Hidden bosonized supersymmetry,
finite-gap systems, and tri-supersymmetry
Mikhail Plyushchay
Universidad de Santiago de Chile
Dubna, SQS-09
1
Supersymmetry as a fundamental symmetry
(Golfand, Likhtman, 1971; Volkov, Akulov, 1972; Wess, Zumino, 1973)
still waits for experimental confirmation: LHC (?)
Supersymmetric quantum mechanics (Witten, 1981): as a toy model
to investigate SUSY breaking in field theory.
⇒ SUSY QM: atomic, nuclear, condensed matter and statistical physics.
Nuclear physics: theoretical prediction of a dynamic supersymmetry
(Iachello, 1980) that links the properties of some bosonic and fermionic nuclei
→ experimental confirmation: 1999.
2
• Are there any physical systems whose peculiar properties could be associated
with a presence of a hidden supersymmetry?
3
Supersymmetry ⇔ Lie superalgebra
Generic structure:
Γ – Z2-grading operator, Γ2 = 1,
Bi – bosonic (Z2-even), Fα – fermionic (Z2-odd) generators:
[Γ, B] = 0, {Γ, F} = 0,
[B, B] ∼ B, [B, F ] ∼ F, {F, F} ∼ B
States: ΓΨ± = ±Ψ±, Ψ+ – bosonic, Ψ− – fermionic
4
D = 1, N = 2 SUSY QM:
H =
(H+ 0
0 H−
), Q1 =
(0 Q−
Q†− 0
), Q2 = iσ3Q1, σ3 =
(1 0
0 −1
),
H – Hamiltonian (second order operator), Q1,2 – supercharges (first order opera-
tors), Γ = σ3 – Z2-grading operator,
[H, Qa] = 0, {Qa, Qb} = 2δabH, [σ3, H ] = {σ3, Qa} = 0.
⇒ energy levels: E ≥ 0,
[H, Qa] = 0 ⇔ Q−H− = H+Q− (Darboux transformation) ⇒ H− and H+, are
(almost) isospectral: all the energy levels (in a non-periodic case) with E > 0 are
doubly degenerate; energy level E = 0, if exists, is nondegenerate.
5
Bosonized D = 1, N = 2 SUSY QM, MP, 1996, Ann. Phys. 245, 339:
H =1
2
(− d2
dx2+ W 2(x)−W ′(x)R
),
Q1 = −i
(d
dx+ W (x)R
), Q2 = iRQ1,
{Qa, Qb} = 2δabH, [Qa, H ] = 0.
W (−x) = −W (x) – odd superpotential, Γ = R – reflection (parity)
operator, Rψ(x) = ψ(−x), H is a nonlocal operator,
R2 = 1, [R, H ] = 0, {R, Qa} = 0.
6
Hidden bosonized SUSY in quantum mechanical systems with local Hamiltonian:
• F. Correa, MP, 2007, Annals Phys, 322, 2493 :
bound states Aharonov-Bohm effect:
H = (pϕ + α)2, Q1 = pϕ + α, Q2 = iΓQ1,
α ∈ Z: Γ = Rϕ, exact SUSY,
α + 12 ∈ Z: Γ = e2iαϕRϕ, broken SUSY;
Dirac delta potential problem,
V (x) = 2βδ(x) + β2: both supercharges are nonlocal, Q1 = −i(
ddx + βRε(x)
),
β < 0 (β > 0) – SUSY is exact (broken);
reflectionless Poschl-Teller potential problem,
V (x) = −n(n + 1) sech2(x): nonlinear SUSY,
{Qa, Qb} = 2δabPn(H), Pn is a polinomial of order n
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• F. Correa, L.-M. Nieto, MP, 2007, Phys.Lett. B 644, 94:
periodic quantum system described by the finite-gap Lame equation,
H = − d2
dx2+ n(n + 1)k2sn2 (x, k) .
