The GSO Projection and Open-String Supersymmetry

The unprojected NSR open string contains the right ingredients but not yet the right theory. The NS sector has a tachyon, while the R sector has spacetime spinors. The missing step is the GSO projection, named after Gliozzi, Scherk, and Olive.

The projection does three things at once:

1. it removes the NS tachyon;
2. it chooses one Ramond chirality;
3. it makes the spectrum compatible with spacetime supersymmetry.

The result is the perturbative open superstring. Its massless spectrum is the ten-dimensional $\mathcal N=1$ vector multiplet: a gauge field $A_\mu$ and a Majorana-Weyl gaugino $\lambda$.

The NS projection removes the tachyon and keeps states of one worldsheet-fermion parity. The R projection keeps one ten-dimensional chirality.

Worldsheet fermion number in the NS sector

The NS sector has half-integer fermion modes. The unprojected mass formula is

$$\alpha' M^2=N-\frac12,$$

with

$$N= \sum_{n=1}^{\infty}\alpha_{-n}\cdot\alpha_n + \sum_{r>0}r\,\psi_{-r}\cdot\psi_r, \qquad r\in\mathbb Z+\frac12.$$

Worldsheet fermion number is measured modulo two. We choose the standard open-superstring convention

$$(-1)^F|0;k\rangle_{\rm NS}=-|0;k\rangle_{\rm NS},$$

and every fermion oscillator changes the sign:

$$(-1)^F\psi_{-r}^\mu=-\psi_{-r}^\mu(-1)^F.$$

The usual open-string GSO projection keeps

$$(-1)^F=+1.$$

Therefore the NS vacuum is removed:

$$|0;k\rangle_{\rm NS} \quad\text{has}\quad (-1)^F=-1, \qquad \alpha'M^2=-\frac12.$$

The first excited state survives:

$$\zeta_\mu\psi_{-1/2}^\mu|0;k\rangle_{\rm NS} \quad\text{has}\quad (-1)^F=+1, \qquad \alpha'M^2=0.$$

This is the massless vector.

With the standard assignment $(-1)^F|0\rangle_{\rm NS}=-|0\rangle_{\rm NS}$, the GSO projection keeps the massless vector and removes the tachyon and the even-fermion $N=1$ level.

The surviving NS levels have odd fermion-oscillator number, hence half-integer $N$:

$$N=\frac12,\frac32,\frac52,\ldots.$$

Their masses are

$$\alpha'M^2=N-\frac12=0,1,2,\ldots.$$

So after the projection the open-string NS spectrum has no tachyon and begins with a massless vector.

The Ramond projection: choosing a chirality

In the R sector, the mass formula is

$$\alpha'M^2=N.$$

The ground state has $N=0$, so it is massless. Its zero modes form the Clifford algebra, so the ground state is a ten-dimensional spinor. The R-sector GSO projection chooses a chirality.

A convenient way to write the R parity operator is

$$(-1)^F_{\rm R} = \Gamma_{11}(-1)^{F_{\rm osc}},$$

where $F_{\rm osc}$ counts nonzero fermion oscillators modulo two. On the ground state, $F_{\rm osc}=0$, so the projection is simply a chirality projection:

$$\Gamma_{11}u=+u \quad\text{or}\quad \Gamma_{11}u=-u.$$

The two choices are related by reversing the chirality convention. What matters for the open superstring is that one keeps a single ten-dimensional Majorana-Weyl spinor.

Before the projection, the R ground state contains both ten-dimensional chiralities. The GSO projection keeps one Majorana-Weyl spinor.

The massless R state obeys

$$k_\mu\Gamma^\mu u(k)=0.$$

A ten-dimensional Majorana-Weyl spinor has $16$ real components off shell, and the massless Dirac equation leaves $8$ physical polarizations. These are an $SO(8)$ chiral spinor representation, either $8_s$ or $8_c$ depending on the chirality choice.

