Representations and particles of osp (1|2n) generalized conformal supersymmetry

Representations and particles of Representations and particles of osp(1|2n) generalized osp(1|2n) generalized

conformal supersymmetryconformal supersymmetry

Igor Salom

Institute of physics, University of Belgrade

Motivation/OutlineMotivation/Outline• Hundreds of papers on osp generalization of conformal

supersymmetry in last 30 years, yet group-theoretical approach is missing

• Almost only the simplest UIR appears in literature (“massless tower of raising helicities”), but there are many more, rich in properties, carrying nontrivial SU(n) numbers

• UIR’s should correspond to particles/fundamental system configurations

• Problems:– Classify UIR’s (already difficult)– Construct/“work” with these UIR’s (a nonstandard approach needed)– Give them physical interpretation

osp(1|2n) as generalized osp(1|2n) as generalized superconformal algebrasuperconformal algebra

• osp generalization of supersymmetry first analyzed by C. Fronsdal back in 1986

• Since then appeared in different context: higher spin fields (simplest UIR corresponds to tower of increasing helicities), BPS particles, branes, M-theory algebra…

• Considered mostly: n=16, 32 (10 or 11 space-time dimensions)

• n = 4 case corresponds to d=4

Generalized Generalized supersymmetrysupersymmetry

• in 4 spacetime dimensions:

• in 11 spacetime dimensions:

this is known as M-theory algebra• can be extended to super conformal case

Tensorial central charges

Generalized conformal Generalized conformal supersymmetrysupersymmetry

• Algebra is defined when we specify commutators:

• This is osp(1|2n,R) superalgebra!

New even symmetry generators, we may name them or not

Relations get much nicer if expressed using:

Everything follows from a single relation:

Positive energy UIR’sPositive energy UIR’s

• Physically most interesting• Positive conformal energy:

• Lowest weight representations, quotients of Verma module

• Labeled by:– SU(n) subrepresentation (on lowest E subspace), i.e.

by a Young diagram – d = E + const, a real parameter

Allowed Allowed dd values values

• Spectrum is dependent on the SU(n) labels• In general, d has continuous and discrete parts of

spectrum: – continuous: d > d1 ← LW Verma module is irreducible

– discrete: d = d1, d2, d3,… dk ← submodules must be factored out

• Discrete part is specially interesting for (additional) equations of motion, continuous part might be nonphysical (as in Poincare case)

• points in discrete spectrum may arrise due to:– singular vectors ← quite understood, at known values of d– subsingular vectors ← exotic, did require computer analysis!

nn = 4 case = 4 case

• UIRs of SU(4) can be labeled by three integers s1, s2, s3:

s1

s2

s3

ss11=s=s22=s=s33=0=0

(zero rows)(zero rows)• d = 0, trivial representation

• d = 1/2,

• d = 1,

• d = 3/2,

• d > 3/2

3 discrete “fundamentally scalar” UIRs

e.g. this one will turn into and massless Dirac equations!

ss11=s=s22=0, s=0, s33>0>0

(1 row)(1 row)

• d = 1 + s3/2,

• d = 3/2 + s3/2,

• d = 2 + s3/2,

• d > 2 + s3/2

3 discrete families of 1-row UIRs, in particular 3 discrete “fundamental spinors” (first, i.e. s3=1 particles).

ss11=0, s=0, s22>0, s>0, s33 ≥0 ≥0

(2 rows)(2 rows)

• d = 2 + s2/2 + s3/2,

• d = 5/2 + s2/2 + s3/2,

• d > 5/2 + s2/2 + s3/2

2 discrete families of 2-rows UIRs

ss11>0, s>0, s22 ≥ 0, s ≥ 0, s33 ≥0 ≥0

(3 rows)(3 rows)

• d = 3 + s1/2 + s2/2 + s3/2,

• d > 3 + s1/2 + s2/2 + s3/2

single discrete familiy of 3-rows UIRs (i.e. discrete UIR is determined by Young diagram alone)

Conjecture: extrapolation Conjecture: extrapolation of n ≤ 4 casesof n ≤ 4 cases

• Classification of positive energy UIR’s for arbitrary n:

How to work with/interpret

these?

