Vector Supersymmetry and Non-linear Realizations 1

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1 1 Based on R. Casalbuoni, J. Gomis, K. Kamimura and G. Longhi, arxiv: , JHEP (2008) R. Casalbuoni, F. Elmetti, J. Gomis, K. Kamimura, L. Tamassia and A. Van Proeyen, arxiv: , JHEP (2009) and unpublished work Vector Supersymmetry and Non-linear Realizations 1 Departament d Estructura i Constituents de la Matéria Universitat de Barcelona Roberto Casalbuoni Fest GGI, September 21, 2012

2 Outline Motivation VSUSY Spinning Particle Action Quantization Representations VSUSY VSUSY and Strings Conclusions

3 Super Poincare group. Generators M µν, P µ, Q α Geometrical dynamical objects invariant under this symmetry Superparticle 2 Green-Schwarz (GS) string 3 An alternative formulation Ramond-Neveu-Schwarz (RNS) string 4 Spinning particle 5 Does Exist an space-time supersymmetry group with odd vector Generators associated to these objects? M µν, P µ, G µ, G 5 2 Casalbuoni (1976), Brink Schwarz (1981) 3 Green, Schwarz (84) 4 Neveu, Schwarz (71), Ramond (71), Gervais, Sakita (1971), Deser, Zumino (76), Brink, Di Vecchia, Howe (1976) 5 Barducci, Casalbuoni, Lusanna (76), Berezin (77), Brink, Deser, Zumino, Di Vecchia, Howe (77)

4 Super Space-time The space-time is identified with the (super)translations of an space-time graded Lie algebra Ordinary Superspace= IIASuperPoincare/Lorentz We have the translations P m, Q α The flat susyalgebra is, m = 0, 1,...9, α = 1,..., 32 [M mn, M rs] = i η nr M ms +... [M mn, P r ] = i η [nr P m] [P m, P r ] = 0 [Q, M mn] = i 2 QΓmn, [Q, P m] = 0, {Q, Q} = 2CΓ m P m where C is the charge conjugation matrix, {Γ m, Γ n } = +2η mn = +2( ; +...+),

5 We parametrize the coset as g = e ipmxm e iqαθα. The MC form Ω= i g 1 d g=p ml m MmnLmn +Q αl α L m =dx m +i θγ m dθ, L α = dθ α Supersymmetry transformations δθ = ɛ, δx m = i ɛγ m θ.

6 Vector Superspace Vector Superspace= Vector Super Poincare/Lorentz We have the translations P µ, G µ, G 5. The vector susy algebra (VSUSY) 6 is [M µν, M ρσ] = iη νρm µσ iη µσm νρ + iη νσm µρ + iη µρm νσ, [M µν, P ρ] = iη µρp ν iη νρp µ, [M µν, G ρ] = iη µρg ν iη νρg µ, [G µ, G ν] + = 0, [G 5, G 5] + = 0, [G µ, G 5] + = P µ. We locally parameterize the coset element as g = e ipµxµ e ig 5ξ 5 e igµξµ. 6 Barducci, Casalbuoni, Lusanna (76)

7 The Maurer-Cartan form (MC) is 7 Ω G A L A = P µl µ x + G 5L 5 + G µl µ ξ MµνLµν with L µ x = dx µ iξ µ dξ 5, L µ ξ = dξµ, L 5 = dξ 5, L µν = 0. MC equation dl µ x = L µ νl ν x i/2l µ ξ L5 ξ, dl µν = L µρ L ρ ν, dl µ ξ = L µ νl ν ξ 7 Casalbuoni, Gomis, Kamimura, Longhi (2007)

8 We can construct two Lorentz scalar closed invariant 2-forms Ω 2 = L µ ξ Lν ξ η µν = dξ µ dξ ν η µν, Ω 2 = dξ 5 dξ 5. These forms are exact, since Ω 2 = dω 1 = d (ξ µ dξ ν η µν), ( Ω 2 = d Ω 1 = d ξ 5 dξ 5). According to Chevalley-Eilenberg cohomology this susy algebra admits two central extensions 8 given by [G µ, G ν] + = η µνz, [G 5, G 5] + = Z. 8 Barducci, Caslabuoni, Lusanna (76)

