If we are willing to accept tachyonic string modes, are there valid projections other than the GSO projections?

Question

One reason superstring theory has often been touted as being an improvement over bosonic string theory is that we can impose GSO projections to remove the tachyonic mode. If we insist upon the no tachyon condition, we are left with only five superstring theories.

But what if we don't mind tachyonic modes? After all, they only signify the background we are expanding about is unstable. Can we have more string models then? Type 0 string theories are one example, but are there others? I realize we can't have supersymmetry in target space if we have negative energies, but there's no theorem stating that string theories have to be supersymmetric.

Answer 1

First, a disclaimer: expanding around an unstable background is... problematic. The actual dynamics of an unstable background dictates a disappearance of the space within a string time. Spacetime supersymmetry at some level is almost certainly needed for some basic consistency.

Now, if you care about theories similar to type 0 string theories, we need to focus on the maximally dimensional ones because there are way too many vacua, even the supersymmetric ones, when some dimensions are compactified (the "landscape").

As far as closed string possibilities go, there is a bosonic string theory in $D=26$. Open strings may be allowed by various D-branes in those theories - the classification of possible D-branes is another step in the classification of backgrounds. And strings may be constrained to be unoriented by the additional of orientifold planes - which are classified analogously to D-branes.

Now, for $D=10$ string theories with the world sheet supersymmetry, type 0A and 0B string theories are the only consistent non-supersymmetric theories that use the superstring both for left-movers and right-movers. It's not hard to see why in the RNS formalism. You always absolutely need the "overall" or "diagonal" GSO projection for all the fermions. Type II theories have separate ones for left-movers and right-movers. Type 0 theories only have the shared one and one may distinguish how the GSO operator acts in the PP sector (odd or even forms, again). Type 0 theories are covered as standard material in chapter 10 of Polchinski's book.

If you consider heterotic string theories, $D=26$ on one side and $D=10$ on the other, there are extra 10-dimensional theories. The extra left-movers may be fermionized and the actions on the supersymmetric fermions and the left-moving $\lambda$ fields may be coupled in various ways. As explained in section 11.3 of Polchinski's book, you may get seven tachyonic heterotic string theories with $SO(16)\times SO(16)$, $SO(16)\times E_8$, $SO(24)\times SO(8)$, $E_7\times E_7\times SO(4)$, $SU(16)\times SO(2)$, single $E_8$, and $SO(32)$ (without SUSY!) gauge groups.

Answer 2

In ten dimensions this question was answered long ago in the three papers listed below. The first one found several new possibilities using orbifold projections including a theory without spacetime supersymmetry and with no tree-level tachyon, the latter two papers classified modular invariant spin structures in a fermionic formulation. Below ten dimensions there are many more possibilities.

1) String Theories in Ten-Dimensions Without Space-Time Supersymmetry. Lance J. Dixon, Jeffrey A. Harvey, (Princeton U.) . PRINT-86-0244 (PRINCETON), Feb 1986. 18pp. Published in Nucl.Phys.B274:93-105,1986.

2) Spin Structures in String Theory. N. Seiberg, (Weizmann Inst. & Princeton, Inst. Advanced Study) , Edward Witten, (Princeton U.) . Print-86-0218 (PRINCETON), Feb 1986. 28pp. Published in Nucl.Phys.B276:272,1986.

3) An O(16) x O(16) Heterotic String. Luis Alvarez-Gaume, Paul H. Ginsparg, Gregory W. Moore, C. Vafa, (Harvard U.) . HUTP-86/A013, Feb 1986. 15pp. Published in Phys.Lett.B171:155,1986.