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.

25

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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