A new approach to (3+1)-dimensional TQFTs via topological modular forms
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911156463992832 |
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| author | Gukov, Sergei Krushkal, Vyacheslav Meier, Lennart Pei, Du |
| author_facet | Gukov, Sergei Krushkal, Vyacheslav Meier, Lennart Pei, Du |
| contents | In this paper, we present a construction toward a new type of TQFTs at the crossroads of low-dimensional topology, algebraic geometry, physics, and homotopy theory. It assigns TMF-modules to closed 3-manifolds and maps of TMF-modules to 4-dimensional cobordisms. This is a mathematical proposal for one of the simplest examples in a family of $π_*({\rm TMF})$-valued invariants of 4-manifolds which are expected to arise from 6-dimensional superconformal field theories. As part of the construction, we define TMF-modules associated with symmetric bilinear forms, using (spectral) derived algebraic geometry. The invariant of unimodular bilinear forms takes values in $π_*({\rm TMF})$, conjecturally generalizing the theta function of a lattice. We discuss gluing properties of the invariants. We also demonstrate some interesting physics applications of the TMF-modules such as distinguishing phases of quantum field theories in various dimensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12402 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new approach to (3+1)-dimensional TQFTs via topological modular forms Gukov, Sergei Krushkal, Vyacheslav Meier, Lennart Pei, Du Algebraic Topology High Energy Physics - Theory Geometric Topology Quantum Algebra In this paper, we present a construction toward a new type of TQFTs at the crossroads of low-dimensional topology, algebraic geometry, physics, and homotopy theory. It assigns TMF-modules to closed 3-manifolds and maps of TMF-modules to 4-dimensional cobordisms. This is a mathematical proposal for one of the simplest examples in a family of $π_*({\rm TMF})$-valued invariants of 4-manifolds which are expected to arise from 6-dimensional superconformal field theories. As part of the construction, we define TMF-modules associated with symmetric bilinear forms, using (spectral) derived algebraic geometry. The invariant of unimodular bilinear forms takes values in $π_*({\rm TMF})$, conjecturally generalizing the theta function of a lattice. We discuss gluing properties of the invariants. We also demonstrate some interesting physics applications of the TMF-modules such as distinguishing phases of quantum field theories in various dimensions. |
| title | A new approach to (3+1)-dimensional TQFTs via topological modular forms |
| topic | Algebraic Topology High Energy Physics - Theory Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2509.12402 |