Consdier the classical singular cohomology ring $H^*(X, \mathbb{Z})$ of a manifold $X$. The $H^n(X, \mathbb{Z})$'s are the cohomology groups of a chain complex and it turns out that the multiplication lifts to an $E_\infty$-ring structure on that chain complex.

For a symplectic manifold $X$, with almost complex structure $J$, it is useful to consider the quantum cohomology ring of $X$, defined using $J$-pseudoholomorphic curves in $X$. We may now ask the same question about this quantum cohomology - is there an $E_\infty$-ring structure?

Does the mirror symmetry perspective tell us anything here?

By the way, my question is motivated by the existence of Quantum Steenrod Operations constructed by Fukaya (which I don't understand yet) which I found out from this arxiv paper today:

Covariant constancy of quantum Steenrod operations

Paul Seidel, Nicholas Wilkins

https://arxiv.org/abs/2102.06432

EDIT - Is there any reference for the $E_2$ structure, or is it still conjectural? The statement in Dustin Clausen's answer is a conjecture of Kontsevich mentioned here and which I guess is still open. But maybe this weaker consequence is known?

Indepedently of that, maybe there is a reference for the $E_1$ or $A_\infty$ structure?

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