Anderson duality of topological modular forms and its differential-geometric manifestations
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866918075252604928 |
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| author | Tachikawa, Yuji Yamashita, Mayuko |
| author_facet | Tachikawa, Yuji Yamashita, Mayuko |
| contents | 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 $π_d\mathrm{TMF}$ with spin manifolds whose boundaries are equipped with string structure. A few negative-degree elements of $π_d\mathrm{TMF}$ 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 $π_d\mathrm{TMF}$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_06196 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Anderson duality of topological modular forms and its differential-geometric manifestations Tachikawa, Yuji Yamashita, Mayuko Algebraic Topology High Energy Physics - Theory K-Theory and Homology 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 $π_d\mathrm{TMF}$ with spin manifolds whose boundaries are equipped with string structure. A few negative-degree elements of $π_d\mathrm{TMF}$ 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 $π_d\mathrm{TMF}$, 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. |
| title | Anderson duality of topological modular forms and its differential-geometric manifestations |
| topic | Algebraic Topology High Energy Physics - Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2305.06196 |