As we discussed, there are two big problems associated with the bosonic string if it is meant to describe our world. The first one is the presence of tachyons in the spectrum, both for closed string and open strings. The other is that there is no spacetime fermions. This is where superstring theory is motivated and cames to the rescue which is obtained by adding a fermionic part on the world-sheet.
The RNS formalism of superstrings The world-sheet action in the conformal gauge hab = ηab
takes the form
S = −Ts 2
Z
d2ξ(∂Xµ∂X¯ µ− i ¯ψµρa∂aψµ) (1.37)
where ρa is the 2D Dirac matrices defined as ρ0 =0 −i i 0 , ρ1=0 i i 0 . (1.38)
1.2. Superstring Theory
and hence ψµ, the 2D world-sheet Majorana fermion with a two-component spinor, is given by ψµ=ψ µ − ψ+µ (1.39) with the reality conditions ψ±∗ = ψ±. These two components are also Weyl spinors satisfying
the massless Dirac equation
∂+ψµ−= ∂−ψ+µ = 0. (1.40)
Here we are still focusing on the flat Minkowski spacetime R1,D−1. The bosonic parts are same to the one discussed in the bosonic strings. Note that the world-sheet Majorana-Weyl spinor ψµ± are vector fields in the spacetime just as the scalar fields Xµ. Further, with this action, they
enjoy the world-sheet 2D N = (2, 2) supersymmetry for closed superstrings. For details on the supersymmetric transformations we refer to the standard textbooks on string theory.
The relevant informations we would like to obtain are the massless spectra. To this end, we shall perform the same mode decomposition as for the bosonic field Xµ before. To do that, we first notice there are also two boundary conditions: Ramond(R) and Neveu-Schwarz(NS) for the WS fermions ψµ± in the open superstrings, which are given by
R : ψ+µ(τ, σ = 0) = ψ−µ(τ, σ = 0), ψ+µ(τ, σ = π) = ψ−µ(τ, σ = π),
N S : ψ+µ(τ, σ = 0) = ψ−µ(τ, σ = 0), ψ+µ(τ, σ = π) = −ψ−µ(τ, σ = π). (1.41) Given these boundary conditions, following the same procedures shown in the bosonic strings, we first obtain the general solutions for the 2D fermions in the open superstrings by mode decomposition as ψ±µ = √1 2 X r bµre−ir(τ ±σ), (bµr)∗ = bµ−r (1.42)
where r is half-integer for NS sector and integer for R sector.
In the same way, we can obtain similar mode expansions in the closed string sector, but now with independent boundary conditions for the left-mover ψ+µ and the right-moverψµ− as
ψµ+=X r ebµre−2ir(τ +σ); ψ−µ = X r bµre−2ir(τ −σ) (1.43)
where for each of ebr, brone can assign half-integers and integers for r corresponding to NS-sectors
and R-sectors. As we shall see later, the R-sector will give rise to spacetime fermions whereas the NS-sector yields spacetime bosons. Depending on different pairing for ψ+, ψ− we can group
four closed string sectors depending on their spacetime states as: • Bosons: NS-NS and R-R
• Fermions: NS-R and R-NS
The V irasoro algebras in the superstring have been extended to super − V irasoro as Ln= 1 2 ∞ X m=−∞ an−m· am+ 1 4 X r (2r − n)bn−r· br Gr= ∞ X m=−∞ am· br−m (1.44)
superstrings. The vanishing of the energy-momentum tensor and the supercurrent impose the constraint Ln= Gr= 0, ∀(n, r), which again is infinite number. This actually reflects that the
2D RNS action of the superstring is a 2D N = (2, 2) superconformal theory.
The superstring spectrum Let us first focus on the case of the open superstrings. Applying the standard canonical quantization, the canonical anti-commutator relations for the WS fermion generators are given by
{bµr, bνs} = ηµνδr+s,0. (1.45)
Given this, we can interpret the bµr with r < 0 as the raising operators and ones with r > 0 as
lowering operators. Note that only R-sector gives a zero mode bµ0. The full Hilbert space is the free tensor product of the bosonic and the fermionic spaces.
By the same token in the bosonic strings, one can obtain the mass formula from the super Virasor algebra constraints, i.e. (L0− cs|physi = 0), which reads
M2= 1
α0(N + Nψ) − cs (1.46)
where Nψ =Pr>0rb−r· br. The normal ordering constant cs turns out to be 1/2 for the NS
sector and cs= 0 for the R sector as the fermionic modes br contribute. The critical dimension
D of the superstring can be shown to be
D = 10 (1.47)
as a consequence of being a consistent theory.
In the sequel, we only focus on analyzing the spectrum of the fermionic part, as the bosonic part essentially is as the one in the pervious section.
NS sector: In the NS sector, the ground state is still a tachyon with the mass given by −1
4p
2 = −1
2 based on (1.46) with N = Nφ= 0. At the first excited level, there is a massless
vector
|µ, pi = bi−1/2|pi, i = 1, ..., 8. (1.48) Here in order to manifest the unitarity, we have employed the light-cone gauge, where only the transverse components of raising operators bi−1/2, i = 1, .., 8 are physical. Thereby bi−1/2|pi transforms as the vector representation of SO(8) denoted as 8v.
