### Heterotic Nostalgia

**(Yup, another MathML-enabled post. The usual caveats apply:
you need the fonts and you need Mozilla to see the equations as they were intended.)**

I gave a talk today in our Brown bag seminar about doublet-triplet splitting (and related matters) in String/M-Theory. It made me feel old because most of the audience were far too young (having been in elementary school at the time) to be familiar with the heterotic version of the story. But it also made me realize what had been (to me, at least) a nagging lack in the M-theory version of the story, as promulgated to date.

One of the great successes of the heterotic story was that one
obtains the “successful” predictions of SU(5) SUSY GUTS (*e.
g.* for ${sin}^{2}\left({\theta}_{W}\right)$), without some of the less-successful
features (the doublet-triplet splitting problem, SU(5) relations
among superpotential couplings, … ).

The origin of this is that, in the heterotic string, unlike in
conventional GUTS, the generations do *not* form SU(5)
multiplets. The vertex operators for quarks and leptons in a single
“generation” are constructed using *distinct* chiral primaries
of the internal SCFT. If there’s just a single $5+\stackrel{\u203e}{5}$ of
Higgses, the corresponding chiral primaries must be *
invariant* under the ${\mathbb{Z}}_{n}$ orbifolding. Hence they
don’t give rise to massless colour triplets^{*}.

This feature of the heterotic story has, so far, been invisible in the M-theory approach. I strongly believe (at least in some cases) that it is still there, but it is a bit harder to tease out of the M-theory formulation.

^{*} If people are curious to see the details, I could put up
some TeXed notes on the subject. I can’t be bothered to do it in
itex.