Quantum Steenrod operations of symplectic resolutions
Jae Hee Lee
TL;DR
This work establishes a precise link between p-curvature and quantum Steenrod operations in mod p equivariant quantum cohomology of conical symplectic resolutions, proving that, for Springer-type examples, the quantum Steenrod operation on divisor classes equals the p-curvature of the mod p quantum connection for almost all primes. A key novelty is a compatibility relation between quantum Steenrod operations and shift operators, enabling a TQFT-style argument to equate two a priori different constructions. The authors extend the Springer result to a broad class of symplectic resolutions under natural discreteness and simple-spectrum assumptions and discuss extensions to Nakajima quiver varieties, hypertoric varieties, and Hilbert schemes. The findings provide an enumerative-geometric interpretation of p-curvature in positive characteristic, with potential implications for 3D mirror symmetry and large-center phenomena in representation theory. Overall, the paper blends Gromov–Witten theory, equivariant quantum cohomology, and p-adic/positive-characteristic insights to connect quantum Steenrod operations with fundamental flat-connection invariants.
Abstract
We study the mod $p$ equivariant quantum cohomology of conical symplectic resolutions. Using symplectic genus zero enumerative geometry, Fukaya and Wilkins defined operations on mod $p$ quantum cohomology deforming the classical Steenrod operations on mod $p$ cohomology. We conjecture that these quantum Steenrod operations on divisor classes agree with the $p$-curvature of the mod $p$ equivariant quantum connection, and verify this in the case of the Springer resolution. The key ingredient is a new compatibility relation between the quantum Steenrod operations and the shift operators.