Smith’s equivalence over fields containing a square root of −1

Determine whether Ivan Smith’s A-infinity quasi-equivalence between the component Fukaya category of the intersection of two quadrics in CP^5 and the Fukaya category of a genus-2 curve holds over every coefficient field that contains a square root of −1; equivalently, ascertain whether the absence of a square root of −1 is the only arithmetic obstruction to such a quasi-equivalence.

Background

Over the complex numbers, Smith proved that a component of the Fukaya category of the intersection of two quadrics in CP^5 is equivalent to the Fukaya category of a genus-2 curve. This paper observes an arithmetic obstruction to extending that equivalence over general coefficient fields: Proposition 5.2 shows that any such field must contain a square root of −1.

The authors raise the question of whether this is the only obstruction, i.e., whether the equivalence should hold over all fields containing √−1. They further show that, if such an equivalence holds over an odd-characteristic field containing √−1, it places strong constraints on quantum cohomology and determines quantum Steenrod operations (Proposition 5.3).