A New Type of Superstring in four dimension

A bosonic string in twenty six dimensions is effectively reduced to four dimensions by eleven Majorana fermions which are vectors in the bosonic represetation SO(d-1,1). By dividing the fermions in two groups, actions can be written down which are world sheet supersymmetric, 2-d local and local 4-d supersymmetric. The novel string is anomally free, free of ghosts and the partition function is modular invariant.

The internal symmetry of the φ µ 's is SO (11). They are real Majorana fermions which are vectors in the bosonic representation SO (3,1). If the action is to be supersymmetric, we must find a supersymmetric partner Ψ µ to the bosonic coordinate X µ . This can only be the linear sum of φ µ 's like Ψ µ = 11 j=1 e j φ φ µ j . With usual anticommutator following from action (1), the Ψ µ will have the same anticommutator rule provided 11 j=1 e j φ e φj = 1. We searched for such coefficients to find a linear combination which will make theaction supersymmetric. After extensive and laborious probing we found only one possibility namely to group φ µ 's into two species ψ µ,j : j=1,2....6 and φ µ,k ; k=1,2,..5. For one group, the positive and negative frequency are ψ µ,j = ψ (+)µ,j + ψ (−)µ,j where as for the other, allowing for the phase uncertainty of the creation operator for Majorana spinors, φ µ,k = φ (+)µ,k − φ (−)µ,k . For the latter species, the equal time anticommutator has a negative sign. The action is taken as The 2d Dirac matrices are and It is worth while noting that the light cone fermionic action ψ µj ρ α ∂ α ψ µj −φ µk ρ α ∂ α φ µk has central charge 11 and S − S lc has central charge 15 as the usual N=1, supersymmetric action. In this model,S l.c. is the equivalent of the Faddeev-Popov super conformal ghost action of the superstring. It is necessary to introduce two unit vectors e ψ and e φ with eleven components e.g. for j=3, and k=3, e 3 ψ = (0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0), e 3 φ = (0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0) with properties e j ψ e ψn = δ j n and e k φ e φm = δ k m . We shall frequently use e j ψ e ψj =6 and e k φ e φk =5. Every where the upper index referes to a column and the lower to a row. The suffixes ψ and φ are for book keeping only. The action S in (2) is found to be invariant under the following infinitesimal supersymmetric transformation. and with ǫ is the infinetesimal constant anticommutating spinor. The commutator of two transformations is the translation where a α = 2iǭ 1 ρ α ǫ 2 Equation (8) is true only and only if The Majorana fermion is such that and and is the right candidate to be the supersymmetric partner of the bosonic coordinates X µ and also is the desired sum of ψ µ . The action in (1) is only world sheet supersymmetric, but not local 2d supersymmetric. Besides the above pair, with the introduction of another pair, the zweibein e α (σ, τ ) and the two dimensional spinor-cum-world sheet vector χ α , δχ α = ∇ α ǫ the local 2d supersymmetric invariant action is [6] This is necessary to derive the vanishing of the energy momentum tensor and Noether current from a gauge principle [6,11]. Varying the field and the zweibein, we derive the vanishing of the supercurrent J α and the energy momentum tensor T αβ on the world sheet, The gravitino field χ α decouples and in the gauge χ α =0, we have Variation of action with respect to 'zweibein' e a α gives T a β = − π 2e δS δe β a = 0 and implies T αβ = e αa T a β . The variation with respect to the factor 'e' gives the traceless part. Since h αβ = e α a e β b η ab , the bosonic term is obtained by finding the variation δS δh αβ . Due to spin connection ∇ α of the fermionic part is simply ∂ α and ρ α ∂ α should be written as ρ a e β a ∂ β so that the variation gives In a basisψ = (ψ + , ψ − ) andφ = (φ + , φ − ) the vanishing of the light cone components are and where . The component constraints are the following equations (17) These eleven constraints eliminate the eleven Lorentz metric ghosts from the physical spectrum. However they follow from one current constraint of equation (17). Besides the conformal ghosts(b,c), it appears, we shall need eleven pairs of the super conformal (β j , γ j ) and (β k , γ k ) ghosts to write down BRST charge. These eleven pairs behave like one pair (β, γ), like the current generator. One can easily construct 'null' physical states as done in reference [6]. For the two groups of Fermions the general constraints are the two equations However action given in equation (13) is not invariant under four dimensional local supersymmetry. To obtain the simplest Green Schwarz action for N=1 local supersymmetry, we note that there are four component Dirac spinor representations of SO(3,1) which we denote by θ jδ and θ kδ respectively with δ=1,2,3,4. A genuine space time fermion can be constructed rather than space time vector as The G.S. action is [6] for N=1 where Γ µ are the Dirac gamma matrices and . The action is space time supersymmetric in four dimensions [6]. We have traded the vector four indices µ for the four spinor components δ to form the genuine space time Dirac Spinors.. The major defect is that the naive covariant quantisation does not work. To proceed with covariant formulation we have to implement NS-R [7] scheme with G.S.O. [8] projection which is simple, elegant and equivalent. The non vanishing equal time commutator and anti commutators follow from the action in equation (1). and Even though A,B refer to the component indices 1,2 or the helicities '+,−'. There are no ghost quanta other than µ = ν = 0 due to negative phase of creation operator. We immediately check that and It will be useful to note b µ,j Here A(m), B(r) and B F (m) are normal ordering constants, since the generators are the sum of the products of normal ordered quantum operators ': :'. A single dot implies sum over all qualifying indices .