The massless open-superstring multiplet

After the GSO projection, the massless open-string states are

$$\text{NS:}\quad \zeta_\mu\psi_{-1/2}^\mu|0;k\rangle_{\rm NS},$$

and

$$\text{R:}\quad |u;k\rangle_{\rm R}, \qquad \Gamma_{11}u=\pm u.$$

The physical conditions are

$$k^2=0, \qquad k\cdot\zeta=0, \qquad \zeta_\mu\sim \zeta_\mu+\lambda k_\mu,$$

and

$$k_\mu\Gamma^\mu u=0.$$

The bosonic state has

$$D-2=8$$

physical polarizations in $D=10$. The fermionic state also has $8$ physical polarizations. In little-group language,

$$A_\mu: 8_v, \qquad \lambda: 8_s\ \text{or}\ 8_c.$$

The massless GSO-projected open string contains $8$ bosonic and $8$ fermionic polarizations. These form the ten-dimensional $\mathcal N=1$ vector multiplet.

In spacetime field theory language, this is the field content of ten-dimensional super Yang-Mills theory:

$$A_\mu \quad\oplus\quad \lambda_\alpha.$$

At low energies, for a stack of open strings with Chan-Paton factors, this becomes nonabelian super Yang-Mills. The schematic supersymmetry transformations are

$$\delta A_\mu=\bar\epsilon\Gamma_\mu\lambda, \qquad \delta\lambda=\frac12F_{\mu\nu}\Gamma^{\mu\nu}\epsilon,$$

where $\epsilon$ is a constant Majorana-Weyl supersymmetry parameter of the same chirality as the gaugino.

First projected levels

The beginning of the projected open-superstring spectrum is as follows.

sector	level $N$	$\alpha'M^2$	representative states	interpretation
NS	$0$	$-1/2$	$\lvert 0;k\rangle_{\rm NS}$	removed
NS	$1/2$	$0$	$\psi_{-1/2}^\mu\lvert 0;k\rangle_{\rm NS}$	gauge boson
NS	$1$	$1/2$	$\alpha_{-1}^\mu\lvert 0;k\rangle$, $\psi_{-1/2}^\mu\psi_{-1/2}^\nu\lvert 0;k\rangle$	removed
NS	$3/2$	$1$	$\psi_{-3/2}^\mu\lvert 0;k\rangle$, $\alpha_{-1}^\mu\psi_{-1/2}^\nu\lvert 0;k\rangle$, $\psi_{-1/2}^\mu\psi_{-1/2}^\nu\psi_{-1/2}^\rho\lvert 0;k\rangle$	first massive bosons
R	$0$	$0$	chiral $\lvert u;k\rangle_{\rm R}$	gaugino
R	$1$	$1$	$\alpha_{-1}^\mu\lvert u;k\rangle$, $\psi_{-1}^\mu\lvert u;k\rangle$	first massive fermions

The physical-state constraints remove unphysical polarizations and assemble the remaining states into irreducible massive representations. The remarkable fact is that, after GSO projection, the number of bosonic and fermionic physical states matches at every mass level.

The GSO-projected open superstring has matching bosonic and fermionic degeneracies level by level. The massless equality is $8_v=8_{s,c}$ as a count of states.

In light-cone gauge this equality can be summarized by Jacobi’s abstruse identity. Schematically, the NS and R oscillator characters are built from theta functions, and the identity

$$\vartheta_3(0|\tau)^4-\vartheta_4(0|\tau)^4-\vartheta_2(0|\tau)^4=0$$

is the partition-function shadow of spacetime supersymmetry.

What has been accomplished

The GSO projection turns the NSR string from a tachyonic theory with worldsheet fermions into a spacetime supersymmetric string theory. The open-string result is

$$\boxed{\text{open superstring massless spectrum} = A_\mu \oplus \lambda_\alpha,}$$

with $A_\mu$ a gauge boson and $\lambda_\alpha$ a ten-dimensional Majorana-Weyl gaugino.

The closed superstring is obtained by taking independent left- and right-moving copies of this construction. Different choices of left and right Ramond chiralities lead to Type IIA and Type IIB strings.

