Modular invariance

We can already anticipate that not all of these theories will be con-sistent. The open string multiplets, with 16 states, are representations of d = 10, N = 1 supersymmetry but not of N = 2 supersymmetry. Thus the open superstring cannot couple to the oriented closed superstring theories, which have two gravitinos.⁵ It can only couple to the unoriented closed string theory (10.6.18) and so the open string theory must also be unoriented for consistent interactions. With the chirality (10.6.18), the massless open string states must be 8_v+ 8. This is required by spacetime supersymmetry, or by conservation of exp(πiF) on the world-sheet. The result is the unoriented type I open plus closed superstring theory, with massless content

[0] + [2] + (2) + 8^+ 56 + (8_v+ 8)S O(n) or S p(k) . (10.6.19) There is a further inconsistency in all but the S O(32) theory. We will see in section 10.8 that for all other groups, as well as the purely closed unoriented theory, there is a one-loop divergence and superconformal anomaly. We will also see, in chapter 12, that the spacetime gauge and coordinate symmetries have an anomaly at one loop for all but the S O(32) theory.

Thus we have found precisely three tachyon-free and nonanomalous string theories in this chapter: type IIA, type IIB, and type I S O(32).

10.7 Modular invariance

one important amplitude that involves no interactions, only the string spectrum. This is the one-loop vacuum amplitude, studied for the bosonic string in chapter 7. We study the vacuum amplitude for the closed super-string in this section and for the open super-string in the next.

We make the guess, correctly it will turn out, that the torus amplitude is again given by the Coleman–Weinberg formula (7.3.24) with the region of integration replaced by the fundamental region for the moduli space of the torus:

ZT₂ = V10

F

d²τ 4τ2

d¹⁰k (2π)¹⁰



i∈H^⊥

(−1)^Fiq^{α(k2+m2i)/4}¯q^{α(k2+ ˜m2i)/4} , (10.7.1) with q = exp(2πiτ). We have included the minus sign for spacetime

5At the world-sheet level the problem is that the total derivative null gravitino vertex operators give rise to nonzero world-sheet boundary terms. Only one linear combination of the two null gravitinos decouples, so we must make the world-sheet parity projection in order to eliminate the other.

Superstring interactions are the subject of chapter 12, but there is

fermions from the Coleman–Weinberg formula, distinguishing the space-time fermion number F from the world-sheet fermion number F. The masses are given in terms of the left- and right-moving parts of the transverse Hamiltonian by

m²= 4H^⊥/α^ , m˜²= 4 ˜H^⊥/α^ . (10.7.2) The trace includes a sum over the different (α, F; ˜α, ˜F) sectors of the superstring Hilbert space. In each sector it breaks up into a product of independent sums over the transverse X, ψ, and ˜ψ oscillators, and the transverse Hamiltonian similarly breaks up into a sum. Each transverse X contributes as in the bosonic string, the total contribution of the oscillator sum and momentum integration being as in eq. (7.2.9),

Z_X(τ) = (4π²α^τ₂)^−1/2(q¯q)^−1/24

For the ψs, the mode sum in each sector depends on the spatial period-icity α and includes a projection operator ¹₂[1± exp(πiF)]. Although for the present we are interested only in R and NS periodicities, let us work out the partition functions for the more general periodicity (10.3.20),

ψ(w + 2π) = exp[πi(1− α)] ψ(w) (10.7.4) where again α = 1− 2ν. By the definition (10.3.23) of the ground state, the raising operators are

ψ_{−m+(1−α)/2} , ψ_{−m+(1+α)/2} , m = 1, 2, . . . . (10.7.5) The ground state weight was found to be α²/8. Then

Trα

To define the general boundary conditions we have joined the fermions into complex pairs. Thus we can define a fermion number Q which is +1 for ψ and−1 for ψ. To be precise, define Q to be the H-momentum in the bosonization (10.3.10) so that it is conserved by the OPE. The bosonization (10.3.25) then gives the charge of the ground state as α/2.

τ

10.7 Modular invariance 33 Thus we can define the more general trace

Z^α_β(τ) = Trα

The notation in the final line was introduced in section 7.2, but our discussion of these functions in the present volume will be self-contained.

