Anderson duality of topological modular forms and its differential-geometric manifestations

TL;DR

This paper explores the Anderson duality of topological modular forms (TMF), introduces a differential version for pairing with spin manifolds, constructs negative-degree elements, and discusses conjectures linking vertex operator algebras.

Contribution

It constructs a morphism implementing Anderson duality for TMF, introduces a differential pairing with spin manifolds, and explores connections to vertex operator algebras and negative-degree elements.

Findings

Constructed a morphism of spectra for Anderson duality of TMF.

Developed a differential pairing with spin manifolds and string structures.

Identified negative-degree elements of π_dTMF using RO(G)-graded TMF.

Abstract

We construct and study a morphism of spectra implementing the Anderson duality of topological modular forms ($\mathrm{TMF}$). Its differential version will then be introduced, allowing us to pair elements of $\pi_d\mathrm{TMF}$ with spin manifolds whose boundaries are equipped with string structure. A few negative-degree elements of $\pi_d\mathrm{TMF}$ will then be constructed using the theory of $\mathrm{RO}(G)$-graded $\mathrm{TMF}$, and will be identified using the differential pairing. We also discuss a conjecture relating vertex operator algebras and negative-degree elements of $\pi_d\mathrm{TMF}$, underlying much of the discussions of this paper. The paper ends with a separate appendix for physicists, in which the contents of the paper are summarized and translated into their language.