Superstring theories in ten dimensions

The observable spectrum is the space of BRST cohomology classes. As in the bosonic theory, we impose the additional conditions

b0|ψ = L0|ψ = 0 . (10.5.25) In addition, in the R sector we impose

β₀|ψ = G0|ψ = 0 , (10.5.26) the logic being the same as for (10.5.25). The reader can again work out the first few levels by hand, the result being exactly the same as for OCQ.

The no-ghost theorem is as in the bosonic case. The BRST cohomology has a positive definite inner product and is isomorphic to OCQ and to the transverse Hilbert spaceH^⊥, which is defined to have no α^0,1, ψ^0,1, b, c, β, or γ excitations. The proof is a direct imitation of the bosonic argument of chapter 4.

We have defined exp(πiF) to commute with QB. We can therefore consider subspaces with definite eigenvalues of exp(πiF) and the no-ghost theorem holds separately in each.

10.6 Superstring theories in ten dimensions

We now focus on the theory in ten flat dimensions. For the four sectors of the open string spectrum we will use in addition to the earlier notation NS±, R± the notation

(α, F) , (10.6.1)

where the combination

α = 1− 2ν (10.6.2)

is 1 in the R sector and 0 in the NS sector. Both α and F are defined only mod 2. The closed string has independent periodicities and fermion numbers on both sides, and so has 16 sectors labeled by

(α, F, ˜α, ˜F) . (10.6.3) Actually, six of these sectors are empty: in the NS− sector the level L₀− α^p²/4 is half-integer, while in the sectors NS+, R+, and R− it is an integer. It is therefore impossible to satisfy the level-matching condition L₀ = ˜L₀ if NS− is paired with one of the other three.

Not all of these states can be present together in a consistent string theory. Consider first the closed string spectrum. We have seen that the spinor fields have branch cuts in the presence of R sector vertex operators. Various pairs of vertex operators will then have branch cuts in their operator products — they are not mutually local. The operator F

counts the number of spinor fields in a vertex operator, so the net phase when one vertex operator circles another is

exp πi^F₁α₂− F2α₁− ˜F1˜α₂+ ˜F₂˜α₁^. (10.6.4) If this phase is not unity, the amplitude with both operators cannot be consistently defined.

A consistent closed string theory will then contain only some subset of the ten sectors. Thus there are potentially 2¹⁰ combinations of sectors, but only a few of these lead to consistent string theories. We impose three consistency conditions:

(a) From the above discussion, all pairs of vertex operators must be mutually local: if both (α1, F₁, ˜α₁, ˜F₁) and (α2, F₂, ˜α₂, ˜F₂) are in the spectrum then

F1α2− F2α1− ˜F1˜α2+ ˜F2˜α1 ∈ 2Z . (10.6.5) (b) The OPE must close. The parameter α is conserved mod 2 under operator products (for example, R × R = NS), as is F. Thus if (α1, F₁, ˜α₁, ˜F₁) and (α2, F₂, ˜α₂, ˜F₂) are in the spectrum then so is

(α1+ α2, F₁+ F2, ˜α₁+ ˜α₂, ˜F₁+ ˜F₂) . (10.6.6) (c) For an arbitrary choice of sectors, the one-loop amplitude will not be modular-invariant. We will study modular invariance in the next section, but in order to reduce the number of possibilities it is useful to extract one simple necessary condition:

There must be at least one left-moving R sector (α = 1) and at least one right-moving R sector (˜α = 1).

We now solve these constraints. Assume first that there is at least one R–NS sector, (α, ˜α) = (1, 0). By the level-matching argument, it must either be (R+,NS+) or (R−,NS+). Further, by (a) only one of these can appear, because the product of the corresponding vertex operators is not single-valued. By (c), there must also be at least one NS–R or R–R sector, and because R–NS × R–R = NS–R, there must in any case be an NS–R sector. Again, this must be either (NS+,R+) or (NS+,R−), but not both.

So we have four possibilities, (R+,NS+) or (R−,NS+) with (NS+,R+) or (NS+,R−). Applying closure and single-valuedness leads to precisely two additional sectors in each case, namely (NS+,NS+) and one R–R sector.

The spectra which solve (a), (b), and (c) with at least one R–NS sector are IIB: (NS+,NS+) (R+,NS+) (NS+,R+) (R+,R+) ,

IIA: (NS+,NS+) (R+,NS+) (NS+,R−) (R+,R−) , IIA^: (NS+,NS+) (R−,NS+) (NS+,R+) (R−,R+) , IIB^: (NS+,NS+) (R−,NS+) (NS+,R−) (R−,R−) .

