1. Institute of physics, University of Belgrade
Representations and particles of osp(1|2n) generalized conformal supersymmetry
Igor Salom
Institute of physics, University of Belgrade

2. Motivation/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

3. osp(1|2n) as generalized superconformal 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

4. Generalized supersymmetry
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

5. Generalized conformal supersymmetry
New even symmetry generators, we may name them or not
Relations get much nicer if expressed using:
Everything follows from a single relation:
Algebra is defined when we specify commutators:
This is osp(1|2n,R) superalgebra!

6. Positive 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

7. Allowed d 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!

8. n = 4 case UIRs of SU(4) can be labeled by three integers s1, s2, s3:

9. s1=s2=s3=0 (zero rows) d = 0, trivial representation d = 1/2, d = 1,
e.g. this one will turn into
and massless Dirac equations!
d = 0, trivial representation
d = 1/2,
d = 1,
d = 3/2,
d > 3/2
 3 discrete “fundamentally scalar” UIRs

10. s1=s2=0, s3>0 (1 row) d = 1 + s3/2, d = 3/2 + 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).

11. s1=0, s2>0, s3 ≥0 (2 rows) d = 2 + s2/2 + s3/2,
 2 discrete families of 2-rows UIRs

12. s1>0, s2 ≥ 0, s3 ≥0 (3 rows) 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)

13. Conjecture: extrapolation of n ≤ 4 cases
Classification of positive energy UIR’s for arbitrary n:
How to work with/interpret these?

14. Case 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…

15. Analogy in the osp(1|2n) case
The simplest representation
Symmetry of tensor product:
“one box” UIR → bosonic oscillator UIR
symmetric group → orthogonal group

16. Tensoring the simplest UIR = Green’s ansatz
[T.D.Palev, 1994.]
Green’s ansatz:
anticommute for different indices:
otherwise satisfy bose commutation relations:
strange combination?!

17. “Covariant” Green’s ansatz
The awkward situation with mixed commutativity and anticommutativity can be fixed by writing:
where are now ordinary bose operators and are elements of a real Clifford algebra:
The obtained ansatz is SO(p) covariant [Greenberg]:

18. “Gauge” symmetry of the ansatz
“orbital” part
“spin” part
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 :

19. Ansatz “gauge” symmetry
odd p
even p
Spin(p)
Pin(p)

20. Representation space Odd osp(1|2n) operators act in
is n-dim oscillator Fock space
is Clifford representation space
“orbital” space
“spin” space

21. Interplay of osp(1|2n) and gauge symmetry
A priori, space decomposes into (half)integer positive energy UIR’s of osp(1|2n):
Gauge group certainly removes some degeneracy:
multiplicity label
labels vector within UIR Λ
labels osp UIR
labels gauge UIR
labels possible multiplicity of UIR
labels vector within UIR M

22. The first important result:
Each pair (Λ, M) appears at most once in reduction of , i.e. there is no additional degeneracy.
There is one-to-one correspondence between UIR's of osp(1|2n) and of the gauge group that appear in the decomposition, i.e. M determines Λ, and vice versa.
The theorem is proved by explicit construction of the bijection Λ ↔ M

23. Root system – osp(1|2n) Cartan subalgebra: Positive roots:
lowest weight:
signature:
where

24. Root system – gauge group
Cartan subalgebra:
highest weight:
satisfies
signature:
where
“spin” highest weight:
Pin(p) when p is even

25. “Spin-orbit” coupling
E.g. in the familiar p=3 case:
– osp(1|2n) lowest weight vector
that transforms as UIR of the gauge group.
From the viewpoint of spin-orbit coupling, there is a number(≤ 2n) of options:
And, where is the l.w.v.? ?

26. “Spin-orbit” coupling
Crucial:
osp(1|2n) lowest weight vector orbital part transforms as such that !
each of the rest contains (at least) one l.w.v. of the sp(2n) subalgebra!
– osp(1|2n) lowest weight vector
that transforms as UIR of the gauge group.
From the viewpoint of spin-orbit coupling, there is a number(≤ 2n) of options:
And, where is the l.w.v.? ?

27. Sketch of an elegant proof
Curious operator :
commutes with even subalgebra sp(2n),
is a “spin-orbit coupling” operator:
if is an osp(1|2n) l.w.v. then
Combinig 2. and 3. yields the proof.

28. Now we know enough about the form of l.w.v. to conclude:
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:
This also reveals osp(1|2n) content of for a given p.

29. Decomposition of the tensor product space
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

30. All (half)integer energy UIR’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”

31. Simplest nontrivial UIR - p=1-
operators act as ordinary bose operators and supersymmetry generators Qa and Sb 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:

32. Simplest nontrivial UIR - p=1-
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:

33. Simplest nontrivial UIR - p=1-
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

34. Next more complex class 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”:

35. Space p=2 in d=4 Fiertz identities, in general give: where:
only the third term vanishes, leaving two mass terms! Dirac equation is affected.

