Type I and type II superstrings
10.5 Physical states
In the bosonic string we started with a (diff×Weyl)-invariant theory.
After fixing to conformal gauge we had to impose the vanishing of the conformal algebra as a constraint on the states. In the present case there is an analogous gauge-invariant form, and the superconformal algebra emerges as a constraint in the gauge-fixed theory. However, it is not necessary to proceed in this way, and it would require us to develop some machinery that in the end we do not need. Rather we can generalize directly in the gauge-fixed form, defining the superconformal symmetry to be a constraint and proceeding in parallel to the bosonic case to construct a consistent theory. We will first impose the constraint in the old covariant formalism, and then in the BRST formalism.
OCQ
In this formalism, developed for the bosonic string in section 4.1, one ignores the ghost excitations. We begin with the open string, imposing the physical state conditions
Lmn|ψ = 0 , n > 0 , Gmr |ψ = 0 , r ≥ 0 . (10.5.1) Only the matter part of any state is nontrivial — the ghosts are in their ground state — and the superscript ‘m’ denotes the matter part of each generator. There are also the equivalence relations
Lmn|χ ∼= 0, n < 0 , Gmr |χ ∼= 0, r < 0 . (10.5.2) The mass-shell condition can always be written in terms of the total matter plus ghost Virasoro generator, which is the same as the world-sheet Hamiltonian H because the total central charge is zero:
L0|ψ = H|ψ = 0 . (10.5.3)
In ten flat dimensions this is H =
αp2+ N−1
2 (NS) αp2+ N (R)
. (10.5.4)
10.5 Physical states 21 The zero-point constants from the ghosts and longitudinal oscillators have canceled as usual, leaving the contribution of the transverse modes,
NS: 8
−1 24 − 1
48
=−1
2 , R: 8
− 1 24+ 1
24
= 0 . (10.5.5) For the tachyonic and massless levels we need only the terms
Gm0 = (2α)1/2pµψµ0+ . . . , (10.5.6a) Gm±1/2 = (2α)1/2pµψµ±1/2+ . . . . (10.5.6b) The NS sector works out much as in the bosonic string. The lowest state is
|0; k NS, labeled by the matter state and momentum. The only nontrivial condition is from L0, giving
m2=−k2=− 1
2α . (10.5.7)
This state is a tachyon. It has exp(πiF) = −1, where F was given in eq. (10.2.24). The first excited state is
|e; k NS = e· ψ−1/2|0; k NS . (10.5.8) The nontrivial physical state conditions are
0 = L0|e; k NS = αk2|e; k NS , (10.5.9a) 0 = Gm1/2|e; k NS = (2α)1/2k· e|0; k NS , (10.5.9b) while
Gm−1/2|0; k NS= (2α)1/2k· ψ−1/2|0; k NS (10.5.10) is null. Thus
k2= 0 , e· k = 0 , eµ∼=eµ+ kµ . (10.5.11) This state is massless, the half-unit of excitation canceling the zero-point energy, and has exp(πiF) = +1. Like the first excited state of the bosonic string it is a massless vector, with D − 2 spacelike polarizations. The constraints have removed the unphysical polarizations of ψµ, just as for Xµ in the bosonic case.
In the R sector the lowest states are
|u; k R=|s; k Rus. (10.5.12) Here us is the polarization, and the sum on s is implicit. The nontrivial physical state conditions are
0 = L0|u; k R = αk2|u; k R , (10.5.13a) 0 = Gm0|u; k R = α1/2|s; k Rk· Γssus . (10.5.13b)
λ
Table 10.2. Massless and tachyonic open string states.
sector S O(8) spin m2
NS+ 8v 0
NS− 1 −1/2α
R+ 8 0
R− 8 0
The ground states are massless because the zero-point energy vanishes in the R sector. The Gm0 condition gives the massless Dirac equation
k· Γssus= 0 , (10.5.14) which was our original goal in introducing the superconformal algebra.
The Gm0 condition implies the L0condition, because G20= L0in the critical dimension and the ghost parts of G0 annihilate the ghost vacuum.