10.6 Superstring theories in ten dimensions 27 Notice that none of these theories contains the tachyon, which lives in the sector (NS−,NS−).

These four solutions represent just two physically distinct theories. In the IIA and IIA^ theories the R–R states have the opposite chirality on the left and the right, and in the IIB and IIB^ theories they have the same chirality. A spacetime reflection on a single axis, say

X²→ −X² , ψ²→ −ψ² , ψ˜²→ −˜ψ² , (10.6.7) leaves the action and the constraints unchanged but reverses the sign of exp(πiF) in the left-moving R sectors and the sign of exp(πi˜F) in the right-moving R sectors. At the massless level this switches the Weyl representations, 16↔ 16^. It therefore turns the IIA^ theory into IIA, and IIB^ into IIB.

Now suppose that there is no R–NS sector. By (c), there must be at least one R–R sector. In fact the combination of (NS+,NS+) with any single R–R sector solves (a), (b), and (c), but these turn out not to be modular-invariant. Proceeding further, one readily finds the only other solutions,

0A: (NS+,NS+) (NS−,NS−) (R+,R−) (R−,R+) , 0B: (NS+,NS+) (NS−,NS−) (R+,R+) (R−,R−) .

These are modular-invariant, but both have a tachyon and there are no spacetime fermions.

In conclusion, we have found two potentially interesting string theories, the type IIA and IIB superstring theories. Referring back to table 10.3, one finds the massless spectra

IIA: [0] + [1] + [2] + [3] + (2) + 8 + 8^+ 56 + 56^ , (10.6.8a) IIB: [0]²+ [2]²+ [4]₊+ (2) + 8^2+ 56² . (10.6.8b) The IIB theory is defined by keeping all sectors with

exp(πiF) = exp(πi˜F) = +1 , (10.6.9) and the IIA theory by keeping all sectors with

exp(πiF) = +1 , exp(πi˜F) = (−1)^˜α . (10.6.10) This projection of the full spectrum down to eigenspaces of exp(πiF) and exp(πi˜F) is known as the Gliozzi–Scherk–Olive (GSO) projection. In the IIA theory the opposite GSO projections are taken in the NS–R and R–

NS sectors, so the spectrum is nonchiral. That is, the spectrum is invariant under spacetime parity, which interchanges 8 ↔ 8^ and 56 ↔ 56^. On the world-sheet, this symmetry is the product of spacetime parity and world-sheet parity. In the IIB theory the same GSO projection is taken in each sector and the spectrum is chiral.

The type 0 theories are formed by a different method: for example, 0B is defined by keeping all sectors with

α = ˜α , exp(πiF) = exp(πi˜F) . (10.6.11) The projections that define the type II theories act separately on the left-and right-moving spinors, while the projections that define the type 0 theory tie the two together. The latter are sometimes called diagonal GSO projections.

The most striking features of the type II theories are the massless vector–spinor gravitinos in the NS–R and R–NS sectors. The terminology type II refers to the fact that these theories each have two gravitinos. In the IIA theory the gravitinos have opposite chiralities (Γ eigenvalues), and in the IIB theory they have the same chirality. The NS–R gravitino state is ψ_−1/2^µ |0; s; k NS–Ruµs . (10.6.12) The physical state conditions are

k²= k^µu_µs = k·Γss^u_µs = 0 , (10.6.13) as well as the equivalence relation

u_µs ∼=u_µs+ kµζ_s . (10.6.14) We have learned that such equivalence relations are the signature of a local spacetime symmetry. Here the symmetry parameter ζsis a spacetime spinor so we have local spacetime supersymmetry. In flat spacetime there will be a conserved spacetime supercharge Q^A_s, where A distinguishes the symmetries associated with the two gravitinos, and s is a spinor index of the same chirality as the corresponding gravitino. Thus the IIA theory has one supercharge transforming as the 16 of S O(9, 1) and one transforming as the 16^, and the IIB theory has two transforming as the 16.

The gravitino vertex operators are

Vse^−˜φψ˜^µe^ik·X , e^−φψ^µV˜se^ik·X . (10.6.15) The operators Vs and ˜Vs, defined in eq. (10.4.25), have weights (1, 0) and (0, 1) and so are world-sheet currents associated with the spacetime supersymmetries.

This is our first encounter with spacetime supersymmetry, and the reader should now study the appropriate sections of appendix B. Section B.2 gives an introduction to spacetime supersymmetry. Section B.4 discusses antisymmetric tensor fields, which we have in the massless IIA and IIB spectra. Section B.5briefly discusses the IIA and IIB supergravity theories which describe the low energy physics of the IIA and IIB superstrings.

