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

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

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

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

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!

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

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!

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

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

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).

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

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)

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

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…

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

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?!

“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]:

“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 :

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

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

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

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

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

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

“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.? ?

“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.? ?

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.

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.

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

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”

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:

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:

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

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”:

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.

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

Thank you.

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

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

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 :

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

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.

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

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.

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!

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

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

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

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

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

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

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 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, 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

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

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:

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:

? 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?

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, ...)? 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 mass (momentum), spin

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:  ,

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

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

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: , , ,

A simple relation in a complicated basis

Algebra of anticommutators Isomorphic to sp(8)

Symmetry breaking {Q,S} operators {Q,Q} operators {S,S} operators D J1 J2 J3 {Q,S} operators Y1 Y2 Y3 N11 N12 N13 N21 N22 N23 N31 N32 N33 P0 K0 P11 P12 P13 P21 P22 P23 P31 P32 P33 K11 K12 K13 K21 K22 K23 K31 K32 K33 {Q,Q} operators {S,S} operators

Symmetry breaking {Q,S} operators {Q,Q} operators {S,S} operators D J1 J2 J3 {Q,S} operators ? C(1,3) conformal algebra Potential ~(Y3)2 Y3 N1 N2 N3 P0 K0 P1 P2 P3 K1 K2 K3 {Q,Q} operators {S,S} operators

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

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!

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

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!