The groups and symmetries of superstrings/SUGRA/M-theory

Question

When giving talks to Laymen, we find out that the M-theory paradigm says that there are 5 (only 5) superstring theory types, 11d SUGRA and M-theory. Curiously, we read regularly the symmetry groups of heterotic superstrings is $SO(32)$ and $E_8\times E_8$ gauge groups. However, reading the bibliography, it is not easy or clear what are the gauge groups (or even if there are always closed/open strings in the spectrum):

• Bosonic 26d string theory (critical and non-critical): $G=?$
• Type I superstrings (open, closed strings). $G=?$
• Type IIA superstrings (closed, open strings).$G=?$
• Type IIB superstrings (closed/open strings). $G=?$
• Heterotic $SO(32)$.$G=?$
• Heterotic $E_8\times E_8$.$G=?$
• 11d maximal SUGRA. $G=?$

Moreover, I am puzzled by the gauge groups of the heterotic superstrings. Do they include fermionic dimensions? Shouldn't they be gauge SUPERgroups? And I am puzzled about the $E_8$ superstrings since it is a exceptional GROUP, not a supergroup. Should't it be more convenient to give also the symmetry supergroups if any? What about the gauge group of 11d SUGRA? What is it? It is not ever mentioned as far as I know even in the celebrated CJS paper.

Summarizing: what are the gauge groups/supergroups of all the above theories and, beyond that, what are the symmetry group/supergroup/symmetry proposals for M-theory at the current time? Are duality groups also subgroups?

Clarification: I presume the heterotic groups above are only for the bosonic sector of these theories, shouldn't we also provide the fermionic sector and the whole gauge supergroup? Or are those undefined/unclear yet for the above theories?

I ask this question because I believe it is an interesting question to test the state-of-art of symmetries in the string/M-theory context

Answer 1

Closed strings are ubiquitous to string theories. Open strings are the objects that encode the weakly coupled description of D-branes, and D-branes are optional, some backgrounds may contain them, some others don't.

Type II theories and M-theory are perfectly consistent without open strings in flat ten and eleven dimensional spacetimes respectively. The same is true for purely bosonic string theory and M-theory in 26 and 27 dimensions respectively.

There is no possibility of open fundamental Heterotic strings in ten dimensional spacetime. Nevertheless, open cosmic heterotic strings are possible.

Type I string theory is pretty interesting.The theory in ten dimensions is in itself the theory of a stack of sixteen space filling $D9$-branes. So, both open and closed strings are required for consistency.

Concerning the gauge groups. Gauge-like degrees of freedom are only present in type I and heterotic strings. The groups $SO(32)$ and $E_{8} \times E_{8}$ are requiered to be purely bosonic; recall that those groups arise from the mismatch between the left(bosonic) and right(superstring) mover degrees of freedom of the fundamental heterotic string. The 16 dimensions from the mismatch are curled up in a 16-dimensional tori and the momentum modes of the left moving degrees of freedom are constrained (by one-loop modular invariance) to lie on the root lattice of the groups $SO(32)$ or $E_{8} \times E_{8}$. In other words, the Cartan algebra of the gauge groups arise from the momentum modes of the left(purely bosonic) moving sector of the heterotic sting; there is no explicit need of fermionic degrees of freedom.

Duality groups are other very different story. The structure of the groups I discussed concern the gauge sector of some string theories. Dualities are symmetries connecting the full spectra of two seemingly different string theories not just its gauge sector (a really tiny part of all the rich physical spectrum of a string theory).

Concerning supergroups: A supergroup cannot be the gauge symmetry group of an ordinary string theory. You know, a gauge boson with fermionic indices has "the wrong statistics"; acts as an uncancelled ghost in the EFT. In any case, supergroups have been studied without producing fully consistent theories; the known theories contain closed time-like curves, violate unitarity and Lorentz-covariance, see for example: https://arxiv.org/abs/1603.05665 and https://arxiv.org/abs/hep-th/0601024.