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.
Categories: math.AT hep-th math.KT