Existence of interesting examples of lax group actions on dg categories

Determine whether there exist interesting examples of finite group actions on small differential graded (dg) categories in a lax, homotopy-coherent sense—specifically, actions in which each group element acts via a quasi-equivalence and the equalities in the coherence conditions of Definition 1 are replaced by specified homotopies.

Background

The paper adopts a strict notion of finite group action on a small dg category, requiring that each group element act by an autoequivalence together with specified natural isomorphisms satisfying strict coherence equalities. This framework captures natural geometric examples, such as actions by pullback on dg categories of perfect complexes or matrix factorizations on varieties with group actions.

In Remark 2, the author notes a more lax, homotopy-coherent variant where group elements act via quasi-equivalences and coherence equalities are replaced by specified homotopies. The author explicitly states not knowing interesting examples of such lax actions, highlighting an unresolved question about whether meaningful instances of these homotopy-coherent group actions occur in practice.

References

One could also consider a more lax notion of group action where each group element acts via a quasi equivalence and the equalities of morphisms in the two conditions above are replaced by specified homotopies. However, we don't know of interesting examples where such group actions arise.

— Finite group actions on dg categories and Hochschild homology  (2406.13866 - Nordstrom, 2024) in Remark 2, Section 2 (Finite group actions on dg categories)