       $3^9, 27$    $9, 29$  
      $2,2,2,4,6$       $1^8, 5^2$   $9, 13$  
      $2,2,2,2,6$        $1^9, 9$    $3, 15$  
      $2,2,2,4,4$       $1^9, 15$    $5,19$   

  : This table lists combinatorial data for anomaly-free (0,2) GLSM’s describing rank 8 bundles over ${\mathbb Z}_2$ gerbes on Calabi-Yau’s.[]{data-label="table:DK-duality-exs"}

For example, the first entry in table \[table:DK-duality-exs\] describes a rank 8 bundle given as a kernel $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \:
\oplus_1^9 {\cal O}(1) \: \longrightarrow \: {\cal O}(9) \:
\longrightarrow \: 0$$ over the stack ${\mathbb P}^3_{[2,2,2,4]}[10]$, a ${\mathbb Z}_2$ gerbe over ${\mathbb P}^3_{[1,1,1,2]}[5]$.

We list a few rank 9 examples over ${\mathbb Z}_3$ gerbes in section \[sect:otherexs-good\]. These rank 8 examples are listed in this section because we are enumerating rank 8 bundles over ${\mathbb Z}_2$ gerbes, and the rank 9 examples are not candidates for the dualities discussed here.

Curiously, we were unable to find solutions of the combinatorial consistency conditions for GLSM’s for bundles of rank less than 8. We do not know whether this reflects a fundamental limitation of GLSM’s, or merely the inadequacy of our parameter space search.

Given a Distler-Kachru model with a phase describing a Landau-Ginzburg model over an orbifold of a vector space, methods exist to compute the massless spectrum in that Landau-Ginzburg phase [@kw; @dk1]. When these methods are applied to, for example, heterotic ${\rm Spin}(32)/{\mathbb Z}_2$ compactifications on typical examples from the table above, we find a large number of single vectors and matter representations which likely combine to form representations of a larger nonabelian gauge symmetry, but unfortunately the corresponding worldsheet global symmetry does not seem to be visible in the UV. We conclude that in these examples, much of the needed worldsheet global symmetry appears in the IR, where we have no direct access. This is atypical of Distler-Kachru mod