R sector: In the R sector, the ground state in the R sector is massless and given by the solution
ain|p; 0iR= bir|p; 0iR= 0, (n, r) > 0, (1.49) together with the massless Dirac equation. Further, there is a zero mode bµ0 which satisfies the aniti-commutantor algebra, generating the 10D Clifford algebra
{bµ0, bν0} ∼ ηµν. (1.50)
Thus the bµ0’s shall be viewed as 11D Dirac matrices. Applying this zero mode on the ground state |p; 0iR, one can see that the ground state gives rise to a massless Spin(1, 9) spinor. It
turns out all states in the R sector are spacetime fermions. In 10D, one can impose both the Majorana and Weyl conditions at the same time for massless spinor, so that the ground state |p; 0iR can be chosen to have a definite spacetime chirality. It turns out that there are two
1.2. Superstring Theory
It seems that the tachyons still remain in the spectrum of NS sectors in the superstring theories. However, it turns out that the above spectrum is not consistent with world-sheet modular invariance, which is required for eliminating the absence of global anomalies under 2D large diffeomorphisms disconnected from the identity in some topologically non-trivial Riemann surfaces. This consistency, on the other hand, implies that one should impose the GSO projection
PGSO=
1
2(1 − (−1)
F) (1.51)
where F =P
r>0b−r· br is the fermion number operator, satisfying {(−1)F, ψµ} = 0. The GSO
projection turns out to eliminate the tachyons from the NS sectors while in the R sector it acts as spacetime chirality. It follows that at the massless level of open superstring there is one massless vector and one Weyl Majorana spinor, both having 8 degrees of freedom. This is a consequence of a resulting spacetime supersymmetry.
Now we are at the position to talk about the massless spectrum of the closed superstring, known as Type II superstrings 4. The spectrum can be obtained by taking tensor products of left- and right-movers, each of which can be viewed as the open superstring described as above and hence there are four possible sectors: R-R, NS-NS and R-NS, NS-R. It turns out there are two different theories of closed superstrings, dubbed as Type IIA and Type IIB, depending on chiralities of left- and right- movers. Type IIA have opposite chralities for the R-sectors of the two independent movers wheras type IIB has same chirality.
1.2.1. Massless spectrum of Type IIA superstring
The massless spectrum of type IIA superstring is obtained by the tensor product
(8v⊕ 8s) ⊗ (8v⊕ 8c), (1.52)
where 8v denotes the NS sector and 8s and 8c are the two representations of Majorana-Weyl spinors for the R sector with different chiralities.
In the NS-NS sector, the massless spectra are given by
8v⊗ 8v= 1 ⊕ 28A⊕ 35S = Φ ⊕ Bµν⊕ gµν, (1.53)
which contains the spacetime metric gµν, the dilaton φ and a 2-form Kalb-Ramond potential
B2. In the RR sector, the Type IIA spectra are given by
8s⊗ 8c= 8 ⊕ 56 = C1⊕ C3, (1.54)
which gives rise to the generalized p-form potentials Ci for i = 1, 3. The Fermions, on the other hand, are given by two NS-R sectors
8v⊗ 8s= 8R⊕ 56L,
8v⊗ 8c= 8L⊕ 56R.
(1.55) Indeed, the fermions contains the two 10D Majorana-Weyl spinors ζα and Weyl gravitino ψµα of opposite chirality.
4
By closed strings, we really means that only the closed string can propagates on the bulk spacetime. In Type II strings, open strings do exist, but they only propagate on the defects of the bulk, i.e. D-branes.
The effective theories describing the massless spectrum of superstrings are supergravities. The type IIA supergravity has N = 2 supersymmetry, and the two supersymmetries are of opposite chirality generated by Q1α∈ 16 and Q2
α∈ 160.
1.2.2. Massless of the Type IIB superstrings
The massless spectrum of type IIB superstring is encoded in the tensor product of various representations of SO(8):
(8v⊕ 8s) ⊗ (8v⊕ 8s). (1.56)
Similarly, the bosonic spectrum is given by the NS-NS sector and the R-R sector. In the NS-NS sector, the massless spectrum is same as in Type IIA, given by
8v⊗ 8v= 1 ⊕ 28 ⊕ 35 = φ ⊕ Bµν ⊕ gµν, (1.57)
which contains the spacetime metric gµν, the dilaton φ and a 2-form Kalb-Ramond potential B2. In the RR sector, the Type IIB spectrum is
8s⊗ 8s= 1 ⊕ 28 ⊕ 35+= C0⊕ C2⊕ C4+, (1.58)
which gives rise to the generalized p-form potential fields Ci for i = 0, 2, 4. The ” + ” on C4 denotes that C4 is self-dual.
The fermions are given by two NS-R sectors
8v⊗ 8s= 8R⊕ 56L,
8v⊗ 8s= 8R⊕ 56L,
(1.59) which contains the two 10D Left-handed Majorana-Weyl spinors ζα and right-handed Weyl gravitino ψµα.
Type IIB string theory can be well described by the ten dimensional type IIB supergravity at low energy, which also has N = 2 supersymmetries, and the two supersymmetries are of same chirality generated by Q1α∈ 16 and Q2
α∈ 16.
Before closing this subsection, we would like to mention that upon compactification to 9 dimension on a circle of radius R, the two type II superstrings are equivalent under T-duality:
R → 1
R, IIA ↔ IIB. (1.60)
Insert 2.1: Some facts on the p + 1-form potential fields Ap in superstring theory A p + 1 form field Ap+1 can be viewed as the higher dimensional generalization of the
gauge vector potential field A1. They enjoy similar gauge transformation as
Ap+1→ Ap+1+ dλp, (1.61)
where λp is an arbitrary closed p form and d denotes the Exterior derivative. In superstring
theories, there are two types of higher p + 1 potential forms: B2 and Cp+1. Like the vector
gauge potential A1 coupling to a (electrically) charged point particle with charge e as e
Z