The charge Q modulo 2 is the fermion number F that appears in the GSO projection. Thus the traces that are relevant for the ten-dimensional superstring are We should emphasize that these traces are for a pair of dimensions.

Tracing over all eight fermions, the GSO projection keeps states with exp(πiF) = +1. This is Z_ψ⁺(τ), where The half is from the projection operator, the minus sign in the second term is from the ghost contribution to exp(πiF), and the minus signs in the third and fourth (R sector) terms are from spacetime spin-statistics.

For ˜ψ in the IIB theory one obtains the conjugate Z_ψ⁺(τ)^∗. In the IIA We know from the discussion of bosonic amplitudes that modular in-variance is necessary for the consistency of string theory. In the superstring this works out in an interesting way. The combination d²τ/τ²₂ is modular-invariant, as is ZX. To understand the modular transformations of the fermionic traces, note that Z^αβ is given by a path integral on the torus

over fermionic fields ψ with periodicities

ψ(w + 2π) = − exp(−πiα) ψ(w) , (10.7.11a) ψ(w + 2πτ) = − exp(−πiβ) ψ(w) . (10.7.11b) This gives

ψ[w + 2π(τ + 1)] = exp[−πi(α + β)] ψ(w) . (10.7.12) Naively then, Z^αβ(τ) = Z^αα+β−1(τ + 1), since both sides are given by the same path integral. Also, defining w^ = w/τ and ψ^(w^) = ψ(w),

ψ^(w^+ 2π) = − exp(−πiβ) ψ^(w^) (10.7.13a) ψ^(w^− 2π/τ) = − exp(πiα) ψ^(w^) , (10.7.13b) so that naively Z^αβ(τ) = Z_β−α(−1/τ). It is easy to see that by these two transformations one can always reach a path integral with α = 1, accounting for rule (c) from the previous section.

The reason these modular transformations are naive is that there is no diff-invariant way to define the phase of the path integral for purely left-moving fermions. For left- plus right-moving fermions with matching boundary conditions, the path integral can be defined by Pauli–Villars or other regulators. This is the same as the absolute square of the left-moving path integral, but leaves a potential phase ambiguity in that path integral separately.⁶ The naive result is correct for τ→ −1/τ, but under τ → τ + 1 there is an additional phase,

Z^αβ(τ) = Z^β_−α(−1/τ)

= exp[−πi(3α²− 1)/12] Z^αα+β−1(τ + 1) . (10.7.14) The τ → τ + 1 transformation follows from the explicit form (10.7.7b), the phase coming from the zero-point energy with the given boundary conditions. The absence of a phase in τ→ −1/τ can be seen at once for τ = i. Note that Z¹1actually vanishes due to cancellation between the two R sector ground states, but we have assigned a formal transformation law for a reason to be explained below.

The phase represents a global gravitational anomaly, an inability to define the phase of the path integral such that it is invariant under large coor-dinate transformations. Of course, a single left-moving fermion has c= ˜c and so has an anomaly even under infinitesimal coordinate transforma-tions, but the global anomaly remains even when a left- and right-moving fermion are combined. For example, the product Z¹0(τ)^∗Z⁰₀(τ) has no infinitesimal anomaly and should come back to itself under τ → τ + 2,

6The phase factor is a holomorphic function of τ, because the Z^αβare. Since it has magnitude 1, this implies that it is actually independent of τ.

10.7 Modular invariance 35 but in fact picks up a phase exp(−πi/2). This phase arises from the level mismatch, the difference of zero-point energies in the NS and R sectors.

The reader can verify that with the transformations (10.7.14), the combi-nations Z_ψ^±are invariant under τ→ −1/τ and are multiplied by exp(2πi/3) under τ → τ + 1. Combined with the conjugates from the right-movers, the result is modular-invariant and the torus amplitude consistent. It is necessary for the construction of this invariant that there be a multiple of eight transverse fermions. Recall from section 7.2 that invariance under τ → τ + 1 requires that L0− ˜L0 be an integer for all states. For a single real fermion in the R–NS sector the difference in ground state energies is

1

16. For eight fermions this becomes ¹₂, so that states with an odd number of NS excitations (as required by the GSO projection) are level-matched.