SUSY is nonlinear ,
Q1 is a local differential operator of order 2n + 1,
Q2 = iΓQ1, Γ = R,
{Qa, Qb} = 2δa,bP2n+1(H),
P2n+1(H) is a spectral polynomial.
8
Finite-gap systems (Drach, 1919; Burchnall, Chaundy, 1922) underly periodic
solutions of nonlinear integrable systems:
Korteweg-de Vries, nonlinear Schrodinger, Kadomtsev-Petviashvili, sine-Gordon
equations, etc. (Novikov, Dubrovin, Matveev, Krichever: algebro-geometric
method)
They are analytically solvable systems. Applications:
the modeling of crystals, the theory of monopoles, instantos and sphalerons,
classical Ginzburg-Landau field theory, Josephson junctions theory, magnetostatic
problems, inhomogeneous cosmologies, preheating after inflation modern theories,
string theory, matrix models, supersymmetric Yang-Mills theory, AdS/CFT duality.
9
♦ F. Correa, V. Jakubsky and M. P., 2008, J. Phys. A41: 485303
Hill’s equation:
HΨ(x) = EΨ(x), H = −D2 + u(x), D =d
dx,
u(x) is a real smooth periodic potential, u(x) = u(x + 2L).
Monodromy (translation for the period) operator
TΨ(x) = Ψ(x + 2L), [T, H ] = 0.
The 2× 2 monodromy matrix M(E),
Tψa(x; E) = ψa(x + 2L; E) = Mab(E)ψb(x; E),
where ψa(x; E), a = 1, 2, is some real basis of solutions.
10
Change of the basis, ψa(x; E) → ψa(x; E) = Aabψb(x; E), det A 6= 0, generates
a conjugation M(E) → M(E) = AM(E)A−1.
⇒ Discriminant
Tr M(E) = Tr M ≡ D(E) ∈ Ris invariant, as well as det M(E) = 1, and its eigenvalues µ, given by
det(M(E)− µI) = 0 ⇒ 1−D(E)µ + µ2 = 0,
µ1,2(E) =1
2D(E)±
√D(E)2/4− 1, µ1µ2 = 1.
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[T, H ] = 0 ⇒ common basis (Bloch-Floquet functions) ψ±(x; E),
Tψ±(x; E) = exp(±iκ(E))ψ±(x; E),
where a quasi-momentum κ(E), µ1,2(E) = exp(±iκ(E)), is
2 cos κ(E) = D(E).
• |D(E)| > 2 — prohibited bands (energy gaps),
E ∈ (E2i−1, E2i), i = 0, . . ., E1 = −∞,
• |D(E)| ≤ 2 — permitted bands,
• D(E) = +2(−2) ⇒ periodic (anti-periodic) states,
D′(E) = 0 — doublet, D′(Ei) 6= 0 — singlet.
12
+2
−2
E
D(E)
E0
0
E1
1
E2
1
E3 = E4
2, 2
E5
3
E6
3
E7
4
E8
4
E9 = E10
5, 5
Figure 1: The discriminant D(E) in a generic situation of a periodic potential
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Any n-gap system has an integral A2n+1 = −A†2n+1,
A2n+1 = D2n+1 + cA2 (x)D2n−1 + cA
3 (x)D2n−2 . . . cA2n(x),
where cAi (c) are real coefficients, [A2n+1, H ] = 0 ⇔ cA
1 (x) = 0.
(A2n+1, H) = Lax pair of the nth order KdV equation.
Burchnall-Chaundy theorem ⇒
−A22n+1 = P2n+1(H), P2n+1(H) =
2n∏j=0
(H − Ej),
where Ej are energies of band-edge singlet states Ψj.
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Consider parity-even finite-gap systems with u(x) = u(−x)
⇒ Reflection (parity) operator R, Rx = −xR, R2 = 1, is a non-local integral.
