Wilson loop expectations as sums over surfaces on the plane
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911285280505856 |
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| author | Park, Minjae Pfeffer, Joshua Sheffield, Scott Yu, Pu |
| author_facet | Park, Minjae Pfeffer, Joshua Sheffield, Scott Yu, Pu |
| contents | Although lattice Yang-Mills theory on finite subgraphs of $\mathbb Z^d$ is easy to rigorously define, the construction of a satisfactory continuum theory on $\mathbb R^d$ is a major open problem when $d \geq 3$. Such a theory should in some sense assign a Wilson loop expectation to each suitable finite collection $\mathcal L$ of loops in $\mathbb R^d$. One classical approach is to try to represent this expectation as a sum over surfaces with boundary $\mathcal L$. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities.
In this paper, we show how to make sense of Yang-Mills integrals as surface sums for $d=2$, where the continuum theory is more accessible. Applications include several new explicit calculations, a new combinatorial interpretation of the master field, and a new probabilistic proof of the Makeenko-Migdal equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_02306 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Wilson loop expectations as sums over surfaces on the plane Park, Minjae Pfeffer, Joshua Sheffield, Scott Yu, Pu Probability Mathematical Physics 60D05 (Primary), 58J65, 60H25, 70S15, 81T13, 81T35, 82B41 (Secondary) Although lattice Yang-Mills theory on finite subgraphs of $\mathbb Z^d$ is easy to rigorously define, the construction of a satisfactory continuum theory on $\mathbb R^d$ is a major open problem when $d \geq 3$. Such a theory should in some sense assign a Wilson loop expectation to each suitable finite collection $\mathcal L$ of loops in $\mathbb R^d$. One classical approach is to try to represent this expectation as a sum over surfaces with boundary $\mathcal L$. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities. In this paper, we show how to make sense of Yang-Mills integrals as surface sums for $d=2$, where the continuum theory is more accessible. Applications include several new explicit calculations, a new combinatorial interpretation of the master field, and a new probabilistic proof of the Makeenko-Migdal equation. |
| title | Wilson loop expectations as sums over surfaces on the plane |
| topic | Probability Mathematical Physics 60D05 (Primary), 58J65, 60H25, 70S15, 81T13, 81T35, 82B41 (Secondary) |
| url | https://arxiv.org/abs/2305.02306 |