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


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?

  • 1
    $\begingroup$ I don't know anything about quantum cohomology, but if it's anything like quantum groups (about which I also know very little), you might expect to get an $E_2$ structure or something but not an $E_\infty$-structure. $\endgroup$
    – Tim Campion
    Feb 15 at 14:58
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    $\begingroup$ Quantum cohomology is not really related to quantum groups at all. $\endgroup$ Feb 15 at 15:40
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    $\begingroup$ It's always bold to claim that two mathematical objects are not related! Anyway, at least Tim's conclusion is correct: according to my understanding, quanutum cohomology should have an $E_2$ structure but not an $E_\infty$ structur in general. This is because quantum cohomology is the Hochschild cohomology of the Fukaya category (derived endomorphisms of the identity functor), and as the Fukaya category only a category not a tensor category this only gets an $E_2$-structure. (Experts please correct me if I'm wrong) $\endgroup$ Feb 15 at 15:54
  • $\begingroup$ Interesting, this makes me wanna ask a simpler question: is there a geometric description of the E_1 structure on quantum cohomology that doesn’t require passing through mirror symmetry? $\endgroup$
    – Bbb
    Feb 16 at 19:22
  • $\begingroup$ The A_\infty structure comes from thinking of QH as the Lagrangian Floer cohomology of the diagonal. You can phrase it in a more "quantum cohomologyish" language (counting spheres with point constraints which are required to live along an equator in the domain) but this is really the same thing. $\endgroup$ Feb 17 at 8:18

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