⇒ Singlet states have definite parities ⇒ {R, A2n+1} = 0 ⇒ Define
Z = Z1 ≡ iA2n+1, Z2 ≡ iRZ.
⇒ Hidden bosonized nonlinear N = 2 supersymmetry of order 2n + 1:
{Za, Zb} = 2δabP2n+1(H), [H, Za] = 0, [R,H ] = {R, Za} = 0.
• Any n-gap parity-even system is characterized by a hidden bosonized nonlinear
supersymmetry of order 2n + 1, for which the parity operator plays a role of
the Z2-grading operator.
15
N = 2 nonlinear SUSY ⇔ Crum-Darboux transformation generated by Ak =
Dk +∑k
j=1 cAj Dk−j, that annihilates a space V = span {ψ1, . . . , ψk}, ψi are
eigenstates (not obligatorily to be physical) of the Hamiltonian H
⇒ there holds the intertwining relation (W – Wronskian):
AkH = HAk, H = H + 2(cA1 )′ = H − 2(ln W (ψ1, . . . , ψk))
′′.
⇒For an n-gap system,
if k = 2r and ψi are r pairs of singlet states (Ψ2j−1, Ψ2j) at the edges of r
prohibited bands,
or if k = 2r + 1, and the set ψi as before plus the ground nodeless singlet state
Ψ0
⇒ H is isospectral to H.
If generating operator is Z, W = const 6= 0, ⇒ H = H ⇒ [Z, H ] = 0.
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⇒ For an n-gap system H, one can construct 2n − 1 isospectral systems.
For every pair (H, H),
Z = Q†−Q+ = Q†
+Q−, Z = Q−Q†+ = Q+Q†
−[Z, H ] = 0, [Z, H ] = 0, where Q+ (Q−) is even (odd) order differential operator
that annihilates even (odd) number 2r (2(n− r) + 1) of band-edge singlet states
chosen in accordance with the specified rules.
17
Super-extended system:
H =
(H 0
0 H
), Q± =
(0 Q±
Q†± 0
), Z =
(Z 0
0 Z
),
[H,Z ] = [H,Q±] = [Q+,Q−] = [Z,Q±] = 0,
Z = Q−Q+ = Q+Q−,
Z2 = PZ(H), Q2+ = P+(H), Q2
− = P−(H),
PZ(H) =
2n+1∏j=1
(H−Ej), P+(H) =
2r∏j=1
(H−E+j ), P−(H) =
2(n−r)+1∏j=1
(H−E−j ),
PZ = P+(H)P−(H).
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Trivial mutually commuting integrals
Γ1 = σ3, Γ2 = R, Γ3 = Rσ3,
Γ2i = 1, i = 1, 2, 3, Γ1Γ2Γ3 = 1.
⇒ Any of Γi can be chosen as a Z2-grading operator Γ.
All nonrivial and trivial integrals generate a tri-supersymmetry = a nonlinear
Grading operator σ3 R σ3RBosonic integral Z Q+ Q−Fermionic integrals Q+,Q− Z, Q− Z, Q+
deformation of su(2|2), with H playing a role of a multiplicative central charge.
⇒ For the first time such a structure was observed in the N = 2 super-extended
Dirac delta potential problem: F.Correa, L.M. Nieto, M.P., 2008, Phys. Lett. B
659, 746
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Γ = σ3: a SUSY subalgebra generated by local integrals H, Z and
Q(1)± = Q±, Q(2)
± = iσ3Q±,
is identified as a centrally extended nonlinear N = 4 supersymmetry,
{Q(a)+ ,Q(b)
+ } = 2δabP+(H), {Q(a)− ,Q(b)
− } = 2δabP−(H),
{Q(a)+ ,Q(b)
− } = 2δabZ,
[H,Q(a)± ] = [H,Z ] = [Z,Q(a)
± ] = 0,
in which Z plays a role of the bosonic central charge.
