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

We construct and study a morphism of spectra implementing the Anderson duality of topological modular forms ($\mathrm{T}\mathrm{M}\mathrm{F}$). Its differential version will then be introduced, allowing us to pair elements of ${\pi }_d\mathrm{T}\mathrm{M}\mathrm{F}$ with spin manifolds whose boundaries are equipped with string structure. A few negative-degree elements of ${\pi }_d\mathrm{T}\mathrm{M}\mathrm{F}$ will then be constructed using the theory of $\mathrm{R}\mathrm{O}(G)$-graded $\mathrm{T}\mathrm{M}\mathrm{F}$, and will be identified using the differential pairing. We also discuss a conjecture relating vertex operator algebras and negative-degree elements of ${\pi }_d\mathrm{T}\mathrm{M}\mathrm{F}$, 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.