Case of SU(n) UIR’sCase of SU(n) UIR’s

• “Canonical procedure” of group theory gives everything, nice and simply: highest weights, weight multiplicities, matrix elements…

• Nevertheless, we prefer to use Young tableaux:

– Permutations (symmetric group) is symmetry of the tensor product of defining UIR’s.

→ Symmetric group labels UIR’s, reduces space, gives basis vectors…

Analogy in the Analogy in the osp(1|osp(1|2n)2n) case case

• The simplest representation

• Symmetry of tensor product:

symmetric group → orthogonal group

“one box” UIR → bosonic oscillator UIR

““Covariant” tensor Covariant” tensor productproduct

Represent odd osp(1|2n) operators as:

where are ordinary bose operators and are elements of a real Clifford algebra:

Pin(n) = “Gauge” symmetry Pin(n) = “Gauge” symmetry of the tensor product of the tensor product

• Operators:

generate Spin(p) group action and commute with entire osp(1|2n) algebra:

• For even values of p symmetry is extended to Pin(p) by the inversion operators :

• gauge group quantum numbers label osp(1|2n, R) and sp(2n, R) UIRs and multiplicity

• osp UIR’s belonging to p-fold tensor product are this way explicitly determined

• lowest weight vectors are explicitly constructed

Decomposition of the tensor product space

All (half)integer energy All (half)integer energy UIR’s can be constructedUIR’s can be constructed

• To get the first UIR with nontrivial SU(n) properties (1-row Y.d.) of the l.w.v. two factors are necessary, i.e. p=2

• “Pairing of factor spaces” occurs: to get 2-row diagram UIR’s we need p=4, 3-row UIR’s p=6, etc. States obtained by antisymmetrizing p=2 charged “subparticles”.

• No need to consider arbitrary large p-fold product. In d=4, i.e. n=4 we need up to 3 “p=2 subparticles”

Simplest nontrivial UIRSimplest nontrivial UIR- p=1-- p=1-

• operators act as ordinary bose operators and supersymmetry generators Q and S satisfy n-dim

Heisenberg algebra.

• Hilbert space is that of n-dim nonrelativistic quantum mechanics. We may introduce equivalent of coordinate or momentum basis, e.g., in d=4:


• In d=4 Fiertz identities in general give:

• where:

• since generators Q mutually commute in p=1, all states are massless:

• states are labeled by 3-momentum and helicity:


• introduce “field states” as vector coherent states:

• derive familiar results (Klain-Godon, Dirac eq):

source of equations of motion can be

traced back to the corresponding singular vector

Next more complex Next more complex class of UIR: p=2class of UIR: p=2

• Hilbert space is mathematically similar to that of two particles in n-dim Euclidean space:

• Gauge group is SO(2) = U(1) that has one dimensional UIR’s → each osp UIR, for any number of boxes, appears only once in this space.

• “Charge”:

Space p=2 in d=4Space p=2 in d=4

• Fiertz identities, in general give:

• where:

• only the third term vanishes, leaving two mass terms! Dirac equation is affected.

Remarks/ConclusionRemarks/Conclusion

• Basic group theoretical results are insensitive to choice of action and treatment of (tensorial) coordinates.

• There are many interesting UIR’s carrying SU(n) numbers

• UIR’s are “made” of finitely many “subparticles”• “Gauge symmetry” crucial in the tensor product

space

Thank you.Thank you.

Motivation/OutlineMotivation/Outline• Hundreds of paper on osp generalization of

conformal supersymmetry in last 30 years• Many Actions/Lagrangians written, yet group-

theoretical approach missing• UIR’s should correspond to particles/fundamental

system configurations• Problems:

– Classify UIR’s (already difficult)– “Work” with these UIR’s (a different approach needed)– Give them physical interpretation

• Apart of massless tower of helicities, there are particles carrying nontrivial SU(n) numbers

osp(1|2n) osp(1|2n) = parabose = parabose algebraalgebra

• Parabose algebra, Green: algebra of n pairs of mutually adjoint operators , satisfying:

and relations following from these.

• Operators

form osp(1|2n,R) superalgebra.