9 Relation to N=2 topological algebra 9 N=2 Susy algebra in 4d has 8 charges. The euclidean formulation has symmetry group SU L (2) SU R (2) SU Rsymmetry (2) Q iα (1/2, 0, 1/2) Q i α (0, 1/2, 1/2) if we perform a twisting identifying the R-symmetry index, i, index with the space-time index,α, the new supersymmetric generators are Q iα (1, 0) (0, 0) H αβ Q Q i α (1/2, 1/2) G α β 9 Witten (88)

10 A massive particle breaks spatial translations and boost symmetry. The coset is G spacetime /Rotations g = g L U, U = e im 0i v i where g L is the coset associated to the space, M 0i are the Lorentz boost generators and v i are the corresponding Goldstone parameters. U represents a finite Lorentz boost. The Maurer-Cartan 1-form is now Ω = ig 1 dg = U 1 Ω L U iu 1 du U 1 P µu = P νλ ν µ(v), where Λ n m(v) is a finite Lorentz transformation. The new MC forms associated to the translations are L ν = Λ ν µ(v)l µ

11 The corresponding part of action (NG) is S NG = m ( L0 ) ( = m Λ 0 m(v)l µ) = L NG dτ where * means pull-back on the world line and Λ 0 µ is a time-like Lorentz vector, Λ 0 µλ 0 νη µν = 1. The canonical momentum conjugate to x µ is p µ = LNG ẋ µ = mλ0 µ(v). Since Λ 0 µ(v) is a time-like vector, the momentum verifies the mass-shell constraint p 2 + m 2 = 0. If we consider as basic bosonic variables x µ, p µ The action becomes L NG = p µ(ẋ µ iξ µ ξ 5 ) e 2 (p2 + m 2 ), where e is a lagrange multiplier (ein-bein).

12 We have also two WZ terms There are two Lorentz scalar 1-forms that are quasi-invariant Ω 1 = ξ µ dξ ν η µν, Ω1 = ξ 5 dξ 5. The WZ action in this case S WZ = β Ω 1 + γ Ω 1 The action for an spinning particle is S = S NG + S WZ the lagrangian becomes L C = p µ(ẋ µ iξ µ ξ 5 ) β i 2 ξµ ξ µ γ i 2 ξ5 ξ 5 e 2 (p2 + m 2 ),

13 The Dirac brackets are {p µ, x ν } = δ µ ν, {ξ µ, ξ ν } = i β ηµν, {π 5, ξ 5 } = 1. If βγ = m 2,there are two first class constraints φ = 1 2 (p2 + m 2 ), χ 5 = (π 5 γ i 2 ξ5 ip µξ µ ) {χ 5, χ 5} = iγ i β pµpνηµν = 2i β φ. χ 5 generates the local kappa variation, world line supersymmetry Gervais, Sakita (1971)

14 The generators of the global supersymmetries are G µ = π µ + β i 2 ξµ + ipµξ5 = iβξ µ + ip µξ 5, G 5 = π 5 + γ i 2 ξ5. They satisfy {G µ, G ν} = iβη µν, {G µ, G 5} = ip µ, {G 5, G 5} = iγ. Z = β, Z = γ, Z Z = m 2.

15 Quantization We quantize in a covariant manner by requiring the first class constraints to hold on the physical states. The Dirac brackets are replaced by the following graded-commutators, [p µ, x ν ] = iδ µ ν, [ξ µ, ξ ν ] + = 1 β ηµν, [π 5, ξ 5 ] + = i. The odd variables define a Clifford algebra. We define λ µ = 2βξ µ, λ 5 2 = i (π 5 i2 ) γ γξ5, λ 6 2 = i (π 5 + i2 ) γ γξ5. Note (λ A ) = λ A. The λ A s define a Clifford algebra C 6, [λ A, λ B ] + = 2 η AB, η AB = (, +, +, +, +, ), (A, B = 0, 1, 2, 3, 5, 6).

16 These variables can be identified as a particular combination of the elements of another C 6 algebra with generators Γ µ s isomorphic to the Dirac matrices ( Γ µ γ µ 0 = 0 γ µ ) (, Γ 5 γ 5 = 0 0 γ 5 ) (, Γ = 1 0 ), [Γ A, Γ B ] + = 2 η AB, η AB = (+,,,, +, ). λ A = Γ A Γ 5, A = 0, 1, 2, 3, Γ 5, A = 5, iγ 5 Γ 6, A = 6.. both Clifford algebras have the same automorphism group SO(4, 2).

