3. Superstring Types (I, IIA, IIB, HO, HE)
Adding worldsheet supersymmetry eliminates the bosonic string tachyon and reduces the critical dimension to $D = 10$. There are exactly five consistent superstring theories, all connected by dualities.
Worldsheet Supersymmetry
Extend the bosonic worldsheet action by adding fermionic partners$\psi^\mu(\sigma, \tau)$ to the bosonic fields $X^\mu$. The superstring action in conformal gauge is:
$$S = -\frac{1}{4\pi\alpha'}\int d^2\sigma\left(\partial_\alpha X^\mu\partial^\alpha X_\mu + \bar{\psi}^\mu\rho^\alpha\partial_\alpha\psi_\mu\right)$$
where $\rho^\alpha$ are 2D Dirac matrices. The fermions can have two types of boundary conditions on the closed string:
$$\text{Ramond (R):}\;\psi^\mu(\sigma + 2\pi) = +\psi^\mu(\sigma) \qquad \text{Neveu-Schwarz (NS):}\;\psi^\mu(\sigma + 2\pi) = -\psi^\mu(\sigma)$$
The closed string sectors are NS-NS, NS-R, R-NS, and R-R, where left and right movers can have independent boundary conditions.
GSO Projection
The raw superstring spectrum still contains a tachyon in the NS sector. The Gliozzi-Scherk-Olive (GSO) projection removes it by keeping only states with definite worldsheet fermion number:
$$(-1)^F|\text{phys}\rangle = +|\text{phys}\rangle \qquad (F = \text{worldsheet fermion number})$$
This projection simultaneously: (1) removes the tachyon, (2) ensures spacetime supersymmetry, and (3) guarantees modular invariance of the one-loop partition function:
$$Z(\tau) = \text{Tr}\left[\frac{1 + (-1)^F}{2}\,q^{L_0 - c/24}\,\bar{q}^{\tilde{L}_0 - c/24}\right]$$
The Five Superstring Theories
Type I — SO(32)
Contains both open and closed unoriented strings. $\mathcal{N} = 1$ SUSY in$D = 10$. The gauge group $SO(32)$ is uniquely fixed by tadpole cancellation. The massless spectrum includes the graviton, dilaton, a 2-form$C_2$, and $SO(32)$ gauge bosons.
Type IIA — Non-Chiral
Closed strings only, $\mathcal{N} = 2$ non-chiral SUSY. Opposite GSO projections for left and right movers. The RR sector contains $C_1$ and $C_3$ form fields. The low-energy limit is Type IIA supergravity. This theory arises from M-theory compactified on $S^1$.
Type IIB — Chiral
Closed strings only, $\mathcal{N} = 2$ chiral SUSY. Same GSO projection for both sectors. RR fields include $C_0$, $C_2$, and a self-dual$C_4$. Type IIB is self-dual under S-duality ($g_s \to 1/g_s$).
Heterotic SO(32) and E8 x E8
A remarkable hybrid: the left-movers are bosonic (26D) and the right-movers are supersymmetric (10D). The extra 16 left-moving dimensions are compactified on a self-dual even lattice$\Gamma_{16}$, which must be either the $SO(32)$ or$E_8 \times E_8$ root lattice:
$$\Gamma_{16} \in \left\{\text{Spin}(32)/\mathbb{Z}_2,\; E_8 \times E_8\right\}$$
Superstring Mass Spectrum
The mass formula for the Type II closed superstring includes both bosonic oscillators$\alpha^\mu_n$ and fermionic oscillators $d^\mu_r$:
$$\frac{\alpha'}{4}M^2 = N_B + N_F - a_{\text{NS/R}} \qquad a_{\text{NS}} = \frac{1}{2}, \quad a_{\text{R}} = 0$$
where the bosonic and fermionic number operators are:
$$N_B = \sum_{n=1}^{\infty}\alpha_{-n}\cdot\alpha_n \qquad N_F = \sum_{r>0} r\,d_{-r}\cdot d_r$$
In the Ramond sector, the zero modes $d^\mu_0$ satisfy the Clifford algebra$\{d^\mu_0, d^\nu_0\} = \eta^{\mu\nu}$, making the ground state a spacetime spinor. This is how spacetime fermions emerge from the string.
Universal Massless Sector
All five theories share a common NS-NS sector containing the graviton, dilaton, and (for oriented strings) the Kalb-Ramond B-field. The low-energy effective action for this universal sector is:
$$S_{\text{eff}} = \frac{1}{2\kappa_{10}^2}\int d^{10}x\,\sqrt{-g}\,e^{-2\Phi}\left(R + 4(\partial\Phi)^2 - \frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}\right)$$
where $H = dB$ is the field strength of the B-field and $\Phi$ is the dilaton. The string coupling constant is set by the dilaton VEV:
$$g_s = e^{\langle\Phi\rangle}$$
Green-Schwarz Anomaly Cancellation
Chiral theories in $D = 10$ are potentially anomalous. Green and Schwarz showed that the gauge and gravitational anomalies cancel via a remarkable mechanism. The anomaly polynomial factorizes:
$$I_{12} = X_4 \wedge X_8$$
where $X_4$ and $X_8$ are specific 4-form and 8-form polynomials in the curvature and gauge field strength. The B-field transforms non-trivially under gauge transformations:
$$\delta B = \text{tr}(\Lambda\,dA) - \text{tr}(\Theta\,d\omega) \qquad H = dB - \omega_3^{\text{YM}} + \omega_3^{\text{grav}}$$
This fixes the gauge group to be $SO(32)$ or $E_8 \times E_8$ — precisely the groups allowed by the heterotic string construction. The explicit forms are:
$$X_4 = \text{tr}\,R^2 - \frac{1}{30}\text{tr}\,F^2 \qquad X_8 = \frac{1}{24}\text{tr}\,R^4 + \frac{1}{7200}(\text{tr}\,R^2)^2 - \cdots$$
This was a landmark result in 1984 that launched the “first superstring revolution.”
Simulation: Massless Spectra Comparison
Comparing the massless bosonic field content of all five superstring theories. All share the graviton and dilaton; the differences arise in the RR and gauge sectors.
Click Run to execute the Python code
Code will be executed with Python 3 on the server