Symplectic Geometry Seminar

I will explain the relationship between the cyclotomic structure on symplectic cohomology and the equivariant pants products. This relationship exists for any cohomology theory (in particular, I will give a definition of the equivariant pants product over all generalized cohomology theories, in the setting of polarized symplectically atorodial domains with contact boundary). In the case of $F_p$ coefficients, this result specializes to a host of interesting number-theoretic phenomena occurring in symplectic topology, which will be made completely explicit in some examples. There is an analog of this relationship in the setting of topological Hochschild (co)homology, which expresses the compatibility conditions between the $E_2$ algebra structure on topological Hochschild cohomology and the cyclotomic structure on topological Hochschild homology. This result, which deserves the name of the `noncommutative Kaledin-Bezrukavnikov formula', can be used to give the first computation of nontrivial quantum Steenrod operations in the setting of any symplectic Calabi-Yau manifold containing any holomorphic spheres, e.g. in the case of the quintic threefold. This computation expresses the quantum Steenrod operations in terms of an arithmetic invariant of the small quantum connection called the `p-curvature', confirming a generalized version of a conjecture of Jae Hee Lee.