B-side mirror description of quantum Steenrod operations via HKR and Frobenius

Prove that, for a smooth commutative algebra A of dimension d<p over a field of characteristic p equipped with a superpotential f, under the identification CC^{S^1}_*(MF(A,f)) ⊕ CC^{S^1}_*(MF(A,f))·θ ≅ CC^{\mathbb{Z}/p}_*(MF(A,f)), the Hochschild–Kostant–Rosenberg-type quasi-isomorphisms (\bigwedge^*TA, ι_{df}) ≃ CC^*(MF(A,f)) and (Ω^*_A[[t]], t d − df∧) ≃ CC^{S^1}_*(MF(A,f)) intertwine the \mathbb{Z}/p-equivariant cap product on HH^{\mathbb{Z}/p}_*(MF(A,f)) with the Frobenius p-linear action on H^*(Ω^*_A[[t]], t d − df∧) in which twisted functions act by p-th powers and twisted vector fields D act by i^{[p]}_D = (ι_{D^p} − ℒ_D^{p−1} ι_D) t^{(p−1)/2}.

Background

The paper’s main A-side result (Theorem 1.2) identifies quantum Steenrod operations with a \mathbb{Z}/p-equivariant cap product action on the Hochschild invariants of the Fukaya category. Section 6 proposes a mirror B-side description in terms of matrix factorizations and differential forms in characteristic p.

The conjecture posits that, after identifying \mathbb{Z}/p-equivariant and S^1-equivariant complexes (Theorem 1.3), HKR-type quasi-isomorphisms transfer the \mathbb{Z}/p-equivariant cap product to a Frobenius p-linear operation on de Rham-type cohomology, where twisted functions act via p-th powers and twisted vector fields act via a p-th power contraction i^{[p]} closely related to restricted Lie structures.