Exercises

Exercise 1. The NS tachyon is projected out

Using $(-1)^F|0;k\rangle_{\rm NS}=-|0;k\rangle_{\rm NS}$ and the projection $(-1)^F=+1$, show that the tachyon is removed but the massless vector survives.

Solution

The tachyon is the NS vacuum itself, so it has

$$(-1)^F=-1.$$

The projection keeps only $(-1)^F=+1$, so the tachyon is removed.

The massless vector is

$$\psi_{-1/2}^\mu|0;k\rangle_{\rm NS}.$$

The fermionic oscillator flips the sign of $(-1)^F$, so this state has

$$(-1)^F=+1.$$

It survives.

Exercise 2. Masses of the first projected NS levels

Show that the surviving NS levels begin at $\alpha'M^2=0$ and then $\alpha'M^2=1$.

Solution

The GSO projection keeps NS states with odd fermion-oscillator number. The first such level is

$$N=\frac12,$$

so

$$\alpha'M^2=N-\frac12=0.$$

The next odd-fermion level is

$$N=\frac32,$$

so

$$\alpha'M^2=\frac32-\frac12=1.$$

The intermediate level $N=1$ has even fermion parity and is removed by the usual projection.

Exercise 3. R-sector chirality and physical polarizations

Show that after the R-sector GSO projection, the massless Ramond ground state has $8$ physical polarizations.

Solution

The R-sector GSO projection keeps one ten-dimensional Majorana-Weyl spinor. Such a spinor has $16$ real off-shell components. The massless physical-state condition is the Dirac equation

$$k_\mu\Gamma^\mu u=0.$$

For a massless spinor this removes half the components, leaving

$$\frac{16}{2}=8$$

physical polarizations.

Exercise 4. Matching the massless vector multiplet

Verify that the massless open-superstring bosons and fermions have the same number of physical polarizations in $D=10$.

Solution

A massless vector in $D$ dimensions has $D-2$ transverse polarizations. For $D=10$,

$$D-2=8.$$

From Exercise 3, the GSO-projected massless Ramond spinor also has $8$ physical polarizations. Thus

$$8_{\rm bosonic}=8_{\rm fermionic}.$$

This is the massless ten-dimensional $\mathcal N=1$ vector multiplet.

Exercise 5. First massive GSO-projected states

List representative NS-sector states at $N=3/2$ and R-sector states at $N=1$. What is their mass?

Solution

At NS level $N=3/2$, representative states are

$$\psi_{-3/2}^\mu|0;k\rangle, \qquad \alpha_{-1}^\mu\psi_{-1/2}^\nu|0;k\rangle, \qquad \psi_{-1/2}^\mu\psi_{-1/2}^\nu\psi_{-1/2}^\rho|0;k\rangle.$$

Their mass is

$$\alpha'M^2=\frac32-\frac12=1.$$

At R level $N=1$, representative states are

$$\alpha_{-1}^\mu|u;k\rangle, \qquad \psi_{-1}^\mu|u;k\rangle.$$

Their mass is

$$\alpha'M^2=1.$$

After imposing constraints, these states organize into matching massive bosonic and fermionic representations.

Exercise 6. A partition-function hint of supersymmetry

Explain why the identity

$$\vartheta_3^4-\vartheta_4^4-\vartheta_2^4=0$$

is naturally interpreted as a sign of spacetime supersymmetry in the open superstring.

Solution

In light-cone gauge, the transverse NS and R oscillator sums can be written in terms of theta functions divided by powers of $\eta$. The two NS spin structures contribute terms involving $\vartheta_3^4$ and $\vartheta_4^4$, while the R sector contributes $\vartheta_2^4$.

After the GSO projection, the combination of bosonic and fermionic contributions is proportional to

$$\vartheta_3^4-\vartheta_4^4-\vartheta_2^4.$$

Jacobi’s abstruse identity says this vanishes. A vanishing one-loop vacuum character is exactly what one expects when bosonic and fermionic states cancel level by level in a supersymmetric spectrum.