$\mathrm{U}(1)^{n}$ Chern-Simons theory: partition function, reciprocity formula and Chern-Simons duality
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913930911154176 |
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| author | Kim, Han-Miru Mathieu, Philippe Tagaris, Michail Thuillier, Frank |
| author_facet | Kim, Han-Miru Mathieu, Philippe Tagaris, Michail Thuillier, Frank |
| contents | The $\mathrm{U}(1)$ Chern-Simons theory can be extended to a topological $\mathrm{U}(1)^n$ theory by taking a combination of Chern-Simons and BF actions, the mixing being achieved with the help of a collection of integer coupling constants. Based on the Deligne-Beilinson cohomology, a partition function can then be computed for such a $\mathrm{U}(1)^n$ Chern-Simons theory. This partition function is clearly a topological invariant of the closed oriented $3$-manifold on which the theory is defined. Then, by applying a reciprocity formula a new expression of this invariant is obtained which should be a Reshetikhin-Turaev invariant. Finally, a duality between $\mathrm{U}(1)^n$ Chern-Simons theories is demonstrated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10734 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathrm{U}(1)^{n}$ Chern-Simons theory: partition function, reciprocity formula and Chern-Simons duality Kim, Han-Miru Mathieu, Philippe Tagaris, Michail Thuillier, Frank Mathematical Physics The $\mathrm{U}(1)$ Chern-Simons theory can be extended to a topological $\mathrm{U}(1)^n$ theory by taking a combination of Chern-Simons and BF actions, the mixing being achieved with the help of a collection of integer coupling constants. Based on the Deligne-Beilinson cohomology, a partition function can then be computed for such a $\mathrm{U}(1)^n$ Chern-Simons theory. This partition function is clearly a topological invariant of the closed oriented $3$-manifold on which the theory is defined. Then, by applying a reciprocity formula a new expression of this invariant is obtained which should be a Reshetikhin-Turaev invariant. Finally, a duality between $\mathrm{U}(1)^n$ Chern-Simons theories is demonstrated. |
| title | $\mathrm{U}(1)^{n}$ Chern-Simons theory: partition function, reciprocity formula and Chern-Simons duality |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2409.10734 |