Quantum Steenrod operations and Fukaya categories

The recent introduction of mod p equivariant operations to symplectic Gromov-Witten theory has fueled exciting developments in the field. In this thesis, we develop new tools for understanding these operations and explore an application to the quantum connection. In one direction, we construct certain operations on the equivariant Hochschild (co)homology of a general A∞-category. We show that when applied to the Fukaya category of a nondegenerated closed monotone symplectic manifold, this construction can be identified with the quantum Steenrod operations via Ganatra’s cyclic open-closed maps. A key ingredient in this identification is a new homotopy theoretic framework for studying various equivariant open-closed maps at once, using a combination of cyclic categories, edgewise subdivision and Abouzaid-Groman-Varolgunes’ operadic Floer theory. In another direction, we utilize quantum Steenrod operations, and Lee’s observation that it is related to the p-curvature of the quantum connection, to study singularities of the quantum connection in characteristic 0, and prove the exponential type conjecture for all closed monotone symplectic manifolds.

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2025

Author dc:creator

• Chen, Zihong

Advisor dc:contributor.advisor

• Seidel, Paul

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright retained by author(s)

Licence dc:rights.uri

• https://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/159911

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/159911