Noncommutative Cartier Formulae

We prove, for every $\mathbb{E}_1$ algebra $A$, a formula describing the interaction of the action of the cap product on topological Hochschild homology of $A$ with the cyclotomic structure map, as well as a variant of this result relative to a ring $R$. Specializing to $R = \mathbb{F}_p$ gives a noncommutative analog of a formula of Cartier which describes the conjugation of interior product action on differential forms by the Cartier isomorphism, and which computes the $p$-curvature of the Getzler-Gauss-Manin connection in terms of an equivariant cap product. The motivation for this formula comes from symplectic geometry, where (in the case $R=\mathbb{F}_p$ or a Novikov analog) the symplectic analog of this formula explains the interaction between the cyclotomic structure on symplectic cohomology and the quantum Steenrod operations. We prove, under standard transversality and nondegeneracy assumptions on the Fukaya category, that for a Calabi-Yau symplectic manifold with rational symplectic form, the $p$-curvature of the quantum connection computes the Quantum Steenrod operations. In particular, the $p$-curvature of the quantum connections of projective Calabi-Yau hypersurfaces, and many other examples in mirror symmetry, can be interpreted in terms of $\mathbb{Z}/p\mathbb{Z}$-equivariant genus zero Gromov-Witten invariants.