,0 (1,0)AD1,-1 ,(0,1)B \ /qn PjA = —idf+ kCab, n“A = + KX“A ,
+ VK J M q 0 ^w [P*a,Pjb] = 2K£jjCab , {noA, nbB} = 2^^ .
It involves two bi-harmonic superfields, q(1,0)A and
0(0,1)B
of the N = 4, d = 1 bi-HSS. Without entering into ^ !
details, the superfields q(1,0)A, 0(0,1)B amount to the Qj0 = -(iCA — ¿A)x“Bdf a +—(iCA + ¿A)/-BdxA
following sets of the off-shell components: 2 * 2 1
,»,0>A =* (/-a, 0aA„)), 00,.A ^ (x»A(f)j ,,A(()) . №“ Qjb} = «T(* j, — ¿j^« Z,
iQ*“ Qjbl = 0
The fields (/jA, x“A) are physical, while (0oA, hjA) are , .
auxiliary. The off-shell component lagrangian reads: Here, Z = Cj|(/jBdf¿a + xoBdx«A) is U(1) gener-
• iA B 0A Bn ator, Tj k,T“ b are generators of two automorphism
L « (2i/ /j — 0 0a )Cab SU(2) groups acting on the doublet indices i and a.
+ (2X“AXB — ih-AhB)£ab The fifteen generators Qj0,QJb,Tj k,T“ b, Z form
2i / ¿iALB . i “A ■ B\ im\ ^he superalgebra su(2|2), a graded version of su(4).
The full target space superalgebra is
----r- (/*Ahf + 0“Axf )eAB . (10)
After eliminating auxiliary fields as hjA = — ^ /*A (P-a, noA) x SU(2|2) = ISU(2|2).
0oA = TK Cab xB » one obtains It is a natural generalization of the target ISU (1|1)
iA B 0A N = 2
L = kCab / /B — iKXoAXoA
1 iA 0A B
+ 2 + ¿CabX“AXB) . (11) 7 Quantization
2 ./ iA 1 - - ab /v. /V.a
The Lagrangian (11) involves the Lorentz force-type We use the complex fields:
z = /11, u = /21, C = X11, £ = X21, and c.c.,
B
This field is self-dual:
FiAjB : = diAAjB — djB^¿A = —2kCAB.
The bosonic sector of (11) corresponds to the model
coupling to the external gauge field: = . 11 = .21 Z = 11 21
z — f , u — f ,z — x , ~
AjB/jB , AjB = —kCbd/iD . L = |z|2 + |u|2 — iK(zZ — Zz + uu — uu)
+ ZZ+el — iK(ZZ+zz+ee+ee).
The quantum Hamiltonian reads
Hq = az az + aUa„ — a^ a^ — a^.a^ .
used in [17] to describe U(1) quotum Hall effect on a = i(— + Kz) a = i( — + Ku)
r4 z dz du
d d
. az = dZ — кZ, a? = dZ — Ke
6 Symmetries of N = 4 model
The action (11) respects N = 4 supersymmetry:
[az , az] aU] 2к,
{az, az} = {a^, a^} = —2k.
J/iA =______— eiaCABXaB, ^X°A =___________— eia/A , The N = 4 supercharges read:
where ej0 are four Grassmann parameters. The corresponding conserved Noether supercharge is
S11 := S2 = —f= (aî az — a^az ),
V k ç
?21 := S = _L (at a + at a
Sia = ^CAB XaA/iB ift ch = Si
/k
S := S1 = (a^ au + az az ),
V k
{S1,Si} = {S2,S|} = —2Hq
(the sign will change after redefining the inner prod- there are negative norms. Let C(lm), l, m = 0,1 be
uct). In the covariant notation: a wave function with one or two fermionic quanta a|
{Sio, Sjb} = 2eijeobHq. and a^ . Then
For Sio and Hq there exists a Sugawara representation (C(I,m)|C(I,m)) - ( — 1)I+m(||D(I,m)||2 + 2K||B(I,m)||2
2Vk(Q*o + Q *o) i Pj noA + 2к||C(г,m)|| + 2k ||A(г,m)|| ),
1 i ||/1|2 := i dzdzdudue 2k|z| 2k|u| /(z,u)/(z, u) .
H = 2PiAP*A + ^CAB nBnoA + 2KiZ j
The states with C(1 0^d C(0 1) have negative norm. One can also define one more N = 4 supersymmetry To cure this> one redefines tlie inner product as in
Sia = 2*V'K(Qia - Qia) + ПВCAB, theN 2caSe
((0ІФ)) := (G0|^),
{S*“' = 2e®j £“b Hq. ((W)) (
The closure of the two N = 4 superalgebras is the G = (1 — 2nC)(1 — 2n5)’ nC£ := 2K .
worldline su(2|2):
The metric operator G possesses the standard proper-{Sia, Sjb} = 8iK (>Ьу' - eijT“b) , (12) ties
and T“b being the й-symmetry SU(2) generators.
[Hq, G] = 0 , G2 = 1.
