Genuine $C_n$-equivariant $\mathrm{TMF}$

TL;DR

This paper explores the structure of equivariant topological modular forms (TMF) for cyclic groups, providing explicit module structures for $C_2$ and $C_3$, and proposing a general approach for higher cyclic groups using $U(1)$-equivariant TMF.

Contribution

It determines the $ ext{TMF}$-module structures for genuine $C_2$ and $C_3$-equivariant $ ext{TMF}$ and introduces a strategy for studying $C_n$-equivariant $ ext{TMF}$ via $U(1)$-equivariant methods.

Findings

Explicit $ ext{TMF}$-module structures for $C_2$ and $C_3$-equivariant $ ext{TMF}$.

A proposed general strategy for higher $C_n$-equivariant $ ext{TMF}$.

Identification of a duality phenomenon in equivariant $ ext{TMF}.

Abstract

We determine the $\mathrm{TMF}$-module structures of the genuine $C_2$-equivariant $\mathrm{TMF}$ with $\mathrm{RO}(C_2)$-gradings and of the $C_3$-equivariant $\mathrm{TMF}$. Moreover, we propose a general strategy for studying $C_n$-equivariant $\mathrm{TMF}$ via $U(1)$-equivariant $\mathrm{TMF}$ and a duality phenomenon in equivariant $\mathrm{TMF}$.