20
The supercharges Q(a)+ annihilate a part of bangde-edge states organized in su-
persymmetric doublets, another part of supersymmetric doublets is annihilated by
Q(a)− . The band-edge states which do not belong to the kernel of the supercharges
Q(a)+ (or Q(a)
− ) are transformed (rotated) by these supercharges within the corre-
sponding supersymmetric doublet. The bosonic central charge Z annihilates all
the band-edge states.
⇒ Spontaneously partially broken centrally extended nonlinear N = 4 supersym-
metry, cf. partial supersymmetry breaking in supersymmetric field theories with
BPS-monopoles.
21
H.W. Braden, A.J. Macfarlane (JPA, 1985); G.V. Dunne, J. Feinberg (PRD, 1998):
N = 2 superextension of some periodic (finite-gap) systems may produce a
super-partner Hamiltonian with a potential to be the original one translated for
the half-period: self-isospetrality. When this happens?
⇒ Self-isospectrality conjecture:
Any finite-gap system having antiperiodic singlet states admits a self-isospectral
tri-supersymmetric extension.
22
⇒ Applying the general theory to a broad class of finite-gap elliptic systems de-
scribed by a two-parametric associated Lame equation with
H−m,l = −D2 − Cmdn2x− Cl
k′2
dn2x, Cm = m(m + 1), m ∈ Z,
the conjecture is supported by the explicit construction of the self-isospectral tri-
supersymmetric pairs.
23
F. Correa, V. Jakubsky, L.M. Nieto and M.P., 2008, Phys. Rev. Lett., 101,
030403 :
• Self-isospectrality can be realized by a non-relativistic electron in periodic
magnetic and electric fields of a special form
He = (px + Ax)2 + (py + Ay)
2 + σ3Bz − φ,
Ax = 0, Ay = w(x), ⇒ Bz = dwdx ;
w(x) = α ddx ln(dnx), φ(x) = βw2(x) + γw(x) + δ ⇒
H±m,l = − d2
dx2 + V ±m,l(x), V +
m,l(x) = V −m,l(x + L),
V −m,l(x) = −Cmdn2x− Cl
k′2
dn2x+ c
24
1
k → 1
L → ∞
Figure 2: Qualitative picture of supersymmetry breaking in self-isospectral system with m = 3, l = 1 in theinfinite-period limit. The form of potentials and bands shown on the left corresponds to the modulus k closeto 1. Two lower horizontal dashed lines on the right show energy levels of singlet bounded states, the upperseparated horizontal continuous line corresponds to a doublet of bounded states, the line at the bottom ofcontinuous spectrum indicates a doublet of the lowest states of the scattering sector.
The spontaneously broken tri-supersymmetry of the self-isospectral periodic system
is recovered in the infinite period limit.
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Infinite-period limit → reflectionless Poschl-Teller system (RPT)
⇒ Hidden bosonized nonlinear supersymmetry of the RPT system and tri-
supersymmetric structure of the pair of RPT systems find a natural explanation
in terms of the Aharonov-Bohm effect for a non-relativistic particle on the AdS2 :
F. Correa, V. Jakubsky, M.P., Ann. Phys. 324 (2009) 1078
⇒Hidden nonlinear supersymmetry in Bogoliubov/de Gennes system related to
the AKNS hierarchy (first order Hamiltonian): F. Correa, G. Dunne, MP, Ann.
Phys., in press, ArXiv:0904.2768 [hep-th]
⇒ Hidden bosonized superconformal symmetry in a spinless Aharonov-Bohm sys-
tem: F. Correa, V. Jakubsky, H. Falomir, MP, ArXiv:0906.4055 [hep-th]
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⇒ Some open questions:
• Hidden superconformal symmetry for a spin-1/2 particle Aharonov-Bohm ef-
fect?
• Hidden superconformal symmetry in a charge-monopole system?
• How tri-supersymmetry reveals itself in nonlinear integrable systems?
• Field theories with hidden bosonized supersymmetry?
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