Noncommutative analogue of the monodromy theorem

Formulate and prove a noncommutative version of the classical monodromy theorem for smooth proper d($\mathbb{Z}/2$)g categories and their categorical t-connections.

Background

The classical monodromy theorem gives two conclusions for isolated hypersurface singularities: monodromy eigenvalues are roots of unity, and the largest Jordan block has size bounded by the ambient dimension. The paper establishes a noncommutative regular-singularity and quasi-unipotence theorem, together with alternative Jordan-block bounds involving Hochschild and diagonal dimensions.

The authors explain that directly transferring the classical dimension bound is problematic because the ambient dimension of a hypersurface is not invariant under Knörrer periodicity for matrix-factorization categories. They therefore leave open the task of identifying and proving an appropriate categorical formulation, possibly involving a graded deformation rather than a single Z/2\mathbb{Z}/2-graded category.

References

Another question which we won't pursue in this paper (but think is worth further investigation) is the formulation and proof of a non-commutative version of Theorem \ref{thm:monodromy theorem}. At first sight, this is slightly mysterious as the dimension of the ambient space of an isolated hypersurface singularity $W$ is not an invariant of $\mathcal{C}=\mathrm{MF}(W)$ due to Knörrer periodicity. We remark that it is possible that a correct formulation requires working with a `{$\mathbb{Z}/2$}-graded smooth and proper deformation of a smooth {$\mathbb{Z}$}-graded dg category' instead of just a single smooth proper d$(\mathbb{Z}/2)$g category (for instance, this perspective is central to the work of on the quantum connection). \n\begin{question} Formulate and prove a non-commutative version of Theorem \ref{thm:monodromy theorem}. \end{question}

— p-curvature in non-commutative Hodge theory and the Kontsevich-Soibelman operad  (2609.26765 - Chen, 22 Sep 2026) in Section 6.2, immediately following the discussion of variants of the monodromy theorem