17 Susy generators G µ = iβξ µ + ip µ ξ 5 = = i ) (mλ µ p µ (λ 5 λ 6 ) 2γ i ( ) Γ 5 mγ µ p µ (1 iγ 6 ) 2γ G 5 = π 5 + i γ γ γ 2 ξ5 = i 2 λ6 = 2 Γ5 Γ 6. The expression for the odd first class constraint χ 5 = π 5 i γ 2 ξ5 ip µξ µ = i (mλ 5 p µλ µ) i = Γ 5 (p µγ µ + m). 2β 2β The requirement that the first class constraint χ 5 holds on the physical states is equivalent to require the Dirac equation on an 8-dimensional spinor Ψ (p µγ µ + m)ψ = 0. The other first class constraint, φ = 1 2 (p2 + m 2 ) = 0, is then automatically satisfied.

18 If we impose a further constraint 11 π 5 + i γ 2 ξ5 = Barducci, Casalbuoni, Lusanna (1976)

19 Relation to other models Consider the lagrangian 12 L B = p ẋ β i 2 ξµ ξ µ + γ i 2 ξ5 ξ5 e 2 (p2 + m 2 ) iρ(p ξ + γξ 5 ) where ρ is a Lagrange multiplier and βγ = m 2 as in L C. The quantization can be done by using only a C 5 algebra which can be realized in a 4-dimensional space and we recover the 4d Dirac equation. 1 χψ(x) = 2β γ5 (p µγ µ m)ψ(x) = 0, where ξ µ = 1 2β γµ γ 5, ξ 5 1 = 2γ γ5. 12 Brink, Di Vecchia, Howe (1976)

20 The lagrangian L B also has the local fermionic invariance generated by the first class χ = 0, δx µ = iξ µ κ 5 (τ), δξ µ = 1 β pµ κ 5 (τ), δξ 5 = κ 5 (τ), δe = 2i β ρκ5 (τ), δρ = κ 5 (τ). and the global vector fermionic invariance, δξ µ = ɛ µ, δξ 5 = 1 γ (pµɛµ ), δx µ = iɛ µ ξ 5, δl B = d ( ) i dτ 2 ɛµ (p µξ 5 βξ µ). The generator is, after removing the second class constraints, and satisfies G µ = i(βξ µ + p µξ 5 ) {G µ, G ν} = iβ(η µν + pµpν m 2 ). The commutator of the gauge symmetry and the rigid transformation by vanishes.

21 If we consider the model with higher order derivates obtained from L B by eliminating ξ 5 and quantize the model 13 ones obtains two Dirac equations for mass ±m like the L C lagrangian, probably this model has VSUSY symmetry. 13 Gomis, Paris, Roca (1991)

22 Casimirs The central charges Z and Z are trivial Casimirs of VSUSY. It is also easy to see that P 2 is a Casimir for VSUSY. The correct VSUSY generalization of the Pauli-Lubanski vector, W µ, is 14 Ŵ µ = 1 2 ɛµνρσ P ν(zm ρσ G ρg σ), (1) whose square Ŵ 2 is a Casimir. By introducing the new vector W µ C = 1 2 ɛµνρσ P νg ρg σ, (2) One can easily prove that W 2 C is also a Casimir. ZW µ = Ŵ µ + W µ C. (3) W 2 C = Z 2 P (4) 14 Casalbuoni, Elmetti, Gomis, Kamimura, Tamassia, Van Proeyen (2008)

23 We are implicitly considering the case of representations with Z 0. In that case Z is just a number and can be divided out. Therefore, in the rest frame of the massive states where P 2 = m 2, they satisfy the rotation algebra [ ] W i m, W j m = ɛ ijk W k m, (5) and define three different spins. The superspin Y labels the eigenvalues m 2 Z 2 Y (Y + 1) of the Casimir Ŵ 2. The spin associated to WC 2 (C-spin) is fixed to 1/2, as one can see from (4). Finally, we denote the usual Lorentz spin by s. On the other hand, only W µ = 1 Ŵ µ commutes with G Z λ, and thus only the superspin Y characterizes a multiplet. Since ] [Ŵ µ, WC ν = 0, one can immediately obtain the particle content of a VSUSY multiplet by using the formal theory of addition of angular momenta applied to (3). A multiplet of superspin Y contains two particles of Lorentz spin Y ± 1/2, for Y > 0 integer or half-integer. In the degenerate case of superspin Y = 0, the multiplet consists of two spin 1/2 states.