Verma module structureVerma module structure

• superalgebra structure: osp(1|2n) root system, positive roots , defined ordering

• – lowest weight vector, annihilated by all negative roots

• Verma module: • some of vectors – singular and subsingular –

again “behave” like LWV and generate submodules

• upon removing these, module is irreducible

osp(1|2n) osp(1|2n) = parabose = parabose algebraalgebra

• Parabose algebra, Green: algebra of n pairs of mutually adjoint operators , satisfying:


,

• Operators

form osp(1|2n) superalgebra.

UIR labelsUIR labels

• states of the lowest E value (span “vacuum” subspace) are annihilated by all , and carry a representation of SU(n) group generated by (traceless) operators .

• thus, each positive energy UIR of osp(1|2n, R) is labeled by an unitary irreducible representation of SU(n) and value of a (continuous) parameter – more often it is so called “conformal weight” d than E.

• allowed values of parameter d depend upon SU(n) labels, and were not precisely known – we had to find them!

ss11=s=s22=s=s33=0=0

(zero rows)(zero rows)• d = 0, trivial representation

• d = 1/2,

• d = 1,

• d = 3/2,

• d > 3/2

3 discrete “fundamentally scalar” UIRs

these vectors are of zero

(Shapovalov) norm, and thus

must be factored out, i.e. set to

zero to get UIR


How to do “work” with How to do “work” with these representations? these representations?

• solution: realize UIRs in Green’s ansatz!

• automatically: (sub)singular vectors vanish, unitarity guaranteed

• for “fundamentally scalar” (unique vacuum) UIRs Greens ansatz was known

• we generalized construction for SU nontrivial UIRs

Ansatz “gauge” Ansatz “gauge” symmetrysymmetry

Spin(p) Pin(p)

odd

peven p

Interplay of Interplay of osp(1|2n)osp(1|2n) and gauge symmetryand gauge symmetry

• A priori, space decomposes into (half)integer positive energy UIR’s of osp(1|2n,R):

• Turns out: gauge group removes all degeneracy:

multiplicity label

labels vector within UIR Λ

labels osp UIR

labels gauge UIR labels vector

within UIR M

Representation spaceRepresentation space

• Odd osp(1|2n) operators

act in

• is n-dim oscillator Fock space

• is Clifford representation space

• 1-1 mapping between osp and gauge UIR’s, that is: osp(1|2n) UIR, but also sp(2n) can be directly read from gauge transformation properties (much easier).

• osp UIR’s belonging to p-fold tensor product are explicitly determined

• lowest weight vectors are explicitly constructed

Decomposition of the tensor product space

Explicitly…Explicitly…

• Representation appear in the decomposition of if and only if signatures satisfy:

where:• The vector which has lowest osp weight and the

highest gauge group weight has the explicit form:

““Fundamentally scalar Fundamentally scalar UIRs” UIRs”

• d = 1/2 p = 1– this parabose UIR is representation of ordinary bose

operators– singular vector

identically vanishes

• d = 1 p = 2– the lowest weight state is multiple of ordinary bose

vacuums in factor spaces:

• d = 3/2 p = 3– l.w.v.:

1-row, 1-row, dd = 1 + s = 1 + s33/2 UIR/2 UIR

• We show that this class of UIRs exactly constitutes p=2 Green’s ansatz!

• Define:• – two independent pairs of

bose operators • are “vacuum generators”:

• All operators will annihilate this state:

s3

• Other “families” are obtained by increasing p:

– d = 3/2 + s3/2, p = 3,

– d = 2 + s3/2, p = 4

• Spaces of these UIRs are only subspaces of p = 3 and p = 4 Green’s ansatz spaces

1-row, other UIRs1-row, other UIRs

s3

s3

• Two “vacuum generating” operators must be antisymmetrized we need product of two p=2 spaces.

• To produce two families of 2-rows UIRs act on a natural vacuum in p=4 and p=5 by:

2-rows UIRs2-rows UIRs

• Three “vacuum generating” operators must be antisymmetrized we need product of three p=2 spaces.

• Single family of 3-rows UIRs is obtained by acting on a natural vacuum in p=6 by:

3-rows UIRs3-rows UIRs

{Q, Q} = -2i ()P

[M, Q] = -1/4 ([, ])Q,

[P, Q] = 0

What is supersymmetry

supersymmetry = symmetry generated by a (Lie) superalgebra?

