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

Ying-Hsuan Lin, Akira Tominaga, Mayuko Yamashita

TL;DR

We develop a general, fiber-sequence–based framework to determine genuine $C_n$-equivariant TMF structures by reducing to the more tractable $U(1)$-equivariant theory of Topological Jacobi Forms. The approach yields concrete integral, $2$-local, and $3$-local decompositions for $C_2$ and $C_3$ cases and exposes level-rank dualities between $C_2$-equivariant TMF twists and small Spin-group equivariant TMF. Key technical ingredients include the stabilization-restriction sequences for TJF and TEJF, the twisted $ ext{TMF}$-modules $ ext{TMF}[koldsymbol{ ho}_n]^{C_n}$, and the equivariant sigma orientation machinery, together with novel lemmas controlling multiplications in TJF. The results provide integral refinements of known $2$-local pictures (Chua) and complete the $3$-local computation for $ ext{TMF}^{C_3}$, extending the understanding of cyclic-group equivariant TMF and its connections to Jacobi forms and their vector-valued analogs. Overall, the paper proposes and validates a robust strategy for analyzing $C_n$-equivariant TMF in terms of $U(1)$-equivariant data, with broad implications for equivariant elliptic cohomology and modular form dualities.

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 $\mathrm{TMF}$.

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

TL;DR

We develop a general, fiber-sequence–based framework to determine genuine -equivariant TMF structures by reducing to the more tractable -equivariant theory of Topological Jacobi Forms. The approach yields concrete integral, -local, and -local decompositions for and cases and exposes level-rank dualities between -equivariant TMF twists and small Spin-group equivariant TMF. Key technical ingredients include the stabilization-restriction sequences for TJF and TEJF, the twisted -modules , and the equivariant sigma orientation machinery, together with novel lemmas controlling multiplications in TJF. The results provide integral refinements of known -local pictures (Chua) and complete the -local computation for , extending the understanding of cyclic-group equivariant TMF and its connections to Jacobi forms and their vector-valued analogs. Overall, the paper proposes and validates a robust strategy for analyzing -equivariant TMF in terms of -equivariant data, with broad implications for equivariant elliptic cohomology and modular form dualities.

Abstract

We determine the -module structures of the genuine -equivariant with -gradings and of the -equivariant . Moreover, we propose a general strategy for studying -equivariant via -equivariant and a duality phenomenon in equivariant .

Paper Structure

This paper contains 22 sections, 25 theorems, 156 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Preliminaries
  4. Gepner-Meier's genuine equivariant $\mathrm{TMF}$
  5. $U(1)$-equivariant $\mathrm{TMF}$ $=$ Topological Jacobi Forms
  6. $Sp(1)$-equivariant $\mathrm{TMF}$ $=$ Topological Even Jacobi Forms
  7. $C_n$-equivariant $\mathrm{TMF}$
  8. Equivariant sigma orientations
  9. General strategy : setting up fiber sequences
  10. Untwisted cases
  11. Twisted cases
  12. Odd twisted case for $n=2$
  13. Application 1 : $C_2$-equivariant $\mathrm{TMF}$
  14. Untwisted and $\pm \lambda$-twisted cases
  15. $k\lambda$-twisted cases for $2 \le k \le 6$: the $C_2$ level-rank duality isomorphisms
  16. ...and 7 more sections

Key Result

Proposition 2.68

We have a non-split short exact sequence The generator in the middle term is given by the fundamental representation $\overline{\mu} \colon BU(1) \to BO$ composed with the truncation $BO \to P^4BO$.

Figures (15)

  • Figure 1: The cell diagrams of $\mathrm{TJF}_k$.
  • Figure 2: The cell diagram of $\mathrm{cofib}(c)$.
  • Figure 3: The cell structure of $C$.
  • Figure 4: The cell diagrams of $\mathrm{TMF}[n\overline{\lambda}]^{C_2}$, $n \in \mathbb{Z}/8$.
  • Figure 5: The cell diagram of $\mathrm{TMF}^{C_3}_{(3)}$.
  • ...and 10 more figures

Theorems & Definitions (56)

  • Definition 2.14: $\mathrm{TJF}_k$
  • Remark 2.31
  • Definition 2.42
  • Remark 2.46
  • Proposition 2.68
  • proof
  • Remark 2.75
  • Lemma 2.77
  • proof
  • Proposition 2.83: Periodicity of the $\mathrm{RO}(C_p)$-graded $\mathrm{TMF}$
  • ...and 46 more