Gross et al. (1985) came up with heterotic string theory by combining left-moving bosonic strings in 26d with right-moving fermionic strings in 10d by compactifying 16 of the 26 dimensions on a self-dual torus of \(\textrm{E}_8 \!\times\! \textrm{E}_8\) or \(\textrm{SO}(32)\). Spin(11,1) bosonic string theory achieves 12d by compactifying 26d on a 14d maximal torus of the subgroup \(\textrm{SU}(8) \!\times\! \textrm{SU}(8)\) of the group \(G^2(10) = \textrm{U}(16)\) generated by multivectors of grade 2 (mod 4) in 10d.
A standard \(\textrm{SU}(n)\) torus is not self-dual (self-dual means determinant of Cartan matrix is 1), but the \(\textrm{SU}(n)\) torus with one of its roots replaced by a spinor root is self-dual, so that winding numbers and momenta on the torus are dual, as required by modular invariance (\(\tau \leftrightarrow \sigma\) symmetry) of string theory. Again, this is similar to heterotic theory, where substituting a root of \(\textrm{SO}(32)\) with a spinor root makes its torus self-dual.
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Standard and spinor
\(\textrm{SU}(3)\) lattices.
● = standard (integral) vertices; ○ = spinor (1/2 integral) vertices. The 3-fold cover of the spinor lattice is self-dual. |
The (periodically identified) vertices are joined by \(\tfrac{1}{2}(n{-}1)n = 3\) distinct lattice lines. Fundamental modes of strings along each line in each direction represent \((n{-}1)n = 6\) massless eigenstates of \(\textrm{SU}(n)\) gauge bosons. These combine with the \(n{-}1 = 2\) Kaluza-Kline modes of the torus to form the \(n^2{-}1 = 8\) gauge bosons of \(\textrm{SU}(n)\).
9/15. Eigenstates of open strings on nonflat complex manifolds
