known that the TMF classes represented by the group manifolds (with their Lie group framings) $$\mathrm{Sp}(2),\qquad G_2,\qquad G_2\times U(1)$$ are nonzero: their exact orders are, respectively, $$3, \qquad 2, \qquad 2.$$ The same logic as in [@WittenTMF] suggests that the ${\mathcal N}=(0,1)$ sigma models for $\mathrm{Sp}(2)$ and $G_2$ flow in the IR to SCFTs consisting purely of ($10$ and $14$, respectively) antichiral free fermions, with supersymmetries that encode the structure constants of the Lie algebras $\mathfrak{sp}(2)$ and $\mathfrak{g}_2$.[^6] The sigma model with target $G_2\times U(1)$ does not flow to a purely-antichiral theory, but rather to the product of the $\mathfrak{g}_2$-theory and the “standard” circle theory studied in §\[subsubsec.S1-nonbounding\]. The elliptic and mod-2 indexes of all three SQFTs vanish. Moreover, the invariant described in this paper vanishes for all three SQFTs — the first two for degree reasons, but the third nontrivially. Due to the expected relationship between TMF and SQFTs, we expect that these SQFTs are not nullhomotopic. We leave the reader with the following puzzles:

\[sp2puzzle\]

1.  Show that the SQFT $\overline{\operatorname{Fer}}(\mathfrak{sp}(2))$ of $10$ antichiral free fermions and supersymmetry encoding the structure constants of $\mathfrak{sp}(2)$ is not nullhomotopic.

2.  Show that the SQFT $\overline{\operatorname{Fer}}(\mathfrak{g}_2)$ of $14$ antichiral free fermions 