24 In particular, we observe that a VSUSY multiplet contains either only particles of half-integer Lorentz spin or only particles of integer Lorentz spin. The spinning particle constructed is a realization of the degenerate case Y = 0. eigenvalue vector superp 1 Z 2 m 2 Y (Y + 1) superspin = Y Casimir Y = 0 1 Z 2 W 2 C m C spin = 1 2 Casimir C = 1 2 W 2 m 2 s(s + 1) Lorentz spin = s = Y ± 1 2 not Casimir s = 1 2 Some field theory representations of VSUSY has been presented Casalbuoni, Elmetti, Tamassia (2010)

25 Odd "Casimir" If Z Z + m 2 = 0. (6) Q = G P + G 5Z. (7) Q acts as an odd constant on the states in the previous representations. Unless the model under consideration has a natural odd constant, Q has to annihilate all states in those representations. As a result, the physical role of the odd Casimir is to give a Dirac-type equation for the particle states.

26 Is the RNS string VSUSY invariant? The fields of the RNS sigma model are the target space bosonic coordinate X µ (µ = 1,, D), being an even vector from the spacetime point of view and D even scalars from the worldsheet point of view, and the fermions ψ µ ±, being two odd vectors from the spacetime point of view and 2D Majorana-Weyl spinors from the worldsheet point of view. The action of RNS string in the conformal gauge is S = 1 d 2 σ ( 2 +X µ X µ + iψ µ π + ψ +µ + iψ µ ) +ψ µ (8) and must be supplemented with the super-virasoro constraints: and similar for the sector. J + = ψ µ + +X µ = 0 T ++ = +X µ +X µ + i 2 ψµ + +ψ +µ = 0 (9)

27 We can then ( promote our VSUSY charges to worldsheet spinors, Q Q µ µ ) + Q µ and construct an N = 2 VSUSY algebra [M µν, M ρσ] = iη νρm µσ iη µσm νρ + iη νσm µρ + iη µρm νσ ; [M µν, P ρ] = iη µρp ν iη νρp µ ; [M µν, Q ±ρ] = iη µρq ±ν iη νρq ±µ ; {Q ±µ, Q ±ν} = Z ±η µν ; {Q ±5, Q ±5} = Z ± ; {Q ±µ, Q ±5} = P (10) µ. The transformations δ V + X µ = 0 δ V X µ = 0 δ V + ψ µ + = ɛ µ + δ V ψ µ + = 0 δ V + ψ µ = 0 δ V ψ µ = ɛ µ (11) δ 5+X µ = iɛ 5+ψ µ + δ 5 X µ = iɛ 5 ψ µ δ 5+ψ µ + = 2ɛ 5+ +X µ δ 5 ψ µ + = 0 δ 5+ψ µ = 0 δ 5 ψ µ = 2ɛ 5 X µ, (12) where the parameters ɛ µ ±, ɛ5 ± are real odd constants, are realization of N=2VSUSY.

28 The previous N=2 VSusy transformations leave the action invariant but the super-virasoro constraints are not Vsusy invariant. The RNS string model is obtained after gauge-fixing from [ L Brink = e 1 2 αx µ β X µg αβ i 2 ψ µ γ α αψ µ + i 2 χαγβ γ α ψ µ β X µ + 1 ] 8 ( χαγβ γ α ψ µ )( χ β ψ µ), (13) which is invariant under local worldsheet supersymmetry and local Weyl transformations (details follow). In this notation α is a curved worldsheet vector index, a is a flat worldsheet vector index and worldsheet spinor indices are not explicitly written. We have been unable to find a type of VSUSY rigid for this action, unfortunately at this moment we can not prove the non-existence of this type of symmetry.