The wave functions are defined as follows: With the new definition of the hermitian conjugation,
A. The lowest Landau level (LLL), HC° = ° Hq becomes manifestly positive-definite:
C0 = e-KK 00, K = |z|2 + |u|2 + ZC + el 00(z, u, Z, e) = A0(z, u) + ZB0(z, u)
+ eC0 (z, u) + ZeD0 (z, u). 8 Generalized N = 4 models
The LLL wave function has a four-fold degeneracy: ^,
A0, B0, C0, D0 are closed under ISU(2|2) It is a sin- = 4 actlon of the
i . r Kr a 4- supernelds q(1,0)A and 0(0,1)A is
N = 4
B. Next LLs, H C(N) = 2kN C(n ), wit h N > 0 Sse" = ^^2'°l2'°((1{1'0)A,u,v)
Introducing the SU(2) covariant notation an = — ^M0,-20(0,1)A0(0,1)B£ab
(az a„),; ao = (a^ a^) „ [a|,aj] = 2kJj, {ao,ab} =1 ,■ ^
2k^ , the wave function C(N) can be constructed as + VK / M-2,0q(1,0)AD1,-10(0,1)B£abJ ,
C(N) = a|. a| ...a1 )e-KK^(ili2"'iN)(z,u,Z,C) u r20- u-. r .■
(i1 *2 -n) where L2t0 is an arbitrary function of its arguments.
. tat at at e-kK0o(i1i2...iN-1)(z u z e) After going to components and eliminating auxiliary
I C-a o ( * (an ...^an , (u, Z , )
(i1 2 N-1) fields one obtains the on-shell Lagrangian
+ (a*)2a|. a1 ...a1 e-KK^(n1j2...jN-2).(z, u, Z, £) 1
(j1 *2 *N-2) r _ 1
L = f AiA~ iKXoAXoA I— f /a
r-f-il /■*]• f* • i^jl J *—A /v /VoA I ¿-v J J nA
2
worldline N = 4. The N = 4 supermultiplet for N-th + i K (g-1 )abxoAxB .
SU(2) 2 o
Here
2, s2 = “,s3 = ~^J . W t _ f _1 dL2
N N -1 N - 2\
si = IT> s2 = , S3 = . (13)
A*a(/) = -^ dudvu
0
^ ^ df(1,0)A 5
The degeneracy of the N-th level is j .
^ K ^ d2L2,0
4[(2s1 + 1) + 2(2s2 + 1) + (2s3 + 1)] = 16N. (14) gab(/) = 2J dudv d/(1,0)Ad/(1,0)B .
Like in other super Landau models, with the stan- Tfae gauge feld ^(/) by construction is self-
dard dennition of the inner product on r4
(Ф|0) = J ^Мф(/,х)0(/,х), = d4/d4X
FiBjA := AjA - djA= -2GAB%' .
In the bosonic sector we obtain some generalization of the Elvang, Polchinski U (1) model in R4, with an arbitrary self-dual external gauge field. All other terms in the Lagrangian can be brought to the same form as in the original N = 4 super Landau model. How to obtain non-trivial target metric as in the general N =2 Landau model? This is an open question. Perhaps one should make use of some nonlinear versions of the N = 4
9 Outlook
Among possible physical applications of the super
N = 4
model), as well as the problems for the further study, we would like to distinguish the following ones.
These models might constitute a basis of possible supersymmetric versions of the Quantum Hall Effect (QHE) in diverse dimensions. For instance, as follows
R4
N = 4 N = 4
Landau model. Also, we expect a close relation of the models considered to integrable structures in the planar N = 4, d = 4 SYM theory and string theory. Indeed, the integrable su(2|2) mid su(3|2) spin chains play an important role in these theories [19,20], and it would be hardly accidental that the supergroups of
similar type appear as the target space symmetries in the quantum-mechanical super Landau models. In this context, it is of clear interest to construct and study N = 4
on the supercoset SU(3|2)/U(2|2) — C(2|2). Our planar model should be reproduced from such a curved system in the contraction limit R ^ to. The SU(3|2) system can be treated as a superextension of one of the SU(3)/U(2)
the four-dimensional QHE.
It is interesting to generalize our consideration to higher N worldline supersymmetries (e.g. N =8) and N = 4
based on other known off-shell N = 4, d =1 multiplets, e.g. (3, 4,1) Mid (2, 4, 2). The supersymmetric Landau-type models with couplings to external non-abelian gauge fields (introduced by methods of ref. [22]) also present an ambitious subject for the future investigations.
Acknowledgement
The author thanks the Organizers of the Tomsk Conference QFTG’2012 for inviting him to present this talk. This research was supported by the RFBR grants Nr. 12-02-00517 and Nr. 11-02-90445.
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Received 01.10.2012
E. Иванов
МОДЕЛИ ЛАНДАУ С СУПЕРСИММЕТРИЯМИ НА МИРОВОЙ ЛИНИИ И В ПРОСТРАНСТВЕ ОТОБРАЖЕНИЯ: N = 2 И N = 4 ПРИМЕРЫ
Представлен синопсис суиеррасширеиных моделей Ландау, обладающих как суперсимметрией на мировой линии, так и градуированной внутренней симметрией, реализованной в пространстве отображения. Основное внимание уделено недавно построенной модели с мировой N = 4 суперсимметрией и ISU(2|2) симметрией пространства отображения.
Ключевые слова: суперсимметрия, модель Ландау, суперпростраиство.
Иванов Е. А., доктор физико-математических наук, профессор.
Объединённый институт ядерных исследований.
Ул. Жолио-Кюри, Об, Дубна, Московская обл., Россия, 141980.
E-mail: [email protected]
