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

1. 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: s3 s2 s1

9. e.g. this one will turn into and massless Dirac equations! s1=s2=s3=0(zero rows) • 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, • 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, • d = 5/2 + s2/2 + s3/2, • d > 5/2 + 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, • 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 • Green’s ansatz: • anticommute for different indices: • otherwise satisfy bose commutation relations: [T.D.Palev, 1994.] 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 even p odd p Pin(p) Spin(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 bijectionΛ ↔ M

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

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

25. “Spin-orbit” coupling • – 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: E.g. in the familiar p=3 case: 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 Qaand 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. e.g. this one will turn into and massless Dirac equations! s1=s2=s3=0(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

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 even p odd p Pin(p) Spin(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