It's not quite a new paper, rather it is a new appendix (pdf of a standalone version), to be included in A-D-E diagrams, Hodge–Tate hyperplane sections and semisimple quantum cohomology by Sergey Galkin, Naichung Conan Leung, Changzheng Li, and Rui Xiong.

It is a short appendix, in which we observe that one of the main results in the Galkin–Leung–Li–Xiong paper is also sufficiently strong to deduce that the small quantum cohomology of certain Kronecker moduli is not semisimple. Here, Kronecker moduli is shorthand for moduli spaces of stable quiver representations of the Kronecker quiver; this class of smooth projective varieties has many extremely nice properties, and many strong tools to study them.

I refer to the introduction of the main paper for additional context motivating the study of quantum cohomology. Let me just point out that

  • combining the Schofield and Dubrovin conjectures predicts that the big quantum cohomology should be generically semisimple;
  • special cases of Kronecker moduli are Grassmannians, for which the small quantum cohomology is generically semisimple;
  • Meng has shown that the small quantum cohomology for the Kronecker moduli space for the 3-Kronecker quiver and dimension vector $(2,3)$ (the smallest case which is not a Grassmannian) has generically simple quantum cohomology.

One can experimentally (using QuiverTools) check that the obstruction from Galkin–Leung–Li–Xiong's Theorem 1.2 vanishes for many examples of Kronecker moduli. But Markus and I noticed that it does not vanish for the following infinite family of Kronecker moduli:

Theorem Let $\mathrm{M}_{(2,m)}^m$ be the Kronecker moduli space for the $m$-Kronecker quiver and dimension vector $(2,m)$, and $m\geq 5$ odd. Then the small quantum cohomology of $\mathrm{M}_{(2,m)}^m$ is not generically semisimple.

It's a fun, combinatorial proof, that fits in 1.5 page, so I won't comment further on it.