36. Remarks/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

38. Thank you.

39. Any analogy in the osp(1|2n) case?
It turns out:
symmetric group → orthogonal group

40. “Covariant” tensor product
Represent odd osp(1|2n) operators as: where are ordinary bose operators and are elements of a real Clifford algebra:

41. Pin(n) = “Gauge” symmetry 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 :

42. Motivation/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

43. osp(1|2n) = parabose algebra
Parabose algebra, Green: algebra of n pairs of mutually adjoint operators , satisfying:
and relations following from these.
Operators
form osp(1|2n,R) superalgebra.

44. Verma 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

45. osp(1|2n) = parabose algebra
Parabose algebra, Green: algebra of n pairs of mutually adjoint operators , satisfying:
and relations following from these.  ,
Operators
form osp(1|2n) superalgebra.

46. UIR 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!

47. s1=s2=s3=0 (zero rows) d = 0, trivial representation d = 1/2, d = 1,
e.g. this one will turn into
and massless Dirac equations!
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

48. How to do “work” with 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

49. Ansatz “gauge” symmetry
odd p
even p
Spin(p)
Pin(p)

50. Interplay of osp(1|2n) and 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

51. Representation space Odd osp(1|2n) operators act in
is n-dim oscillator Fock space
is Clifford representation space

52. Decomposition of the tensor product 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

53. 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:

54. “Fundamentally scalar 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.:

55. 1-row, d = 1 + s3/2 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

56. 1-row, other UIRs 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 s3 s3

57. 2-rows UIRs
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:

58. 3-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:

60. ? What is supersymmetry supersymmetry = = ruled out in LHC?
{Qa, Qb} = -2i (gm)ab Pm,
[Mmn, Qa] = -1/4 ([gm, gn])ab Qb,
[Pm, Qa] = 0 ?
supersymmetry =
symmetry generated by a (Lie) superalgebra
ruled out in LHC?
Poincaré supersymmetry! =
HLS theorem – source of confusion?

61. Simplicity as motivation?
+ supersymmetry:
[Mmn, Qa] = -1/4 ([gm, gn])ab Qb,
[Pm, Qa] = 0,
{Qa, Qb} = -2i (gm)ab Pm
+ conformal symmetry:
[Mmn, Sa] = -1/4 ([gm, gn])ab Sb,
{Sa, Sb} = -2i (gm)ab Km,
[Km, Sa] = 0,
+ tens of additional relations
Simplicity as motivation?
Poincaré space-time:
Something else?
Parabose algebra:
[Mmn, Mlr] = i (hnl Mmr + hmr Mnl - hml Mnr - hnr Mml),
[Mmn, Pl] = i (- hnl Pm + hml Pn),
[Pm, Pn] = 0
hmn =
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, ...)?
mass (momentum), spin

62. Parabose algebra
Algebra of n pairs of mutually adjoint operators satisfying:
and relations following from these.
Generally, but not here, it is related to parastatistics.
It is generalization of bose (Heisenberg) algebra:  ,

63. Close relation to orthosymplectic 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

64. Change of basis - step 1 of 2 -
Switch to hermitian combinations
consequently satisfying “para-Heisenberg” algebra:

65. Change of basis - step 2 of 2 -
define new basis for expressing parabose anticommutators:
we used the following basis of 4x4 real matrices:
6 antisymmetric:
10 symmetric matrices: , , ,

66. A simple relation in a complicated basis

67. Algebra of anticommutators
Isomorphic to sp(8)

68. Symmetry breaking {Q,S} operators {Q,Q} operators {S,S} operators D
J J J3
{Q,S} operators Y1 Y2 Y3
N N N13
N N N23
N N N33 P0 K0
P P P13
P P P23
P P P33
K K K13
K K K23
K K K33
{Q,Q} operators
{S,S} operators

69. Symmetry breaking {Q,S} operators {Q,Q} operators {S,S} operators D
J J J3
{Q,S} operators ?
C(1,3) conformal algebra
Potential ~(Y3)2 Y3
N N N3 P0 K0
P P P3
K K K3
{Q,Q} operators
{S,S} operators

70. Generalized conformal superalgebra
Connection with standard conformal algebra:
Y1 = Y2 = N11 = N21 = P11 = P21 = K11 = K21 ≡ 0
{Qa,Qb }={Qa,Qb }={Sa,Sb }={Sa,Sb }= 0
Choice of basis
+ bosonic part of algebra

71. 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”
e.g. this one will turn into
and massless Dirac equations!

72. Green’s ansatz representations
Now we have only ordinary bose operators and everything commutes!
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

73. Conclusion 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!