Note also that modular invariance forces the minus signs in the combi-nation (10.7.9), in particular the relative sign of (Z⁰0)⁴ and (Z¹0)⁴ which corresponds to Fermi statistics for the R sector states.

In the type 0 superstrings the fermionic trace is 1

2

|Z⁰0(τ)|^N +|Z⁰1(τ)|^N+|Z¹0(τ)|^N ∓ |Z¹1(τ)|^N (10.7.15)

with N = 8. This is known as the diagonal modular invariant, and it is invariant for any N because the phases cancel in the absolute values.

The type II theories have spacetime supersymmetry. This implies equal numbers of bosons and fermions at each mass level, and so ZT2 should vanish in these theories by cancellation between bosons and fermions.

Indeed it does, as a consequence of Z¹1 = 0 and the ‘abstruse identity’ of Jacobi,

Z⁰₀(τ)⁴− Z⁰1(τ)⁴− Z¹0(τ)⁴= 0 . (10.7.16) The same cancellation occurs in the open and unoriented theories.

Although we have focused on the path integral without vertex operators, amplitudes with vertex operators must also be modular-invariant. In the present case the essential issue is the path integral measure, and one can show by explicit calculation (or by indirect arguments) that the modular properties are the same with or without vertex operators. However, with a general vertex operator insertion the α = β = 1 path integral will no longer vanish, nor will the sum of the other three. The general amplitude will then be modular-invariant provided that the vacuum is modular-invariant without using the vanishing of Z¹₁ or the abstruse identity (10.7.16) — as we have required.

More on c = 1 CFT

The equality of the bosonic and fermionic partition functions (10.3.17) and (10.3.18) was one consequence of bosonization. These partition

func-tions are not modular-invariant and so do not define a sensible string background. The fermionic spectrum consists of all NS–NS states. The bosonic spectrum consists of all states with integer kR and kL; this is not the spectrum of toroidal compactification at any radius. The simplest modular-invariant fermionic partition function is the diagonal invariant, taking common periodicities for the left- and right-movers. In terms of the states, this amounts to projecting

α = ˜α , exp(πiF) = exp(πi˜F) . (10.7.17) The NS–NS sector consists of the local operators we have been consider-ing, and the chirality projection exp(πiF) = exp(πi˜F) means that on the bosonic side kR = kL mod 2. The bosonic equivalents for the R–R sector states have half-integral kR and kL with again kR = kLmod 2. In all,

(kR, k_L) = (n1, n₂) or (n1+ ¹₂, n₂+ ¹₂) (10.7.18) for integers n1 and n2 such that n1− n2 ∈ 2Z. This is the spectrum of a boson on a circle of radius 2, or 1 by T -duality, which we see is equivalent to a complex fermion with the diagonal modular-invariant projection.

(The dimensionless radius r for the H scalar corresponds to the radius R = r(α^/2)^1/2 for X^µ, so r = 2^1/2 is self-dual.)

To obtain an equivalent fermionic theory at arbitrary radius, add

∂H ¯∂H ∼=−ψψ ˜ψ ˜ψ (10.7.19) to the world-sheet Lagrangian density. The H theory is still free, but the equivalent fermionic theory is now an interacting field theory known as the Thirring model. The Thirring model has a nontrivial perturbation series but is solvable precisely because of its equivalence to a free boson.

Actually, for any rational r, the bosonic theory is also equivalent to a free fermion theory with a more complicated twist (exercise 10.15).

Another interesting CFT consists of the set of vertex operators with kR = m/3^1/2 , kL= n/3^1/2 , m− n ∈ 3Z . (10.7.20) (This discussion should actually be read after section 11.1.) It is easy to check that this has the same properties as the set of vertex operators with integer kR,L. That is, it is a single-valued operator algebra, but does not correspond to the spectrum of the string for any value of r, and does not have a modular-invariant partition function. Its special property is the existence of the operators

exp^±i3^1/2H(z)^ , exp^±i3^1/2H(¯˜ z)^ . (10.7.21) These have weights (³₂, 0) and (0,³₂): they are world-sheet supercurrents!