There are currently five distinct string theories, which are related by dualities to M-theory. In addition, bosonic string theory also has many variants though it is not considered a valid candidate due to the tachyon and exclusion of fermions from the spectrum.
Bosonic String
The variants of the bosonic string theory can be summarised as:
- Closed oriented strings: $G_{\mu\nu}, B_{\mu\nu}, \Phi$.
- Closed unoriented strings: $G_{\mu\nu}, \Phi$.
- Closed and open oriented strings: $G_{\mu\nu}, B_{\mu\nu}, \Phi, A_{\mu}$.
- Closed and open unoriented strings: $G_{\mu\nu}, \Phi$.
Based on which configurations we include in the definition of the S-matrix for the bosonic string, one will have different amplitudes, and the spectrum of each theory can be different.
Superstring
The type $\mathrm I$ and type $\mathrm{II}$ superstring degrees of freedom on the worldsheet can be divided into left-movers and right-movers (though for open strings, these combine).
For type $\mathrm{II}$, the left-moving and right-moving modes give rise to independent conserved super-symmetry charges, each being a Majorana-Weyl spinor of $16$ real components. Thus, for type $\mathrm{II}$ we have $\mathcal N = 2$ supersymmetry with a total of $32$ components. The subtypes are:
- Type $\mathrm{IIA}$: Two Majorana-Weyl spinors have opposite chirality.
- Type $\mathrm{IIB}$: Two Majorana-Weyl spinors have equal chirality.
For the type $\mathrm I$ superstring, only the sum of the supercharges of the type $\mathrm{IIB}$ theory remains after projection, and so type $\mathrm I$ has $\mathcal N= 1$ supersymmetry in ten dimensions.
Finally, there are two heterotic string theories. In these theories, the left-moving modes are treated as being for a bosonic string in $d=26$ and right-moving for a superstring in $d=10$.
Compactification on the required sixteen dimensional lattices (which must be self-dual and even) gives rise to two variants, namely $SO(32)$ and $E_8 \times E_8$.
This requirement can also be derived from the fact that for $\mathcal N = 1$ supersymmetry, to cancel all anomalies,$^{\dagger}$ there must be massless super Yang-Mills multiplets based on $SO(32)$ or $E_8 \times E_8$ gauge group in the spectrum.
Although we have five distinct superstring theories as candidates, there are an extremely high number of possible compactification schemes we could choose to recover a theory that describes the Standard Model in $d=4$. These give rise to different low energy effective field theories, but it should be understood these all come from one of the five main candidate theories.
$\dagger$ A general demonstration of these anomaly cancellations is too elaborate to present here, as it necessitates the use of characteristic classes and representation theory.