29 A N=1 VSUSY string from Non-linear Realizations Here we are going to use the MC form of VSUSY to construct an VSUSY invariant string action. We consider the coset G/O(3, 1) which we locally parameterize the coset as g = e ipµxµ e ig 5ξ 5 e igµξµ e izc e i Z c, therefore the coordinates of the superspace are x µ, ξ µ, ξ 5. The MC 1-form is Ω = ig 1 dg = P µ (dx µ iξ µ dξ 5) + G 5dξ 5 + G µdξ µ + + Z (dc i2 ) dξµξµ + (d c Z i2 ) dξ5 ξ 5. The even MC 1-forms are L µ x = dx µ iξ µ dξ 5, L Z = dc + i 2 ξµ dξ µ, L Z = d c + i 2 ξ5 dξ 5 whereas the odd ones are given by L µ ξ = dξµ, L 5 ξ = dξ 5.

30 One way to construct a VSUSY string action is to conside the two dimensional Polyakov g αβ world sheet metric with and the the MC 1-forms of vecorsuperspace. det g L 1 = g αβ η µνv αv µ β ν, v α µ αx µ iξ µ αξ 5. 2 Upon integration over g αβ we obtain the NG forms, L 1 = det G αβ, G αβ η µνv µ αv ν β. A possible WZ term for N=1 is given The WZ term verifies an invariant three form. L WZ = i ɛ αβ αx µ ξ µ β ξ 5 2i c ɛ αβ αξ 5 β ξ 5. dl WZ = i L µ P dξµdξ5 2i L c dξ 5 dξ 5,

31 We consider as complete lagrangian L = L 1 + L WZ = det G + i ɛ αβ αx µ ξ µ β ξ 5 2i c ɛ αβ αξ 5 β ξ 5. If we perform the analysis of constraints and we work in the reduced space (x µ, p ν, ξ 5, π 5). The canonical lagrangian is where L c = pẋ + π 5 ξ 5 H = pẋ + π 5( 0 λ + 1 )ξ 5 λ+ 4 (p + x ) 2 + λ 4 (p x ) 2. H + = 1 4 (p + x ) 2 + π 5 ξ 5, H = 1 4 (p x ) 2. and λ, λ + are arbitrary functions. Notice that we have the ordinary bosonic lagragian of the string plus a "bc" like system.

32 Therefeore in the conformal gauge λ + = λ = 1 we have a conformal theory L con = pẋ 1 2 (p2 + x 2 ) + π 5( 0 1 )ξ 5. = 1 2 (ẋ 2 x 2 ) + π 5( 0 1 )ξ 5. Notice that this model is chiral, we have only right movers fermions

33 The VSUSY generators in the reduced space are ( ) G µ = dσ iξ 5 (p µ + x µ ), G 5 = dσ (π 5). They verify {G 5, G 5} = 0, {G µ, G 5} = {G µ, G ν} = η µν dσ ( i(p µ + x µ) ) = i ( dσ 2 ξ 5 ξ 5 ) η µν iz. dσ p µ, where Z = ( dσ 2i ξ 5 ξ 5 ), Z = 0. The VSUSY transformations are δx µ = i ξ 5 ɛ µ, δp µ = i ξ 5 ɛ µ, δπ 5 = i (p + x ) µ ɛ µ, δξ 5 = ɛ 5.

34 If consider a non-chiral model L conc = pẋ 1 2 (p2 + x 2 ) + π( 0 1 )ξ + π( )ξ = 1 2 (ẋ 2 x 2 ) + π( 0 1 )ξ + π( )ξ. It is invariant under N=2 VSUSY, δx µ = i ξ ɛ µ i ξ ɛ µ, δπ = i (ẋ + x ) µ ɛ µ, δπ = i (ẋ x ) µ ɛ µ, δξ = ɛ 5, δξ = ɛ 5.

35 If we introduce a 2D notaion ( ξ Ψ = ξ ) ( π, Π = π ) L conc = 1 2 ( αx µ α x µ) + i Π γ α α Ψ, Π Π C. Note this action is similar to the NSR string but the Ψ does not have Lorentz index µ. In addition Π and Ψ are independent and can transform differently under VSUSY.

36 Conclusions Vector supersymmetry appears naturally in the spinning particle model of 16 It seems RNS string is not VSUSY invariant We can construct VSUSY string theories, whose physical meaning should be further analyzed. Grazie Roberto 16 Barducci, Casalbuoni, Lusanna (1976)