Discrete Approximations to U(1) Principal Bundles in Abelian Gauge Theory
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915869447159808 |
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| author | Borsten, Leron Kim, Hyungrok |
| author_facet | Borsten, Leron Kim, Hyungrok |
| contents | A $(d+1)$-dimensional field theory with a periodic spatial dimension may be approximated by a $d$-dimensional theory with a truncated Kaluza-Klein tower of $k$ fields; as $k\to\infty$, one recovers the original $(d+1)$-dimensional theory. One may similarly expect that $\operatorname U(1)$-valued Maxwell theory may be approximated by $\mathbb Z_k$-valued gauge theory and that, as $k\to\infty$, one recovers the original Maxwell theory. However, this fails: the $k\to\infty$ limit of $\mathbb Z_k$-valued gauge theory is flat Maxwell theory with no local degrees of freedom. We instead construct field theories $\mathcal T_k$ such that, with appropriate matter couplings, the $k\to\infty$ limit does recover Maxwell theory in the absence of magnetic monopoles (but with possible Wilson loops), and show that $\mathcal T_k$ can be understood as Maxwell theory with the insertion of a certain nonlocal operator that projects out principal $\operatorname U(1)$-bundles that do not arise from principal $\mathbb Z_k$-bundles sectors (in particular, projecting out sectors with monopole charges). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete Approximations to U(1) Principal Bundles in Abelian Gauge Theory Borsten, Leron Kim, Hyungrok High Energy Physics - Theory Mathematical Physics 81T27 (Primary) 70S15, 55R10, 53C05 (Secondary) A $(d+1)$-dimensional field theory with a periodic spatial dimension may be approximated by a $d$-dimensional theory with a truncated Kaluza-Klein tower of $k$ fields; as $k\to\infty$, one recovers the original $(d+1)$-dimensional theory. One may similarly expect that $\operatorname U(1)$-valued Maxwell theory may be approximated by $\mathbb Z_k$-valued gauge theory and that, as $k\to\infty$, one recovers the original Maxwell theory. However, this fails: the $k\to\infty$ limit of $\mathbb Z_k$-valued gauge theory is flat Maxwell theory with no local degrees of freedom. We instead construct field theories $\mathcal T_k$ such that, with appropriate matter couplings, the $k\to\infty$ limit does recover Maxwell theory in the absence of magnetic monopoles (but with possible Wilson loops), and show that $\mathcal T_k$ can be understood as Maxwell theory with the insertion of a certain nonlocal operator that projects out principal $\operatorname U(1)$-bundles that do not arise from principal $\mathbb Z_k$-bundles sectors (in particular, projecting out sectors with monopole charges). |
| title | Discrete Approximations to U(1) Principal Bundles in Abelian Gauge Theory |
| topic | High Energy Physics - Theory Mathematical Physics 81T27 (Primary) 70S15, 55R10, 53C05 (Secondary) |
| url | https://arxiv.org/abs/2512.22114 |