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In Zwiebach's book, in the first chapter it was written,

"The lack of adjustable dimensionless parameters is a sign of uniqueness of string theory: it means that the theory cannot be deformed or changed continuously by changing these parameters. But there could be other theories that cannot be reached by continuous deformations. So how many string theories are there?"

So my question is that if we do not have any adjustable dimensionless parameters in string theory, how can we get different string theories?

I think that I am missing some idea or might be the idea is not clear to me.

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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:

  1. Closed oriented strings: $G_{\mu\nu}, B_{\mu\nu}, \Phi$.
  2. Closed unoriented strings: $G_{\mu\nu}, \Phi$.
  3. Closed and open oriented strings: $G_{\mu\nu}, B_{\mu\nu}, \Phi, A_{\mu}$.
  4. 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.

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