In each of the type II theories, there is a unique massless representation, which has 2⁸ = 256 states. The massless superstring spectra are the

10.6 Superstring theories in ten dimensions 29 massless representations of IIA and IIB d = 10 spacetime supersymmetry respectively. This is to be expected: if all requirements for a consistent string theory are met (and they are) then the existence of the gravitinos implies that the corresponding supersymmetries must be present.

The reader may feel that the construction in this section, which is the Ramond–Neveu–Schwarz (RNS) form of the superstring, is somewhat ad hoc. In particular one might expect that the spacetime supersymmetry should be manifest from the start. There is certainly truth to this, but the existing supersymmetric formulation (the Green–Schwarz superstring) seems to be even more unwieldy.

Note that the world-sheet and spacetime supersymmetries are distinct, and that the connection between them is indirect. The world-sheet super-symmetry parameter η(z) is a spacetime scalar and world-sheet spinor, while the spacetime supersymmetry parameter ζs is a spacetime spinor and world-sheet scalar. The world-sheet supersymmetry is a constraint in the world-sheet theory, annihilating physical states. The spacetime super-symmetry is a global super-symmetry of the world-sheet theory, giving relations between masses and amplitudes, though it becomes a local symmetry in spacetime.

Let us note one more feature of the GSO projection. In bosonized form, all the R sector vertex operators have odd length-squared and all the NS sector vertex operators have even length-squared, in terms of the

◦ product defined in section 10.4. This can be seen at the lowest levels for the operators (10.4.22) and (10.4.25), the tachyon having been removed by the GSO projection. By the remark at the end of section 10.4, the space-time spin is then correlated with the world-sheet statistics. In fact, this is the same as the space-time statistics. The world-sheet statistics governs the behavior of the world-sheet amplitude under simultaneous exchange of world-sheet position, spacetime momentum, and other quantum num-bers. After integrating over position, this determines the symmetry of the spacetime S-matrix. The result is the expected spacetime spin-statistics connection. Note that operators with the wrong spin-statistics connection, such as ψ^µ and e^−φ, appear at intermediate stages but the projections that produce a consistent theory also give the spin-statistics connection.

This is certainly a rather technical way for the spin-statistics theorem to arise, but it is worth noting that all string theories seem to obey the usual spin-statistics relation.

Unoriented and open superstrings

The IIB superstring, with the same chiralities on both sides, has a world-sheet parity symmetry Ω. We can gauge this symmetry to obtain an

unoriented closed string theory.⁴ In the NS–NS sector, this eliminates the [2], leaving [0] + (2), just as it does in the unoriented bosonic theory.

The fermionic NS–R and R–NS sectors of the IIB theory have the same spectra, so the Ω projection picks out the linear combination (NS–R) + (R–NS), with massless states 8^+ 56. In particular, one gravitino survives the projection. Finally, the existence of the gravitino means that there must be equal numbers of massless bosons and fermions, so a consistent definition of the world-sheet parity operator must select the [2] from the R–R sector to give 64 of each. One can understand this as follows. The R–R vertex operators

VsV˜s^ (10.6.16)

transform as 8× 8 = [0] + [2] + [4]+. The [0] and [4]₊ are symmetric under interchange of s and s^ and the [2] antisymmetric (one can see this by counting states, 36 versus 28, or in more detail by considering the Sa

eigenvalues of the representations). World-sheet parity adds or subtracts a tilde to give

V˜sVs^ =−Vs^V˜s, (10.6.17) where the final sign comes from the fermionic nature of the R vertex operators. Thus, projecting onto Ω = +1 picks out the antisymmetric [2].

The result is the type I closed unoriented theory, with spectrum

[0] + [2] + (2) + 8^+ 56 = 1 + 28 + 35 + 8^+ 56 . (10.6.18) However, this theory by itself is inconsistent, as we will explain further below.

Now consider open string theories. Closure of the OPE in open + open

→ closed scattering implies that any open string that couples consistently to type I or type II closed superstrings must have a GSO projection in the open string sector. The two possibilities and their massless spectra are

I: NS+, R+ = 8v+ 8 ,

˜I: NS+, R− = 8v+ 8^ .

Adding Chan–Paton factors, the gauge group will again be U(n) in the oriented case and S O(n) or S p(k) in the unoriented case. The 8 or 8^spinors are known as gauginos because they are related to the gauge bosons by supersymmetry. They must be in the adjoint representation of the gauge group, like the gauge bosons, because supersymmetry commutes with the gauge symmetry.

4The analogous operation in the IIA theory would be to gauge the symmetry which is the product of world-sheet and spacetime parity, but this breaks some of the Poincar´e invariance. We will encounter this in section 13.2.