El hombre fuera de los Ecosistemas pero dentro de la Naturaleza

NN)

As in the bosonic theory, the physical spectrum is most easily determined in lightcone quanti- sation, where the residual gauge symmetries - here the super-conformal symmetry - is exploited at a classical level to solve for the super-Virasoro constraints explicitly. We briefly outline the procedure here - focussing for simplicity to the open string with Neumann-Neumann boundary conditions - and then move on to a detailed discussion of the spectrum.

• The residual conformal symmetry allows us to set

X+(τ, σ) = x++ p+τ (6.91) in light-cone coordinates X±= √1

2(X

• In addition it is simple to see that the chiral SUSY transformations (6.15) on ψµ_Aallow us to set

ψ+_A(τ, σ) = 0. (6.92)

by a suitable choice of chiral SUSY parameter ∓.

• One can then solve for X−_{(τ, σ) and ψ}−

A(τ, σ) by exploiting

T±±= 0 and J±= 0. (6.93)

• In particular one can solve for α−

m and b−r in terms of (d − 2) transverse modes and plug

these into the Hamiltonian to find the mass formula for physical states. i) NS-sector

The result of this procedure is the mass-shell condition

α0M2= ∞ X n=1 αi_−nαi_n | {z } =:N(X) + ∞ X r=1 2 r bi_−rbi_r | {z } =:N(ψ) −aNS. (6.94)

The spectrum thus organises as follows:

• The ground state |0; k iNS is a spacetime scalar state with momentum kµ and of mass

α0M2= −aNS.

• The first level excitations arise by acting not with αi

−1, but with bi−1 2 on the vacuum, |ψ i = ζibi₋1 2|0; k iNS (6.95) with mass α0M2= 1 2 − aNS. (6.96)

This is a transverse vector of SO(d−2) and must thus be massless in a Lorentz invariant theory. Therefore

aNS=

1

2. (6.97)

On the other hand, in the NS sector aNS= (d − 2)( 1 24 |{z} X-CFT + 1 48 |{z} ψ-CFT ), (6.98)

which fixes the critical dimension of the superstring to be

d = 10. (6.99)

Note that the ground state is still tachyonic. We will see how to construct a consis- tent theory without the tachyon in the context of the GSO-projection the subsequent chapters.

• The states at the second excited level, |ψ i = (ζiαi−1+ ζ[ij]bi₋1 2 bj −1 2 ) |0; k iNS, (6.100)

comprise 8 + 8₂
= 36 transverse components. Since this is a massive state of mass α0M2=1

2 (6.101)

it must organise into an irreducible representation of the little group SO(9). Indeed the 36-component representation of SO(9) is just the anti-symmetric as follows by counting

numbers of d.o.f. of antisymmetric of SO(9) = 9 2 

= 36. (6.102)

To summarise the first few level of the open NS-string tower consist of • a tachyonic 1 of SO(9),

• a massless 8V of SO(8),

• massive bosons in tensor representation of SO(9). ii) R-sector:

The mass-shell condition is

α0M2= ∞ X n=1 αi_−nαin+ ∞ X n=1 nbi_−nbin. (6.103)

Indeed, as noted already, the normal ordering constant vanishes because aR= (d − 2)(₂₄1 − 1

24) = 0.

• The ground state |ua_{, k i}

R is a massless (k2 = 0) spacetime spinor. Based on the

notation |s i = |s0, . . . , ski for a spacetime spinor as in section (6.4.2) we introduce the

symbol usfor the wavefunction or polarisation of the various spinor components

|ua_{, k i = |s, k i u} s

|{z}

polarisation

. (6.104)

As a consequence of the Majorana condition on the R-sector zero modes bµ₀, |s i is a priori a Majorana spinor of SO(10). As discussed at the end of section (6.4.2), in 10 dimensions Majorana spinors can be decomposed further into Weyl spinors, corre- sponding to the splitting

32 = 16 ⊕ 160 with real components. The prime denotes a negative chirality Weyl spinor.

Under the decomposition

SO(1, 9) → SO(1, 1) | {z } x± × SO(8) | {z } trans. direct. xi (6.105)

induced by going to spacetime lightcone gauge, the Weyl spinors decompose as 16 → (1₂, 8) ⊕ (−1 2, 8 0_{), (6.106)} 160 → (1 2, 8 0_{) ⊕ (−}1 2, 8). (6.107)

Furthermore |s i must satisfy the Dirac equation (6.83) due to the supercurrent zero- mode constraint,

G0|0 iR= 0.

Since k2 _{= 0 we can pick w.l.o.g. k}

0 = k1, ki = 0. The Dirac equation reads 0 =

kµΓµ|0 i = (k0Γ0+ k1Γ1) |0 i, where

0 = kµΓµ|0 i = k0Γ0+ k1Γ1= −k1Γ0(Γ0Γ1− 1). (6.108)

Here we used that (Γ0)2= −1.

Recalling from section that (6.4.2) S0= Γ0,+Γ0,−−1₂ we rewrite (6.108) as

0 = −2 k1Γ0(S0−

1

2). (6.109)

Thus the Dirac equation implies (S0−1₂) |0 iR= 0, i.e. only the components s0 = 1₂

are kept for the on-shell vacuum. In all, we have

|0 iR= (1₂, 8) ⊕ (1₂, 80). (6.110)

• All higher excitations form massive spinors in irreducible representations of SO(9). For later purposes we define the fermion number

F =  P∞ r=1 2b i −rbir (NS) P∞ n=1b i −nbin (R)  (6.111)

with the property that (−1)F =



1 for even number of b-excitations P∞

n=1b i

−nbin for odd number of b-excitations



(6.112) We will also need the G-parity operator

(NS) : G = (−1)F +1, (6.113)

(R) : G = Γ(−1)F, Γ = Γ0Γ1. . . Γ9. (6.114)

The lowest level open spectrum can be summarised as follows: Sector G-parity SO(8) m2 _Statistics

(NS) + 8v 0 boson

(NS) - 1 − 1

2α0 boson

(R) + 8s 0 fermion

(R) - 8c 0 fermion