In ten dimensions, massless particle states are classified by their behavior under the S O(8) rotations that leave the momentum invariant. Take a frame with k0 = k1. In the NS sector, the massless physical states are the eight transverse polarizations forming the vector representation 8v of S O(8). In the R sector, the massless Dirac operator becomes
k0Γ0+ k1Γ1=−k1Γ0(Γ0Γ1− 1) = −2k1Γ0(S0− 12) . (10.5.15) The physical state condition is then
(S0−12)|s, 0; k Rus= 0 , (10.5.16) so precisely the states with s0 = +12 survive. As discussed in section B.1, we have under S O(9, 1)→ SO(1, 1) × SO(8) the decompositions
16 → (+12, 8) + (−12, 8) , (10.5.17a) 16 → (+12, 8) + (−12, 8) . (10.5.17b) Thus the Dirac equation leaves an 8 with exp(πiF) = +1 and an 8 with exp(πiF) =−1.
The tachyonic and massless states are summarized in table 10.2. The open string spectrum has four sectors, according to the periodicity ν and the world-sheet fermion number exp(πiF). We will use the notation NS± and R± to label these sectors. We will see in the next section that consistency requires us to keep only certain subsets of sectors, and that there are consistent string theories without the tachyon.
10.5 Physical states 23 Table 10.3. Products of S O(8) representations appearing at the massless level of the closed string.The R–NS sector has the same content as the NS–R sector.
sector S O(8) spin tensors dimensions
(NS+,NS+) 8v× 8v = [0] + [2] + (2) = 1 + 28 + 35 (R+,R+) 8× 8 = [0] + [2] + [4]+ = 1 + 28 + 35+
(R+,R−) 8× 8 = [1] + [3] = 8v+ 56t (R−,R−) 8× 8 = [0] + [2] + [4]− = 1 + 28 + 35−
(NS+,R+) 8v× 8 = 8+ 56
(NS+,R−) 8v× 8 = 8 + 56
Closed string spectrum
The closed string is two copies of the open string, with the momentum rescaled k→ 12k in the generators. With ν, ˜ν taking the values 0 and 12, the mass-shell condition can be summarized as
α
4m2= N− ν = ˜N− ˜ν . (10.5.18) The tachyonic and massless closed string spectrum is obtained by com-bining one left-moving and one right-moving state, subject to the equal-ity (10.5.18).
The (NS−,NS−) sector contains a closed string tachyon with m2 =
−2/α. At the massless level, combining the various massless left- and right-moving states from table 10.2 leads to the S O(8) representations shown in table 10.3. Note that level matching prevents pairing of the NS− sector with any of the other three. As in the bosonic string, vector times vector decomposes into scalar, antisymmetric tensor, and traceless symmetric tensor denoted (2). The products of spinors are discussed in section B.1.
The 64 states in 8v × 8 and 8v × 8 each separate into two irreducible representations. Denoting a state in 8v× 8 by |i, s , we can form the eight linear combinations
|i, s Γiss . (10.5.19) These states transform among themselves under S O(8), and they are in the 8 representation because the chirality of the loose index s is opposite to that of s. The other 56 states form an irreducible representation 56. The product 8v×8 works in the same way. Note that there are several cases of distinct representations with identical dimensions: at dimension 8 a vector and two spinors, at dimension 56 an antisymmetric rank 3 tensor and two vector-spinors, at dimension 35a traceless symmetric rank 2 tensor and self-dual and anti-self-dual rank 4 tensors.
BRST quantization
From the general structure discussed in chapter 4, in particular the expres-sion (4.3.14) for the BRST operator for a general constraint algebra, the BRST operator can be constructed as a simple extension of the bosonic one: and the same on the antiholomorphic side. As in the bosonic case, this is a tensor up to an unimportant total derivative term.
The BRST current has the essential property jB(z)b(0)∼ . . . +1
zTB(0) , jB(z)β(0)∼ . . . + 1
zTF(0) , (10.5.22) so that the commutators of QBwith the b, β ghosts give the corresponding constraints.3 In modes,
{QB, bn} = Ln , [QB, βr] = Gr . (10.5.23) From these one can verify nilpotence by the same steps as in the bosonic case (exercise 4.3) whenever the total central charge vanishes. Thus, we can replace some of the spacelike Xµψµ SCFTs with any positive-norm SCFT such that the total matter central charge is cm = ˜cm = 15. The BRST current must be periodic for the BRST charge to be well defined.
The supercurrent of the SCFT must therefore have the same periodicity, R or NS, as the ψµ, β, and γ. The expansion of the BRST operator is where m and n run over integers and r over (integers + ν). The ghost normal ordering constant is as in eq. (10.4.5).
3The bcβγ theory actually has a one-parameter family of superconformal symmetries, related by rescaling β → xβ and γ → x−1γ. The general BRST construction (4.3.14) singles out the symmetry (10.1.21); this is most easily verified by noting that it correctly leads to the OPEs (10.5.22).