Noncommutative Scherk-type conjugacy statement

Prove that for a smooth proper Calabi–Yau d($\mathbb{Z}/2$)g category whose noncommutative Hodge-to-de Rham spectral sequence degenerates, the nilpotent part of the monodromy of the categorical t-connection lies in the closure of the conjugacy class of the endomorphism given by cup product with the Hochschild cohomology class of the differential.

Background

The paper proves an upper bound on the size of the largest Jordan block of the monodromy in terms of the nilpotence order of the Hochschild class [d] and, more generally, in terms of categorical dimensions. It then compares this result with refinements for quantum connections and with Scherk’s conjecture in classical singularity theory.

The proposed conjecture seeks a stronger description of the monodromy: not merely a bound on Jordan-block sizes, but a relation between the nilpotent monodromy operator and cup product by [d] under the Calabi–Yau identification of Hochschild cohomology and homology. The paper does not prove this assertion.

References

Let $\mathcal{C}$ be a smooth proper Calabi-Yau d$(\mathbb{Z}/2)$g category such that the non-commutative Hodge-to-de-Rham spectral sequence degenerates\footnote{Such assumptions are satisfied for $\mathcal{A}=\mathrm{MF(W)}$ where $W$ is an isolated hypersurface singularity and $\mathcal{A}=\mathrm{Fuk}(X)$ for a large class of closed monotone symplectic manifolds $X$ .}. Then, under the identification $HH*(\mathcal{C})\cong HH_(\mathcal{C})$ via the Calabi-Yau structure, the nilpotent part of the monodromy of $\nabla{\mathcal{C}_{\partial_t}$ lies in the closure of the conjugacy class of the operator $[d]\cup- \in \mathrm{End}(HH^(\mathcal{C}))$.

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