N S : Both in NS and R sectors, there are the light cone ghosts corresponding to the fermionic components µ = 0 and 3 which are present in L m . By standard technique, we calculate that C=26 is the central charge. Using Jacobi Identity The central charge C can also be deduced from the v.e.v. of the product of two energy momentum tensors e.g. with The normal order anomally in Ramond sector must note that F o has no ambiguity. So F 2 o = L o . Using the algebraic relation for F's and the Jacobi Identity leads to The anomalies are have to be cancelled out and one needs the Faddeev Popov ghosts b ±± , c ± fields given by the action The c ± , b ±± field quanta b n , c n satisfy the anticommutator relation where a is the normal ordering constant. If G gh r and F gh r are the ghost current generators of conformal dimension 3 2 , the super Virasoro algebras for the ghosts for the two sectors follow. The anomaly term is deduced to be To be anomaly free, A gh (m) +A(m)=0. This gives the value of a=1 for the normal ordering constant and -26 for the central charge of the ghost. Due to Jacobi Identity, the anomaly terms for the ghosts for both the sectors cancels as well. Superconformal ghosts which contributes central charge 11 in a normal superstring are not necessary. To explain this we observe that the light cone gauge is ghost free.Dropping the helicity suffixes, the lightcone vectors, have the anticommutators with negative sign and are the ghost modes. The total ghost energy-momentum tensor comes from these (0,3) coordinates.
But the vacuum correlation function is Thus the contribution of these ghosts to the central charge 26 is 11 like the superconformal ghosts. Without these ghosts the central charge is 15 as is the case for normal ten dimensional superstring. With no anomalies and ghosts, the physical states can be constructed as . We proceed to prove that these physical state conditions eliminate all the ghosts. We begin with Gupta Bleuler postulate where L −m is given by equation (42) and L −m is given by the equation (44) and Q' takes care of the constraints due to the vanishing of the current generator operators acting on the physical states (57) and (58). In constructing the nilpotent charge we strictly followed the procedure for construction from operators satisfying Lie and graded Lie algebra [6].It is to be noted that Q BRST − Q ′ and Q BRST are separately nilpotent. Actually there are eleven subsidiary physical state conditions like (57) and (58) having the property following the fourier transformation and definition with G r = e j ψ G r,j − e k φ G r,k and F m = e j ψ F m,j − e k φ F m,k . We can follow up by writing (γ j r , β j r ) and (γ k r , β k r ) with similar definitions. But, in the NS sector, for instance, G j −r γ r,j − G k −r γ r,k ≡ G −r γ r where γ r = e j ψ γ r,j − e k φ γ r,k . So that the expressions for Q and Q ′ contain the conformal ghosts (b,c) and the pair of (β, γ) super conformal ghosts unitedly allowing for a BRST charge Q 2 BRST = 0. The theory is ghost free and unitary.
The self energy of the scalar tachyonic vacuum is cancelled by the contribution of the fermionic loop of the Ramond sector because the self energy of the fermion loop is The partition function of the forty member assembly will be the partition function of eight fermions raised to the power of five. The path integral function for the eight fermions is given by Seiberg and Witten [10]. In their notation A k ((− +), τ ) = A k ((+ −), −1 τ ) = −( Θ 4 (τ ) η(τ ) ) 4 (69) A k ((+ +), τ ) = 0 (70) The sum of all spin structure is The total partition function Z = |A k (τ )| 10 is not only modular invariant but vanishes due to the famous Jacobi relation which is the necessary condition for space time supersymmetry.
Thus we have been able to construct a novel modular invariant superstring which is free of anomalies and ghosts. Novel features of the model are that this is the only superstring known and constructed whose central charge is 26 and there is no need for superconformal ghosts in covariant formulation except in the construction of BRST Charge. These ghosts are usually contributing 11 to the central charge and are replaced by 11 Lorentz metric ghost of vector fermions. This linkage needs further study. There are useful and harmless tachyons in NS and R sector, but the self energies of the tachyonic vacua cancel.This cancellation mechanism, perhaps, makes the vacua unique. Since bosonic string theory is simpler and well understood in general, a derivation of the four dimensional space time supersymmetric action will positively help in studying the dynamics of the superstring theory in greater depth and insight.