Poincaré supersymmetry!

=

HLS theorem – source of confusion?

ruled out in LHC?

Simplicity as motivation?Simplicity as motivation?

[M, M] = i ( M+

M- M- M),

[M, P] = i ( P P),

[P, P] = 0

1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1

Poincaré space-time:

mass (momentum), spin

Something else?

• mass (momentum), spin

• usual massless particles

• “charged” particles carrying SU(2) x U(1) numbers

• “elementary” composite particles from up to 3 charged subparticles

• a sort of parity asymmetry

• ….(flavors, ...)?

Parabose algebra:

+ supersymmetry:

[M, Q] = -1/4 ([, ])Q,

[P, Q] = 0,

{Q, Q} = -2i ()P

+ conformal symmetry:

[M, S] = -1/4 ([, ])S,

{S, S} = -2i ()K,

[K, S] = 0, + tens of additional relations

Parabose algebraParabose algebra

• Algebra of n pairs of mutually adjoint operators satisfying:


• Generally, but not here, it is related to parastatistics.

• It is generalization of bose (Heisenberg) algebra:

,

Close relation to Close relation to orthosymplectic superalgebraorthosymplectic superalgebra

• Operators

form osp(1|2n) superalgebra.• osp generalization of supersymmetry first analyzed by C.

Fronsdal back in 1986• Since then appeared in different context: higher spin

fields, bps particles, branes, M-theory algebra • mostly n=16, 32 (mostly in 10 or 11 space-time

dimensions)• we are interested in n = 4 case that corresponds to d=4.

From now on n = 4

Change of basisChange of basis- step 1 of 2 -- step 1 of 2 -

• Switch to hermitian combinations

consequently satisfying “para-Heisenberg” algebra:

• define new basis for expressing parabose anticommutators:

• we used the following basis of 4x4 real matrices:

– 6 antisymmetric:

– 10 symmetric matrices:

Change of basis Change of basis - step 2 of 2 -- step 2 of 2 -

, ,

,

A simple relation in a A simple relation in a complicated basiscomplicated basis

Algebra of anticommutatorsAlgebra of anticommutators

Isomorphic to sp(8)

Symmetry breakingSymmetry breaking

N11 N12 N13

N21 N22 N23

N31 N32 N33

J1 J2 J3

P0

D

K0

Y1

Y2

Y3

P11 P12 P13

P21 P22 P23

P31 P32 P33

K11 K12 K13

K21 K22 K23

K31 K32 K33

{Q,S} operators

{S,S} operators

{Q,Q} operators

Symmetry breakingSymmetry breaking

P0 K0

{Q,S} operators

{S,S} operators

{Q,Q} operators

C(1,3) conformal algebra

N1 N2 N3

J1 J2 J3

P1 P2 P3

D

K1 K2 K3

Y3Potential

~(Y3)2

?

Generalized conformal Generalized conformal superalgebrasuperalgebra

Choice of basis

+ bosonic part of algebra

Connection with standard conformal algebra:

Y1 = Y2 = N11 = N21 = P11 = P21 = K11 = K21 ≡ 0

{Q,Q}={Q,Q}={S,S}={S,S}= 0

Technical problems…Technical problems…• Complicated Verma module structure

(subsingular vectors, many descendant singular vectors…)

• For n ≤ 4 detailed analysis carried out using computers

• Singular vector examples:– nice, directly turns into motion eq.:

– “not so nice”


Green’s ansatz representationsGreen’s ansatz representations

• Green’s ansatz of order p (combined with Klain’s transformation):

• we introduced 4p pairs of ordinary bose operators:

• and “spinor inversion” operators that can be constructed as 2 pi rotations in the factor space:

• all live in product of p ordinary 4-dim LHO Hilbert spaces:

• p = 1 is representation of bose operators

Now we have only ordinary bose operators and

everything commutes!

ConclusionConclusion

• Simple in statement but rich in properties

• Symmetry breaking of a nice type

• Promising particle structure

• Many predictions but yet to be calculated

Promising type of supersymmetry!

Representations and particles of osp (1|2n) generalized conformal supersymmetry

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