We construct and study various properties of a negative spin version of the Witten $r $ -spin class. By taking the top Chern class of a certain vector bundle on the moduli space of spin curves that parametrises $r $ -th roots of the anticanonical bundle, we construct a non-semisimple cohomological field theory (CohFT) that we call the Theta class $\Theta ^{r} $ . This CohFT does not have a flat unit and its associated Dubrovin–Frobenius manifold is nowhere semisimple. Despite this, we construct a semisimple deformation of the Theta class, and using the Teleman reconstruction theorem, we obtain tautological relations on $\overline{\mathcal{M}}_{g,n} $ . Furthermore, we prove that the descendant potential of the Theta class is the unique solution to a set of $\mathcal{W} $ -algebra constraints, which implies a recursive formula for all the descendant integrals. Using this result for $r = 2 $ , we prove Norbury’s conjecture which states that the descendant potential of $\Theta ^{2} $ coincides with the Brézin–Gross–Witten tau function of the KdV hierarchy. Furthermore, we conjecture that the descendant potential of $\Theta ^{r} $ is the $r $ -BGW tau function of the $r $ -KdV hierarchy and prove the conjecture for $r = 3 $ .
